# B U L L E T I N

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B U L L E T I N

B U L L E T I N DE LA SOCIÉTÉ DES SCIENCES ET DES LETTRES DE £ÓD SÉRIE: RECHERCHES SUR LES DÉFORMATIONS Volume LX, no. 1 Rédacteur en chef et de la Série: JULIAN £AWRYNOWICZ Comité de Rédaction de la Série P. DOLBEAULT (Paris), H. GRAUERT (Göttingen), O. MARTIO (Helsinki), W.A. RODRIGUES, Jr. (Campinas, SP), B. SENDOV (Sofia), C. SURRY (Font Romeu), P.M. TAMRAZOV (Kyiv), E. VESENTINI (Torino), L. WOJTCZAK (£ódŸ), Ilona ZASADA (£ódŸ) Secrétaire de la Série: JERZY RUTKOWSKI N £ÓD 2010 !"# $%&"' (")*+ , -"+ ).)&$ /(") & $-*0" 1)")0" )%" 2" &)*" /('&3 42 5 , /(") 2"" 3& " -$%(+ 61" 7"%*&( 8% $%""+ * 9: ;""" <, ,9 BULLETIN DE LA SOCIÉTÉ DES SCIENCES ET DES LETTRES DE L ÓDŹ 2010 Vol. LX Recherches sur les déformations no. 1, 174 pp. FROM COMPLEX ANALYSIS TO ASYMMETRIC RANDERS-INGARDEN STRUCTURES III Editors of the Volume Julian L awrynowicz, Dariusz Partyka and Józef Zaja̧c Julian L awrynowicz Institute of Physics, University of L ódź Pomorska 149/153, PL-90-236 L ódź, Poland Institute of Mathematics, Polish Academy of Sciences L ódź Branch, Banacha 22, PL-90-238 L ódź, Poland Dariusz Partyka Faculty of Mathematics and Natural Sciences The John Paul II Catholic University of Lublin Al. Raclawickie 14, P.O. Box 129, PL-20-950 Lublin, Poland State University of Applied Science in Chelm, Pocztowa 54 PL-22-100 Chelm, Poland, e-mail: [email protected] Józef Zaja̧c State University of Applied Science in Chelm Pocztowa 54, PL-22-100 Chelm, Poland Chair of Applied Mathematics The John Paul II Catholic University of Lublin Al. Raclawickie 14, P.O. Box 129, PL-20-950 Lublin, Poland e-mail: [email protected] Note. Since this year the numbering of volumes of the series Recherches coincides with that of the Bulletin. This permits to simplify the citations: Bull. Soc. Sci. Lettres L ódź Sér. Rech. Déform. 60, no. 1 (2010) etc. 4 INSTRUCTION AUX AUTEURS 1. La présente Série du Bulletin de la Société des Sciences et des Lettres de L ódź comprend des communications du domaine des mathématiques, de la physique ainsi que de leurs applications liées aux déformations au sense large. 2. 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Description 1 Description 2 Description 3 Description 4 Row 1, Col 1 Row 2, Col 1 Row 1, Col 2 Row 2, Col 2 Row 1, Col 3 Row 2, Col 3 Row 1, Col 4 Row 2, Col 4 3. How to submit a manuscript Manuscripts have to be submitted in electronic form, preferably via e-mail as attachment ﬁles sent to the address zoﬁ[email protected] If a whole manuscript exceeds 2 MB composed of more than one ﬁle, all parts of the manuscript, i.e. the text (including equations, tables, acknowledgements and references) and ﬁgures, should be ZIP-compressed to one ﬁle prior to transfer. If authors are unable to send their manuscript electronically, it should be provided on a disk (DOS format ﬂoppy or CD-ROM), containing the text and all electronic ﬁgures, and may be sent by regular mail to the address: Department of Solid State Physics, University of Lodz, Bulletin de la Société des Sciences et des Lettres de L ódź, Pomorska 149/153, 90-236 Lodz, Poland. References [1] Aﬃliation/Address [5] TABLE DES MATIÈRES 1. P. Dolbeault, Complex Plateau problem: old and new results and prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–31 2. D. Mierzejewski, Spheres in sets of solutions of quadratic quaternionic equations of some types . . . . . . . . . . . . . . . . . . . . . . . . . 33–43 3. A. K. Kwaśniewski, Cobweb posets and KoDAG digraphs are representing natural join of relations, their di-bigraphs and the corresponding adjacency matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45–65 4. C. Boloşteanu, The Riemann-Hilbert problem with isolated points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67–75 5. D. Partyka and J. Zaja̧c, Generalized problem of regression . 77–94 6. J. Dziok, Extremal problems in a generalized class of uniformly convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95–108 7. B. Bochorishvili and H. M. Polatoglou, Electronic properties of quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109–121 8. D. Georgakaki, Ch. Mitsas, and H. M. Polatoglou, Time series analysis of the response of measurement instruments . . . . 123–135 9. R. S. Ingarden and J. L awrynowicz, Model of magnetic electron microscope including the scanning microscope III. Variational approach and calculation of the focal length in a Randerstype geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137–154 10. R. S. Ingarden and J. L awrynowicz, Finsler-geometrical model of quantum electrodynamics I. External field vs. Finsler geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155–174 Left to right (sitting): Julian L awrynowicz (Lódź), Bogdan Bojarski (Warszawa), Ilpo Laine (Joensuu), and Luis Manuel Tovar Sánchez (México, D.F.) during opening ceremony of the XV International Conference on Analytic Functions in Chelm (Poland), July 2009. Behind, in the centre (standing): Józef Zaja̧c (Chelm/Lublin) 9 PL ISSN 0459-6854 BULLETIN DE LA SOCIÉTÉ DES SCIENCES ET 2010 DES LETTRES DE L ÓDŹ Vol. LX Recherches sur les déformations no. 1 pp. 11–31 Pierre Dolbeault COMPLEX PLATEAU PROBLEM: OLD AND NEW RESULTS AND PROSPECTS Summary The Plateau problem is the research of a surface of minimal area, in the 3-dimensional Euclidean space, whose boundary is a given continuous closed curve. The complex Plateau problem is analogous in a Hermitian complex manifold: it is a geometrical problem of extension of a closed real curve or manifold into a complex analytic subvariety, or into a Levi-ﬂat subvariety. Wirtinger’s inequality in Cn is recalled. Minimality of complex analytic subvarieties and analogous properties of Levi-ﬂat subvarieties, in Kähler manifolds, are given. Known results in Cn and CP n are recalled. Extensions to real parametric problems are solved or proposed, leading to the construction of Levi-ﬂat hypersurfaces with prescribed boundary in some complex manifolds. 1. Introduction Given a Hermitian manifold X, the complex Plateau problem is the research of an even dimensional subvariety with negligible singularities, with given boundary, and of minimal volume in X. We will call mixed Plateau problem the research of a real hypersurface with given boundary, and of minimal volume in X. More brieﬂy, both problems will be called complex Plateau problem. First we shall recall or show that complex analytic subvarieties, resp. Levi-ﬂat hypersurfaces are solutions of the Plateau problem when X is Kähler (Sect. 2). Then we will consider the complex Plateau problem as the research of the extension of an odd dimensional, compact, oriented, connected submanifold into a complex analytic subvariety, and recall known solutions (Sects. 3, 4). To solve the mixed Plateau problem as the research of the extension of an oriented, compact, connected, 2-codimensional submanifold into a Levi-ﬂat hypersurface, we will need solutions of the complex Plateau problem with real parameter, in Cn and CPn ; in CPn , it is an open problem to explicit satisfactory conditions on 12 P. Dolbeault the boundary. In this way, we get very peculiar solutions of mixed Plateau problems (Sect. 5). Finally, the mixed Plateau problem is solved in Cn , in particular cases, as a projection of a Levi-ﬂat variety, and set up in CPn (Sects. 6, 7): known solutions are recalled in Cn when the complex points of the boundary are elliptic; special elliptic and hyperbolic points of the boundary are deﬁned, and a solution when the boundary is a ”horned sphere” is described; this will be the opportunity to precise and complete results announced in ([D 08], Sect. 4). Problems when the boundary has general hyperbolic points are still open. Proofs of the results in Sects. 6 and 7 will appear in detail elsewhere [D 09]. 2. Volume minimality of complex analytic subvarieties and of Levi-ﬂat hypersurfaces in Kähler manifolds 2.1. Wirtinger’s inequality (1936) [H 77] In Cn ,with complex coordinates (z1 , . . . , zn ), we have the Hermitian metric H= n dzj ⊗ dz j j=1 and the exterior form (standard Kähler) n n i ω= dzj ∧ dz j = dxj ∧ dyj . 2 j=1 j=1 From the real vector space IR2n ∼ = Cn , we consider the real vector space Λ2p IR2n of the 2p-vectors with the associated norm |.|; every decomposable vector (exterior product of elements of IR2n ) deﬁnes a real 2p-plane of Cn i.e. an element of the Grassmannian G2p 2n . We deﬁne the norm || ζ ||= inf | ζj | where ζ = ζj , ζj j j is decomposable. N Let Ppp = { λj ζj ; ζj decomposable deﬁning a complex p-plane of Cn ; λj ≥ 0; N ∈ IN ∗ }. j=1 2.1.1. Theorem. For every ζ ∈ Λ2p Cn , we have: equality uniquely for ζ ∈ Ppp 1 p ω (ζ) ≤|| ζ ||; p! [W 36]. 2.1.2. Corollary. Let V be a smooth real oriented 2p-dimensional submanifold of a Hermitian manifold X = (X, ω) of complex dimension n. Then Complex Plateau problem: old and new results and prospects V 13 ω p /p! ≤ vol2p (V ) with equality iﬀ V is complex. 2.2. Currents with measure coeﬃcients [H 77] 2.2.1. Comass of an r-form; mass of a current with measure coeﬃcients. Let ϕ ∈ Λr IR2n , the comass of ϕ is deﬁned as || ϕ ||∗ = sup{ϕ(ζ) : ζ ∈ Gr2n ⊂ Λ2p IR2n }. Let Ω be an open subset of Cn , for every diﬀerential form ϕ of degree r on Ω, let || ϕ ||∗ = sup{|| ϕ(z) ||∗ : z ∈ Ω} where || ϕ(z) ||∗ is the comass of ϕ(z). Let T be a current with measure coeﬃcients on Ω, K be any compact subset of Ω and χK the characteristic function of K, MK (T ) = sup |χK T (ϕ)| ||ϕ||∗≤1 is, by deﬁnition, the mass of T on K. The measure which assigns the number MK (T ) to each compact set K ⊂ Ω is called the mass or volume measure of T and denoted || T ||, so that MK (T ) =|| T || (K). 2.3. Complex Plateau problem 2.3.1. [H 77] On Ω ⊂ Cn , or more generally, on a Hermitian manifold (X, ω), let B be a d-closed current of dimension 2p − 1 with compact support, and let T be a (2p)-current with compact support and measure coeﬃcients such that dT = B. The complex Plateau problem is to ﬁnd such a T with minimal mass, i.e. for every compactly supported current S, with measure coeﬃcients such that dS = B, to have M (T ) ≤ M (S), or equivalently, for every compactly supported, d-closed (2p)-current with measure coeﬃcients R, M (T ) ≤ M (T + R). Such a T is said absolutely volume minimizing on X. Let T be a d-closed (2p)-current with measure coeﬃcients on X. If, for each compact subset K of X, MK (T ) ≤ M (χK T + R) for all compactly supported d-closed (2p)-current R with measure coeﬃcients on X, then T is said to be absolutely volume minimizing on X. 14 P. Dolbeault 2.3.2. Theorem. [H 77] Let T be a 2p-current with measure coeﬃcients on a Hermitian manifold (X, ω) and K be a compact subset of X. Then (χK T )(ω p /p!) ≤ MK (T ) and equality holds iﬀ χK T is strongly positive. 2.3.3. [H 77] Volume minimality of complex analytic sets in a Kähler manifold. 2.3.4. Corollary to Theorem 2.3.2. Assume that X = (X, ω) is a Käkler manifold and does not contain compact p-dimensional complex subvarieties. Let V be a p-dimensional complex subvatiety, and T = [V ], then T is absolutely volume minimizing on X. Proof. T is strongly positive. Let K be a compact subset of X and R be a compactly supported d-closed (2p)-current with measure coeﬃcients. From Theorem 2.3.2, MK (T ) = (χK T )(ω p /p!). But locally ω = ddc ψ, then ω p = ω p−1 ∧ddc ψ = d(ω p−1 ∧dc ψ), so in the neighborhood of any point of X, R(ω p ) = R(d(ω p−1 ∧ dc ψ)). Let (αj )j∈J be a partition C ∞ of unity subordinate to a locally ﬁnite open covering (Uj )j∈J of X such that for every j, ω|Uj = ddc ψj . Then αj R(d(ω p−1 ∧ dc ψj )) = ± d(αj R)(ω p−1 ∧ dc ψj ) = 0, R(ω p ) = j because: j j d(αj R) = j dαj ∧ R + αj ∧ dR = 0 j and, as in the proof of ([H 77], Corollary 1.25), in an open set of the Hermitian Cn , MK (T ) = (χK T )(ω p /p!) = (χK T + R)(ω p /p!) ≤ MK (T + R). 2.3.5. Remark. If X contains a compact p-dimensional complex subvariety W , d[V ] = 0, but MK ([V ] > 0; then T is relatively volume minimizing on X. 2.4. Volume minimality of Levi-ﬂat hypersurfaces in Kähler manifolds We suppose to be in the category of currents with measure coeﬃcients. Recall the deﬁnition: A Levi-ﬂat subvariety (with negligible singularities), of odd dimension, is, outside of the singularities, a submanifold with Levi form ≡ 0, or, equivalently, is foliated by complex analytic hypersurfaces. Let M be a C ∞ Levi-ﬂat hypersurface of a C ∞ Kähler manifold X = (X, ω) bearing a foliation L by complex hypersurfaces Ml and let L be the space of the foliation L assumed to be a C ∞ real curve. Let M be a C ∞ hypersurface of X bearing a foliation L with the same space L; the leaves of L being C ∞ subvarieties with negligible singularities. Complex Plateau problem: old and new results and prospects 15 Let S be a C ∞ compact submanifold of codimension 2 of X. We denote by the same notation the hypersurfaces and submanifold and the integration currents they deﬁne. 2.5. Mixed Plateau problem Given S to ﬁnd a C ∞ hypersurface in X \ S whose boundary is S in the category H of foliated hypersurfaces with the same space of foliation, a real curve. If M is such a hypersurface whose space of foliation is L and the leaves (Ml , l ∈ L), then vol(M ) = L vol(Ml )dl. From Sect. 2, for every l ∈ L, vol(Ml ) ≥ vol(Ml ) then M is relatively volume minimizing in the category H and, by deﬁnition, M is solution of the mixed Plateau problem. 2.6. Research of solutions of the complex Plateau problem The present method of resolution consists in ﬁnding complex analytic, resp. Levi-ﬂat subvarieties, in X \ S, whose boundary S (in the sense of currents) is a submanifold of X with convenient properties. 3. Possible origin: holomorphic extension; polynomial envelope of a real curve 3.1. The extension theorem of Hartogs, obtained at the beginning of the 20th century, has been completely proved by Bochner and Martinelli, independently, in 1943. The simplest version is: Let Ω be a bounded open set of Cn , n ≥ 2. Suppose that ∂Ω be of class C k (1 ≤ k ≤ ∞) or of class C ω (i.e. real analytic). Let f be a function in C l (∂Ω), 1 ≤ l ≤ k. Then the two conditions are equivalent: (i) f is a CR function, i.e. the diﬀerential of f restricted to the complex subspaces of the tangent space to ∂Ω, at every point, is C-linear; (ii) there exists F ∈ C l (Ω) ∩ O(Ω) such that F |∂Ω = f . Then the graph of f is the boundary of the complex analytic submanifold deﬁned by the graph of F in Cn+1 . 3.2. Let M be a compact submanifold of dimension 1 of Cn , we call polynomial envelope of M , the compact set {z ∈ Cn ; | P (z) |≤ max | P (ζ) |; P ∈ C[z], the polynomial ζ∈M ring with complex coeﬃcients }. Then (J. Wermer (1958)), the polynomial envelope of M is either M , or the union of M with the support of a complex analytic variety T , of complex dimension 1, whose boundary is M [We 58]. 16 P. Dolbeault 4. Solutions of the complex Plateau problem (or boundary problem) in diﬀerent spaces 4.1. The ﬁrst result has been obtained in 1958, by J. Wermer, in Cn , for p = 1 and M holomorphic image of the unit circle in C [We 58]; this result has been generalized to the case where M is a union of C 1 real connected curves by Bishop and Stolzenberg (1966), looking for the polynomial envelope of M according to Sect. 3.2. In Cn , after preliminary results by Rothstein (1959) [Rs 59], the boundary problem has been solved by Harvey and Lawson (1975), for p ≥ 2, under the necessary and suﬃcient condition: M is compact, maximally complex and, for p = 1, under the moment condition: M ϕ = 0, for every holomorphic 1-form ϕ on Cn [ HL 75]. For n = p + 1, the method, inspired by the Hartogs’ theorem consists in building T as the divisor of a meromorphic function the deﬁning function R; this function itself is constructed, step by step, from solutions of ∂-problems with compact support. T can also be viewed as graph (with multiplicities on the irreducible components) of an analytic function with a ﬁnite number of determinations. For any p, we come back to the particular case using projections. In CPn \ CPn−r , 1 ≤ r ≤ n, for compact M , the problem has a une solution if and only if, for p ≥ r + 1, M is maximally complex and if, for p = r, M satisﬁes the moment condition: M ϕ = 0, for every ∂-closed (p, p − 1)-forme ϕ. The method consists in solving the boundary problem, in Cn+1 \ Cn−r+1 , for the inverse image of M by the canonical projection [HL 77]. In both cases, the solution is unique. Harvey et Lawson assume the given M to be, except for a closed set of Hausdorﬀ (2p − 1)-dimensional measure zero, an oriented manifold of class C 1 ; we will say: M is a variety C 1 with negligible singularities. The boundary problem in CPn has been set up, for the ﬁrst time, by J. King [Ki 79]; uniqueness of the solution is no more possible, since two solutions diﬀer by an algebraic p-chain. 4.2. In CPn , a solution of the boundary problem has been obtained by P. Dolbeault et G. Henkin for p = 1, (1994), then for every p (1997) and more generally, in a qlinearly concave domain X of CPn , i.e. a union of projective subspaces of dimension q [DH 97]. The necessary and suﬃcient condition for the existence of a solution is an extension of the moment condition: it uses a Cauchy residue formula in one variable and a non linear diﬀerential condition which appears in many questions of Geometry or Mathematical Physics. In the simplest case: p = 1, n = 2, this is the shock wave equation for a local holomorphic function in 2 variables ξ, η, f ∂f /∂ξ = ∂f /∂η. From a local condition, the above relation allows to construct, by extension Complex Plateau problem: old and new results and prospects 17 ot the coeﬃcients, a meromorphic function playing, in Cn , the same part as the Harvey-Lawson deﬁning function described above; it deﬁnes a holomorphic p-chain extendable to CPn using the classical Bishop-Stoll theorem. 4.2.1. The conditions of regularity of M have been weakened, ﬁrst in Cn , and for p = 1, to a condition, a little stronger than the rectiﬁability, by H. Alexander [Al 88] who, moreover, has given an essential counter-example [Al 87], then by Lawrence [Lce 95] and ﬁnally, and for any p, in Cn and CPn , by T. C. Dinh [Di 98]: M is a rectiﬁable current whose tangent cone is a vector subspace almost everywhere. Moreover, Dinh has obtained the reduction of the boundary problem in CPn to the case p = 1, with weaker conditions than above and by an elementary analytic procedure [Di 98]. All the previous results are obtained as corollaries. New progress by Harvey and Lawson [HL 04]. 5. Extension to real parametric problems 5.1. ∼ R × Cn−1 , and k : R × Cn−1 → R be the projection. Let N ⊂ E be a 5.1.1. Let E = compact, (oriented) CR subvariety of Cn of real dimension 2n − 4 and CR dimension n − 3, (n ≥ 4), of class C ∞ , with negligible singularities (i.e. there exists a closed subset τ ⊂ N of (2n − 4)-dimensional Hausdorﬀ measure 0 such that N \ τ is a CR submanifold). Let τ be the set of all points z ∈ N such that either z ∈ τ or z ∈ N \ τ and N is not transversal to the complex hyperplane k −1 (k(z)) at z. Assume that N , as a current of integration, is d-closed and satisﬁes: (H) There exists a closed subset L ⊂ Rx1 with H 1 (L) = 0 such that for every x ∈ k(N ) \ L, the ﬁber k −1 (x) ∩ N is connected and does not intersect τ . 5.1.2. Theorem [DTZ 09] (see also [DTZ 05]). Let N satisfy (H) with L chosen accordingly. Then, there exists, in E = E \ k −1 (L), a unique C ∞ Levi-ﬂat (2n − 3)subvariety M with negligible singularities in E \ N , foliated by complex (n − 2)subvarieties, with the properties that M simply (or trivially) extends to E as a (2n − 3)-current (still denoted M ) such that dM = N in E . The leaves are the sections by the hyperplanes Ex01 , x01 ∈ k(N )\L, and are the solutions of the “HarveyLawson problem” for ﬁnding a holomorphic subvariety in Ex01 ∼ = Cn−1 with prescribed boundary N ∩ Ex01 . 5.2. In a real hyperplane of CPn+1 5.2.1. The simplest signiﬁcant case is the boundary problem in CP 3 . For the boundary problem with real parameter in C3 , we considered a boundary problem in IR×C3 , i.e. in the subspace of C4 , in which the ﬁrst coordinate is real. In the same way, we will consider in CP 4 , with homogeneous coordinates (w0 , w1 , . . . , w4 ), a boundary problem in the subspace E deﬁned by w1 = λw0 , with λ ∈ IR. Then, for personal 18 P. Dolbeault convenience, we will follow, step by step, the known construction in CP 3 in the oldest version [DH 97]. Particularly, the coeﬃcients Cm of the deﬁning function R of the solution are estimated as for the problem in CP 3 . The end of the proof of the main theorem seems analogous to the known case in IR × C3 . 5.2.2. The projective space CP 3 has homogeneous coordinates w = (w0 , w2 , . . . , w4 ); denote Q = {w0 = 0} the hyperplane at inﬁnity of CP 3 . For w0 = 0, let k be the projection: E → IRλ , (w0 , w1 = λw0 , w2 , w3 , w4 ) → λ; π : E → CP 3 , for w0 = 0, λ is indeterminate.. We also have the projection: (w0 , λw0 , w2 , w3 , w4 ) → (w0 , w2 , w3 , w4 ). In the same way, (E \ {w0 = 0}) ∼ = IR × C3 . 5.2.3. Let N ⊂ E ⊂ CP 4 be a submanifold of class C ∞ , CR, oriented, compact of E, of dimension 4, of CR dimension 1, with negligible singularities. N being compact in E, k(N ) is compact in IR, i.e. in N , the parameter λ varies in a closed, bounded interval Λ of IR. Assume that N satisﬁes the same properties as in Sect. 5.1.1. 5.2.4. Consider the complex hyperplanes of CP 4 , whose equation is h̃(w) = w4 − ξ2 w0 − η2 w1 − η2 w2 = 0 and, in E, the subspaces Pνλ whose equation is h̃1 (w , λ) = w4 − ξ2 w0 − η2 λw0 − η2 w2 = w4 − (ξ2 + η2 λ)w0 − η2 w2 = 0, of real dimension 5. Their restrictions to (E \ IR × Q) ∼ = IR × C3 are real aﬃne subspaces of dimension 5. We note νλ the 1 × 2-matrix (ξ2 + η2 λ) η2 . Generically, Γνλ = N ∩ Pνλ is of dimension 2. / L, N ∩ Eλ is of dimension 3 and For z ∈ N, λ = k(z). let Eλ = k −1 k(z); for λ ∈ is contained in Eλ ∼ = CP 3 . Consider the linear forms h(w) = w3 − ξ1 w0 − η1 w1 − η1 w2 , h̃0 (w , λ) = w3 − ξ1 w0 − η1 λw0 − η1 w2 = w3 − (ξ1 + η1 λ)w0 − η1 w2 . Denote by νλ = (ξλ , η) the 2 × 2-matrix (ξ1 + η1 λ) (ξ2 + η2 λ) η1 η2 . For ﬁxed λ, νλ is a coordinate system of a chart of the Grassmannian GC (2, 4), i.e. νλ is a coordinate system of a chart of GC (2, 4) × IR. and we identify νλ with the point of GC (2,4) × IR having these coordinates. Let ξλ =t (ξ1 + η1 λ) (ξ2 + η2 λ) ; η =t (η1 η2 ). Remark that ξλ depends on (ξ1 , ξ2 , η1 , η2 ); to get eﬀective dependance on the parameter λ, it suﬃces to ﬁx η1 = 0, .η2 = 0. Recall: ξλ =t (ξλ1 ξλ2 ), ξλl = ξl + ηl λ, l = 1, 2, η =t (η1 η2 ). Complex Plateau problem: old and new results and prospects 19 Let zj = wj /w0 , j = 2, 3, 4, be the non homogeneous coordinates; h̃0 deﬁnes the aﬃne function: h = z3 − (ξ1 + η1 λ) − η1 z2 . The two forms h̃0 et h̃1 are linearly independent, then the set of their common zeros Dνλ is of real dimension 3, is contained in Pνλ ; in general, Dνλ ∩N is a ﬁnite set Zνλ ; then, for general enough ﬁxed λ and νλ , Zνλ = ∅. For every ﬁxed real number λ∈ / L, the situation in Eλ is the classical situation in CP 3 . 5.2.5. Boundary problem. Given N , ﬁnd a complex analytic subvariety M depending on the real parameter λ such that dM = N in the sense of currents, under a necessary and suﬃcient condition on N . To do this, we can check, step by step, the solution of the boundary problem in CP 3 [HL 97], introducing the parameter λ. For λ ∈ / L, γνλ = N ∩ Pνλ ∩ Eλ is of dimension 1. Under the notations of the Sect. 5.2.4, consider the function 1 dh z2 . (1) G(νλ ) = 2πi γν h λ 5.2.6. Tentative statement. The following two conditions are equivalent: (i) There exists, in E = E \ k −1 L, a C ∞ Levi-ﬂat subvariety M , (with negligible singularities), of dimension 5, foliated by complex analytic subvarieties Mλ of complex dimension 2, such that M extends simply (or trivially) to E as a current of dimension 5 (still denoted M ) such that dM = N in E . The leaves are the sections by the subspaces Eλ , λ ∈ k(N ) \ L, and are the solutions of the boundary problem for ﬁnding complex analytic subvarieties in Eλ ∼ = CP 3 with given boundary N ∩ Eλ . (ii) N is a submanifold CR, oriented, of CR dimension 1 outside a closed set of 4-dimensional Hausdorﬀ measure 0. There exists a matrix νλ∗∗ in the neighborhood of which Dξ2λ G(νλ ) = Dξ2λ N fj (νλ ) j=1 where fj , j = 1, . . . , N , is a holomorphic function in νλ , C ∞ en λ, and satisﬁes the system of P.D.E. (2) fj ∂fj ∂fj = , l = 2, 3. ∂ξλl ∂ηl 5.2.7. Remark. This result is not satisfactory because the relation of the analytic conditions with the geometry of the submanifold N is not explicit. 20 P. Dolbeault 5.3. Boundary problem in a real hyperplane of Cn+1 or CP n+1 Cn+1 and CP n+1 are both Kähler. The solutions of the above boundary problems are both Levi ﬂat, hence, from a plain extension of Sect. 2.5, volume minimal, i.e. solution, of codimension 3, of mixed Plateau problems. 6. Levi-ﬂat hypersurfaces with prescribed boundary: preliminaries 6.1. Introduction Let S ⊂ Cn be a compact connected 2-codimensional submanifold. Find a Levi-ﬂat hypersurface M ⊂ Cn \ S such that dM = S (i.e. whose boundary is S, possibly as a current). For n = 2, near an elliptic complex point p ∈ S, S \ {p} is foliated by smooth compact real curves which bound analytic discs (Bishop [Bi 65]). The family of these discs ﬁlls a smooth Levi-ﬂat hypersurface. In 1983, Bedford-Gaveau considered the case of a particular sphere with two elliptic complex points. If S is contained in the boundary of a strictly pseudoconvex bounded domain, then the families of analytic discs in the neighborhood of each elliptic point extend to a global family ﬁlling a 3-dimensional ball M bounded by S. In 1991, Bedford-Klingenberg [BeK 91] and Kruzhilin extended the result when there exist hyperbolic complex points on S with the same global condition. Results of increasing generality have been obtained by Chirka, Shcherbina, Slodowski, G. Tomassini until 1999. The global suﬃcient condition of embedding of S in the boundary of a strictly pseudoconvex domain is still required in these papers. A ﬁrst result for n ≥ 3 (in the sense of currents), and for elliptic points only, has been obtained four years ago ([DTZ 05] and [DTZ 09] in detailed form); we got new results when S is homeomorphic to a sphere, with three elliptic and one hyperbolic special points (see [D 08] for a ﬁrst draft), or a torus, with two elliptic and two hyperbolic special points and, more generally, a manifold which is obtained by gluing together elementary models. A local condition is required because, in general, S is not locally the boundary of a Levi-ﬂat hypersurface. The proof uses the construction of a foliation of S by CR orbits, Thurston’s stability theorem for foliations on S, and a parametric version of the Harvey-Lawson theorem on boundaries of complex analytic varieties. There is no global condition. 6.2. Preliminaries and deﬁnitions 6.2.1. A smooth, connected, CR submanifold M ⊂ Cn is called minimal at a point p if there does not exist a submanifold N of M of lower dimension through p such that HN = HM |N . By a theorem of Sussman, all possible submanifolds N such Complex Plateau problem: old and new results and prospects 21 that HN = HM |N contain, at p, one of the minimal possible dimension, called a CR orbit of p in M whose germ at p is uniquely determined. 6.2.2. S is said to be a locally ﬂat boundary at a point p if it locally bounds a Leviﬂat hypersurface near p. Assume that S is CR in a small enough neighborhood U of p ∈ S. If all CR orbits of S are 1-codimensional (which will appear as a necessary condition for our problem), the following two conditions are equivalent [DTZ 05]: (i) S is a locally ﬂat boundary on U ; (ii) S is nowhere minimal on U . 6.2.3. Complex points of S [DTZ 05]. At such a point p ∈ S, Tp S is a complex hyperplane in Tp Cn . In suitable holomorphic coordinates (z, w) ∈ Cn−1 × C vanishing at p, S satisﬁes (1) w = Q(z) + O(|z|3 ), Q(z) = (aij zi zj + bij zi z j + cij z i z j ) 1≤i,j≤n−1 S is said ﬂat at a complex point p ∈ S if bij zi z j ∈ λR, λ ∈ C. We also say that p is ﬂat. Let S ⊂ Cn be a locally ﬂat boundary with a complex point p. Then p is ﬂat. By making the change of coordinates (z, w) → (z, λ−1 w), we make bij zi z j ∈ IR for all z. By a change of coordinates (z, w) → (z, w + aij zi zj ) we can choose the holomorphic term in (1) to be the conjugate of the antiholomorphic one and so make the whole form Q real-valued. We say that S is in a ﬂat normal form at p if the coordinates (z, w) as in (1) are chosen such that Q(z) ∈ R for all z ∈ Cn−1 . 6.2.4. Properties of Q. Assume that S is in a ﬂat normal form; then, the quadratic form Q is real valued. Only holomorphic linear changes of coordinates are allowed. If Q is positive deﬁnite or negative deﬁnite, the point p ∈ S is said to be elliptic; if the point p ∈ S is is not elliptic, and if Q is non degenerate, p is said to be hyperbolic. From Sect. 6.4, we will only consider particular cases of the quadratic form Q. From [Bi 65], for n = 2, in suitable holomorphic coordinates, Q(z) = (zz + λRe z 2 ), λ ≥ 0, under the notations of [BeK 91]; for 0 ≤ λ < 1, p is said to be elliptic, and for 1 < λ, it is said to be hyperbolic. The parabolic case λ = 1, not generic, is omitted [BeK 91]. When n ≥ 3, the Bishop’s result is not valid in general. 6.3. Elliptic points 6.3.2. Proposition ([DTZ 05], [DTZ 09]). Assume that S ⊂ Cn , (n ≥ 3) is nowhere minimal at all its CR points and has an elliptic ﬂat complex point p. Then there exists a neighborhood V of p such that V \ {p} is foliated by compact real (2n − 3)dimensional CR orbits diﬀeomorphic to the sphere S2n−3 and there exists a smooth function ν, having the CR orbits as the level surfaces. 22 P. Dolbeault Sketch of Proof (see [DTZ 09]). In the case of a quadric S0 (w = Q(z)), the CR orbits are deﬁned by w0 = Q(z), where w0 is constant. Using (1), we approximate the tangent space to S by the tangent space to S0 at a point with the same coordinate z; the same is done for the tangent spaces to the CR orbits on S and S0 ; then we construct the global CR orbit on S through any given point close enough to p. 6.4. Special ﬂat complex points We say that the ﬂat complex point p ∈ S is special if in convenient holomorphic coordinates, (2) Q(z) = n−1 (zj z j + λj Re zj2 ), λj ≥ 0. j=1 Let zj = xj + iyj , xj , yj real, j = 1, . . . , n − 1, then: n−1 (3) Q(z) = l=1 (1 + λl )x2l + (1 − λl )yl2 + O(|z|3 ). A ﬂat point p ∈ S is said to be special elliptic if 0 ≤ λj < 1 for any j. A ﬂat point p ∈ S is said to be special k-hyperbolic if 1 < λj for j ∈ J ⊂ {1, . . . , n − 1} and 0 ≤ λj < 1 for j ∈ {1, . . . , n − 1} \ J = ∅, where k denotes the number of elements of J. Special elliptic (resp. k-hyperbolic) points are elliptic (resp. hyperbolic). 6.5. Special hyperbolic points 6.5.1. We will not consider special parabolic points (one λj = 1 at least) which don’t appear generically. S being given by (1), let S0 be the quadric of equation w = Q(z). Suppose that S0 is ﬂat at 0 and that 0 is a special k-hyperbolic point. Then, in a neighborhood of 0, and with the above local coordinates, it is CR and nowhere minimal outside 0, and the CR orbits of S0 are the (2n − 3)-dimensional submanifolds given by w = const. = 0. The section w = 0 of S0 is a real quadratic cone Σ0 in R2n whose vertex is 0 and, outside 0, it is a CR orbit Σ0 in the neighborhood of 0. 6.6. Foliation by CR-orbits in the neighborhood of a special 1-hyperbolic point We mimic the begining of the proof of 2.4.2. in ([DTZ 05], [DTZ 09]). 6.6.1. Local 2-codimensional submanifolds. In C3 , consider the 4-dimensional submanifold S locally deﬁned by the equation (1) w = ϕ(z) = Q(z) + O(|z|3 ) and the 4-dimensional submanifold S0 of equation (4) w = Q(z) Complex Plateau problem: old and new results and prospects 23 with Q = (λ1 + 1)x21 − (λ1 − 1)y12 + (1 + λ2 )x22 + (1 − λ2 )y22 having a special 1-hyperbolic point at 0, (λ1 > 1, 0 ≤ λ2 < 1), and the cone Σ0 whose equation is: Q = 0. On S0 , a CR orbit is the 3-dimensional submanifold Kw0 whose equation is w0 = Q(z). If w0 > 0, Kw0 does not cut the line L = {x1 = x2 = y2 = 0}; if w0 < 0, Kw0 cuts L at two points. 6.6.2. Remark. Σ0 = Σ0 \ 0 has two connected components in a neighborhood of 0. Proof. The equation of Σ0 ∩ {y1 = 0} is (λ1 + 1)x21 + (1 + λ2 )x22 + (1 − λ2 )y22 = 0 whose only zero , in the neighborhood of 0, is {0}: the connected components are obtained for y1 > 0 and y1 < 0 respectively. 6.6.3. Behaviour of local CR orbits. Under the notations of [DTZ 09], follow the construction of the complex tangent space E(z, ϕ(z)) to the CR orbit at z; compare with E0 (z, Q(z)). We know the integral manifold, the orbit of E0 (z, Q(z)); deduce an evaluation of the integral manifold of E(z, ϕ(z)). 6.6.4. Lemma. Under the above hypotheses, if k = 1, the local orbit Σ corresponding to Σ0 has two connected components in the neighborhood of 0. Proof. Use Remark 6.6.2 and the adaptation of the technique of [DTZ 09]. 6.7. CR-orbits near a subvariety containing a special 1-hyperbolic point 6.7.2. Proposition. Assume that S ⊂ Cn (n ≥ 3), is a locally closed (2n − 2)submanifold, nowhere minimal at all its CR points, which has a unique spcial 1hyperbolic ﬂat complex point p, and such that: (i) the orbit Σ whose closure Σ contains p is compact; (ii) Σ has two connected components σ1 , σ2 , whose closures are homeomorphic to spheres of dimension 2n − 3. Then, there exists a neighborhood V of Σ such that V \Σ is foliated by compact real (2n − 3)-dimensional CR orbits whose equation, in a neighborhood of p is (3), and, the w(= xn )-axis being assumed to be vertical, each orbit being diﬀeomorphic to the sphere S2n−3 above Σ , the union of two spheres S2n−3 under Σ , and there exists a smooth function ν, having the CR orbits as the level surfaces. 6.8. Geometry of the complex points of S 6.8.1. Let G be the manifold of the oriented real linear (2n − 2)-subspaces of Cn . The submanifold S of Cn has a given orientation which deﬁnes an orientation of the 24 P. Dolbeault tangent space to S at any point p ∈ S. By mapping each point of S into its oriented tangent space, we get a smooth Gauss map t : S → G. 6.8.2. Dimension of G. dim G = 2(2n − 2). 6.8.3. Proposition. For n ≥ 2, in general, S has isolated complex points. Proof. Let π ∈ G be a complex hyperplane of Cn whose orientation is induced by C = CPn−1∗ ⊂ G, as real its complex structure; the set of such π is H = Gn−1,n submanifold. If p is a complex point of S, then t(p) ∈ H or −t(p) ∈ H. The set of complex points of S is the inverse image by t of the intersections t(S) ∩ H and −t(S) ∩ H in G. Since dim t(S) = 2n − 2, dim H = 2(n − 1), dim G = 2(2n − 2), the intersection is 0-dimensional, in general. 6.8.4. Homology of G. (cf [P 08]). G has the structure of a complex quadric; let S1 , S2 be generators of H2n−2 (G, ZZ); we assume that S1 and S2 are fundamental cycles of complex projective subspaces of complex dimension (n−1) of G. Then, denoting also S, the fundamental cycle of the submanifold S and t∗ the homomorphism deﬁned by t, we have: t∗ (S) ∼ u1 S1 + u2 S2 where ∼ means homologous to. 6.8.5. Lemma (proved for n = 2 in [CS 51]). With the notations of 6.8.1, we have: u1 = u2 ; u1 + u2 = χ(S), Euler-Poincaré characteristic of S. The proof for n = 2 works for any n ≥ 3. 6.8.6. Local intersection numbers of H and t(S) when all complex points are ﬂat. Proposition (known for n = 2 [Bi 65], here for n ≥ 3). Let S be a smooth, oriented, compact, 2-codimensional, real submanifold of Cn whose all complex points are ﬂat and special. Then, on S, (special elliptic points) + (special k-hyperbolic points, with k even) - (special k-hyperbolic points, with k odd) = χ(S). If S is a sphere, this number is 2. 7. Levi-ﬂat hypersurfaces with prescribed boundary: particular cases 7.1. To solve the boundary problem by Levi-ﬂlat hypersurfaces, S has to satisfy necessary and suﬃcient local conditions. A way to prove that these conditions can occur is to construct an example for which the solution is obvious. Complex Plateau problem: old and new results and prospects 25 7.2. Sphere with elliptic points 7.2.1. Example. In C3 , Let S be deﬁned by the equations: z1 z 1 + z2 z 2 + z3 z 3 = 1 (S) = z3 z3 We have CR-dim S = 1 except at the points z1 = z2 = 0; z3 = ±1 where CR-dim S = 2. S is the unit sphere in C2 × IR; it bounds the unit ball M in C2 × IR, which is foliated by the complex balls C2 × {x3 } ∩ M . The leaves are relatively compact of real dimension 4 and are bounded by compact leaves (3-spheres) of a foliation of M . 7.2.2. Theorem [DTZ 05]. Let S ⊂ Cn , n ≥ 3, be a compact connected smooth real 2-codimensional submanifold satisfying the conditions (i) S is nonminimal at every CR point; (ii)every complex point of S is ﬂat and elliptic and there exists at least one such point; (iii) S does not contain complex manifold of dimension (n − 2). Then S is a topological sphere, and there exists a Levi-ﬂat (2n − 1)-subvariety M̃ ⊂ C × Cn with boundary S̃ (in the sense of currents) such that the natural projection π : C × Cn → Cn restricts to a bijection which is a CR diﬀeomorphism between S̃ and S outside the complex points of S. 7.3. Sphere with one special 1-hyperbolic point (sphere with two horns) 7.3.1. Example. In C3 , let (zj ), j = 1, 2, 3, be the complex coordinates and zj = xj + iyj . In R6 ∼ = C3 , consider the 4-dimensional subvariety (with negligible singularities) S deﬁned by: y3 = 0, x3 (x21 + y12 + x22 + y22 + x23 − 1) + (1 − x3 )(x41 + y14 + x42 + y24 + 4x21 0 ≤ x3 ≤ 1; −2y12 + x22 + y22 ) = 0, −1 ≤ x3 ≤ 0; x3 = x41 + y14 + x42 + y24 + 4x21 − 2y12 + x22 + y22 . The singular set of S is the 3-dimensional section x3 = 0 along which the tangent space is not everywhere (uniquely) deﬁned. S being in the real hyperplane {y3 = 0}, the complex tangent spaces to S are {x3 = x0 } for convenient x0 . The set S will be smoothed along the complement of 0 (origin of C3 ) in its section by the hyperplane {x3 = 0} by a small deformation leaving h unchanged. In the following S will denote this smooth submanifold. From elementary analytic geometry, complex points of S are deﬁned by their coordinates: e3 : xj = 0, yj = 0 (j = 1, 2), x3 = 1; h: xj = 0, yj = 0 (j = 1, 2), x3 = 0; e1 , e2 : x1 = 0, y1 = ±1, x2 = 0, y2 = 0, x3 = −1. 26 P. Dolbeault Lemma. The complex points are ﬂat and special. The points e1 , e2 , e3 are special elliptic; the point h is special 1-hyperbolic. Remark that the numbers of special elliptic and special hyperbolic points satisfy the conclusion of Proposition 6.8.6. 7.3.1’. Shape of Σ = S ∩ {x3 = 0} in the neighborhood of the origin 0 of C3 . Lemma. Under the above hypotheses and notations: (i) Σ = Σ \ 0 has two connected components σ1 , σ2 ; (ii) The closures of the three connected components of S \ Σ are submanifolds with boundaries and corners. Proof. (i) The only singular point of Σ is 0. We work in the ball B(0, A) of C2 (x1 , y1 , x2 , y2 ) for small A and in the 3-space πλ = {y2 = λx2 }, λ ∈ IR. For λ ﬁxed, πλ ∼ = IR3 (x1 , y1 , x2 ), and Σ ∩ πλ is the cone of equation 4x21 − 2y12 + (1 + λ2 )x22 + O(|z|3 ) = 0 with vertex 0 and basis in the plane x2 = x02 the hyperboloid Hλ of equation 3 4x21 − 2y12 + (1 + λ2 )x02 2 + O(|z| ) = 0; the curves Hλ have no common point outside 0. So, when λ varies, the surfaces Σ ∩ πλ are disjoint outside 0. The set Σ is clearly connected; Σ ∩ {y1 = 0} = {0}, the origin of C3 ; from above: σ1 = Σ ∩ {y1 > 0}; σ2 = Σ ∩ {y1 < 0}. (ii) The three connected components of S \ Σ are the components which contain, respectively e1 , e2 , e3 and whose boundaries are σ 1 , σ 2 , σ 1 ∪ σ 2 ; these boundaries have corners as shown in the ﬁrst part of the proof. The connected component of C2 ×IR\S containing the point (0, 0, 0, 0, 1/2) is the Levi-ﬂat solution, the complex leaves being the sections by the hyperplanes x3 = x03 , −1 < x03 < 1. The sections by the hyperplanes x3 = x03 are diﬀeomorphic to a 3-sphere for 0 < x03 < 1 and to the union of two disjoint 3-spheres for −1 < x03 < 0, as can be shown intersecting S by lines through the origin in the hyperplane x3 = x03 ; Σ is homeomorphic to the union of two 3-spheres with a common point. 7.3.2. Proposition (cf [D 08], Proposition 2.6.1). Let S ⊂ Cn be a compact connected real 2-codimensional manifold such that the following holds: (i) S is a topological sphere; S is nonminimal at every CR point; Complex Plateau problem: old and new results and prospects 27 (ii) every complex point of S is ﬂat; there exist three special elliptic points ej , j = 1, 2, 3 and one special 1-hyperbolic point h; (iii) S does not contain complex manifolds of dimension (n − 2); (iv) the singular CR orbit Σ through h on S is compact and Σ \ {h} has two connected components σ1 and σ2 whose closures are homeomorphic to spheres of dimension 2n − 3; (v) the closures S1 , S2 , S3 of the three connected components S1 , S2 , S3 of S \ Σ are submanifolds with (singular) boundary. Then each Sj \ {ej ∪ Σ }, j = 1, 2, 3 carries a foliation Fj of class C ∞ with 1-codimensional CR orbits as compact leaves. Proof. From conditions (i) and (ii), S satisfying the hypotheses of Proposition 6.3.2, near any elliptic ﬂat point ej , and of Proposition 6.7.2 near Σ , all CR orbits are diﬀeomorphic to the sphere S2n−3 . The assumption (iii) guarantees that all CR orbits in S must be of real dimension 2n − 3. Hence, by removing small connected open saturated neighborhoods of all special elliptic points, and of Σ , we obtain, from S \ Σ , three compact manifolds Sj ”, j = 1, 2, 3, with boundary and with the foliation Fj of codimension 1 given by its CR orbits whose ﬁrst cohomology group with values in R is 0, near ej . It is easy to show that this foliation is transversely oriented. 7.3.2’. Recall the Thurston’s Stability Theorem ([CaC], Theorem 6.2.1). Let (M, F ) be a compact, connected, transversely-orientable, foliated manifold with boundary or corners, of codimension 1, of class C 1 . If there is a compact leaf L with H 1 (L, R) = 0, then every leaf is homeomorphic to L and M is homeomorphic to L×[0, 1], foliated as a product, Then, from the above theorem, Sj ” is homeomorphic to S2n−3 × [0, 1] with CR orbits being of the form S2n−3 × {x} for x ∈ [0, 1]. Then the full manifold Sj is homeomorphic to a half-sphere supported by S2n−2 and Fj extends to Sj ; S3 having its boundary pinched at the point h. 7.3.3. Theorem. Let S ⊂ Cn , n ≥ 3, be a compact connected smooth real 2codimensional submanifold satisfying the conditions (i) to (v) of Proposition 7.3.2. Then there exists a Levi-ﬂat (2n − 1)-subvariety M̃ ⊂ C × Cn with boundary S̃ (in the sense of currents) such that the natural projection π : C × Cn → Cn restricts to a bijection which is a CR diﬀeomorphism between S̃ and S outside the complex points of S. Proof. By Proposition 6.3.2 , for every ej , a continuous function νj , C ∞ outside ej , can be constructed in a neighborhood Uj of ej , j = 1, 2, 3, and by Proposition 6.7.2, we have an analogous result in a neighborhood of Σ . Furthermore, from Sect. 7.3.2’, a smooth function ν”j whose level sets are the leaves of Fj can be obtained globally on Sj \ {ej ∪ Σ }. With the functions νj and ν”j , and analogous functions near Σ , then using a partition of unity, we obtain a 28 P. Dolbeault global smooth function νj : Sj → R without critical points away from the complex points ej and from Σ . Let σ1 , resp. σ2 the two connected, relatively compact components of Σ \ {h}, according to condition (iv); σ 1 , resp. σ 2 are the boundary of S1 , resp. S2 , and σ 1 ∪σ 2 the boundary of S3 . We can assume that the three functions νj are ﬁnite valued and get the same values on σ 1 and σ 2 . Hence a function ν : S → R. The submanifold S being, locally, a boundary of a Levi-ﬂat hypersurface, is orientable. We now set S̃ = N = gr ν = {(ν(z), z) : z ∈ S}. Let Ss = {e1 , e2 , e3 , σ1 ∪ σ2 }. λ : S → S̃ z → ν((z), z) is bicontinuous; λ|S\Ss is a diﬀeomorphism; moreover λ is a CR map. Choose an orientation on S. Then N is an (oriented) CR subvariety with the negligible set of singularities τ = λ(Ss ). At every point of S \ Ss , dx1 ν = 0, then condition (H) (Sect. 5.1.1) is satisﬁed at every point of N \ τ . Then all the assumptions of Theorem 5.1.2 being satisﬁed by N = S̃, in a particular case, we conclude that N is the boundary of a Levi-ﬂat (2n − 2)-variety (with negligible singularities) M̃ in R × Cn . Taking π : C × Cn → Cn to be the standard projection, we obtain the conclusion. 7.4. Case of a torus 7.4.1. Euler-Poincaré characteristic of a torus is χ(Tk ) = 0. 7.4.2. Example. In C 3 , let (zj ), j = 1, 2, 3, be the complex coordinates and zj = xj + iyj . In R6 ∼ = C3 , consider the 4-dimensional subvariety (with negligible singularities) S deﬁned by: y3 = 0 0 ≤ x3 ≤ 1; x3 (x21 + y12 + x22 + y22 + x23 − 1) + (1 − x3)(x41 + y14 + x42 + y24 + 4x21 − 2y12 + x22 + y22 ) = 0 − 12 ≤ x3 ≤ 0; x3 = x41 + y14 + x42 + y24 + 4x21 − 2y12 + x22 + y22 , glue it with the symmetric with respect to the real hyperplane x3 = − 12 , and and smooth along {x3 = 0}, {x3 = ± 12 }. The complex points are ﬂat and special. 7.4.3. Theorem. Let S ⊂ Cn , n ≥ 3, be a compact connected smooth real 2codimensional submanifold satisfying the following conditions: (i) S is a topological torus; S is nonminimal at every CR point; (ii) every complex point of S is ﬂat; there exist two special elliptic points e1 , e2 and two special 1-hyperbolic points h1 , h2 ; Complex Plateau problem: old and new results and prospects 29 (iii) S does not contain complex manifolds of dimension (n − 2); (iv) the singular CR orbits Σ1 , Σ2 through h1 and h2 on S are compact and, for j = 1, 2, Σj \ {hj } have two connected components σj1 and σj2 ; (v) the closures S1 , S2 , S3 , S4 of the four connected components S1 , S2 , S3 , S4 of S \ Σ1 ∪ Σ2 are submanifolds with (singular) boundary. Then there exists a Levi-ﬂat (2n − 1)-subvariety M̃ ⊂ C × Cn with boundary S̃ (in the sense of currents) such that the natural projection π : C × Cn → Cn restricts to a bijection which is a CR diﬀeomorphism between S̃ and S outside the complex points. 7.5. Generalizations 7.5.1. Elementary models and their gluing. The examples and the proofs of the theorems when S is homeomorphic to a sphere (Sect. 7.3) or a torus (Sect. 7.4) suggest the following deﬁnitions. 7.5.2. Deﬁnitions. Let T be a smooth, locally closed (i.e. closed in an open set), connected submanifold of Cn , n ≥ 3. We assume that T has the following properties: (i) T is relatively compact, non necessarily compact, and of codimension 2; (ii) T is nonminimal at every CR point; (iii) T has exactly 2 complex points which are ﬂat and either special elliptic or special 1-hyperbolic; (iv) If p ∈ T is 1-hyperbolic, the singular orbit Σ through p is compact, Σ \ p has two connected components σ1 , σ2 , whose closures are homeomorphic to spheres of dimension 2n − 3; (v) If p ∈ T is 1-hyperbolic, in the neighborhood of p, with convenient coordinates, the equation of T , up to third order terms is zn = n−1 (zj z j + λj Re zj2 ); λ1 > 1; 0 ≤ λj < 1 for j = 1 j=1 or in real coordinates xj , yj with zj = xj + iyj , n−1 (1 + λj )x2j + (1 − λj )yj2 + O(|z|3 ). xn = (λ1 + 1)x21 − (λ1 − 1)y12 + j=2 Other conﬁgurations are easily imagined. up- and down- 1-hyperbolic points. Let T be the (2n− 2)-submanifold with (singular) boundary contained into T such that either σ 1 (resp. σ 2 ) is the boundary of T near p, or Σ is the boundary of T near p. In the ﬁrst case, we say that p is 1-up, (resp. 2-up), in the second that p is down. Such a T will be called an elementary model. For instance, T is 1-up and has one special elliptic point, we solve the boundary problem as in S1 in the proof of Theorem 7.3.3. 7.5.3. The gluing (to be precised) happens between two compatible elementary models along boundaries, for instance down and 1-up. 30 P. Dolbeault 7.6. Other possible generalizations The mixed Plateau problem can be set up in projective space CPn and in subspaces of CPn on which the complex Plateau problem can be solved, using Statement 5.2.6, its gemeralisation to any n ≥ 3 and a better geometric condition on the given boundary. Acknowledgments I thank G. Tomassini and D. Zaitsev for discussions, corrections and remarks about several parts of this paper. 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F-75005 Paris France Presented by Wlodzimierz Waliszewski at the Session of the Mathematical-Physical Commission of the L ódź Society of Sciences and Arts on March 2, 2010 ZESPOLONY PROBLEM PLATEAU: STARE I NOWE WYNIKI ORAZ PERSPEKTYWY Streszczenie Problem Plateau polega na badaniu powierzchni o minimalnym polu w 3-wymiarowej przestrzeni euklidesowej, przy czym brzeg powierzchni jest dana̧ cia̧gla̧ krzywa̧ zamkniȩta̧. Zespolony problem Plateau jest analogiczny na hermitowskiej rozmaitości zespolonej: jest to geometryczny problem uogólnienia zamkniȩtej krzywej lub rozmaitości rzeczywistej na analityczna̧ podrozmaitość zespolona̧, lub podrozmaitość plaska̧ w sensie Leviego. Przypominamy nierówność Wirtingera w przestrzeni Cn . Uzyskujemy minimalność analitycznych podrozmaitości zespolonych i analogiczne wlasności podrozmaitości plaskich w sensie Leviego na rozmaitościach Kählera. Przypominamy znane wyniki w przestrzeni Cn i przestrzeni rzutowej CP n . Rozwia̧zujemy lub proponujemy rozszerzenia do rzeczywistych zagadnień parametrycznych, co prowadzi do konstrukcji hiperpowierzchni plaskich w sensie Leviego o danym z góry brzegu w przypadku pewnych rozmaitości zespolonych. PL ISSN 0459-6854 BULLETIN DE LA SOCIÉTÉ DES SCIENCES ET DES LETTRES DE L ÓDŹ 2010 Vol. LX Recherches sur les déformations no. 1 pp. 33–43 Dmytro Mierzejewski SPHERES IN SETS OF SOLUTIONS OF QUADRATIC QUATERNIONIC EQUATIONS OF SOME TYPES Summary We study sets of solutions of quadratic quaternionic equations of some types by the method of sections by hyperplanes perpendicular to the real axis. Namely, we look for spheres in such sections. We get necessary and suﬃcient conditions for such section to be spherical for a quaternionic equation of the form ax2 + x2 b = c and also (as a simple corollary) of the form ax2 + x2 b + apx + xpb + axp + pxb = q. We prove that for any quaternionic equation of the form ax2 + x2 b + xcx + m p() xq () = d =1 with c ∈ R any such section is not spherical. 1. Notations and terminology of the paper We use H as the standard notation for the set of all (real) quaternions. (We do not deal with so-called complex quaternions whose components are complex.) The notation R has its usual sense (the set of real numbers). We use the standard notations i, j, k for the quaternionic imaginary units; recall that i2 = j 2 = k 2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j. We always use subindices to denote the components of a quaternion as follows: ξ = ξ0 + ξ1 i + ξ2 j + ξ3 k, 34 D. Mierzejewski where ξ0 , ξ1 , ξ2 , ξ3 ∈ R. We often treat sets of some quaternions geometrically using the well-known interpretation of any quaternion ξ0 + ξ1 i + ξ2 j + ξ3 k as the point (ξ0 , ξ1 , ξ2 , ξ3 ) of the four-dimensional space (here ξ0 , ξ1 , ξ2 , ξ3 ∈ R). We name a quaternionic equation any one in which every known parameter is a quaternion; as for solutions of such equations, we always consider solutions being quaternions. Analogously, real equations are ones with real parameters and real solutions being considered. In every quaternionic equation of this paper the letter x denotes the unknown and other letters denote given parameters, if there is no other explanation. 2. Introduction Investigations of solutions of polynomial quaternionic equations (or, by other words, zeros of quaternionic polynomials) were performed in many works during 20th and especially 21st century. Some examples of these works are in References. The topic turns out to be much more diﬃcult than similar investigations of real or complex polynomials. For readers that are not very familiar with the topic we would like to note that due to the absence of commutativity in the system of quaternions even an arbitrary quaternionic monomial has comparatively complicated general form, namely a(1) xa(2) x . . . a(n) xa(n+1) ; moreover a quaternionic polynomial written the simplest way can contain any number of terms of the same degree. For example, the general form of an arbitrary linear quaternionic polynomial is m a() xb() + c, =1 and the general form of quadratic one is m =1 a () xb () xc () + q d(p) xf (p) + g. p=1 Note that if in the last expression there is such for which b() = 1 then the corresponding term can be rewritten as a() x2 c() ; in this case we say that here is a non-split square, while any term of the form a() xb() xc() is referred to as one with a split square. In fact if b() is a real number then it is easy to make the square non-split because every real number commutes with every quaternion. It turns out that, in contrast to real and complex polynomials, it occurs often that a quaternionic polynomial has inﬁnitely many zeros. In particular, the set of the zeros of a quaternionic polynomial or a part of this set may constitute a sphere (see [7], [5], [3]). This paper is devoted to just this shape. We look for spheres in Spheres in sets of solutions of quadratic quaternionic equations of some types 35 the sets of the solutions of polynomial (namely, quadratic) quaternionic equations of several certain types. More precisely, we consider sections of the set of the solutions of an equation by (three-dimensional) hyperplanes perpendicular to the real axis; and we look for the sections being spheres. As for method of investigations, we pass from a quaternionic equation to a system of four real equations with four unknowns, that is, we equate coeﬃcients at the same quaternionic units after fulﬁlment arithmetical operations with the unknown and all parameters written in the form ξ0 + ξ1 i + ξ2 j + ξ3 k. This method may be called very simple and simultaneously it often leads to very complicated expressions. Nevertheless it was successfully used in [8], [3], [9], [4]. By the way, consideration of a section of the set of the solutions by a hyperplane perpendicular to the real axis now means a ﬁxed x0 in a system of equations, so that only x1 , x2 , x3 are considered as unknowns; just this point of view was used in [3] and [4]. In Section 3 we investigate an arbitrary quaternionic equation of the form ax2 + x2 b = c. This is a particular case of a quadratic quaternionic equation with only non-split squares. It was shown in [4] that for every such equation every above-mentioned section is of one of the following shapes: a linear manifold, a sphere, a circle, a set of two points, or the empty set. Now, according to the task of this paper, we look only for spheres, and in the main theorem of Section 3 we get necessary and suﬃcient conditions for such spherical section. Moreover a remark and a corollary at the end of the section provide some generalizations of the main theorem. In Section 4 we investigate some quaternionic quadratic equations with one split square, namely: m p() xq () = d, ax2 + x2 b + xcx + =1 where c ∈ R (to ensure that the split square is “really” split). In particular, here c = 0, while it is possible that a or b (or even both a and b) equals 0. Thus this type of equations is very wide. But by some simple reasonings performed in Section 4 we conclude that any equation of this type has no sphere in any from the considered sections. 3. Some quadratic equations with two non-split squares Theorem 1. Let a quaternionic equation of the following form be given: (1) ax2 + x2 b = c. For each real number ξ0 consider the set Sξ0 of such solutions x of (1) that x0 = ξ0 . Then Sξ0 can be a sphere only in the case ξ0 = 0. Namely, S0 is a sphere if and only if either c0 c1 c2 c3 (2) = = = < 0, a0 + b 0 a1 + b 1 a2 + b 2 a3 + b 3 36 D. Mierzejewski or these proportions and inequality hold true in wide sense, namely: for at most three values of ∈ {0, 1, 2, 3} it is allowed c = a + b = 0, and (2) must get true after deleting every fraction with zeros in both the numerator and the denominator. The centre of the corresponding sphere is always in the origin, and its radius equals −c /(a + b ) with any ∈ {0, 1, 2, 3} for which this fraction is well-defined. Proof. Direct calculations give the following system of real equations equivalent to (1): ⎧ (a0 + b0 )(x21 + x22 + x23 ) + 2(a1 + b1 )x0 x1 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2(a2 + b2 )x0 x2 + 2(a3 + b3 )x0 x3 = (a0 + b0 )x20 − c0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a1 + b1 )(x21 + x22 + x23 ) − 2(a0 + b0 )x0 x1 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2(a3 − b3 )x0 x2 + 2(b2 − a2 )x0 x3 = (a1 + b1 )x20 − c1 , ⎨ (3) ⎪ ⎪ ⎪ (a2 + b2 )(x21 + x22 + x23 ) + 2(b3 − a3 )x0 x1 − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ 2(a0 + b0 )x0 x2 + 2(a1 − b1 )x0 x3 = (a2 + b2 )x0 − c2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a3 + b3 )(x21 + x22 + x23 ) + 2(a2 − b2 )x0 x1 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2(b1 − a1 )x0 x2 − 2(a0 + b0 )x0 x3 = (a3 + b3 )x20 − c3 . We have to investigate this system treating x0 as a ﬁxed number, so that the unknowns are x1 , x2 , x3 and we consider shapes in the tree-dimensional space of points (x1 , x2 , x3 ). The equations from (3) have similar structure. About each of them it is essential to know whether its coeﬃcient at the expression x21 + x22 + x23 equals 0. If yes then the equation generates, as a rule, a plane, but sometimes the whole three-dimensional space (hyperplane) or the empty set; if no then the equation generates a sphere, a point, or the empty set. We are interesting in the cases where the intersection of the four ﬁgures generated by the equations is a sphere. Obviously, it occurs if and only if one of the following four situations takes place: 1) every equation generates the same sphere; 2) one equation generates the hyperplane, and each from other three equations generates the same sphere; 3) two equations generate the hyperplane, and each from other two equations generates the same sphere; 4) three equations generate the hyperplane, and other equation generates a sphere. Spheres in sets of solutions of quadratic quaternionic equations of some types 37 We will investigate every situation. But ﬁrstly we will rewrite every equation from (3) in such a way that it will be convenient to determine the centres and the radii of the spheres: ⎧ 2 2 2 a1 + b 1 a2 + b 2 a3 + b 3 ⎪ ⎪ x + x + x + x + x + x = ⎪ 1 0 2 0 3 0 ⎪ ⎪ a0 + b 0 a0 + b 0 a0 + b 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a0 + b0 )2 + (a1 + b1 )2 + (a2 + b2 )2 + (a3 + b3 )2 2 c0 ⎪ ⎪ x0 − , ⎪ 2 ⎪ (a + b ) a ⎪ 0 0 0 + b0 ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎪ a0 + b 0 a3 − b 3 b 2 − a2 ⎪ ⎪ − x + x + x + x + x = x 1 0 2 0 3 0 ⎪ ⎪ a1 + b 1 a1 + b 1 a1 + b 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a0 + b0 )2 + (a1 + b1 )2 + (a2 − b2 )2 + (a3 − b3 )2 2 c1 ⎪ ⎪ x0 − , ⎪ ⎪ ⎨ (a1 + b1 )2 a1 + b 1 (4) ⎪ 2 2 2 ⎪ ⎪ b 3 − a3 a0 + b 0 a1 − b 1 ⎪ ⎪ + x + x − x + x + x = x ⎪ 1 0 2 0 3 0 ⎪ ⎪ a2 + b 2 a2 + b 2 a2 + b 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a0 + b0 )2 + (a1 − b1 )2 + (a2 + b2 )2 + (a3 − b3 )2 2 c2 ⎪ ⎪ ⎪ x0 − , ⎪ 2 ⎪ (a + b ) a 2 2 2 + b2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ a2 − b 2 b 1 − a1 a0 + b 0 ⎪ ⎪ ⎪ + x + x + x + x − x = x 1 0 2 0 3 0 ⎪ ⎪ a3 + b 3 a3 + b 3 a3 + b 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 2 ⎪ ⎪ ⎩ (a0 + b0 ) + (a1 − b1 ) + (a2 − b2 ) + (a3 + b3 ) x20 − c3 . (a3 + b3 )2 a3 + b 3 Of course, each equation from (4) is well-deﬁned only if any denominator in it does not equal 0, but it is just a condition under which the equation can generate a sphere and cannot generate any hyperplane. Thus we have to use an equation from (4) when it generates a sphere, but one from (3) when it generates a hyperplane. Let us investigate ﬁrstly the case where all the four equations generate spheres. We have to look for the situation where these spheres are identical. It means that they have the same centre and the same radius. The coordinates of the centre are subtracted from x1 , x2 , x3 in the brackets in the left-hand side of an equation from (4), and the radius equals the square root of the right-hand side. By the way, the right-hand side has to be positive for the case of the sphere (if it is negative then the equation generates the empty set, and if it equals 0 then the equation generates a point). So, the following conditions arise: (5a) a0 + b 0 b 3 − a3 a2 − b 2 a1 + b 1 x0 = − x0 = x0 = x0 , a0 + b 0 a1 + b 1 a2 + b 2 a3 + b 3 (5b) a3 − b 3 a0 + b 0 b 1 − a1 a2 + b 2 x0 = x0 = − x0 = x0 , a0 + b 0 a1 + b 1 a2 + b 2 a3 + b 3 38 D. Mierzejewski a3 + b 3 b 2 − a2 a1 − b 1 a0 + b 0 x0 = x0 = x0 = − x0 , a0 + b 0 a1 + b 1 a2 + b 2 a3 + b 3 (5c) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (5d) (a0 + b0 )2 + (a1 + b1 )2 + (a2 + b2 )2 + (a3 + b3 )2 2 c0 x0 − (a0 + b0 )2 a0 + b 0 = (a0 + b0 )2 + (a1 + b1 )2 + (a2 − b2 )2 + (a3 − b3 )2 2 c1 x0 − 2 (a1 + b1 ) a1 + b 1 ⎪ ⎪ (a0 + b0 )2 + (a1 − b1 )2 + (a2 + b2 )2 + (a3 − b3 )2 2 c2 ⎪ ⎪ ⎪ = x0 − ⎪ 2 ⎪ (a2 + b2 ) a2 + b 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a + b0 )2 + (a1 − b1 )2 + (a2 − b2 )2 + (a3 + b3 )2 2 c3 ⎪ ⎩ = 0 x0 − > 0. (a3 + b3 )2 a3 + b 3 Note that the condition ∀ ∈ {0, 1, 2, 3} a + b = 0 (necessary for the case of four spheres) follows from (5a-d) because otherwise (5a-d) has no sense. Supposing that x0 = 0 we at once conclude from the ﬁrst equation of (5a-d) that a1 + b 1 a0 + b 0 =− , a0 + b 0 a1 + b 1 but it is impossible for real numbers. Therefore x0 = 0. Then (5a-d) can be suﬃciently reduced as follows: ⎧ ⎪ ⎨ x0 = 0, ⎪ ⎩ c1 c2 c3 c0 = = = < 0. a0 + b 0 a1 + b 1 a2 + b 2 a3 + b 3 This is just what is claimed in the theorem, and moreover it is easy to see that now the centre and the radius of the sphere are just as claimed in the theorem. Now we pass to the case where three equations generate spheres and one generates the hyperplane. Firstly let the ﬁrst equation of (3) generate the hyperplane. It means that a0 + b0 = 0 and the equation can be rewritten as 2x0 ((a1 + b1 )x1 + (a2 + b2 )x2 + (a3 + b3 )x3 ) = −c0 . Moreover the case of the hyperplane occurs if and only if every coeﬃcient of this linear equation equals 0, that is, x0 (a1 + b1 ) = x0 (a2 + b2 ) = x0 (a3 + b3 ) = c0 = 0. If x0 = 0 then a1 + b1 = a2 + b2 = a3 + b3 = 0, but this is in contradiction with the fact that the last three equations of (3) generate spheres. Thus we conclude that x0 = 0. Then, analogously to the previous case, taking into attention that the spheres have to be identical we get: c2 c3 c1 = = < 0, a1 + b 1 a2 + b 2 a3 + b 3 Spheres in sets of solutions of quadratic quaternionic equations of some types 39 that, along with the obtained in the previous paragraph equalities x0 = a0 + b0 = c0 = 0, means that the theorem again holds true in this case. The cases where the only second, third, or fourth equation of (3) generates the hyperplane are analogous. The most essential diﬀerence is the fact that supposing x0 = 0 one refutes the case of a sphere for only the ﬁrst equation, not for all the three. But the result is the same. Passing to the cases of two spheres and two hyperplanes we again obtain analogous pictures. The condition x0 = 0 arises every time due to the presence of a1 + b1 , a2 + b2 , and a3 + b3 in the ﬁrst equation and of a0 + b0 in every other equation. Other conditions are of the form a + b = am + bm = c = cm = 0, cn cp = <0 an + b n ap + b p and show that the theorem holds true. And at last the reader can easily verify that the cases of one sphere and three hyperplanes are also analogous. In these cases the conditions look as follows: x0 = a + b = am + bm = an + bn = c = cm = cn = 0, cp < 0. ap + b p By the way, here is already no need to equate the radii (since here is only one sphere) and the last inequality may be rewritten also as cp (ap + bp ) < 0. Remark 1. In fact Theorem 1 provides also information about every quaternionic equation of the form (6) αx2 β + γx2 δ = λ, because (6) can be easily reduced to the form (1) by multiplication by β −1 on the right and by γ −1 on the left (and if β or γ equals 0 then the corresponding multiplication is simply unnecessary). Corollary 1. Let a quaternionic equation of the following form be given: (7) ax2 + x2 b + apx + xpb + axp + pxb = q. For each real number ξ0 consider the set Sξ0 of such solutions x of (7) that x0 = ξ0 . Then Sξ0 can be a sphere only in the case ξ0 = −p0 . Moreover putting c := q + ap2 + p2 b one obtains that S−p0 is a sphere if and only if either (2) holds true, or those proportions and inequality hold true in wide sense, namely: for at most three values of ∈ {0, 1, 2, 3} it is allowed c = a +b = 0, and (2) must get true after deleting every 40 D. Mierzejewski fraction with zeros in both the numerator and the denominator. The centre of the corresponding sphere is always in the point −p, and its radius equals −c /(a + b ) with any ∈ {0, 1, 2, 3} for which this fraction is well-defined. Proof. The point is that if one changes x by x + p in (1) then one gets just (7) with q = c − ap2 − p2 b. So, the set of the solutions of (7) can be obtained by the corresponding shifting of the set of the solutions of (1), and that is all. 4. Some quadratic equations with one split square The main result of this section is Theorem 2 situated at the end. Every other proposition of this section gives information about a particular case of an equation considered in Theorem 2. There is no formal reason to prove other propositions before the main theorem, but we do this for convenience of the reader; namely, it is easier ﬁrstly to understand a proof with more or less simple expressions and then to think over how it changes after a certain complicating of the expressions. Proposition 1. Let a quaternionic equation of the following form be given: (8) xax = b, where a ∈ R. For each real number ξ0 consider the set Sξ0 of such solutions x of (8) that x0 = ξ0 . Then for any ξ0 ∈ R Sξ0 is not a sphere. Proof. Direct calculations give the following system of real equations equivalent to (8): ⎧ −a0 x21 − a0 x22 − a0 x23 − 2a1 x0 x1 − 2a2 x0 x2 − 2a3 x0 x3 = b0 − a0 x20 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −a x2 + a x2 + a x2 + 2a x x − 2a x x − 2a x x = b − a x2 , ⎪ ⎨ 1 1 1 2 1 3 0 0 1 2 1 2 3 1 3 1 1 0 (9) ⎪ ⎪ ⎪ a2 x21 − a2 x22 + a2 x23 + 2a0 x0 x2 − 2a1 x1 x2 − 2a3 x2 x3 = b2 − a2 x20 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a3 x21 + a3 x22 − a3 x23 + 2a0 x0 x3 − 2a1 x1 x3 − 2a2 x2 x3 = b3 − a3 x20 . Obviously, with any ﬁxed x0 each equation of (9) generates a surface of at most second degree (in particular, it may be the empty set) or the whole hyperplane (in very special cases). Obviously, a sphere in the intersection of such ﬁgures can arise only if at least one of these ﬁgures is a sphere; moreover for this aim each equation has to generate either a sphere, or the hyperplane. From simple geometrical considerations we see that among these for equations only the ﬁrst one can generate a sphere. Therefore in order to obtain a sphere in the intersection it is necessary to obtain the hyperplane by every equation excepting the Spheres in sets of solutions of quadratic quaternionic equations of some types 41 ﬁrst one. But it can occur only if every coeﬃcient from the equations equals 0. In particular it means that a1 = a2 = a3 = 0, but it is in contradiction with the fact that a ∈ R. (Note that here it is impossible to get a0 = 0 because a0 appears only along with x0 , which is considered as a constant now.) So, any sphere in the intersection is impossible, and the proposition is proved. Proposition 2. Let a quaternionic equation of the following form be given: (10) xax + m p() xq () = b, =1 where a ∈ R. For each real number ξ0 consider the set Sξ0 of such solutions x of (10) that x0 = ξ0 . Then for any ξ0 ∈ R Sξ0 is not a sphere. Proof. Constituting the system of real equations equivalent to (10) one gets a system diﬀering from (9) by only presence of some additional terms being linear with respect to x0 , x1 , x2 , x3 . It is easy to verify that under the circumstances all considerations from the proof of Proposition 1 work as well for the current proof. Proposition 3. Let a quaternionic equation of the following form be given: (11) ax2 + x2 b + xcx = d, where c ∈ R. For each real number ξ0 consider the set Sξ0 of such solutions x of (11) that x0 = ξ0 . Then for any ξ0 ∈ R Sξ0 is not a sphere. Proof. Now the corresponding system of real equations is the following: ⎧ −(a0 + b0 + c0 )(x21 + x22 + x23 ) − 2(a1 + b1 + c1 )x0 x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − 2(a2 + b2 + c2 )x0 x2 − 2(a3 + b3 + c3 )x0 x3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = d0 − (a0 + b0 + c0 )x20 , ⎨ (12a) ⎪ 2 2 2 ⎪ ⎪ ⎪ −(a1 + b1 + c1 )x1 − (a1 + b1 − c1 )(x2 + x3 ) − 2c2 x1 x2 − 2c3 x1 x3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 2(a0 + b0 + c0 )x0 x1 + 2(b3 − a3 )x0 x2 + 2(a2 − b2 )x0 x3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ = d1 − (a1 + b1 + c1 )x20 , 42 D. Mierzejewski ⎧ −(a2 + b2 + c2 )x22 − (a2 + b2 − c2 )(x21 + x23 ) − 2c1 x1 x2 − 2c3 x2 x3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 2(a3 − b3 )x0 x1 + 2(a0 + b0 + c0 )x0 x2 + 2(b1 − a1 )x0 x3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = d2 − (a2 + b2 + c2 )x20 , ⎨ (12b) ⎪ ⎪ ⎪ −(a3 + b3 + c3 )x23 − (a3 + b3 − c3 )(x21 + x22 ) − 2c1 x1 x3 − 2c2 x2 x3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 2(b2 − a2 )x0 x1 + 2(a1 − b1 )x0 x2 + 2(a0 + b0 + c0 )x0 x3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ = d3 − (a3 + b3 + c3 )x20 . Analogously to the proof of Proposition 1 we can see that for the presence of a sphere in the section each equation of (12a, b) (with a ﬁxed x0 ) has to generate either a sphere, or the whole hyperplane. Supposing that the second equation generates a sphere we obtain: a 1 + b 1 + c1 = a 1 + b 1 − c1 , c2 = c3 = 0, but it is the same that c1 = c2 = c3 = 0, so that c ∈ R, a contradiction with the condition of the proposition. Analogously it is easy to show that the third and fourth equations cannot generate any sphere. So, the only possibility to get a sphere in the intersection is to provide for all the last three equations to generate the hyperplane. But for this aim all coeﬃcients of the corresponding polynomials (with respect to x1 , x2 , x3 ) have to equal 0; in particular, we obtain again the impossible equalities c1 = c2 = c3 = 0. So, it is impossible to obtain any sphere in the intersection, and the proposition is proved. Theorem 2. Let a quaternionic equation of the following form be given: (13) ax2 + x2 b + xcx + m p() xq () = d, =1 where c ∈ R. For each real number ξ0 consider the set Sξ0 of such solutions x of (13) that x0 = ξ0 . Then for any ξ0 ∈ R Sξ0 is not a sphere. Proof. Analogously to the proof of Proposition 2, it is easy to understand that this theorem can be proved by practically the same considerations as in the proof of Proposition 3 (that is, additional linear terms do not disturb). Spheres in sets of solutions of quadratic quaternionic equations of some types 43 References [1] S. Eilenberg, I. Niven, The “fundamental theorem of algebra” for quaternions, Bull. Amer. Math. Soc. 50, no. 4 (1944), 246–248. [2] D. Janovská, G. Opfer, Linear equations in quaternionic variables, Mitt. Math. Ges. Hamburg 27, (2008), 223–234. [3] D. Mierzejewski, Investigation of quaternionic quadratic equations I. Factorization and passing to a system of real equations, Bull. Soc. Sci. Lettres L ódź. 58 Sér. Rech. Déform. 56 (2008), 17–26. [4] D. Mierzejewski, Quasi-spherical and multi-quasi-spherical polynomial quaternionic equations: introduction of the notions and some examples, Advances in Applied Clifford Algebras, submitted. [5] D. Mierzejewski, V. Szpakowski, On solutions of some types of quaternionic quadratic equations, Bull. Soc. Sci. Lettres L ódź. 58 Sér. Rech. Déform. 55 (2008), 49–58. [6] I. Niven, The roots of a quaternion, Amer. Math. Monthly 49 (1942), 386–388. [7] A. Pogorui, M. Shapiro, On the structure of the set of the zeros of quaternionic polynomials, Complex Variables and Elliptic Equations 49, no. 6 (2004), 379–389. [8] V. Szpakowski, Solution of general linear quaternionic equations, The XI Kravchuk International Scientiﬁc Conference, Kyiv (Kiev), Ukraine, 2006, p. 661 [in Ukrainian]. [9] V. Szpakowski, Solution of quadratic quaternionic equations, Bull. Soc. Sci. Lettres L ódź. Sér. Rech. Déform. 60 (2010) – to appear. Teatralna Street, 5-b, ﬂat 6 UA-10-014 Zhytomyr Ukraine e-mail: [email protected] Chair od Mathematics Kielce University of Technology (Politechnika Świȩtokrzyska) Al. Tysia̧clecia Państwa Polskiego 7, bud. C PL-25-314 Kielce, Poland Presented by Julian L awrynowicz at the Session of the Mathematical-Physical Commission of the L ódź Society of Sciences and Arts on March 2, 2010 SFERY W ZBIORACH ROZWIA̧ZAŃ ROWNAŃ KWADRATOWYCH KWATERNIONOWYCH NIEKTORYCH TYPÓW Streszczenie Badamy zbiory rozwia̧zań równań kwadratowych kwaternionowych niektórych typów metoda̧ przekrojów hiperplaszczyznami prostopadlymi do osi rzeczywistej. Mianowicie, szukamy sfer w takich przekrojach. Otrzymujemy warunki konieczne i wystarczaja̧ce dla sferyczności takich przekrojów dla równania kwaternionowego postaci ax2 + x2 b = c oraz (jako prosty wniosek) postaci ax2 + x2 b + apx + xpb + axp + pxb = q. Udowadniamy, że dla jakiegokolwiek równania kwaternionowego postaci ax2 + x2 b + xcx + m =1 z c ∈ R żaden taki przekrój nie jest sferyczny. p() xq () = d PL ISSN 0459-6854 BULLETIN DE LA SOCIÉTÉ DES SCIENCES ET 2010 Recherches sur les déformations DES LETTRES DE L ÓDŹ Vol. LX no. 1 pp. 45–65 Andrzej Krzysztof Kwaśniewski COBWEB POSETS AND KoDAG DIGRAPHS ARE REPRESENTING NATURAL JOIN OF RELATIONS, THEIR DI-BIGRAPHS AND THE CORRESPONDING ADJACENCY MATRICES Summary Natural join of di-bigraphs (directed bi-parted graphs) and their corresponding adjacency matrices is deﬁned and then applied to investigate the so-called cobweb posets and their Hasse digraphs called KoDAGs. KoDAGs are special orderable Directed Acyclic Graphs which are cover relation digraphs of cobweb posets introduced by the author few years ago. KoDAGs appear to be distinguished family of F errers digraphs which are natural join of a corresponding ordering chain of one direction directed cliques called di-bicliques. These digraphs serve to represent faithfully corresponding relations of arbitrary arity so that all relations of arbitrary arity are their subrelations. Being this chain − way complete (compare with Kompletne, Kuratowski Kn,m bipartite graphs) their DAG denotation is accompanied with the letter K in front of descriptive abbreviation oDAG. The way to join bipartite digraphs of binary into multi-ary relations is the natural join operation either on relations or their digraph representatives. This natural join operation is denoted here by ⊕→ symbol deliberately referring – in a reminiscent manner – to the direct sum ⊕ of adjacency matrices as it becomes the case for disjoint di-bigraphs. 1. Introduction to cobweb posets 1.1. Notation One may identify and interpret some classes of digraphs in terms of their associated posets. (see [1] Interpretations in terms of posets Section 9). Deﬁnition 1 (see [1]). Let D = (Φ, ≺) be a digraph. w, v ∈ Φ are said to be equivalent iﬀ there exists a directed path containing both w and v vertices. We then write: v ∼ w for such pairs and denote by [v] the ∼ equivalence class of v ∈ Φ. 46 A. K. Kwaśniewski Deﬁnition 2 (see [1]). The poset P (D) associated to D = (Φ, ≺) is the poset P (D) = (Φ/ ∼, ≤) where [v] ≤ [w] iﬀ there exists a directed path from a vertex x ∈ [v] to a vertex y ∈ [w]. The graded digraphs case: If D = (Φ, ≺) is graded digraph then D = (Φ, ≺) is necessarily acyclic. Then no two elements of D = (Φ, ≺) are ∼ equivalent and thereby P (D) = (Φ/ ∼, ≤) associated to D = (V, ≺) is equivalent to: P (D) ≡ (Φ, ≤) = transitive, reﬂexive closure of D = (Φ, ≺). The cobweb posets where introduced in several paper (see [2–6] and references therein) in terms of their poset [Hasse] diagrams. Here we deliver their equivalent deﬁnition preceded by preliminary notation and nomenclature. Notation: nomenclature, di-bicliques and natural join In order to proceed proﬁciently we adopt the following. Deﬁnition 3. A digraph D = (Φ, ≺·) is transitive irreducible iﬀ its transitive reduction (D) equals D. Deﬁnition 4. A poset P (D) = (Φ, ≤) is associated to a graded digraph D = (Φ, ≺) iﬀ P (D) is the transitive, reﬂexive closure of D = (Φ, ≺). Obvious D = (Φ, ≺ ·) is transitive irreducible iﬀ its transitive reduction (D) = D iﬀ D = (Φ, ≺·) is Hasse diagram of the poset P (D) = (Φ, ≤) associated to D ≡ D = (Φ, ≺·) is cover relation ≺· digraph ≡ D = (Φ, ≺·) is P (D) = (Φ, ≤) poset diagram. 1.2. Further on we adopt also the following nomenclature We shall use until stated otherwise the convention: N = {1, 2, ..., k, ...}. n ∈ N ∪{∞}. The Cartesian product Φ1 × ... × Φk of pairwise disjoint sets Φ1 , ..., Φk is a k-ary relation, called sometimes the universal relation and here now on Kompletna relation or K-relation, (in Polish this means complete). The purpose of introducing the letter K is to distinguish in what follows [for k = 2] from complete digraphs notions established content. Convention 1. [identiﬁcation]. The binary relation E ⊆ X × Y is being here identiﬁed with its bipartite digraph representation B = (X ∪ Y, E). → Notation Km,n ≡ B = (X ∪ Y, E) if |X| = m, |Y | = n. Colligate with Kuratowski and Km,n . Comment 1. Complete n-vertex graphs for which all pairs of vertices are adjacent are denoted by Kn , The letter K had been chosen in honor of Professor Kazimierz Kuratowski, a Cobweb posets and KoDAG digraphs are representing natural join of relations 47 distinguished pioneer in graph theory. The corresponding two widely used concepts for digraphs are called complete digraphs or complete symmetric digraph in which every two diﬀerent vertices are joined by an arc and complete oriented graphs i.e. tournament graphs. The binary K-relation E = X × Y equivalent to bipartite digraph B = (X ∪ → Y, E) ≡Km,n is called from now on a di-biclique following [6]. Example of di-bicliques obtained from bicliques: See Fig. 1. If you imagine arrows → left to the right - you would see two examples of dibicliques Fig. 1: Examples of di-bicliques if edges are replaced by arrows of join direction if you imagine arrows ← right to the left, you would see another examples of dibicliques. Convention 2. [recall] The binary relation E ⊆ X ×Y is identiﬁed with its bipartite digraph B = (X ∪ Y, E) unless otherwise denoted distinctively deliberately. The natural join The natural join operation is a binary operation like Θ operator in computer science denoted here by ⊕→ symbol deliberately referring – in a quite reminiscent manner – to direct sum ⊕ of adjacency Boolean matrices and – as a matter of fact and in eﬀect – to direct the sum ⊕ of corresponding biadjacency [reduced] matrices of digraphs under natural join. ⊕→ is a natural operator for sequences construction. ⊕→ operates on multi-ary relations according to the scheme: (n + k)ary ⊕→ (k + m)ary = (n + k + m)ary . For example: (1 + 1)ary ⊕→ (1 + 1)ary = (1 + 1 + 1)ary , binary ⊕→ binary = ternary. Accordingly an action of ⊕→ on these multi-ary relations’ digraphs adjacency matrices is to be designed soon in what follows. Domain-Codomain F -sequence condition dom(Rk+1 ) = ran(Rk ), k = 0, 1, 2, ... Consider any natural number valued sequence F = {Fn }n≥0 . Consider then any chain of binary relations deﬁned on pairwise disjoint ﬁnite sets with cardinalities appointed by F -sequence elements values. For that to start we specify at ﬁrst a relations’ domain-co-domain F -sequence. 48 A. K. Kwaśniewski Domain-Codomain F -sequence (|Φn | = Fn ) Φ0 , Φ1 , ...Φi , ... Φk ∩ Φn = ∅ for k = n, |Φn | = Fn ; i, k, n = 0, 1, 2, ... n Let Φ = k=0 Φk be the corresponding ordered partition [anticipating-Φ is the vertex set of D = (Φ, ≺· ) and its transitive, reﬂexive closure (Φ, ≤)]. Impose dom(Rk+1 ) = ran(Rk ) condition, k ∈ N ∪ {∞}. What we get is binary relations chain. Deﬁnition 5 [Relation‘s chain]. Let Φ = nk=0 Φk , Φk ∩ Φn = ∅ for k = n be the ordered partition of the set Φ. Let a sequence of binary relations be given such that R0 , R1 , ..., Ri , ..., Ri+n , ..., Rk ⊆ Φk × Φk+1 , dom(Rk+1 ) = ran(Rk ). Then the sequence Rk k≥0 is called natural join (binary) relation’s chain. Extension to varying arity relations’ natural join chains is straightforward. As necessarily dom(Rk+1 ) = ran(Rk ) for relations’ natural join chain any given binary relation’s chain is not just a sequence; therefore we use “link to link” notation for k, i, n = 1, 2, 3, ... ready for relational data basis applications: R0 ⊕→ R1 ⊕→ ...⊕→ Ri ⊕→ ...⊕→ Ri+n , ...is an F − chain of binary relations where ⊕ → denotes natural join of relations as well as both natural join of their bipartite digraphs and the natural join of their representative adjacency matrices (see Section 3). Relation’s F -chain naturally represented by [identiﬁed with] the chain of their bipartite digraphs R0 ⊕→ R1 ⊕→ ...⊕→ Ri ⊕→ ...⊕→ Ri+n , ... ⇔ ⇔ B0 ⊕→ B1 ⊕→ ...⊕→ Bi ⊕→ ...⊕→ Bi+n , ... results in F -partial ordered set Φ, ≤ with its Hasse digraph representation looking-like speciﬁc “cobweb” image [see ﬁgures below]. 1.3. Partial order ≤ The partial order relation ≤ in the set of all points-vertices is determined uniquely by the above equivalent F -chains. Let x, y ∈ Φ = nk=0 Φk and let k, i = 0, 1, 2, ... . Then (1) c R c i+k−1 )y x ≤ y ⇔ ∀x∈Φ : x ≤ x ∨ Φi x < y ∈ Φi+k if f x(Ri ... c stays for [Boolean] composition of binary relations. where “” Relation (≤) deﬁned equivalently: x ≤ y in (Φ, ≤) iﬀ either x = y or there exist a directed path from x to y; x, y ∈ Φ. Let now Rk = Φk × Φk+1 , k ∈ N ∪ {0}. For “historical” reasons [2–6] we shall call such partial ordered set Π = Φ, ≤ the cobweb poset as theirs Hasse digraph representation looks like speciﬁc “cobweb” image (imagine and/or draw also their transitive and reﬂexive cover digraph Φ, ≤. Cobweb? Super-cobweb!... – with fog droplets loops?). Cobweb posets and KoDAG digraphs are representing natural join of relations 49 1.4. Cobweb posets (Π = Φ, ≤) Convention 3. [recall]. The binary relation E ⊆ X×Y is identiﬁed with its bipartite → digraph B = (X ∪ Y, E) ≡Km,n where |X| = m, |Y | = n. Deﬁnition 6 [cobweb poset]. Let D = (Φ, ≺·) be a transitive irreducible digraph. Let n ∈ N ∪ {∞}. Let D be a natural join D = ⊕→nk=0 Bk of di-bicliques Bk = (Φk ∪ Φk+1 , Φk × Φk+1 ), n ∈ N ∪ {∞}. Hence the digraph D = (Φ, ≺·) is graded. The poset Π(D) associated to this graded digraph D = (Φ, ≺·) is called a cobweb poset. Convention 4. In a case we want to underline that we deal with ﬁnite cobweb poset (a subposet of appropriate – for example inﬁnite F -cobweb poset Π(D)) we shall use a subscript and write Pn . See: [2–6], [10], [13], [18]. Comment 2. Graded graph is a natural join of bipartite graphs that form a chain of consecutive levels [i.e. graded graphs’ antichains]. Graded digraph is a natural join of bipartite digraphs that form a chain of consecutive levels [i.e. graded digraphs’ antichains]. Comment 3. (Deﬁnition 6. Recapitulation in brief.) Cobweb poset is the poset Π = Φ, ≤, where n Φ= and k=0 ≺· = ⊕→n−1 k=0 Φk × Φk+1 , n ∈ N ∪ {∞}. Cobweb poset is the poset Π = Φ, ≤, where Φ = n k=0 and → ≺· = ⊕→n−1 k=0 Kk,k+1 , n ∈ N ∪ {∞}, where ≤ is the transitive, reﬂexive cover of ≺·. Comment 4. (F -partial ordered set) Cobweb poset Π = Φ, ≤ is naturally graded and sequence F – denominated thereby we call it sometimes F -partial ordered set Φ, ≤. 2. Dimension of cobweb posets – revisited 2.1. oDAG [7] Observation 1. [cobwebs are oDAGs]. In [2] it was observed that cobweb posets’ Hasse diagrams are the members of the so-called oDAGs family i.e. cobweb posets’ 50 A. K. Kwaśniewski Hasse diagrams are orderable Directed Acyclic Graphs which is equivalent to say that the associated poset P (D) = (Φ, ≤) of D = (Φ, ≺·) of is of dimension 2. Recall: DAGs – hence graded digraphs with minimal elements always might be considered – up to digraphs isomorphism – as natural digraphs [8] i.e. digraphs with natural labeling (i.e. xi < xj ⇒ i < j). Deﬁnition 7 [Plotnikov – see [7], [2] and then below]. A digraph D = (Φ, ≺) is called the orderable digraph (oDAG) if there exists a dim 2 poset such that its Hasse diagram coincides with the digraph G . The statement from [2] may be now restated as follows: Observation 2. [oDAG]. Cobweb P (D) = (Φ, ≤) posets’ Hasse diagrams D = (Φ, ≺·) are oDAGs. Proof. Obvious. Cobweb posets are posets with minimal elements set Φ0 . Cobweb posets Hasse diagrams are DAGs. Cobweb posets representing the natural join of are then dim 2 posets as their Hasse digraphs are intersection of a natural labeling linear order L1 and its “dual” L2 denominated correspondingly in a standard way by: L1 = natural labeling: chose for the topological ordering L1 the labeling of minimal elements set Φ0 with labels 1, 2, ..., from the left to the right (see Fig. 2) then proceed up to the next level Φ1 and continue the labeling “→” from the left to the right [Φ1 is now treated as the set o minimal elements if Φ0 is removed] and so on. Apply the procedure of subsequent removal of minimal elements i.e. removal of subsequent labeled levels Fk – labeling the vertices along the levels from the left to the right. L2 = “dual” natural labeling: chose for the topological ordering L2 the labeling of minimal elements set F0 with labels 1, 2, ..., from the right to the left to (see Fig. 1) then proceed up to the next level F1 and continue the labeling “←” from the right to the left [Φ1 is now treated as the set o minimal elements if Φ0 is removed] and so on. Apply the procedure of subsequent removal of minimal elements i.e. removal of subsequent labeled levels Φk – labeling now the vertices along the levels from the right to the left q.e.d. 2.2. Brief history of the short oDAG’s name life On the history of oDAG nomenclature with David Halitsky and others input one is expected to see more in [15]. See also the December 2008 subject of The Internet Gian Carlo Rota Polish Seminar (http://ii.uwb.edu.pl/akk/sem/sem rota.htm). Here we present its sub-history leading the author to note that cobweb posets are oDAGs. According to Anatoly Plotnikov the concept and the name of oDAG was introduced by David Halitsky from Cumulative Inquiry in 2004. Cobweb posets and KoDAG digraphs are representing natural join of relations 51 oDAG-2004 (Plotnikov) Quote 1. A digraph G ∈ Dn will be called orderable (oDAG) if there exists are dim 2 poset such that its Hasse diagram coincide with the digraph G. The Quote 1 comes from [9] in [2] i.e. from A. D. Plotnikov, A formal approach to the oDAG/POSET problem (2004) html://www.cumulativeinquiry.com/Problems/ solut2.pdf (submitted to publication – March 2005). The quote of the Quote 1 is to be found in [9]: oDAG-2005 [2] Quote 2. A digraph G is called the orderable digraph (oDAG) if there exists a dim 2 poset such that its Hasse diagram coincides with the digraph G [2]. oDAG-2006 [7] Quote 3. A digraph G is called the orderable if there exists a dim 2 poset such that its Hasse diagram coincides with the digraph G [7]. For further use of oDAG nomenclature see [6], and references therein. For further references and recent results on cobweb posets see [10] and [11]. Deﬁnition 8 [KoDAG]. The transitive and reﬂexive reduction of cobweb poset Π = Φ, ≤ i.e. posets’ Π cover relation digraph [Hasse diagram] D = (Φ, ≺·) is called KoDAG; see [11–14]. Comment 5. Apply Comment 1. Why do we stick to call KoDAGs graded digraphs with associated poset Π = Φ, ≤ the orderable DAGs on their own independently of the nomenclature quoted? Let D = (Φ, ≺·) denote any transitive irreducible DAG [for example any graded digraph including KoDAG digraph for example as above]. Let a poset P (D) = (Φ, ≤) be associated to D = (Φ, ≺·). Deﬁnition 9 [Ferrers dimension]. We say that the poset P (D) = (Φ, ≤) is of Ferrers dimension k iﬀ it is associated to D = (Φ, ≺·) of Ferrers dimension k. Observation 3. [Ferrers dimension]. Cobweb posets are posets of Ferrers dimension equal to one. Proof. Apply any of many characterizations of Ferrers digraphs to see that cobweb posets are posets’ cover relation digraphs [Hasse diagrams] are Ferrers digraphs. For example consult Section 3 and see that biadjacency matrix does not contain any of two 2 × 2 permutation matrices. Comment 6. Any KoDAG digraph D = (Φ, ≺ ·) is the digraph stable under the transitive and reﬂexive reduction i.e. [“irreducible”] Hasse portrait of Ferrers relation ≺·. The positions of 1’s in biadjacecy [reduced adjacency] matrix display the support of Ferrers relation ≺·. D = (Φ, ≺·) is then interval order relation digraph. The digraph (Φ, ≤) of the cobweb poset P (D) = (Φ, ≤) associated to KoDAG 52 A. K. Kwaśniewski digraph D = (Φ, ≤) is the portrait of Ferrers relation ≤. The positions of 1’s in biadjacecy [reduced adjacency] matrix display the support of Ferrers relation ≤. Note: for F -denominated cobweb posets the nomenclature identiﬁes: biajacency [reduced adjacency] matrix ≡ zeta matrix i.e. the incidence matrix ζF of the F - poset (see: Fig. ζN and Fig. ζF ). Recall that this F -partial ordered set Φ, ≤ is a natural join of F -chain of binary K-relations (complete or universal relations as called sometimes). → These relations are represented by di-bicliques Kk,k+1 which are on their own the Ferrers dimension one digraphs. As for the other – not necessarily K-relations’ chains we may end up with Ferrers or not digraphs in corresponding di-bigraphs’ chain. See below, then Section 4 and more in [15]. 3. The natural join ⊕→ operation We deﬁne here the adjacency matrices representation of the natural join ⊕→ operation. 3.1. Recall Let D(R) = (V (R) ∪ W (R), E(R)) ≡ (V ∪ W, E); V ∩ W = ∅, E(R) ⊆ V × W . Let D(R) denotes here down the bipartite digraph of binary relation R with dom(R) = V and rang(R) = W . Colligate with the anticipated examples → R = Rk ⊆ Φk × Φk+1 ≡Kk,k+1 , V (R) ∪ W (R) = Φk ∪ Φk+1 . 3.2. The adjacency matrices and their natural join The adjacency matrix A of a bipartite graph with biadjacency (reduced adjacency [16]) matrix B is given by 0 B A= . BT 0 Deﬁnition 10. The adjacency matrix A[D] of a bipartite digraph D(R) = (P ∪L, E ⊆ P × L) with biadjacency matrix B is given by 0k,k B(k × m) . A[D] = 0m,m 0m,k where k = |P |, m = |L|. c = composition of binary relations S and Convention 5. S R c S = BR B R ⇔ BRS c where (|V | = k, |W | = m) BR (k × m) ≡ BR is the (k × m). biadjacency [or another name: reduced adjacency] matrix of the bipartite relac apart from relations composition denotes also Boolean tions’ R digraph B(R) and Cobweb posets and KoDAG digraphs are representing natural join of relations 53 multiplication of these rectangular biadjacency Boolean matrices BR , BS . What is their form? The answer is in the block structure of the standard square (n × n) adjacency matrix A[D(R)]; n = k + m. The form of standard square adjacency matrix A[G(R)] of bipartite digraph D(R) has the following apparently recognizable block reduced structure [Os×s stays for (k × m) zero matrix]: Ok×k AR (k × m) A[D(R)] = . Om×k Om×m Let D(S) = (W (S) ∪ T (S), E(S)); W ∩ T = ∅, E(S) ⊆ W × T ; (|W | = m, |T | = s); hence Om×m AS (m × s) A[D(S)] = . Os×m Os×s Deﬁnition 11 [natural join condition]. The ordered pair of matrices A1 , A2 is said to satisfy the natural join condition iﬀ these matrices have the block structure of A[D(R)] and A[D(S)] as above i.e. iﬀ they might be identiﬁed accordingly: A1 = A[D(R)] and A2 = A[D(S)]. Correspondingly, if two given digraphs G1 and G2 are such that their adjacency matrices A1 = A[G1 ] and A2 = A[G2 ] do satisfy the natural join condition, we shall say that G1 and G2 satisfy the natural join condition. For matrices satisfying the natural join condition one may deﬁne what follows. c → First we deﬁne the Boolean reduced or natural join composition and secondly the natural join ⊕→ of adjacent matrices satisfying the natural join condition. c Deﬁnition 12 (→ composition). c c A[D(RS)] =: A[D(R)]→ A[D(S)] = Ok×k Os×k ARS c (k × s) Os×s c S (m × s). where ARS c (k × s) = AR (k × m)A According to the scheme: c [(m + s) × (m + s)] = [(k + s) × (k + s)]. [(k + m) × (k + m)]→ Comment 7. The adequate projection makes out the intermediate, joint in common dom(S) = rang(R) = W , |W | = m. c The above Boolean reduced composition → of adjacent matrices technically reduces then to the calculation of just Boolean product of the reduced rectangular adjacency matrices of the bipartite relations‘ graphs. We are however now in need of the Boolean natural join product ⊕→ of adjacent matrices already announced at the beginning of this presentation. Let us now deﬁne it. 54 A. K. Kwaśniewski As for the natural join notion we aim at the morphism correspondence: S⊕→ R ⇔ MS⊕→R = MR ⊕→ MS , where S⊕→ R = natural join of binary relations S and R while MS⊕→R = MR ⊕→ MS = natural join of standard square adjacency matrices (with customary convention: M [G(R)] ≡ MR adapted). Attention: recall here that the natural join of the above binary relations R⊕→ S is the ternary relation – and on one results in k-ary relations if with more factors undergo the ⊕→ product. As a matter of fact ⊕→ operates on multi-ary relations according to the scheme: (n + k)ary ⊕→ (k + m)ary = (n + k + m)ary . For example: (1 + 1)ary ⊕→ (1 + 1)ary = (1 + 1 + 1)ary , binary⊕→ binary = ternary. Technically – the natural join of the k-ary and n-ary relations is deﬁned accordingly the same way via ⊕→ natural join product of adjacency matrices – the adjacency matrices of these relations’ Hasse digraphs. With the notation established above we ﬁnally deﬁne the natural join ⊕→ of two adjacency matrices as follows: Deﬁnition 13. Natural join ⊕→ of biadjacency matrices: = A[D(R⊕→ S)] =: A[D(R)]⊕→ A[D(S)] = Om×m AS (m × s) Ok×k AR (k × m) ⊕→ = Om×k Om×m Os×m Os×s ⎤ ⎡ Ok×k AR (k × m) Ok×s = ⎣ Om×k Om×m AS (m × s) ⎦ . Os×s Os×k Os×m Comment 8. The adequate projection used in natural join operation lefts one copy of the joint in common “intermediate” submatrix Om×m and consequently lefts one copy of “intermediate” joint in common m according to the scheme: [(k + m) × (k + m)]⊕→ [(m + s) × (m + s)] = [(k + m + s) × (k + m + s)]. 3.3. The biadjacency matrices of the natural join of adjacency matrices Denote by B(A) the biadjacency matrix of the adjacency matrix A. Let A(G) denote the adjacency matrix of the digraph G, for example a di-biclique relation digraph. Let A(Gk ), k = 0, 1, 2, ... be the sequence adjacency matrices of the sequence Gk , k = 0, 1, 2, ... of digraphs. Let us identify B(A) ≡ B(G) as a convention. Deﬁnition 14 [digraphs natural join]. Let digraphs G1 and G2 satisfy the natural join condition. Let us make then the identiﬁcation A(G1 ⊕→ G2 ) ≡ A1 ⊕→ A2 as deﬁnition. The digraph G1 ⊕→ G2 is called the digraphs natural join of digraphs G1 and G2 . Note that the order is essential. We observe at once what follows. Cobweb posets and KoDAG digraphs are representing natural join of relations 55 Observation 4. B(G1 ⊕→ G2 ) ≡ B(A1 ⊕→ A2 ) = B(A1 ) ⊕ B(A2 ) ≡ B(G1 ) ⊕ B(G2 ). Comment 9. Observation 4 justiﬁes the notation ⊕→ for the natural join of relations digraphs and equivalently for the natural join of their adjacency matrices and equivalently for the natural join of relations that these are faithful representatives of. As a consequence we have Observation 5. B (⊕→ni=1 ) ≡ B[⊕→ni=1 A(Gi )] = ⊕ni=1 B[A(Gi )] ≡ diag(B1 , B2 , ..., Bn ) = ⎤ ⎡ B1 ⎥ ⎢ B2 ⎥ ⎢ ⎥ , n ∈ N ∪ {∞}. ⎢ B3 =⎢ ⎥ ⎦ ⎣ ... ... ... Bn 3.4. Applications Once any positive integer-valued sequence F = {Fn }n≥1 is being chosen its KoDAG digraph is identiﬁed with Hasse cover relation digraph. Its adjacency matrix AF is sometimes called Hasse matrix and is given in a plausible form and impressively straightforward way. Just use the fact that the Hasse digraph which is displaying cover relation ≺· is an F -chain of coined bipartite digraphs – coined each preceding with a subsequent one by natural join operator ⊕ → [resemblance of ⊕ → to direct matrix sum is not naive - compare “natural join” of disjoint digraphs with no common set of marked nodes (“attributes”) ]. Note: I(s×k) stays for (s×k) matrix of ones i.e. [I(s×k)]ij = 1; 1 ≤ i ≤ s, 1 ≤ j ≤ k. Let us start ﬁrst with F = {Fn }n≥1 = N ; see Fig. 2. Then its associated F partial ordered set Φ, ≤ has the following Hasse digraph displaying cover relation of the ≤ partial order. The Hasse matrix AN i.e. adjacency matrix of cover relation digraph i.e. adjacency matrix of the Hasse diagram of the N -denominated cobweb poset Φ, ≤ is given by upper triangular matrix AN of the form ⎡ AN ⎢ ⎢ =⎢ ⎢ ⎣ O1×1 O2×1 O3×1 O4×1 ...etc. I(1 × 2) O2×2 O3×2 O4×2 ... O1×∞ I(2 × 3) O3×3 O4×3 and ⎤ O2×∞ I(3 × 4) O3×∞ O4×4 I(4 × 5) O4×∞ so on... ⎥ ⎥ ⎥. ⎥ ⎦ One may see that the zeta function matrix of the F = N choice is geometrical series in AN i.e. the geometrical series in the poset Φ, ≤ Hasse matrix AN : 56 A. K. Kwaśniewski Fig. 2: Display of a ﬁnite subposet Π6 of the N natural numbers cobweb poset c ζ = (1 − AN )−1 . Explicitly: c c ≡ I∞ + AN + A2 ζ = (1 − AN )−1 N + ... = ⎡ ⎢ ⎢ =⎢ ⎢ ⎣ I1×1 O2×1 O3×1 O4×1 ...etc I(1 × ∞) I2×2 O3×2 O4×2 ... ⎤ I(2 × ∞) I3×3 I(3 × ∞) O4×3 I4×4 and so ⎥ ⎥ ⎥, ⎥ I(4 × ∞) ⎦ on... c . Indeed, let AN = A with Akij being the number of maximal ζ = (1 − AN )−1 k-chains (k > 0) from the x0 ∈ Φi to xk ∈ Φj , i.e. here 0 k = j − i 1 k = j − i, c = Akij = , and hence Ak ij j!/i! k = j − k 0 k= j − k, c c and Am are disjoint for k = m. and the supports (nonzero matrices blocks) of Ak Indeed: the entry in row i and column j of the inverse (I − A)−1 gives the number of directed paths from vertex xi to vertex xj . This can be seen from geometric series with adjacency matrix as an argument (I − A)−1 = I + A + A2 + A3 + ... taking care of the fact that the number of paths from i to j equals the number of paths of length 0 plus the number of paths of length 1 plus the number of paths of length 2, etc. c gives the Therefore the entry in row i and column j of the inverse (I − A)−1 answer whether there exist directed paths from a vertex i to vertex j (Boolean value 57 Cobweb posets and KoDAG digraphs are representing natural join of relations 1) or not (Boolean value 0), i.e. whether these vertices are comparable, i.e. whether xi < xj , or not. Remark: In the cases – Boolean poset 2N and the “Ferrand-Zeckendorf” poset of ﬁnite subsets of N without two consecutive elements considered in [17] one has c c ≡ I∞×∞ + A + A2 + ... ζ = exp[A] = (1 − A)−1 because in those cases 1 k 0 k = j − i 1 k =j−i c Aij = Ak , and hence = . Akij = ij k! k = j − k 0 k = j − k k! How does it go in our F -case? Just see For example: ⎡ O1×1 O1×2 I(1 × 3) ⎢ O2×1 O2×2 O2×3 ⎢ c ⎢ A2 O3×2 O3×3 N = ⎢ O3×1 ⎣ O4×1 O4×2 O4×3 ...etc. ... and c c c c 0 1 2 A2 N and then add AN ∨ AN ∨ AN ∨ ... . O1×∞ I(2 × 4) O3×4 O4×4 so ⎤ O2×∞ I(3 × 5) O3×∞ O4×5 I(4 × 6) O4×∞ on... ⎥ ⎥ ⎥, ⎥ ⎦ Consequently we arrive at the incidence matrix ζ = exp[AN ] for the positive integers cobweb poset displayed by Fig. 3. Note that incidence matrix ζ representing uniquely its corresponding cobweb poset exhibits (see below) a staircase structure of zeros above the diagonal which is characteristic to Hasse diagrams of all cobweb posets. ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 . 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 . 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 . 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 . 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 . 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 . 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 . 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 . 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 . 1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 . 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 . 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 . 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 . 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· . .··· ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Matrix ζN : The incidence matrix ζ for the natural numbers i.e. the N-cobweb poset. 58 A. K. Kwaśniewski Fig. 3: Display of the F -Fibonacci numbers cobweb poset Comment 9. The given F -denominated staircase zeros structure above on the diagonal of zeta matrix zeta is the unique characteristic of its corresponding F -KoDAG Hasse digraphs. For example see Fig. ζF . below (from [6]). ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 . 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 . 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 . 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 . 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 . 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 . 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 . 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 . 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 . 1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 . 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 . 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 . 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 . 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 1 ··· 0 ··· 0 ··· 1 ··· . .··· ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Matrix ζF : The incidence matrix ζ for the Fibonacci cobweb poset associated to F -KoDAG Hasse digraph. Cobweb posets and KoDAG digraphs are representing natural join of relations 59 The zeta matrix, i.e. the incidence matrix ζF for the Fibonacci numbers cobweb poset [F – KoDAG], determines completely its incidence algebra and corresponds to the poset with Hasse diagram displayed by the Fig. 3. The explicit expression for zeta matrix ζF via known blocks of zeros and ones for arbitrary natural numbers valued F -sequence is readily found due to brilliant mnemonic eﬃciency of the authors up-side-down notation (see Appendix in [13]). With this notation inspired by Gauss and the reasoning just repeated with “kF ” numbers replacing k – positive integers one gets, in the spirit of Knuth [18], the clean result: ⎡ ⎢ ⎢ AF = ⎢ ⎢ ⎣ 01F ×1F 02F ×1F 03F ×1F 04F ×1F ... I(1F × 2F ) 02F ×2F 03F ×2F 04F ×2F etc. 01F ×∞ I(2F × 3F ) 03F ×3F 04F ×3F ... ⎤ 02F ×∞ I(3F × 4F ) 03F ×∞ 04F ×4F I(4F × 5F ) 04F ×∞ and so on ... ⎥ ⎥ ⎥ ⎥ ⎦ and c c −1 ζF = exp ≡ I∞×∞ + AF + A2 c [AF ] ≡ (1 − AF ) F + ... = ⎡ ⎢ ⎢ =⎢ ⎢ ⎣ I1F ×1F O2F ×1F O3F ×1F O4F ×1F ... I(1F × ∞) I2F ×2F O3F ×2F O4F ×2F etc. ⎤ I(2F × ∞) I3F ×3F I(3F × ∞) O4F ×3F I4F ×4F I(4F × ∞) ... and so on ... ⎥ ⎥ ⎥. ⎥ ⎦ Comment 10. (ad “upside down notation”) Concerning Gauss and Knuth – see remarks in [18] on Gaussian binomial coeﬃcients. Observation 6. Let us denote by Φk → Φk+1 (see the papers quoted) the dibicliques denominated by subsequent levels Φk , Φk+1 of the graded F -poset P (D) = (Φ, ≤) i.e. levels Φk , Φk+1 of its cover relation graded digraph D = (Φ, ≺·) [Hasse diagram]. Then B (⊕→nk=1 Φk → Φk+1 ) = diag(I1 , I2 , ..., In ) = ⎡ ⎢ ⎢ =⎢ ⎢ ⎣ I(1F × 2F ) ⎤ I(2F × 3F ) ⎥ ⎥ ⎥, ⎥ ⎦ I(3F × 4F ) ... I(nF × (n + 1)F ) where Ik ≡ I(kF × (k + 1)F ), k = 1, ..., n and where – recall – I(s × k) stays for the (s × k)-matrix with [I(s × k)]ij = 1; 1 ≤ i ≤ s, 1 ≤ j ≤ k. and n ∈ N ∪ {∞}. 60 A. K. Kwaśniewski Observation 7. Consider bigraphs chain obtained from the above di-bicliques chain via deleting or no arcs making thus (if deleting arcs) some or all of the di-bicliques Φk → Φk+1 not di-bicliques; denote them by Gk . Let Bk = B(Gk ) denote their biadjacency matrices correspondingly. Then for any such F -denominated chain (hence any chain) of bipartite digraphs Gk the general formula reads: B (⊕→ni=1 Gi ) ≡ B[⊕→ni=1 A(Gi )] = ⊕ni=1 B[A(Gi )] ≡ diag(B1 , B2 , ..., Bn ) = ⎡ ⎢ ⎢ =⎢ ⎢ ⎣ ⎤ B1 ⎥ ⎥ ⎥, ⎥ ⎦ B2 B3 ... n ∈ N ∪ {∞}. Bn Observation 8. The F -poset P (G) = (Φ, ≤), i.e. its cover relation graded digraph m G = (Φ, ≺·) =⊕→ Gk k=0 is of Ferrers dimension one iﬀ in the process of deleting arcs from the cobweb poset Hasse diagram n D = (Φ, ≺·) =⊕→ Φk → Φk+1 , k=0 does not produce 2 × 2 permutation submatrices in any bigraphs Gk biadjacency matrix Bk = B(Gk ). Examples (ﬁnite subposets of cobweb posets) Fig. 4 and Fig. 5 display a Hasse diagram portraits of ﬁnite subposets of cobweb posets. In view of Observation 2 these subposets are naturally Ferrers digraphs i.e. of Ferrers dimension equal one. Fig. 4: Display of the subposet P5 of the F = Fibonacci sequence F -cobweb poset and σP5 subposet of the σ permuted Fibonacci F -cobweb poset. Cobweb posets and KoDAG digraphs are representing natural join of relations 61 Fig. 5: Display of the subposet P4 of the F = Gaussian integers sequence (q = 2) F -cobweb poset and σP4 subposet of the σ permuted Gaussian (q = 2) F -cobweb poset. 4. Summary 4.1. Principal – natural identiﬁcations Any KoDAG is a di-bicliques chain ⇔ Any KoDAG is a natural join of complete bipartite graphs [di-bicliques] = Φk , Ek ) ≡ D(Φ, E), (Φ0 ∪ Φ1 ∪ ... ∪ Φn ∪ ..., E0 ∪ E1 ∪ ... ∪ En ∪ ...) ≡ D( k≥0 k≥0 → Φk+1 ≡Kk,k+1 and E = k≥0 Ek . where Ek = Φk × Naturally, as indicated earlier, any graded posets’ Hasse diagram with ﬁnite width including KoDAGs is of the form Φk , Ek ) ⇔ Φ, ≤, D(Φ, E) ≡ D( k≥0 k≥0 → Φk+1 ≡Kk,k+1 where Ek ⊆ Φk × and the deﬁnition of ≤ from 1.3. is applied. In front of all the above presentation the following is clear. Observation 9. “Many” graded digraphs with ﬁnite width including KoDAGs D = (V, ≺·) encode bijectively their correspondent n-ary relation (n ∈ N ∪ {∞} as seen from its following deﬁnition: → Ek ⊆ Φk × Φk+1 ≡Kk,k+1 where (n-ary relation) n−1 n k=0 k=0 E(n) =⊕→ Ek ⊂ × Φk i.e. E(n)(n ∈ N {∞}) i.e. identiﬁed with the graded poset Vn , E obtained via, i.e. E(n) is an from natural join obtained (n + 1)-ary relation E which is a subset of the Cartesian product achieved by approapriate deleting (Observation 8) the universal (n + 1)-ary relation identiﬁed with cobweb poset digraph Vn , ≺·); V∞ ≡ V . Which are those “many”? The characterization is arrived at with au rebour point of view. Any n-ary relation (n ∈ N ∪ {∞}) determines uniquely (may be identiﬁed 62 A. K. Kwaśniewski Fig. 6: Display of the example ternary = Binary1 ⊕→ Binary2 . with) its correspondent graded digraph with minimal elements set Φ0 given by the (n-ary rel.) formula n E = ⊕→n−1 k=0 Ek ⊂ ×k=0 Φk , → where the sequence of binary relations Ek ⊆ Φk × Φk+1 ≡Kk,k+1 is denominated by the source n-ary relation as the following example shows. Example (ternary = Binary1 ⊕ → Binary2 ). Let T ⊂ X × Z × Y where X = {x1 , x2 , x3 }, Z = {z1 , z2 , z3 , z4 }, Y = {y1 , y2 }, and T = {x1 , z1 , y1 , x1 , z2 , y1 , x1 , z4 , y2 , x2 , z3 , y2 , x3 , z3 , y2 }. Let X × Z ⊃ E1 = {x1 , z1 , x1 , z2 , x1 , z4 , x2 , z3 , x3 , z3 }, and Z × Y ⊃ E2 = {z1 , y1 , z2 , y1 , z3 , y1 , z4 , y2 }. Then T = E1 ⊕→ E2 . More on that – see [15] and see references to the recent papers therein. Comment 11. As a comment to Observation 9 and Observation 3 consider Fig. 7 which was the source of inspiration for cobweb posets birth [2–6] and here serves as Hasse diagram DFib ≡ (Φ, ≺·Fib ) of the poset P (DFib ) = (Φ, ≤Fib ) associated to DFib . Obviuosly, P (DFib ) is a subposet of the Fibonacci cobweb poset P (D) and DFib is a subgraph of the Fibonacci cobweb poset P (D) Hasse diagram D ≡ (Φ, ≺·). The Ferrers dimension of DF ib is obviously not equal one. Exercise. Find the Ferrers dimension of DFib . What is the dimension of the poset P (DF ib ) = (Φ, ≤Fib )? (Compare with Observation 2). Find the chain Ek ⊂ Φk × Φk+1 , k = 0, 1, 2, ... of binary relations such that DFib,n = ⊕→nk=0 Ek , n ∈ N ∪ {∞}. Find the Ferrers dimension of DFib,n . Cobweb posets and KoDAG digraphs are representing natural join of relations 63 Fig. 7: Display of Hasse diagram of the form of the Fibonacci tree. Ad Bibliography Remark On the history of oDAG nomenclature with David Halitsky and others input one is expected to see more in [15]. See also the December 2008 subject of the Internet Gian Carlo Rota Polish Seminar (http : //ii.uwb.edu.pl/akk/sem/sem rota.htm). Recommended readings on Ferrers digraphs of immediate use here are [19–25]. For example see pp. 61 and 85 in [19], see page 2 in [20]. The J. Riguet paper [21] is the source paper including also equivalent characterizations of Ferrers digraphs as well as the other paper [22–24]. The now classic reference on interval orders and interval graphs is [25]. Acknowledgments Thanks are expressed here to the student of Gdańsk University Maciej Dziemiańczuk for applying his skillful TeX-nology with respect to the present work as well as for his general assistance and cooperation on KoDAGs investigation. References [1] J. Jonsson, Simplicial Complexes of Graphs Series, Lecture Notes in Mathematics, 1928, XIV (2008), 382 (see [JJ] Interpretations in terms of posets Section 9). [2] A. K. Kwaśniewski, Cobweb posets as noncommutative prefabs, Adv. Stud. Contemp. Math. 14 (1) (2007), 37–47. arXiv:math/0503286v4, [v1] Tue, 15 Mar 2005 04:26:45 GMT. [3] A. K. Kwaśniewski, The logarithmic Fib-binomial formula, Adv. Stud. Contemp. Math. 9, no.1 (2004), 19–26 arXiv:math/0406258v1 [v1] Sun, 13 Jun 2004 17:24:54 GMT. 64 A. K. Kwaśniewski [4] A. K. Kwaśniewski, Fibonomial cumulative connection constants, Bulletin of the ICA 44 (2005), 81–92, see ArXiv:math/0406006v2 [v6] Fri, 20 Feb 2009 02:26:21 GMT, upgrade of Bulletin of the ICA 44 (2005), 81–92. [5] A. K. Kwaśniewski, First Observations on Prefab Posets Whitney Numbers, Advances in Applied Cliﬀord Algebras 18, no. 1 (2008), 57–73, arXiv:0802.1696v1, [v1] Tue, 12 Feb 2008 19:47:18 GMT. [6] A. K. Kwaśniewski, On cobweb posets and their combinatorially admissible sequences, Adv. Studies Contemp. Math. 18 (1), (2009), – in press (upgraded version of arXiv:math/0512578v4 [v4] Sun, 21 Oct 2007 15:02:09 GMT). [7] A. D. Plotnikov, About presentation of a digraph by dim 2 poset., Adv. Stud. Contemp. Math., Kyungshang 12, no. 1 (2006), 55–60). [8] R. P. Stanley, A matrix for counting paths in acyclic digraphs, J. Combinatorial Theory (A) 74 (1996), 169–172. [9] Ewa Krot, Characterization of the Fibonacci Cobweb Poset as oDAG, arXiv:math/0503295v1 Tue, 15 Mar 2005 11:52:45 GMT. [10] A. K. Kwaśniewski and M. Dziemiańczuk, Cobweb posets - Recent Results, Adv. Stud. Contemp. Math. 16 (2) (2008), 197–218; arXiv: math. /0801.3985 Fri, 25 Jan 2008 17:01:28 GMT. [11] Ewa Krot-Sieniawska, Reduced Incidence algebras description of cobweb posets and KoDAGs, arXiv:0802.4293 Fri, 29 Feb 2008. [12] Ewa Krot-Sieniawska, Characterization of Cobweb Posets as KoDAGs, arXiv:0802.2980v1 [v1] Thu, 21 Feb 2008 05:32:10 GMT. [13] A. K. Kwaśniewski and M. Dziemiańczuk, On cobweb posets’ most relevant codings, arXiv:0804.1728v1 [v1] Thu, 10 Apr 2008 15:09:26 GMT. [14] http://www.faces-of-nature.art.pl/cobwebposets.html [15] A. K. Kwaśniewski, On natural join of posets properties and first applications arXiv:0908.1375v1 [v2] Sat, 22 Aug 2009 10:42:44 GMT. [16] V. E. Levit and E. Mandrescu, Matrices and ?-Stable Bipartite Graphs, Journal of Universal Computer Science, 13, no. 11 (2007), 1692–1706 [17] E. Ferrand, An analogue of the Thue-Morse sequence, The Electronic Journal of Combinatorics 14 (2007), #R 30. [18] D. E. Knuth, Two notes on notation, American Mathematical Monthly 99, (5) (1992), 403–422. [19] T. A. McKee and F. R. McMorris, Topics in intersection graph theory, [SIAM Monographs on Discrete Mathematics and Applications #2] Philadelphia 1999. [20] S. Chatterjee and S. Ghosh, Ferrers Dimension and Boxicity, arXiv:0811.1882v1 [v1] Wed, 12 Nov 2008 12:32:12 GMT. [21] J. Riguet, Les Relations des Ferrers, C. R. Acad. Sci. Paris 232 (1951), 1729. [22] O. Cogis, A characterization of digraphs with Ferrers dimension 2, Rapport de Recherche, 19, G. R. CNRS no. 22, Paris, 1979. [23] M. Sen, S. Das, A. B. Roy, and D. B. West, Interval Digraphs: An Analogue of Interval Graphs, J. Graph Theory 13 (1989), 189–202. [24] I.-J. Lin, M. K. Sen, and D. B. West, Classes of interval digraphs and 0, 1-matrices (with). Proc. 28th SE Conf., Congressus Numer. 125 (1997), 201–209. [25] P. C. Fishburn, Interval Orders and Interval Graphs: A Study of Partialy Ordered Sets, John Wiley & Sons, New York, 1985. Cobweb posets and KoDAG digraphs are representing natural join of relations 65 Institute of Combinatorics and its Applications High School of Mathematics and Applied Informatics Kamienna 17, PL-15-021 Bialystok Poland e-mail: [email protected] Presented by Julian L awrynowicz at the Session of the Mathematical-Physical Commission of the L ódź Society of Sciences and Arts on July 16, 2010 SKIEROWANE GRAFY COBWEB POSETÓW OKREŚLAJA̧ ZLA̧CZENIE NATURALNE DI-GRAFÓW RELACJI CZYLI RÓWNOWAŻNIE OPERACJȨ ZLA̧CZENIA NATURALNEGO MACIERZY SA̧SIEDZTWA Streszczenie Zdeﬁniowano zla̧czenie naturalne (“natural join”) skierowanych grafów dwudzielnych oraz odpowiadaja̧ce im zla̧czenie naturalne ich macierzy sa̧siedztwa. Operacjȩ natural join zastosowano do wprowadzonych tutaj szczególnych czȩściowo uporza̧dkowanych zbiorów ze stopniowaniem zwanych już wcześniej “cobweb posets”. Operacjȩ “natural join” zastosowano zatem i do ich di-grafów tworza̧cych lańcuch tzw. “bi-di-klik”. Stanowia̧ one w zla̧czaniu naturalnym cia̧gi Kompletnych Grafów dwudzielnych – uporza̧dkowanych (ordered) oraz skierowanych i acyklicznych (DAG’s). Na cześć Profesora Kazimierza Kuratowskiego – wspóltwórcy wspólczesnej teorii grafów – grafy Hassego nazwano tu KoDAGs. Jak wykazano – KoDAGs stanowia̧ wyróżniona̧ rodzin di-grafów Ferrersa. Mianowicie ich wymiar Ferrersa wynosi jeden, przy czym podano warunki dostateczne i wystarczaja̧ce na to, by “natural join” lańcucha di-grafów dwudzielnych tworzyl di-graf o wymiarze Ferrersa równym jeden. Wyróżnione di-grafy w zla̧czaniu naturalnym tworza̧ cia̧gi di-grafów określonych dla odpowiednio licznej rodziny relacji k-narnych – w tym np. binarnych. Operacjȩ “natural join” świadomie i z uzasadnieniem oznaczono symbolem nawizuja̧cym do symbolu operacji sumy prostej. PL ISSN 0459-6854 BULLETIN DE LA SOCIÉTÉ DES SCIENCES ET DES LETTRES DE L ÓDŹ 2010 Vol. LX Recherches sur les déformations no. 1 pp. 67–75 Carmen Boloşteanu THE RIEMANN-HILBERT PROBLEM WITH ISOLATED POLES Summary Let L be a simple smooth closed contour dividing the plane Ĉ = C ∪ {∞} into the interior domain D+ and the exterior domain D− and a non-vanished function of position on the contour, G(t) which satisﬁes the Hölder condition. We have to ﬁnd a meromorphic function Φ which has given isolated poles in D+ and D− respectively, which satisﬁes on L the relation Φ+ (t) = G(t)Φ− (t). The same problem is solved on the real projective plane, which is the factor manifold of Ĉ with respect to the group {1Ĉ , k}, where k(z) = −1/z̄. 1. Introduction The Riemann-Hilbert problem for dianalytic functions on the Möbius strip was solved in [6] using the point of view introduced by I. Bârză and D. Ghişa in [1], [2], [3] and the method used by Gakhov [7] and Muskelishvili [8]. In this paper we solve the same problem, but for meromorphic functions on the complex plane and then for N-meromorphic functions on the real projective plane. Some deﬁnitions introduced in other speciﬁed papers are used. 2. The case of complex plane Let D+ ⊂ Ĉ be a relative compact domain, 0 ∈ / D+ . Suppose that ∂D+ = L, where − + L is a smooth curve. Denote by D = Ĉ \ {D ∪ L}. On L a Hölder function G(t) is deﬁned which does not vanish. We have to ﬁnd a meromorphic function Φ such that (1) Φ+ (t) = G(t)Φ− (t) on L, where Φ+ (z) = Φ(z) for z ∈ D+ and Φ− (z) = Φ(z) for z ∈ D− . 68 C. Boloşteanu Suppose that Φ+ (z) = Φ1 (z) (z − z1 )k1 Φ− (z) = Φ2 (z) , (z − z2 )k2 and where Φ1 and Φ2 are holomorphic functions in D+ and D− , respectively, and z1 , z2 are given poles of orders k1 , k2 > 0, with z1 ∈ D+ and z2 ∈ D− . The relation (1) becomes Φ2 (t) Φ1 (t) = G(t) . (t − z1 )k1 (t − z2 )k2 We have (2) Φ1 (t) = G(t) (t − z1 )k1 Φ2 (t). (t − z2 )k2 Denote by G1 (t) = G(t) (t − z1 )k1 . (t − z2 )k2 If the index of G with respect to the curve L is n, then the index of G1 is n + k1 − k2 . Let Π(z) = (z − z1 )n+k1 −k2 . Multiplying the relation (2) by Π−1 (t), we obtain Φ1 (t)Π−1 (t) = G(t) and Φ1 (t) = G(t) (t − z1 )k2 −n Φ2 (t) (t − z2 )k2 (t − z1 )k2 −n Φ2 (t)Π(t). (t − z2 )k2 Now we deﬁne a new unknown holomorphic function Ψ(z), where ⎧ + if z ∈ D+ , ⎨ Ψ (z) = Φ1 (z) Ψ(z) = ⎩ − Ψ (z) = Φ2 (z)Π(z) if z ∈ D− . The problem is reduced to a classic Riemann-Hilbert problem. We have to ﬁnd the sectional holomorphic function Ψ such that (3) Ψ+ (t) = G0 (t)Ψ− (t), where G0 (t) = G(t) (t − z1 )k2 −n (t − z2 )k2 and its index is zero. Obviously, G0 is a Hölder function and it does not vanish on L. By taking logarithms, the relation (3) becomes log Ψ+ (t) − log Ψ− (t) = log G0 (t) The Riemann-Hilbert problem with isolated poles 69 and we can apply the Sokhotski-Plemelj formulae. We obtain log G0 (ζ) 1 dζ log Ψ(z) = 2πi ζ −z L = 1 2πi L log G(ζ) + (k2 − n) log(ζ − z1 ) − k2 log(ζ − z2 ) dζ. ζ −z Γ(z) , where 1 log G(ζ) + (k2 − n) log(ζ − z1 ) − k2 log(ζ − z2 ) Γ(z) = dζ. 2πi ζ −z Then Ψ(z) = e L + For z ∈ D , we have + Φ1 (z) = Ψ+ (z) = eΓ and for z ∈ D− we have (z) − eΓ (z) Ψ− (z) = . Φ2 (z) = Π(z) Π(z) The solution to the problem (1) is the meromorphic function Φ(z), where ⎧ + eΓ (z) ⎪ ⎪ , if z ∈ D+ ⎨ (z − z1 )k1 Φ(z) = (4) − ⎪ eΓ (z) ⎪ ⎩ , if z ∈ D− . (z − z1 )n+k1 −k2 (z − z2 )k2 According with the Sokhotski-Plemelj formulae, we obtain 1 Γ+ (t) = (log G(t) + (k2 − n) log(t − z1 ) − k2 log(t − z2 )) 2 ⎛ ⎞ 1 log G(ζ) + (k − n) log(ζ − z ) − k log(ζ − z ) 2 1 2 2 + CP V ⎝ dζ ⎠ ; 2πi ζ −t L 1 Γ− (t) = − (log G(t) + (k2 − n) log(t − z1 ) − k2 log(t − z2 )) 2 ⎛ ⎞ 1 log G(ζ) + (k2 − n) log(ζ − z1 ) − k2 log(ζ − z2 ) ⎠ + CP V ⎝ dζ , 2πi ζ −t L where CP V is the principal value of the Cauchy integral from the brackets. Making simple calculus, one can see that Φ+ , Φ− and G satisﬁes the relation (1). 3. The case of real projective plane The real plane P2 appears as a factor manifold of Ĉ with respect to the projective group 1Ĉ , k , where k(z) = −1/z̄. P2 is a nonorientable Riemann surface, Ĉ is an 70 C. Boloşteanu orientable Riemann surface and the double cover of P2 , and k is an antianalytic involution without ﬁxed points: P2 = [z] | z ∈ Ĉ , where [z] = {z, −1/z̄} . Let the function p : Ĉ → P2 be the canonical projection with p(z) = [z]: In this paper we use the symmetry in the sense of Klein. A set D ⊂ Ĉ is called symmetric with respect to k if k(D) = D [1]. Let D ⊂ Ĉ be a symmetric set and f : D → Ĉ a function. f is called symmetric function (with respect to k) if f (z) = f (k(z)) for every z ∈ D [1]. Let D̃ be a subset of P2 and let f˜ : D̃ → Ĉ be a function. One can deﬁne a symmetric function f : p−1 (D̃) → Ĉ by f (z) = f (k(z)) := f˜([z]) for every [z] = {z; k(z)} ∈ D̃. If one considers only the functions f˜ : D̃ → C, then it is obvious that the algebra of these functions is canonically isomorphic with the algebra of the symmetric functions f : p−1 (D̃) → C. The study of functions on P2 is canonically reduced to the study of functions on Ĉ. Definition 3.1. [4], [5] A function ϕ̃ fulfills the Hölder condition on a smooth curve L̃ ∈ P2 if |ϕ̃([t1 ]) − ϕ̃([t2 ])| ≤ C δ([t1 ], [t2 ])α for all [t1 ], [t2 ] ∈ L̃ and 0 < α ≤ 1. Here δ([z1 ], [z2 ]) is the distance between two points [z1 ] and [z2 ], which was introduced in [5]. It is given by 1 , δ([z1 ], [z2 ]) = min d(z1 , z2 ), d z1 , − z¯2 where 1 1 1 d(z1 , z2 ) = (|z1 − z2 | + |k(z1 ) − k(z2 )|) = |z1 − z2 | 1 + . 2 2 |z1 z2 | The Riemann-Hilbert problem with isolated poles 71 Let g : Ĉ → Ĉ be a meromorphic function. Deﬁne f = g + g ◦ k and F : P2 → Ĉ with (5) F ([z]) := f (z). Definition 3.2. [1] The function F is called N-meromorphic function if for every [z] ∈ P2 its value is given by a function f as in the relation (5). Let D̃+ ⊂ P2 be a relative compact domain bounded by a smooth contour and [0] ∈ / D̃+ . Suppose that on ∂ D̃+ = L̃ a non-vanishing function G̃ is deﬁned which satisﬁes the Hölder condition. We have to ﬁnd an N-meromorphic function Φ̃ with given poles [z1 ] ∈ D̃+ and [z2 ] ∈ D̃− = P2 \ (D̃+ ∪ L̃) such that (6) Φ̃+ ([t]) = G̃([t])Φ̃− ([t]) on L̃, where Φ̃+ ([z]) = Φ̃([z]) for [z] ∈ D̃+ and Φ̃− ([z]) = Φ̃([z]) for [z] ∈ D̃− . Let us denote: D+ = p−1 (D̃+ ), D− = p−1 (D̃− ), L = p−1 (L̃). Obviously, D+ , D− , and L are symmetric sets. We can deﬁne the function G = G̃ ◦ p on L. Theorem 3.1. The function G fulfils the Hölder condition on L. Besides, G is symmetric with respect to k (i.e. G(t) = G(k(t)), for t ∈ L). Proof. See in [5]. The conclusion is G̃([t]) = G(k(t)) = G(t). Now the problem is transferred to the complex plane, where we have a Hölder function G(t) deﬁned on the boundary of the set D+ . Solving the problem for the symmetric set D+ and the function G on L, we obtain a meromorphic function Φ which fulﬁls (7) Φ+ (t) = G(t)Φ− (t) for t ∈ L. Theorem 3.2. If Φ is the solution to the problem (7), then the N-meromorphic function 1 Φ̃([z]) = {Φ(z) + Φ(k(z))} 2 is the solution to the problem (6). Proof (cf. [6]). For [z] ∈ D̃, we have Φ̃+ ([z]) = 1 + {Φ (z) + Φ+ (k(z))}. 2 72 C. Boloşteanu Taking the limit [z] → [t] ∈ L̃ in the above relation and using the relation (7) together with the symmetry of the function G with respect to k, we obtain: 1 1 Φ̃+ ([t]) = {Φ+ (t) + Φ+ (k(t))} = {G(t)Φ− (t) + G(k(t))Φ− (k(t))} 2 2 1 − = G(t) Φ (t) + Φ− (k(t)) = G̃([t])Φ̃− ([t]), 2 as desired. 3.1. Explicit solution We have to solve the problem on Ĉ for the set D+ and the non vanishing function G(t) deﬁned on ∂D+ , which satisﬁes a Hölder condition. 3.1.1. Solution for the complex plane Suppose that D+ = D1 ∪ D2 , where D̄1 ∩ D̄2 = ∅ and D1 = k(D2 ). Let Li = ∂Di , i = 1, 2. Obviously, D− = k(D− ), D+ = k(D+ ) and L1 = k(L2 ). Moreover, p−1 ([z1 ]) = {z1 , k(z1 )} and p−1 ([z2 ]) = {z2 , k(z2 )}. We have to ﬁnd an N-meromorphic function Φ(z) with given poles z1 , k(z1 ) ∈ D+ and z2 , k(z2 ) ∈ D− which fulﬁls the condition Φ+ (t) = G(t)Φ− (t) on L1 ∪ k(L1 ). (8) We can write Φ+ (z) = Φ1 (z) (z − z1 )(z − k(z1 )) Φ− (z) = Φ2 (z) , (z − z2 )(z − k(z2 )) and where Φ1 and Φ2 are holomorphic functions on D+ and D− , respectively. It was proved in [6] that the index of a function with respect to a curve L is equal to the index of the same function with respect to k(L). Let n be the index of G on L1 . Then the index of G(t) given on L1 ∪ k(L1 ) is 2n. The relation (8) becomes Φ1 (t) = G(t) (t − z1 )(t − k(z1 )) Φ2 (t). (t − z2 )(t − k(z2 )) Introduce the function Π(z) = (z − z1 )n (z − k(z1 ))n . Then the argument of (9) G0 (t) = Π−1 (t)G(t) (t − z1 )1−n (t − k(z1 ))1−n (t − z1 )(t − k(z1 )) = G(t) (t − z2 )(t − k(z2 )) (t − z2 )(t − k(z2 )) 73 The Riemann-Hilbert problem with isolated poles will return to its initial value after any circuit of the contours L1 , k(L1 ) and hence log G0 (t) is a deﬁnite function, one-valued, continuous and satisfying the Hölder condition on L1 ∪ k(L1 ). Introduce a new unknown holomorphic function Ψ(z) with + if z ∈ D+ Ψ (z) = Φ1 (z) Ψ(z) = Ψ− (z) = Φ2 (z)Π(z) if z ∈ D− . Using the same method as in Section 2, we obtain the solution of a new problem. Theorem 3.3. The solution of (8) is the meromorphic function ⎧ + eΓ (z) ⎪ ⎪ if z ∈ D+ ⎨ (z − z1 )(z − k(z1 )) Φ(z) = (10) − ⎪ eΓ (z) ⎪ ⎩ if z ∈ D− , Π(z)(z − z2 )(z − k(z2 )) where (11) Γ(z) = 1 2πi log G(t) + log L1 ∪k(L1 ) (t − z1 )1−n (t − k(z1 ))1−n (t − z2 )(t − k(z2 )) dt. t−z 3.1.2. Solution for the real projective plane Theorem 3.4. If the function Φ(z) is the solution on the complex plane, then the function + Φ ([z]) if [z] ∈ D̃+ Φ̃([z]) = Φ̃− ([z]) if [z] ∈ D̃− with 1 Φ+ ([z]) = 2 and 1 Φ̃− ([z]) = 2 + + eΓ (z) eΓ (k(z)) + (z − z1 )(z − k(z1 )) (k(z) − z1 )(k(z) − k(z1 )) − − eΓ (z) eΓ (k(z)) + Π(z)(z − z2 )(z − k(z2 )) Π(k(z))(k(z) − z2 )(k(z) − k(z2 )) is the solution of the problem on P2 , where Γ(z) is given by (11). Proof. Obviously, [z] ∈ D̃+ ⇐⇒ z ∈ D+ and [z] ∈ D̃− ⇐⇒ z ∈ D− . By Theorem 3.2 we can write the formula for 1 Φ̃([z]) = {Φ(z) + Φ(k(z))}. 2 74 C. Boloşteanu Taking limiting values in the formula (11) and using the Sokhotski-Plemelj formulae for multiply-connected domains, we ﬁnd 1 (t − z1 )1−n (t − k(z1 ))1−n Γ+ (t) = log G(t) + log + Γ(t), 2 (t − z2 )(t − k(z2 )) 1 (t − z1 )1−n (t − k(z1 ))1−n Γ− (t) = − log G(t) + log + Γ(t) 2 (t − z2 )(t − k(z2 )) where Γ(t) is the Cauchy principal value of the integral (11). For Γ+ (k(t)) and Γ− (k(t)) the formulae are analogous. Then we obtain Φ̃+ ([t]) = lim Φ̃+ ([z]), Φ̃− ([t]) = lim Φ̃− ([z]). [z]→[t] [z]→[t] Using the symmetry of G(t) (Theorem 3.1) and making a simple calculus, one can check that Φ̃+ ([t]), Φ̃− ([t]), and G̃([t]) fulﬁl the relation Φ̃+ ([t]) = G̃([t])Φ̃− ([t]). Acknowledgements The author would like to express her gratitude to Professor Bogdan Bojarski for his constant kind help and to Professor Jaroslav Zemanek for the ﬁnancial support in the frame of the European Community project TODEQ (MTKD-CT-2005-030042). References [1] I. Bârză, Calculus on Nonorientable Riemann Surfaces, Libertas Mathematica 15 (1995), 1–45. [2] I. Bârză, Integration on Nonorientable Riemann Surfaces. In: Almost Complex Structures. (Eds. K. Sekigawa and S. Dimiev). World Scientiﬁc, Singapore-New JerseyLondon-Hong Kong, 1995, 63–97. [3] I. Bârză and D. Ghişa, Boundary value problems on non orientable surfaces, Rev. Roumaine Math. Pures Appl. 43 (1998), 1–2, 67–79. [4] C. Boloşteanu, The Sokhotski-Plemelj formulae on the Möbius strip, Complex Variables and Elliptic Equations 53, no. 7 (2008), 657–666. [5] C. Boloşteanu and M. Boloşteanu, Distances and Hölder functions on fonorientable Riemann surfaces, Buletin Ştiinţiﬁc – Universitatea din Piteşti 13 (2007), 15–26. [6] C. Boloşteanu, The Riemann-Hilbert problem on the Möbius strip, Complex Variables and Elliptic Equations, to appear. [7] F. D. Gakhov, Boundary Value Problems. Moscow, 1963 (Russian); English translation, Pergamon Press, Oxford 1966. [8] N. I. Muskhelishvili, Singular Integral Equations. Moscow, 1946 (Russian); English translation, Noordhoﬀ, Groningen 1953. Faculty of Accounting and Finance “Spiru Haret” University Bucharest R-115100 Câmpulung Muscel, 223 Traian Street Romania e-mail: [email protected] The Riemann-Hilbert problem with isolated poles 75 Presented by Leon Mikolajczyk at the Session of the Mathematical-Physical Commission of the L ódź Society of Sciences and Arts on March 2, 2010 ZAGADNIENIE RIEMANNA-HILBERTA Z IZOLOWANYMI BIEGUNAMI Streszczenie Niech L bȩdzie prostym gladkim konturem zamkniȩtym dziela̧cym plaszczyznȩ Ĉ = C∪{∞} na obszar wewnȩtrzny D+ i obszar zewnȩtrzny D− , G(t) zaś – nieznikaja̧ca̧ funkcja̧ polożenia na konturze spelniaja̧ca̧ warunek Höldera. Poszukujemy funkcji meromorﬁcznej Φ posiadaja̧cej dane bieguny w obszarach D+ i D− , spelniaja̧cej na konturze L relacjȩ Φ+ (t) = G(t)Φ− (t). Takie samo zagadnienie jest rozwia̧zane na rzeczywistej plaszczyźnie rzutowej, która jest czynnikiem plaszczyzny Ĉ ze wzglȩdu na grupȩ {1Ĉ , k}, gdzie k(z) = −1/z̄. PL ISSN 0459-6854 BULLETIN DE LA SOCIÉTÉ DES SCIENCES ET DES 2010 LETTRES DE L ÓDŹ Vol. LX Recherches sur les déformations no. 1 pp. 77–94 Dedicated to Professor Julian L awrynowicz on the occasion of his 70-th birthday Dariusz Partyka and Józef Zaja̧c GENERALIZED PROBLEM OF REGRESSION Summary The authors present a generalization of the classical regression idea by studying and then solving certain extremal problem, well deﬁned on the ground of ﬁnite or inﬁnite dimensional Hilbert space. Given empiric data, discrete or continuous, the class of solutions is determined and uniquely expressed in a new form of the regression functions sequences (RFS). A large variety of observed phenomena, in diﬀerent areas of practical and theoretical sciences, can be described and researched, with certain precision and a help of the RFS technique. Introduction Contemporary world is characterized by the increasing inﬂux of information. The observation of even simple processes requires numerical data methods of analysis which take individual parameters, distinguished by these processes. Some of them are possible to achieve in applied selected moments of time, whereas others can be observed in continuous time system. In the avalanche of pouring information one has to know how to ﬁnd interesting dependence of both relational and functional type. The latter are often too complicated to capture and describe by means of simple mathematical expression. Approximate functional dependences which would describe interesting phenomena with assigned accuracy should be sought. Studying of appropriately constructed approaching functions can lead to detection of so far unknown dependences as well as assess separate and combined eﬀects caused by several observed variables. It has 78 D. Partyka and J. Zaja̧c a huge signiﬁcance, especially in situation when dependence, expressed in physical, chemical and biological laws, between observed parameters is unknown. Particular, although simpliﬁed, example of solution to such set scientiﬁc problem is a method of linear regression with its various modiﬁcations, formulated on the basis of probability calculus; cf. [13], [11], [9], [1] and [7]. The method found series of implementations and experienced numerous theoretical modiﬁcations, crucial due to seriousness of implementation problem; cf. [12], [11], [3], [8], [4] and [2]. The generalized regression problem considered in this paper has a form of solution of properly formulated extremal problem well stated in the Hilbert space environment, both ﬁnite and inﬁnite dimensional. A short presentation of the classical approach to the regression problem can be formulated as follows. For any p, q ∈ R set Zp,q := {k ∈ Z : p ≤ k ≤ q}. Let F be the family of all functions R t → at + b, where a, b ∈ R, and let x, y : Z0,n → R be arbitrarily given sequences. It is well known that if x is not a constant sequence, then there exists the unique f0 ∈ F satisfying the following condition n n (0.1) (f (xk ) − yk )2 ≥ (f0 (xk ) − yk )2 , f ∈ F . k=0 k=0 In fact, the function f0 is of the form f0 (t) = a0 t + b0 as t ∈ R, where n n n n n xk yk − xk yk y k − a0 xk (n + 1) k=0 k=0 k=0 k=0 k=0 a0 := (0.2) ; and b := 0 n n n+1 (n + 1) x2k − ( xk )2 k=0 k=0 cf. e.g. [13] and [2]. The function f0 is usually said to be the regression line for the empiric sequences x, y : Z0,n → R. In view of (0.1), the function f0 has a natural interpretation as an optimal function with the smallest quadratic deviation from the empiric observations {(xk , yk ) : k ∈ Z0,n }. The function f0 plays an essential role in diﬀerent areas of applied mathematics; cf. e.g. [4] and [2]. It shows that the mentioned above extremal problem, can be considerably qeneralized and solved, which is a subject of this paper. To this end we introduce the regression structures R := (A, B, δ; x, y), where: I.1. A, B are nonempty sets; I.2. x : Ω1 → A and y : Ω2 → B for some nonempty sets Ω1 and Ω2 ; I.3. δ : (Ω1 → B) × (Ω2 → B) → R. A set F is called the functional model of R if F ⊂ (A → B), where A → B means the class of all functions acting from A to B. According to the extremal problem (0.1), the components δ, x and y as well as a functional model F of R have the following interpretations: – F is a theoretic functional model of the considered phenomena, i.e. F consists of all functions describing theoretically the considered phenomena; Generalized problem of regression 79 – x : Ω1 → A and y : Ω2 → B are empirical functions derived from an experiment or observation, called the empirical data functions in the sequel; – δ is a deviation criterion of theoretic functions from empirical ones. Given a regression structure R and a functional model F of R we seek the optimal theoretic functions f0 ∈ F which are the best ﬁtted to the empirical data – represented by the empirical data functions x and y – with respect to the criterion δ. To be more precise, we consider the extremal problem of determining all functions f0 ∈ F minimizing the functional (0.3) F f → F (f ) := δ(f ◦ x, y) ∈ R , i.e. all functions f0 ∈ F satisfying the following inequality (0.4) F (f ) ≥ F (f0 ) , f ∈F . The set of all f0 ∈ F satisfying the inequality (0.4) will be denoted by M(F , R). Each function f0 ∈ M(F , R) is said to be the regression function in F with respect to R. The problem of describing all regression functions in F with respect to R we shall call the regression problem for F and R. Example 0.1. Consider an electric circuit with direct current. According to Ohm’s law the voltage V depends on the intensity I by the equality V = RI, where the multiplier R is the resistance of the circuit. We want to determine the parameter R by means of measurements samples of intensity and voltage represented by a sequence Z0,n k → (ik , vk ). To this end we consider the regression structure R, where A := R, B := R, the empiric data functions are deﬁned by Z0,n k → x(k) := ik and Z0,n k → y(k) := vk , and, as a criterion of deviation δ, we take the smallest squares method, i.e. n (0.5) (f (k) − g(k))2 , f, g : Z0,n → R . δ(f, g) := k=0 The theoretic functional model F is represented by linear functions R t → rt for n r ∈ R. Calculating the critical point of the function R r → k=0 (rik − vk )2 we obtain n n (0.6) ik vk i2k . R= k=0 k=0 In what follows we shall study the regression problem for a wide range of theoretic functional models F and regression structures R involving a generalized variant of quadratic deviation applied in (0.1). We aim at showing the main idea of our approach. Therefore we conﬁne ourselves to the basic concepts. In particular we focus our attention on the case where the theoretic functional model F is a ﬁnite-dimensional linear set with respect to the standard operations of adding and multiplying complex-valued functions. The complete version of this article will be published elsewhere. 80 D. Partyka and J. Zaja̧c Most of results in this paper were presented by the ﬁrst named author during the seminar “Hypercomplex Seminar 2009: From Schauder Basis to Hypercomplex, Randers-Ingarden and Fractal Structures, and Nanostructures”, Mathematical Conference Center at Bȩdlewo (Poland), July 24 – July 31, 2009. 1. The generalized quadratic deviation From now on we shall study the family of regression structures R := (A, B, δ; x, y) satisfying additional assumptions: II.1. B = R or B = C; II.2. There exist a σ-ﬁeld B of subsets of the cartesian product Ω1 × Ω2 and a measure μ : B → [0; +∞] such that the function δ satisﬁes the following equality (1.1) |u(t1 ) − v(t2 )|2 d μ(t1 , t2 ) , δ(u, v) = Ω1 ×Ω2 provided both the functions Ω1 × Ω2 (t1 , t2 ) → u(t1 ) and Ω1 × Ω2 (t1 , t2 ) → v(t2 ) are B-measureable, and δ(u, v) = +∞ otherwise; II.3. The function Ω1 × Ω2 (t1 , t2 ) → y(t2 ) is B-measureable. Example 1.1. Consider a regression structure R deﬁned as follows. Given n, m ∈ N let Ω1 := Z0,n and Ω2 := Z0,m . Let B be the set of all subsets of the cartesian product Ω1 × Ω2 . Obviously, the set B is a σ-ﬁeld of subsets of Ω1 × Ω2 , and hence we can deﬁne a unique measure μ : B → [0; +∞) satisfying the condition (1.2) μ({(k, l)}) = ρk,l , k ∈ Ω1 , l ∈ Ω2 , where Ω1 × Ω2 (k, l) → ρk,l ∈ R is a given non-negative function. Then, for any functions Ω1 t → u(t) ∈ B and Ω2 t → v(t) ∈ B, we derive from (1.1), δ(u, v) = (1.3) |u(t1 ) − v(t2 )|2 d μ(t1 , t2 ) Ω1 ×Ω2 = |u(t1 ) − v(t2 )|2 d μ(t1 , t2 ) (k,l)∈Ω1 ×Ω2{(k,l)} = m n ρk,l |u(k) − v(l)|2 . k=0 l=0 In particular, assuming m = n and setting 1 as k = l , (1.4) ρk,l := 0 as k = l , 81 Generalized problem of regression we conclude from (1.3) that δ(u, v) = (1.5) n |u(k) − v(k)|2 , k=0 which is exactly the classical square deviation used in (0.1). Namely, combining (0.3) with (1.3) and (1.4) we obtain (1.6) F (f ) = n 2 |f ◦ x(k) − y(k)| = k=0 n |f (xk ) − yk |2 , f ∈F , k=0 for given empirical data functions Z0,n k → xk and Z0,n k → yk , where F is the set of all functions R t → at + b as a, b ∈ R. From now on we may conﬁne ourselves to the case where B = C, which naturally embrace the case where B = R. Fix a regression structure R satisfying the properties II.1–II.3. We consider the set L1 (R) of all functions f : A → B such that Ω1 × Ω2 (t1 , t2 ) → f ◦ x(t1 ) is a B-measureable function and (1.7) |f ◦ x(t1 )|2 d μ(t1 , t2 ) < +∞ . Ω1 ×Ω2 We shall also consider the set L2 (R) of all functions g : Ω2 → B such that Ω1 × Ω2 (t1 , t2 ) → g(t2 ) is a B-measureable function and (1.8) |g(t2 )|2 d μ(t1 , t2 ) < +∞ . Ω1 ×Ω2 From (1.7) and the inequality (1.9) |zw| ≤ 1 (|z|2 + |w|2 ) , 2 it follows that the functional (1.10) z, w ∈ C , L1 (R) × L1 (R) (u, v) → u|v := u ◦ x(t1 )v ◦ x(t1 ) d μ(t1 , t2 ) Ω1 ×Ω2 is well deﬁned. Hence, u|u ≥ 0 as u ∈ L1 (R), and so the functional ⎛ ⎞1/2 L1 (R) u → u := u|u = ⎝ (1.11) |u ◦ x(t1 )|2 d μ(t1 , t2 )⎠ , Ω1 ×Ω2 is also well deﬁned. It can be shown, in much the standard way, that the structure H(R) := (L1 (R), +, ·, · | · ) is a complex (resp. real in case B = R) pseudo-Hilbert space. Here the symbols “+” and “·” denote the standard operations of adding and multiplying functions. This means that the structure (L1 (R), +, ·) is a linear space, the following properties: 82 D. Partyka and J. Zaja̧c αu + βv|w = αu|w + βv|w ; u|v = v|u ; (1.12) u|u ≥ 0 , hold for all α, β ∈ B and u, v, w ∈ L1 (R) as well as every Cauchy sequence from L1 (R) is convergent to certain function in L1 (R) with respect to the pseudo-norm · . A more delicate treatment needs the proof of its completeness. Remark 1.2. The properties (1.12) yield the well known Schwarz inequality (1.13) |u|v| ≤ uv , u, v ∈ L1 (R) ; cf. e.g. [6] and [10]. By (1.8) and (1.9) we see that for each y ∈ L2 (R) the functional ∗ (1.14) u ◦ x(t1 )y(t2 ) d μ(t1 , t2 ) L1 (R) u → y (u) := Ω1 ×Ω2 is well deﬁned. It is clear that the structure (L2 (R), +, ·) is a complex (resp. real in case B = R) linear space. Moreover, from the algebraic properties of the Lebesgue integral and Schwarz’s integral inequality it follows that for each y ∈ L2 (R) the functional y ∗ is linear and bounded on H(R) and the supremum norm of y ∗ satisﬁes the following inequality ⎛ ⎞1/2 (1.15) sup{|y ∗ (f )| : f ∈ L1 (R) and f ≤ 1} ≤ ⎝ |y(t2 )|2 d μ(t1 , t2 )⎠ . Ω1 ×Ω2 2. Solution of the regression problem From now on we shall study the regression problem for F and R, where R is a given regression structure satisfying the assumptions II.1-II.3 and F is a linear functional model of R with respect to the standard operations of adding and multiplying functions, i.e. f + g ∈ F and λf ∈ F for f, g ∈ F and λ ∈ B. If additionally F ⊂ L1 (R), then the regression problem means the extremal problem determining all functions f0 ∈ F which are minimizing the functional F . It satisﬁes, according to (0.3) and (1.1), the following equality F (f ) = (2.1) |f ◦ x(t1 ) − y(t2 )|2 d μ(t1 , t2 ) , f ∈ F . Ω1 ×Ω2 We shall start with the following basic characterization of the regression functions. Lemma 2.1. If F = ∅ is a linear set in H(R) and y ∈ L2 (R), then for every f ∈ F the following property holds: 83 Generalized problem of regression (2.2) f ∈ M(F , R) ⇔ h|f = y ∗ (h) , h∈F . Proof. Fix f, h ∈ F and λ ∈ C. By the deﬁnitions of the functions F and δ we have F (f + λh) = δ((f + λh) ◦ x, y) |(f + λh) ◦ x(t1 ) − y(t2 )|2 d μ(t1 , t2 ) = Ω1 ×Ω2 |f ◦ x(t1 ) − y(t2 ) + λh ◦ x(t1 )|2 d μ(t1 , t2 ) = Ω1 ×Ω2 |f ◦ x(t1 ) − y(t2 )|2 d μ(t1 , t2 ) = Ω1 ×Ω2 Re(f ◦ x(t1 ) − y(t2 ))λh ◦ x(t1 ) d μ(t1 , t2 ) +2 Ω1 ×Ω2 + |λ|2 |h ◦ x(t1 )|2 d μ(t1 , t2 ) . Ω1 ×Ω2 Hence, and by (2.1), (1.14) as well as by (1.11) we get F (f + λh) = F (f ) + |λ|2 h2 − 2 Re λ y ∗ (h) ⎤ ⎡ f ◦ x(t1 )h ◦ x(t1 ) d μ(t1 , t2 )⎦ + 2 Re ⎣λ Ω1 ×Ω2 = F (f ) + |λ|2 h2 − 2 Re λ y ∗ (h) + 2 Re λf |h . Therefore (2.3) F (f + λh) − F (f ) = 2 Re [λ(h|f − y ∗ (h))] + |λ|2 h2 , f, h ∈ F , λ ∈ C . Assume now that f ∈ F satisﬁes the right hand side condition in (2.2). Then setting λ := 1 we deduce from (2.3) that F (f + h) − F (f ) = h2 ≥ 0, and so f ∈ M(F , R). Conversely, suppose that f ∈ M(F , R). Then (2.3) yields (2.4) 2 Re[λ(h|f − y ∗ (h))] + |λ|2 h2 ≥ 0 , h∈F,λ∈C. Replacing h by (−h) in (2.4) we get (2.5) −2 Re[λ(h|f − y ∗ (h))] + |λ|2 h2 ≥ 0 , h∈F,λ∈C. Combining (2.4) and (2.5) we have 1 1 − |λ|2 h2 ≤ Re[λ(h|f − y ∗ (h))] ≤ |λ|2 h2 , 2 2 h∈F,λ∈C, 84 D. Partyka and J. Zaja̧c and consequently, 1 1 (2.6) − |λ|h2 ≤ Re[ei α(λ) (h|f − y ∗ (h))] ≤ |λ|h2 , 2 2 h ∈ F , λ ∈ C \ {0} , where α(λ) ∈ [0; 2π) is a unique number satisfying the equality λ = |λ| ei α(λ) . Thus (2.6) yields, in the limiting case as |λ| → 0, the following equality Re[ei α (h|f − y ∗ (h))] = 0 , h∈F,α∈R. Choosing appropriately α we see that h|f − y ∗ (h) = 0 for h ∈ F, which completes the proof. By the properties of a pseudo-norm we see that the set Θ := {h ∈ L1 (R) : h = 0} is linear. We call it the null set of H(R). As a matter of fact Θ is the closed ball with radius 0 and center at the zero function θ, deﬁned by θ(t) := 0 for t ∈ A. We may extend the standard operations of adding and multiplying functions by a constant to any sets F1 , F2 ∈ (A → B) as follows: F1 + F2 := {f1 + f2 : f1 ∈ F1 , f2 ∈ F2 } ; λ · F1 := {λf1 : f1 ∈ F1 } , λ∈C; f + F1 := {f } + F1 and F1 + f := F1 + {f } , f ∈ (A → B) . Corollary 2.2. If F = ∅ is a linear set in H(R) and y ∈ L2 (R), then M(F , R) = F ∩ M(Θ + F , R) . (2.7) If additionally F ⊂ Θ, then M(F , R) = F . Proof. Fix f, h ∈ L1 (R). If h = 0, then from Schwarz’s inequality (1.13) and (1.15) it follows that ⎛ |h|f | ≤ hf = 0 and |y ∗ (h)| ≤ ⎝ ⎞1/2 |y(t2 )|2 d μ(t1 , t2 )⎠ h = 0 . Ω1 ×Ω2 Hence (2.8) h|f = 0 = y ∗ (h) , f ∈ L1 (R) , h ∈ Θ . Assume that f ∈ M(F , R) and h ∈ Θ + F . Then h = h0 + h1 for some h0 ∈ Θ and h1 ∈ F. Applying now Lemma 2.1 and (2.8) we can see that h|f = h0 |f + h1 |f = 0 + y ∗ (h1 ) = y ∗ (h0 ) + y ∗ (h1 ) = y ∗ (h) , h∈Θ+F . By deﬁnition, f ∈ F ⊂ Θ + F . Thus, applying Lemma 2.1 once more, we get f ∈ F ∩ M(Θ + F , R), and so (2.9) M(F , R) ⊂ F ∩ M(Θ + F , R) . Generalized problem of regression 85 Conversely, assume now that f ∈ F ∩ M(Θ + F , R) and h ∈ F. Since h ∈ Θ + F , we conclude from Lemma 2.1 that h|f = y ∗ (h) , h∈F . Thus applying Lemma 2.1 once more we get f ∈ M(F , R), and so F ∩M(Θ+F , R) ⊂ M(F , R). Combining this inclusion with the inclusion (2.9) we derive the equality (2.7). Since Θ ⊂ L1 (R), the equalities in (2.8) hold for all f, h ∈ Θ. Then Lemma 2.1 yields M(Θ, R) = Θ. If now F ⊂ Θ, then the equality (2.7) takes the form M(F , R) = F , which proves the theorem. Given a nonempty set S ⊂ L1 (R) we denote by lin(S) the set of all linear combinations nk=1 λk vk where n ∈ N, Z1,n k → λk ∈ C and Z1,n k → vk ∈ S. It is easy to check that lin(S) is the smallest linear subset of L1 (R) containing S. By S ⊥ we denote the orthogonal complement of S in the space H(R), i.e. S ⊥ := {f ∈ L1 (R) : f |h = 0 for h ∈ S} . The following theorem is motivated by Lemma 2.1 and the well known representation of a linear and continuous functional in a Hilbert space by Riesz and Fréchet; cf. e.g. [5] and [6]. Theorem 2.3. If F is a closed and linear set in H(R) and y ∈ L2 (R), then M(F , R) = ∅ and M(F , R) = Θ + f for each f ∈ M(F , R). Moreover, if F ⊂ S := (y ∗ )−1 (0), then M(F , R) = Θ, and otherwise (F ∩ S)⊥ ∩ F \ Θ = ∅ and M(F , R) = Θ + (2.10) y ∗ (h) h, h2 h ∈ (F ∩ S)⊥ ∩ F \ Θ . Proof. Assume that M(F , R) = ∅ and choose arbitrarily f ∈ M(F , R) and f ∈ L1 (R). If f ∈ M(F , R), then, by Lemma 2.1, g|f = y ∗ (g) , (2.11) g∈F , ∗ and g|f = y (g) for g ∈ F. Hence, setting h := f − f we conclude from (2.11) that h2 = h|f − f = h|f − h|f = y ∗ (h) − y ∗ (h) = 0 . Thus f ∈ Θ + f for f ∈ M(F , R), and so M(F , R) ⊂ Θ + f . Conversely, suppose that f ∈ Θ + f . Then by Schwarz’s inequality (1.13) we see that for every h ∈ F, |h|f − h|f | = |h|f − f | ≤ hf − f = 0 . Hence, and by (2.11), we get h|f = h|f = y ∗ (h) for h ∈ F. Applying Lemma 2.1 once again we see that f ∈ M(F , R) for f ∈ Θ + f , and so Θ + f ⊂ M(F , R). Both the inclusions yield the equality M(F , R) = Θ + f , provided that M(F , R) = ∅, and so we obtain the following implication 86 D. Partyka and J. Zaja̧c M(F , R) = ∅ =⇒ M(F , R) = Θ + f . (2.12) Assume now that F ⊂ S. Then h|θ = 0 = y ∗ (h) , h∈F , which shows, by Lemma 2.1, that θ ∈ M(F , R). Hence, and by (2.12), we see that M(F , R) = Θ + θ = Θ. It remains to consider the case where the inclusion F ⊂ S does not hold. Then F ∩ S ⊂ F = F ∩ S. By the assumption F is a closed set in H(R). Since y ∈ L2 (R), y ∗ is a continuous functional on H(R), and so S is also closed set in H(R). Therefore F ∩ S is a closed set in H(R), and consequently Θ ⊂ F ∩ S = F . (2.13) Hence F \ (F ∩ S) = ∅. Since F ∩ S is a closed set in H(R), it follows that each h ∈ F \ (F ∩ S) has an orthogonal projection hS on F ∩ S, i.e. hS ∈ F ∩ S (2.14) and h − hS |g = 0 , g ∈F ∩S ; ⊥ cf. [6]. Hence h − hS ∈ (F ∩ S) ∩ F. If h − hS ∈ Θ, then from (2.13) and (2.14) it follows that h = hS + (h − hS ) ∈ F ∩ S + Θ = F ∩ S, which is impossible. Therefore / Θ, and so h − hS ∈ (F ∩ S)⊥ ∩ F \ Θ. Thus (F ∩ S)⊥ ∩ F \ Θ = ∅. Given h − hS ∈ h ∈ (F ∩ S)⊥ ∩ F \ Θ we see that h = 0, and so y ∗ (h) = 0. Hence, for each g ∈ F, (2.15) Since (2.16) gS := g − y ∗ (h) h2 h y ∗ (g) h∈F ∩S y ∗ (h) and g − gS = y ∗ (g) h ∈ (F ∩ S)⊥ ∩ F . y ∗ (h) ∈ (F ∩ S)⊥ , we conclude from (2.15) that y ∗ (h) y ∗ (g) y ∗ (h) y ∗ (h) g h h = g − g h = h S h2 h2 y ∗ (h) h2 y ∗ (g) y ∗ (h) = ∗ h|h = y ∗ (g) , g ∈ F . y (h) h2 Applying now Lemma 2.1, we see that (2.17) f := y ∗ (h) h ∈ M(F , R) , h2 h ∈ (F ∩ S)⊥ ∩ F \ Θ . Hence M(F , R) = ∅, and, combining (2.17) with (2.12), we derive the equality (2.10) provided the inclusion F ⊂ S does not hold. In the both cases M(F , R) = ∅, which completes the proof. 3. Calculating regression functions Assume that F is arbitrarily chosen linear and closed set in the space H(R) and y ∈ L2 (R). Then Theorem 2.3 yields M(F , R) = ∅. Moreover, Theorem 2.3 enables us to ﬁnd regression functions in F with respect to R provided we can determine 87 Generalized problem of regression the linear set (F ∩ S)⊥ ∩ F. This is rather diﬃcult task in general. However in the case where the quotient linear space F /F ∩ Θ is ﬁnite dimensional we can eﬀectively calculate all the regression functions in F with respect to R in terms of any base of this space. Obviously, this case is the most essential from practical point of view. Given f, g ∈ L1 (R) we will write f ⊥ g iﬀ f |g = 0. Given p, q ∈ Z, p ≤ q, and a sequence Zp,q k → Fk of nonempty sets in the space H(R), we write q Fk for the set of all k=p Obviously, 2 k=1 q fk where Zp,q k → fk ∈ Fk . k=p Fk = F1 + F2 . We have Theorem 3.1. Given p ∈ N let Z1,p k → hk ∈ F \ Θ be a sequence satisfying the following two conditions (3.1) lin({hk : k ∈ Z1,p }) = F as well as (3.2) hk ⊥ hl , k, l ∈ Z1,p , k = l . If y ∈ L2 (R), then (3.3) M(F , R) = (Θ ∩ F) + p y ∗ (hk ) k=1 hk 2 hk . Proof. Fix p ∈ N and a sequence Z1,p k → hk ∈ F satisfying the assumptions. From (3.1) and (3.2) it follows that F0 := Θ + F is a closed set in H(R). Therefore M(F0 , R) = ∅ by the assumption y ∈ L2 (R) and Theorem 2.3. If y ∗ (hk ) = 0 for k ∈ Z1,p , then by (3.1), F0 ⊂ S := (y ∗ )−1 (0). From Theorem 2.3 it follows that M(F0 , R) = Θ. Hence, and by Corollary 2.2, we obtain M(F , R) = Θ ∩ F, and so the equality (3.3) obviously holds. Assume in contrary, that y ∗ (hk ) = 0 for some k ∈ Z1,p . Then F0 \ S = ∅ and applying again Theorem 2.3 we can see that (F0 ∩ S)⊥ ∩ F0 \ Θ = ∅ as well as that the equality (2.10) holds. Thus we have to ﬁnd an element h ∈ (F0 ∩ S)⊥ ∩ F0 \ Θ. If p = 1, then y ∗ (h1 ) = 0. Hence, and by (3.1), (F0 ∩ S)⊥ ∩ F0 = Θ⊥ ∩ F0 = F0 , and so h1 ∈ (F0 ∩ S)⊥ ∩ F0 \ Θ. Then Theorem 2.3 leads to (3.4) M(F0 , R) = Θ + y ∗ (h1 ) h1 . h1 2 It remains to consider the case where p > 1. Without lost of generality we may assume that y ∗ (h1 ) = 0. Then by (3.1) there exists a sequence Z1,p k → λk ∈ C such that p (3.5) λk hk . h= k=1 88 D. Partyka and J. Zaja̧c Since hk − y ∗ (hk ) h1 ∈ S y ∗ (h1 ) for k ∈ Z1,p and h ∈ (F0 ∩ S)⊥ we have h ⊥ hk − (3.6) y ∗ (hk ) h1 , y ∗ (h1 ) k ∈ Z1,p . Combining this with (3.2) and (3.5) we see that for each l ∈ Z1,p , y ∗ (h ) y ∗ (hl ) l 0 = hhl − ∗ h1 = h|hl − h ∗ h1 y (h1 ) y (h1 ) p p y ∗ (h ) l λk hk hl − ∗ λk hk h1 = y (h1 ) k=1 = p k=1 k=1 p l) λk hk |hl − ∗ λk hk |h1 y (h1 ) y ∗ (h k=1 y ∗ (hl ) y ∗ (hl ) h1 |h1 = λl hl 2 − λ1 ∗ h1 2 . = λl hl |hl − λ1 ∗ y (h1 ) y (h1 ) Using this we can see that λl = (3.7) λ1 y ∗ (hl ) h1 2 . hl 2 y ∗ (h1 ) Thus h= p λk hk = λ1 h1 + k=1 p p λ1 y ∗ (hk ) y ∗ (hk ) λ1 2 2 h h = hk , h 1 k 1 2 ∗ hk y (h1 ) hk 2 y ∗ (h1 ) k=2 k=1 and, consequently, λ1 = 0 and (3.8) h̃ := p y ∗ (hk ) k=1 hk 2 hk = y ∗ (h1 ) h ∈ (F0 ∩ S)⊥ ∩ F0 . λ1 h1 2 Applying now (2.10) and (3.2) we obtain p y∗ (hk ) ∗ h y p hk 2 k y ∗ (h̃) y ∗ (hk ) · hk M(F0 , R) = Θ + h̃ = Θ + k=1 2 2 p hk 2 y∗ (hk ) h̃ k=1 hk 2 hk p =Θ+ p k=1 k=1 k=1 ∗ |y (hk )| hk 2 2 |y ∗ (hk )|2 2 hk 4 hk · p p y ∗ (hk ) y ∗ (hk ) h = Θ + hk . k 2 hk hk 2 k=1 Hence, and by (3.4) we see that for each p ∈ N, (3.9) M(F0 , R) = Θ + f , k=1 89 Generalized problem of regression where, in view of (3.1), (3.10) f := p y ∗ (hk ) hk ∈ lin({hk : k ∈ Z1,p }) = F . hk 2 k=1 From Corollary 2.2, (3.9) and (3.10) it follows that M(F , R) = F ∩ M(F0 , R) = F ∩ (Θ + f ) = (Θ ∩ F) + f . Combining this with (3.10) we derive the equality (3.3), which completes the proof. As far as applications are concerned we will study theoretic models F = lin({hk : k ∈ Z1,p }) spanned by sequences Z1,p k → hk which are not orthogonal in the space H(R) in general, because the pseudo-inner product · | · depends on the empirical data function x : Ω1 → A and measure μ. Therefore we will not apply Theorem 3.1 directly. However, in such a case we shall ortogonalize such sequences. To this end we recall that for a given p ∈ N, a sequence Z1,p k → hk ∈ L1 (R) is said to be an orthogonalization of a sequence Z1,p k → hk ∈ L1 (R), provided ⊥ \Θ , hk ∈ Hk ∩ Hk−1 (3.11) k ∈ Z1,p , where H0 := Θ and Hk := lin({h1 , h2 , . . . hk }), k ∈ Z1,p . Every linearly independent sequence Z1,p k → hk ∈ L1 (R) \ Θ has a sequence being its ortogonalization result. A sequence Z1,p k → hk may be constructed by using the Gramm-Schmidt recursive method, i.e. (3.12) h1 := h1 and hn := hn − n−1 k=1 hn |hk h , hk 2 k n ∈ Z2,p . Corollary 3.2. Given p ∈ N let Z1,p k → hk ∈ F \ Θ be a linearly independent sequence satisfying the condition (3.1) and let a sequence Z1,p k → hk ∈ L1 (R) be its orthogonalization result. If y ∈ L2 (R), then (3.13) M(F , R) = (Θ ∩ F) + p y ∗ (hk ) h . hk 2 k k=1 In particular, the above equality holds for the sequence Z1,p k → hk , deﬁned by the formulas (3.12). Proof. Given p ∈ N ﬁx sequences Z1,p k → hk ∈ F \ Θ and Z1,p k → hk ∈ L1 (R) satisfying the assumptions. From the property (3.11) it follows that hk = 0 for k ∈ Z1,p and hk ⊥ hl for k, l ∈ Z1,p such that k = l. Moreover, by the condition (3.1) we obtain lin({h1 , h2 , . . . hp }) = lin({h1 , h2 , . . . hp }) = F . Thus, applying Theorem 3.1 for the sequence Z1,p k → hk replaced by its ortogonalized associate Z1,p k → hk we derive the equality (3.13), which is our claim. 90 D. Partyka and J. Zaja̧c 4. Complementary remarks In this section we present comments and examples which complete and illustrate our consideration from previous sections. In the following remark we gather a few simple observations from Corollary 3.2. Remark 4.1. Under the assumptions of Corollary 3.2, we have θ ∈ Θ ∩ F, and so the equality (3.13) yields (4.1) f := p y ∗ (h ) k k=1 hk 2 hk ∈ M(F , R) and M(F , R) = (Θ ∩ F) + f . Since Θ ∩F is a linear set, the second equality in (4.1) shows that the class M(F , R) forms an aﬃne variety in the space H(R). Moreover from (4.1) we easily deduce that the following properties are pairwise equivalent: (i) f is a unique regression function in F with respect to R; (ii) Θ ∩ F = {θ}; (iii) h > 0 for every h ∈ F \ {θ}; (iv) h = 0 ⇒ h = θ for every h ∈ F. If additionally the sequence Z1,p k → hk ∈ F \ Θ satisﬁes the orthogonality condition (3.2), then the formulas (3.12) yield hk = hk , k ∈ Z1,p , and consequently the property (4.1) remains valid after replacing hk by hk as k ∈ Z1,p . According to (4.1) the class M(F , R) is determined by the sequence Z1,p k → y ∗ (hk )hk −2 hk . We will call it the regression functions sequence (RFS) generated by a linearly independent sequence Z1,p k → hk ∈ F \ Θ satisfying the condition (3.1). It is worth noting that our approach to the regression theory is very ﬂexible. We provide an universal and simple theory covering classical cases of regressions where the theoretic functional model F is spanned by polynomials, trigonometric polynomials and other speciﬁc functions; cf. e.g. [13] and [2]. Moreover, we study regression functions with respect to the wide range of regression structures R involving the generalized quadratic deviation (1.1) by means of certain measures μ. This simpliﬁes much theoretical considerations on the ground of Hilbert spaces. On the other hand side we gain the possibility of using the modiﬁed smallest square method which can be more adequate in speciﬁc situations. In Example 0.1 the classical smallest square method was used. According to the equality (1.3) in Example 1 this is a special case of the criterion δ with the measure μ satisfying (1.2) and (1.4). In what follows we present an example which motivate using a more sophisticated measure μ. Example 4.2. Following Example 0.1 we wish this time to determine the electric circuit resistance R by means of measurements samples of intensity and voltage Generalized problem of regression 91 represented by two sequences Z0,n k → ik and Z0,m k → vk for some n, m ∈ N. Assume that all measurements were made independently. Given a precision rate let ρk be the probability that the intensity sample ik satisﬁes the precision rate for k ∈ Ω1 := Z0,n and let ρl be the probability that the voltage sample vl satisﬁes the precision rate for l ∈ Ω2 := Z0,m . As in Example 0.1 we consider the regression structure R, where A := R, B := R, the empiric data functions are deﬁned by Z0,n k → x(k) := ik and Z0,n k → y(k) := vk , and as the deviation criterion δ we take the generalized quadratic deviation given by (1.1). Following Example 1 we deﬁne the measure μ as a unique measure satisfying the equalities (1.2). We now need to deﬁne the numbers ρk,l for k ∈ Ω1 and l ∈ Ω2 . We can do it in many ways. In our case all measurements were made independently, so it seems to be natural to set (4.2) ρk,l := ρk · ρl , k ∈ Ω1 , l ∈ Ω2 . Then each coeﬃcient ρk,l is equal to the probability of the event that both the measurement samples ik and vl satisfy simultaneously the prescribed precision rate. As a matter of fact the coeﬃcient ρk,l reﬂects accuracy of the measurement samples ik and vl , and thereby reﬂects accuracy of the measurements devices used for getting these samples. If the coeﬃcient ρk,l is closer to the value 1 then intuitively the corresponding pair (ik , vl ) of samples is more valuable for us. Therefore the generalized quadratic deviation criterion δ deﬁned by (4.2) seems to be more natural in this case as compared to the classical least squares method, where all samples (ik , vk ) are treated equivalently and samples of the form (ik , vl ), k = l, are not considered at all. As in the Example 0.1, we consider the theoretic functional model F represented by linear functions R t → rt for r ∈ R. Then F = lin({h1 }) where h1 is the identity mapping on R, i.e. h1 (t) = t for t ∈ R. Thus we can apply our theory from previous sections in order to determine all regression functions in F with respect to R. The condition (1.7) obviously holds for every function f : A → B, which means that L1 (R) = (R → R). From (1.11) we have m n 2 h1 = (4.3) |(h1 ◦ x)(t1 )|2 d μ(t1 , t2 ) = i2k ρk,l . Ω1 ×Ω2 k=0 l=0 It is also easily seen that each function g : Z0,m → B satisﬁes the condition (1.8), and so L2 (R) = (Z0,m → R). From (1.14) it follows that n m y ∗ (h1 ) = (4.4) (h1 ◦ x)(t1 )y(t2 ) d μ(t1 , t2 ) = ik vl ρk,l . Ω1 ×Ω2 k=0 l=0 Assume that h1 = 0. Then F ⊂ Θ, which implies, by Corollary 2.2, that M(F , R) = F . Suppose for simplicity that ρk,l > 0 for k ∈ Ω1 and l ∈ Ω2 . From (4.3) we see that h1 = 0 iﬀ ik = 0 for k ∈ Ω1 . Thus the equality h1 = 0 means that the current intensity vanishes (current does not ﬂow) or the current intensity 92 D. Partyka and J. Zaja̧c is below the sensitivity of intensity measurements devices. In both the cases we are not able to determine the resistance R. This provides a natural interpretation of the equality M(F , R) = F . Assume now that h1 = 0. Then h1 ∈ F \ Θ, and so Θ ∩ F = {θ}. Theorem 3.1 now leads to ∗ y ∗ (h1 ) y (h1 ) M(F , R) = (Θ ∩ F) + h1 = h1 , (4.5) h1 2 h1 2 which means that the set M(F , R) consists of the unique regression function Rt→ y ∗ (h1 ) y ∗ (h1 ) h1 (t) = t. 2 h1 h1 2 Hence, and by (4.3) and (4.4), we can uniquely determine the resistance n m n m y ∗ (h1 ) R= (4.6) = i v ρ i2k ρk,l . k l k,l h1 2 k=0 l=0 k=0 l=0 Note that if n = m and the coeﬃcients ρk,l are deﬁned by (1.4), then (4.6) yields (0.6). Such a situation naturally corresponds to the sequence Z0,n k → (ik , vk ) of n + 1 simultaneous measurements of current intension and voltage with the same precision. The following example illustrates the usage of Corollary 3.2 in the case where the theoretic functional model F is spanned by two functions. Example 4.2. Given a regression structure R with y ∈ L2 (R) consider the case where the functional model F is spanned by two arbitrary and linearly independent ﬁxed functions h1 , h2 ∈ L1 (R) \ Θ, i.e. F = lin({h1 , h2 }). Applying Corollary 3.2 we can see that 2 y ∗ (hk ) M(F , R) = (Θ ∩ F) + (4.7) h , hk 2 k k=1 where according to (3.12), (4.8) h1 := h1 and h2 := h2 − h2 |h1 h2 |h1 h = h2 − h1 . h1 2 1 h1 2 Hence, h2 ⊥ h1 , and consequently 2 h2 |h1 2 2 2 = h2 2 − |h2 |h1 | . h2 = h2 − (4.9) h 1 h1 2 h1 2 Setting (4.10) a2 := y ∗ (h2 ) h2 2 and a1 := y ∗ (h1 ) h2 |h1 − a2 h1 2 h1 2 we conclude from (4.7) and (4.8) that (4.11) M(F , R) = (Θ ∩ F) + a2 h2 + a1 h1 . Generalized problem of regression 93 Combining (4.10) with (4.8) and (4.9) we obtain (4.12) a2 = y ∗ (h2 )h1 2 − y ∗ (h1 )h2 |h1 h2 2 h1 2 − |h2 |h1 |2 and a1 = y ∗ (h1 ) − h2 |h1 a2 . h1 2 In particular, if h2 (t) = t and h1 (t) = 1 as t ∈ R, and the regression structure R is deﬁned as in Example 1 under the assumption that m = n and the coeﬃcients ρk,l satisfy (1.4), then the equalities in (4.12) yield a0 = a2 and b0 = a1 where a0 and b0 are deﬁned in (0.2). Acknowledgement Presented results were obtained during the Monday Mathematical Seminar, gained in the State University of Applied Science in Chelm by both the authors. References [1] T. W. Anderson, Last squares and best unbiased estimates, Ann. Math. Statist. 33 (1982), 266–272. [2] N. R. Draper and H. Smith, Applied Regression Analysis, John Wiley and Sons, Inc., New York - London - Sydney, 1966. [3] I. Durbin, A note on regression when there is extraneous information about one of the coeﬃcients, IASA 48 (1953), 799–808. [4] N. I. Fisher and A. I. Lee, Regression models for an angular response, Biometrics 48 (1992), no. 3, 665–677. [5] P. R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea, New York, 1957. [6] W. Mlak, Hilbert Spaces and Operator Theory, Mathematics and Its Applications, Kluwer Academic Publishers and PWN-Polish Scientiﬁc Publishers, Dordrecht, Boston, London and Warsaw, 1991. [7] T. Naes and I. S. Helland, Relevant components in regression, Scand. Journal of Statistics 20 (1993), no. 3, 239–250. [8] P. C. B. Philips and Y. Sun, Regression with an evaporating logarithmic trend, Econometric Theory 19 (2003), no. 4, 692–701. [9] R. E. Quandt, The estimation of the parameter of a linear regression system obeying two separate regimes, IASA 53 (1968), 873–890. [10] W. Rudin, Functional Analysis, McGraw-Hill, New York 1991. [11] L. G. Teser, Iterative estimation of a set linear regression equations, IASA 59 (1964), 845–862. [12] W. B. Whiston, Sequential Selection of Variables in Multiple Regression, University of Cincinnati, 1964. [13] E. I. Wiliams, Regression Analysis, John Wiley and Sons, New York, 1959. 94 D. Partyka and J. Zaja̧c Faculty of Mathematics and Natural Sciences The John Paul II Catholic University of Lublin Al. Raclawickie 14, P.O. Box 129 PL-20-950 Lublin, Poland State University of Applied Science in Chelm PL-22-100 Chelm, Pocztowa 54, Poland e-mail: [email protected] State University of Applied Science in Chelm PL-22-100 Chelm, Pocztowa 54, Poland Chair of Applied Mathematics The John Paul II Catholic University of Lublin Al. Raclawickie 14, P.O. Box 129 PL-20-950 Lublin, Poland e-mail: [email protected] Presented by Adam Paszkiewicz at the Session of the Mathematical-Physical Commission of the L ódź Society of Sciences and Arts on March 2, 2010 UOGÓLNIONE ZAGADNIENIE REGRESJI Streszczenie Autorzy przedstawiaja̧ uogólnienie klasycznej idei regresji przez studium i rozwia̧zanie pewnych zagdnień ekstremalnych, określonych poprawnie w ramach skończenie- lub nieskończenie-wymiarowej przestrzeni Hilberta. Przy danych empirycznych, dyskretnych lub cia̧glych, określamy i jednoznacznie wyrażamy klasȩ wszystkich rozwia̧zań zagdnienia w nowej postaci cia̧gów funkcji regresji (RFS). Znacza̧ca rozmaitość obserwowanych zjawisk, w różnych dziedzinach nauk praktycznych i teoretycznych, może być przy użyciu techniki RFS opisana i zbadana z istotna̧ precyzja̧. PL ISSN 0459-6854 BULLETIN DE LA SOCIÉTÉ DES SCIENCES ET DES LETTRES DE L ÓDŹ 2010 Vol. LX Recherches sur les déformations no. 1 pp. 95–108 Jacek Dziok EXTREMAL PROBLEMS IN A GENERALIZED CLASS OF UNIFORMLY CONVEX FUNCTIONS Summary The object of the present paper is to investigate classes of analytic functions with varying argument of coeﬃcients deﬁned by subordination. The classes generalize the wellknown class of uniformly convex functions. By using the extreme points theory we obtain coeﬃcient inequalities and distortion theorems in the classes of functions with varying coeﬃcients. Some integral mean inequalities are also pointed out. 1. Introduction Let B denote the class of functions f : U → C, where U= {z ∈ C : |z| < 1}, and by A we denote the class of functions f ∈ B which are analytic in U. Let F be a subclass of the class A. A function f ∈ F is called an extreme point of F if the condition f = γg + (1 − γ) h (g, h ∈ F, 0 < γ < 1) implies g = h. We shall use the notation EF to denote the set of all extreme points of F . It is clear that EF ⊂ F. We say that F is locally uniformly bounded if for each r, 0 < r < 1, there is a real constant M so that |f (z)| ≤ M (f ∈ F, |z| ≤ r) . We say that a class F is convex if γf + (1 − γ)g ∈ F (f, g ∈ F, 0 ≤ γ ≤ 1). Moreover, we deﬁne the closed convex hull of F as the intersection of all closed convex subsets of A that contain F . We denote the closed convex hull of F by HF . 96 J. Dziok If F = {fn ∈ A : n ∈ N = {1, 2, ...}} , then (1) HF = ∞ γn fn : n=1 ∞ γn = 1, γn ≥ 0 (n ∈ N) . n=1 A real-valued functional J : A → R is called continuous on F if for any sequence {fn } in F which locally uniformly converges to f the sequence {J (fn )} converges to J (f ) . Furthermore, a real-valued functional J : A → R is called convex on a convex class F ⊂ A if J (γf + (1 − γ) g) ≤ γJ (f ) + (1 − γ) J (g) (f, g ∈ F, 0 ≤ γ ≤ 1) . For each ﬁxed value of m, n ∈ N, z ∈ U, the following real-valued functionals are continuous and convex on A: (n) (2) J (f ) = f (0)/n! , J (f ) = |f (z)| , J (f ) = f (m) (z) z ∈ U, f ∈ A . Moreover, for λ > 0, 0 < r < 1, the real-valued functional ⎛ ⎞1/λ 2π λ 1 (n) J (f ) = ⎝ (3) f ∈ A reiθ dθ⎠ f 2π 0 For λ ≥ 1, by Minkowski’s inequality it is also convex on A. is continuous on A. The extreme points theory for analytic functions was intensively investigated by Hallenbeck and MacGregor [1] (see also [2] and [3]). We say that a function f ∈ B is subordinate to a function F ∈ B, and write f (z) ≺ F (z) (or simply f ≺ F ), if and only if there exists a function ω ∈ B, |ω(z)| ≤ |z| (z ∈ U) , such that f (z) = F (ω(z)) (z ∈ U ) . In particular, if F, F ∈ B, is univalent in U, in analogy to the case of analytic functions, we have the following equivalence: f (z) ≺ F (z) ⇐⇒ f (0) = F (0) and f (U) ⊂ F (U). For functions f, g ∈ A of the form ∞ ∞ f (z) = an z n and g(z) = bn z n , n=0 n=0 by f ∗ g we denote the Hadamard product (or convolution) of f and g, deﬁned by (f ∗ g) (z) = ∞ n=0 an b n z n (z ∈ U) . Extremal problems in a generalized class of uniformly convex functions 97 We denote by A the class of functions f ∈ B of the form ∞ f (z) = z + (4) an z n (z ∈ U). n=2 Also, by Tη (η ∈ R) we denote the class of functions f ∈ A of the form (4) for which arg(an ) = π + (1 − n)η (5) (n = 2, 3, ...). For η = 0 we obtain the class T0 of functions with negative coeﬃcients. Moreover, we deﬁne T := (6) Tη . η∈R The class T was introduced by Silverman [4] (see also [5]). It is called the class of functions with varying argument of coeﬃcients. Let k, A, B be real parameters, k ≥ 0, 0 ≤ B ≤ 1, −1 ≤ A < B, and let ϕ, φ ∈ A. By W (φ, ϕ; A, B; k) we denote the class of functions f ∈ A such that (ϕ ∗ f ) (z) = 0 (z ∈ U {0}) and (φ ∗ f ) (z) 1 + Az (φ ∗ f ) (z) (7) − k − 1 ≺ . (ϕ ∗ f ) (z) (ϕ ∗ f ) (z) 1 + Bz Here we use, of course, the deﬁnition of subordination in the class B. If −1 < B < 1, then the condition (4) is equivalent to the following: (φ ∗ f ) (z) 1 − AB (φ ∗ f ) (z) B−A (8) (ϕ ∗ f ) (z) − k (ϕ ∗ f ) (z) − 1 − 1 − B 2 < 1 − B 2 (z ∈ U), and if B = 1, then we have 1+A (φ ∗ f ) (z) (φ ∗ f ) (z) (9) − 1 > Re − k (ϕ ∗ f ) (z) (ϕ ∗ f ) (z) 2 (z ∈ U) . In relation to the classes T and Tη , we deﬁne the following two classes: T W (φ, ϕ; A, B; k) := T ∩ W (φ, ϕ; A, B; k) , T W η (φ, ϕ; A, B; k) := Tη ∩ W (φ, ϕ; A, B; k) . For the present investigations we assume that ϕ, φ are functions of the form ∞ ∞ n (10) αn z , φ(z) = z + βn z n (z ∈ U), ϕ(z) = z + n=2 n=2 where the sequences {αn } , {βn } are real, and 0 ≤ αn < βn (n = 2, 3, . . . ) . Moreover, let us set (11) dn := (k + 1) (1 + B) βn − (kB + A + k + 1) αn (n = 2, 3, . . . ) . The families W (φ, ϕ; A, B; k) and Wη (φ, ϕ; A, B; k) unify various new and wellknown classes of analytic functions. We list a few of them in the last section. 98 J. Dziok The object of the present paper is to investigate extreme points of the class Wη (φ, ϕ; A, B; k) . By using the extreme points theory we obtain coeﬃcients inequalities and distortion theorems in the classes of functions. Some integral means inequalities are also pointed out. 2. Extreme points Since A is a complete metric space Montel’s theorem [6] implies following lemma: Lemma 1. A class F contained in A is compact if and only if F is closed and locally uniformly bounded. First we mention a suﬃcient condition for the function to belong to the class W (φ, ϕ; A, B; k). Theorem 1. Let {dn } be deﬁned by (11), 0 ≤ B ≤ 1, −1 ≤ A < B. If a function f of the form (4), (ϕ ∗ f ) (z) = 0(z ∈ U {0}), satisﬁes the condition ∞ (12) dn |an | ≤ B − A, n=2 then f belongs to the class W (φ, ϕ; A, B; k). Proof. If 0 ≤ B < 1, then for a function f of the form (4) we have 1− ∞ αn |an ||z|n−1 > 0 n=2 and ≤ ≤ (φ ∗ f ) (z) 1 − AB (φ ∗ f ) (z) (ϕ ∗ f ) (z) − k (ϕ ∗ f ) (z) − 1 − 1 − B 2 (φ ∗ f ) (z) B (B − A) (k + 1) − 1 + (ϕ ∗ f ) (z) 1 − B2 ∞ (βn − αn ) |an ||z|n−1 B (B − A) n=2 (k + 1) + ∞ 1 − B2 1− αn |an ||z|n−1 (z ∈ U ). n=2 Thus, by (11) and (12), we obtain (8) and, consequently, f ∈ W (φ, ϕ; A, B; k) . Let now B = 1. Then simple calculations give (φ ∗ f ) (z) (φ ∗ f ) (z) 1 + A − 1 − Re − k (ϕ ∗ f ) (z) (ϕ ∗ f ) (z) 2 ∞ (βn − αn ) |an ||z|n−1 (φ ∗ f ) (z) n=2 − 1 ≤ (k + 1) ≤ (k + 1) ∞ (ϕ ∗ f ) (z) 1− αn |an ||z|n−1 n=2 (z ∈ U ), Extremal problems in a generalized class of uniformly convex functions 99 and, by (11) and (12) again, we obtain (9). Hence f ∈ W (φ, ϕ; A, B; k) and the proof is complete. The next theorem shows that the condition (12) is necessary as well for functions of the form (4), with (5) to belong to the class T W η (φ, ϕ; A, B; k). Theorem 2. Let f be a function of the form (4), with (5). Then f belongs to the class T W η (φ, ϕ; A, B; k) if and only if the condition (12) holds true. Proof. In view of Theorem 1 we need only show that each function f from the class T W η (φ, ϕ; γ, k) satisﬁes the coeﬃcient inequality (12). Let f be a function of the form (4), satisfying the condition (5) and belonging to the class T W η (φ, ϕ; γ, k). Then, putting z = riη in the conditions (8) and (9), we obtain ∞ (βn − αn ) |an |rn−1 B−A n=2 . (k + 1) < ∞ 1+B 1− αn |an |rn−1 n=2 Thus we have ∞ [(k + 1) (1 + B) βn − (k (1 + B) + 1 + A) αn ] |an |rn−1 < B − A, n=2 which, upon letting r → 1 − , readily yields the assertion (12). Since the condition (12) is independent of η, Theorem 2 yields the following theorem. Theorem 3. Let f be a function of the form (4), with (5). Then f ∈ T W (φ, ϕ; A, B; k) if and only if the condition (12) holds true. Theorem 4. The class T W η (φ, ϕ; A, B; k) is a convex and compact subclass of A. Proof. Let functions f, g belong to the class T W η (φ, ϕ; A, B; k) , 0 ≤ γ ≤ 1. Since ∞ ∞ n n an z bn z (13) γf (z) + (1 − γ)g (z) = γ z + + (1 − γ) z + = z+ ∞ n=2 n=2 (γan + (1 − γ) bn ) z n , n=2 by Theorem 2 we have ∞ dn |γan + (1 − γ) bn | ≤ n=2 γ ∞ n=2 ≤ dn |an | + (1 − γ) ∞ dn |bn | n=2 γ (B − A) + (1 − γ) (B − A) = B − A, and consequently the function h = γf + (1 − γ)g belongs to the class T W η (φ, ϕ; A, B; k). Hence the class is convex. 100 J. Dziok Furthermore, for f ∈ T W η (φ, ϕ; A, B; k) , |z| ≤ r < 1, we have |f (z)| ≤ r + (14) ∞ dn an n=2 Since ∞ rn rn ≤ r + (B − A) . dn d n=2 n n1 1 = r lim sup (dn )− n ≤ r < 1, lim sup rn d−1 n (15) n→∞ the power series ∞ n=2 n→∞ rn d−1 converges. Thus, by (14) we conclude that the class n T W η (φ, ϕ; A, B; k) is locally uniformly bounded. By Lemma 1, we only need to show that it is closed i.e. if fm ∈ T W η (φ, ϕ; A, B; k) (m ∈ N) and fm → f, then f ∈ T W η (φ, ϕ; A, B; k) . Suppose that fm (z) = z + ∞ an,m z n (m ∈ N; z ∈ U) n=2 and f is given by (4). Using Theorem 2 we have ∞ dn |an,m | ≤ B − A (m ∈ N) . n=2 Since fm → f, we conclude that an,m → an as m → ∞ (n ∈ N). This gives the condition (12), and, in consequence, f ∈ T W η (φ, ϕ; A, B; k) , which completes the proof. Theorem 5. Let f1 (z) = z and (16) fn (z) = fn,η (z) = z − B − A i(1−n)η n e z dn (n = 2, 3, ...; z ∈ U) , where {dn } is deﬁned by (11). Then ET W (φ, ϕ; A, B; k) = {fn ; n ∈ N} . Proof. By using (12) we verify easily, that the functions of the form (16) are extreme points of the class T W η (φ, ϕ; A, B; k). Now suppose that a function f belongs to the set ET W η (φ, ϕ; A, B; k) and f is not of the form (16). If f (z) = z − γ (B − A) n z dn (0 < γ < 1, n = 2, 3, ...; z ∈ U) , then f (z) = (1 − γ)f1 (z) + γfn (z), z ∈ U and f is not the extreme point of T W η (φ, ϕ; A, B; k) . In other way there exist m, l ∈ N, m = l, so that the coeﬃcients am and al do not vanish in the power series (4). Setting g(z) = f (z) − al z l + al m am l dm am z , h(z) = f (z) − am z m + z, γ= , dm dl dm am + dl al Extremal problems in a generalized class of uniformly convex functions 101 we have g, h ∈ T W (φ, ϕ; A, B; k) , g = h, 0 < γ < 1 and f = γg + (1 − γ) h. It follows that f ∈ / ET W η (φ, ϕ; A, B; k) , and the proof is complete. 3. Applications The following lemmas will be useful later on. If f ≺ g, then Lemma 2. (cf. [7]). Let f, g ∈ A. 2π 0 f (reiθ )λ dθ ≤ 2π g(reiθ )λ dθ (0 < r < 1, λ > 0) . 0 Lemma 3. (The Krein-Milman theorem). If F is a compact convex subclass of the then HEF = F . class A, The Krein-Milman theorem (see [8] and [9]) is fundamental in the theory of extreme points. In particular, it implies the following lemma: Lemma 4. (cf. [1]). Let F be a compact convex subclass of the class A and J : A → R be a real-valued, continuous and convex functional on F . Then max {J (f ) : f ∈ HF } = max {J (f ) : f ∈ F } = max {J (f ) : f ∈ EHF } . Using the extreme points of the class T W η (φ, ϕ; A, B; k) we obtain results listed below. By (1) the Krein-Milman theorem gives Corollary 1. Let f1 (z) = z and fn,η be deﬁned by (16). Then ∞ ∞ γn fn,η ; γn = 1, γn ≥ 0 ( n ∈ N) . T W η (φ, ϕ; A, B; k) = n=1 n=1 Combining (2) with Lemma 4 we get Corollary 2. If a function f of the form (4) belongs to the class T W η (φ, ϕ; A, B; k), then (17) |an | ≤ B−A dn (n = 2, 3, . . . ), where dn is deﬁned by (11). The result is sharp. The functions fn,η of the form (16) are the extremal functions. 102 J. Dziok For the extreme points fn,η of the form (16) we have (18) fn,η (z) = 1 − (19) (l) (z) = − fn,η (l) (z) = 0 fn,η (B − A) n i(1−n)η n−1 e z , dn (B − A) n! i(1−n)η n−l e z (l = 2, 3, ..., n), (n − l)!dn (l > n) . (l) Let l ∈ N0 , 0 < r < 1, and the sequence δn be deﬁned by δn(l) = (20) (B − A) n! n−l r (n − l)!dn (n ≥ max {l, 2}) . Applying (15) we obtain lim sup δn(l) = 0 (l ∈ N0 ) . n→∞ Thus there exist nl ∈ N (l ∈ N0 ) , such that δn(l)l = max δn(l) : n ≥ max {l, 2} (21) (l ∈ N0 ) . Therefore, by Lemma 3 we have Corollary 3. If a function f belongs to the class T W η (φ, ϕ; A, B; k) , |z| = r < 1, then B − A n0 B − A n0 (22) r ≤ |f (z)| ≤ r + r , r− dn0 dn0 (23) 1− (B − A) n1 n1 r dn1 (l) f (z) (24) ≤ |f (z)| ≤ 1 + ≤ (B − A) n1 n1 r , dn1 (B − A) (nl )! nl −l r (nl − l)!dnl (l ≥ 2) , where nl is deﬁned by (21). The result is sharp. The functions fnl ,η of the form (16) are the extremal functions. From Corollary 3 we get Corollary 4. Let a function f belong to the class T W η (φ, ϕ; A, B; k) , |z| = r < (l) (l ∈ N0 ) deﬁned by (20) is nonincreasing with respect to 1. If the sequence δn n, then B−A 2 B−A 2 (25) r ≤ |f (z)| ≤ r + r (l = 0) , r− d2 d2 (26) 1− 2 (B − A) 2 2 (B − A) 2 r ≤ |f (z)| ≤ 1 + r d2 d2 (l = 1) , 103 Extremal problems in a generalized class of uniformly convex functions (l) (B − A) (l)! f (z) ≤ dl (27) (l ≥ 2) , where nl is deﬁned by (21). The result is sharp. The functions fnl ,η of the form (16) are the extremal functions. Now, we consider some integral means inequalities. Corollary 5. Let 0 < r < 1, λ ≥ 1, l ∈ N0 and assume that the sequence (l) δn deﬁned by (20) is nonincreasing with respect to n. If f ∈ T W η (φ, ϕ; A, B, k) , then (28) 2π (l) iθ λ f (re ) dθ ≤ 0 2π (29) 2π (l) iθ λ f2,η (re ) dθ (l = 0, 1) , 0 (l) iθ λ f (re ) dθ ≤ 0 2π (l) iθ λ fl,η (re ) dθ (l = 2, 3, . . . ) , 0 where fl,η are the functions deﬁned by (16). Proof. Since f2,η fn,η ≺ and fn,η ≺ f2,η (n ∈ N) , z z then using Lemma 2 we have ⎧ 2π ⎫ ⎨ λ ⎬ 2π λ (l) (l) iθ max fn,η (re ) dθ : n ∈ N = f2,η (reiθ ) dθ ⎩ ⎭ 0 (l = 0, 1) . 0 Thus, Lemma 4 yields (28). The inequality (29) is an immediate consequence of (27) and (19). Making use of (6) and Corollaries 2, 3, 4, and 5, we obtain the corollaries listed below. Corollary 6. If a function f of the form (4) belongs to the class T W (φ, ϕ; A, B; k), then the coeﬃcients estimates (17) hold true. The result is sharp. The functions fn,η (η ∈ R) of the form (16) are the extremal functions. Corollary 7. If a function f of the form (4) belongs to the class T W (φ, ϕ; A, B; k), |z| = r < 1, then the bounds (22), (23), and (24), hold true. The results are sharp, the functions fnl ,η (η ∈ R) of the form (16) are the extremal functions. Corollary 8. Let a function f of the form (4) belong to the class T W (φ, ϕ; A, B; k), |z| = r < 1. If the sequence (l) δn (l ∈ N0 ) deﬁned by (20) is nonincreasing with 104 J. Dziok respect to n, then the inequalities (25), (26), and (27) hold true. The result is sharp, the functions fn,η (η ∈ R) of the form (16) are the extremal functions. l ∈ N0 and assume that the sequence Corollary 9. Let 0 < r < 1, λ ≥ 1, (l) δn deﬁned by (20) is nonincreasing with respect to n. If a function f belongs to the class T W (φ, ϕ; A, B; k), then the inequalities (28) and (29), hold true. 4. Concluding remarks We conclude this paper by observing that, in view of the subordination relation (7) and so also (8), (9), choosing the functions φ and ϕ, we can consider new and well-known classes of functions. Let n−1 k ϕ x z ; A, B; k , Wn (ϕ; A, B; k) := W zϕ (z) , k=0 where n ∈ N, xn = 1. In particular, the class Wn (ϕ; A, B) := Wn (ϕ; A, B; 0) , contains functions f ∈ A, which satisfy the condition z (ϕ ∗ f ) (z) n−1 k=0 (ϕ ∗ f ) (xk z) ≺ 1 + Az . 1 + Bz It is related to the class of starlike functions with respect to n-symmetric points. Moreover putting n = 1, we obtain the class W (ϕ; A, B) := W1 (ϕ; A, B) deﬁned by 1 + Az z (ϕ ∗ f ) (z) ≺ . (ϕ ∗ f ) (z) 1 + Bz The class is related to the class of starlike function. Analogously, the class Hn (ϕ; γ, k) := Wn (ϕ; 2γ − 1, 1; k) (0 ≤ γ < 1) contains functions f ∈ A, which satisfy the condition z (ϕ ∗ f ) (z) z (ϕ ∗ f ) (z) Re n−1 − γ > k n−1 − 1 k k k=0 (ϕ ∗ f ) (x z) k=0 (ϕ ∗ f ) (x z) (z ∈ U) . It is related to the class of k-uniformly convex function of order γ with respect to n-symmetric points. Moreover setting n = 1, we obtain the class H (ϕ; γ, k) := H1 (ϕ; γ, k) deﬁned by Re z (ϕ ∗ f ) (z) −γ (ϕ ∗ f ) (z) z (ϕ ∗ f ) (z) > k − 1 (ϕ ∗ f ) (z) (z ∈ U) . Extremal problems in a generalized class of uniformly convex functions 105 The class is related to the class of k-uniformly convex function of order γ. The classes z ; γ, k , U ST (γ, k) := H 1−z z U CV (γ, k) := H 2 ; γ, k , (1 − z) are the well-known classes of of k-starlike function of order γ and k-uniformly convex function of order γ, respectively. In particular, the classes U CV := U CV (1, 0) , k − U CV := U CV (k, 0) were introduced by Goodman [10] (see also [11, 12]), and Kanas and Wiśniowska [13] (see also [14]), respectively. We note that the class HT (ϕ; γ, k) := T0 ∩H (ϕ; γ, k) was introduced and studied by Raina and Bansal [15]. If we set h(α1 , z) := z q Fs (α1 , . . . , αq ; β1 , . . . , βs ; z), where q Fs is the generalized hypergeometric function (for details see [16] and [17]), then we obtain the class UH (q, s, λ, γ, k) := HT (λh(α1 + 1, z) + (1 − λ) h(α1 , z); γ, k) (0 ≤ λ ≤ 1) deﬁned by Srivastava et al. [18]. Let λ be a convex parameter. A function f ∈ A belongs to the class ϕ (z) + (1 − λ) ϕ (z) , z; A, B; 0 Vλ (ϕ; A, B) := W λ z if it satisﬁes the condition (ϕ ∗ f ) (z) 1 + Az λ + (1 − λ) (ϕ ∗ f ) (z) ≺ . z 1 + Bz Moreover, a function f ∈ A belongs to the class ϕ (z) + (1 − λ) ϕ (z) ; A, B; 0 Uλ (ϕ; A, B) := W λ z if it satisﬁes the condition z (ϕ ∗ f ) (z) + (1 − λ) z 2 (ϕ ∗ f ) (z) 1 + Az . ≺ 1 + Bz λ (ϕ ∗ f ) (z) + (1 − λ) z (ϕ ∗ f ) (z) The classes Wn (ϕ; A, B) , Hn (ϕ; γ, k) , Uλ (ϕ; A, B) and Vλ (ϕ; A, B) generalize well-known important classes, which were investigated in earlier works, see for example [19–36]. If we apply the results presented in this paper to the classes discussed above, we can obtain several additional new results. Some of these were obtained in earlier works, see for example [31–36]. 106 J. Dziok References [1] D. J. Hallenbeck and T. H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory, Pitman Advanced Publishing Program, Boston, Pitman, 1984. [2] D. J. 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Sivasubramanian, Hypergeometric functions in the parabolic starlike and uniformly convex domains, Integral Transforms Spec. Funct. 18 (2007), 511–520. [34] B. A. Frasin, Comprehensive family of uniformly analytic functions, Tamkang J. Math. 36 (2005), 243–254. [35] K. Vijaya and G. Murugusundaramoorthy, Some uniformly starlike functions with varying arguments, Tamkang. J. Math. 35 (2004), 23–28. [36] G. Murugusundaramoorthy and N. Magesh, On applications of Jung-Kim-Srivastava integral operator to a subclass of starlike functions with negative coeﬃcients, J. Comput. Anal. Appl. 9 (2007), 221–232. Institute of Mathematics University of Rzeszów Rejtana 16A, PL-35-310 Rzeszów Poland e-mail: [email protected] Presented by Zbigniew Jakubowski at the Session of the Mathematical-Physical Commission of the L ódź Society of Sciences and Arts on March 2, 2010 108 J. Dziok ZADANIA EKSTREMALNE W UOGÓLNIONEJ KLASIE FUNKCJI JEDNOSTAJNIE WYPUKLYCH Streszczenie Zadaniem obecnej pracy jest badanie funkcji analitycznych o zmieniaja̧cym siȩ argumencie wspólczynników określonych przez podporza̧dkowanie. Klasy te uogólniaja̧ dobrze znane klasy funkcji jednostajnie wypuklych. Przez zastosowanie teorii punktów krańcowych uzyskujemy nierówności miȩdzy wspólczynnikami i twierdzenia o dystorsji w klasach funkcji o zmieniaja̧cych siȩ wspólczynnikach. Wskazujemy też na pewne nierówności dotycza̧ce średnich calkowych. BULLETIN !" !# #$" # %&'"()*+,# , -- . !"!# $ # % & ' ( ' ! ) ' ( % * + %*+# & &' # * ' ' # ' & $& # , # +, # *! *"( /,),0 %,*# #*"$ *$"# *!)* !)1 )* )#* , %+(,#+, 2!+ ! ()#$"% +, ,),(*"# . 3++,*! ' ) (*" 2)%)4# ,),* !,54 +# ) -)"* ' # +, 2!" %+6",* %+# +-+,# +,*"# * , *! 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""#-,%+,8 #*)*+#*+ ) -"-"*+# % ,* ,7"8 * ),1 -)"*+ $)" 7)$ , *!+# -)-"4 *! K *# ' *!# *",%# !)7 ;, "(7% ;1 *! )--+ )*+, ' *! %+K", +,8 (*!%4 *"),#'"(+,8 *! ,, #*)*+,)"1 *+( #"+# * #*)*+,)"1 ,# ,#+%"+,8 *! G"#* %+K", # ' *! #"+# {Xt} 1+%# *! #)( "#$* )# )--1+,8 ) !+8! -)## G*" * *! $*-$* #+8,)I 23 ∇Xt = Xt − Xt−1 . F"7"4 *! )--+ )*+, ' *! # ,% %+K", # G*"4 +(+,)*# ),1 -"+%+ !)") *"+#*+ # ' *! *+( #"+#I 23 ∇2 Xt = Xt − 2 · Xt−1 + Xt−2 . ! -$"-# ' *! );7 *"),#'"()*+, +# *! "(7) ' ),1 %*"(+,+#*+ )**"+;$*# *!)* 6$% # ", -##+; #* !)#*+ !)") *"+#*+ # ' *! #1#*( *!)* -"%$ # *! *+( #"+# ! *"),#'"(% #"+# +# *!")'*" 0)(+,% 6+*! *+( %()+, )# 6 )# '"<$, 1 %()+, * !,+<$# +, "%" * "7) +*# #* !)#*+ !)") *" ! #+(-#* *#* ' 0*") *+,8 *! "")*+, ' *6 7)"+);# +# *! # )**" -* ' *! %-,%,* 7)"+); )# ) '$, *+, ' *! +,%-,%,* , ! 0+#*, " ,* ' ) *",% +, #$ ! ) # )**" -* +,%+ )*# *! 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(), ' 8* 9.: 23 7)"[X̄] = N ;#"7)*+,#4 6 N N N 1 2 7)" (x ) + R(xj , xk ). i N 2 i=1 N 2 j=1 k>j 6!" *! # ,% *"( ' *! <$)*+, "-"#,*# *! )$* 7)"+), ;*6, )%M) ,* ()#$"(,*# ' *! $,+7)"+)* *+( #"+# ' ,8*! N *",)*+71 6 !)7 23 ⎛ ⎞ N N 1 2 ρ(xj , xk )⎠ . 7)"[X̄] = 7)"[X] · ⎝1 + N N j=1 k>j 6!" ρ(xj , xk ) "-"#,*# *! '") *+, ' )$* 7)"+), 7" 7)"+), ),% +* +# )% )$* "")*+, ,#+%"+,8 *!)* )$* 7)"+), ),% *!$# )$* "")*+, ' ),1 *6 ;#"7)*+,# %-,%# ,1 , *!+" +, ;*6, *+( )8 τ ),% ,* , *! "%"+,8 j, k 6 ), #+(-1 %"+7 *! )$* "")*+, '$, *+, 2?3 ρ(τ ) ),% +*# #*+()*" r(τ ) 23 N −τ r(τ ) = i=1 (x(ti ) − x)(x(ti + τ ) − x) . N 2 i=1 (x(ti ) − x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f )) ),% +()8+,)"1 ((X(f )) -)"*# ' *! ?$"+" H +,*# )" %"+7% ! -"+%8")( +# ) -* ' *! ()8,+*$% #<$)"% ' *! H +,*# S(f ) 7"#$# '"<$, 1 f 4 ,#*+*$*+,8 *! 6" - *") ,#+*1 2 3 F"7"4 *! +# *! ?$"+" "),#'"( ' *! $* """)*+, ?$, *+, ρ(τ ) ),% *!)* ")*+,4 8+7, ;1 <$)*+, 2C34 ()D# *!# * !,+<$# <$+7),* +, *! ),)1#+# ' *+( #"+#I 2C3 +∞ S(f ) = ρ(τ ) · exp(−2πif τ )dτ. −∞ C 51 -**+,8 S(f ) − f +, ) 8 8 %+)8")(4 *! %+K",* ,+# 7# +, *! -6" #- *"$( ), ; #*+()*% '"( *! )%M$#*(,* ' <$)*+, 23 * *! 8 8 -* ! ) $)*+, ' *! #- *") 0-,,* ,4 )%# * *! +%,*+G )*+, ' 6!+* ,+# (n = 0)4 J+ D" ,+# (n = 1)4 "),%( 6)D ,+# (n = 2) " ),1 +,*"(%+)* )# 0 < n < 2 ' "% ,+#I 23 S(f ) = hn , 0 ≤ n ≤ 2. fn & ' ! ), )"+), " 6 #)(- )"+), 6)# G"#* +,*"%$ % ;1 )7+% > ), '" *! 7)$)*+, ' *! #*);++*1 ' *+( ),% '"<$, 1 #*),%)"%# 9: ! ()+, +%) ;!+,% *!+# (*!% +# +, *! ")*+, ' *! %+#-"#+, *!)* ) #* ' ()#$"(,*# !)#4 6+*! *! ,+# 6 0- * * G,% +, *! $*-$* #+8,) ! 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T%U +*1 ' +, # ),% "*# , =$1 4 ( )(* ) +), ( (./* * ) ( ( C () (( ( G ) () ( ( H H (H IH ( ) H JK H ) ( )H ( (( L J ( () ) C ((GI ) GM G ) 6 ) ) H I ) H J ( H G 6 ( ( (G ( G ( G H H ( LH ( ( ( ( ( ( N H I G ( ( L 6 L ( ) ) L B J (H ) ( ( ) ( )H G H ) ( ( (H ( ( I G I ((G L ( ( J L ( J K J ( L PL ISSN 0459-6854 BULLETIN DE LA SOCIÉTÉ DES SCIENCES ET DES LETTRES DE L ÓDŹ 2010 Vol. LX Recherches sur les déformations no. 1 pp. 137–154 Roman Stanislaw Ingarden and Julian L awrynowicz MODEL OF MAGNETIC ELECTRON MICROSCOPE INCLUDING THE SCANNING MICROSCOPE III VARIATIONAL APPROACH AND CALCULATION OF THE FOCAL LENGTH IN A RANDERS-TYPE GEOMETRY Summary In the ﬁrst part of the paper we have recalled general properties of electromagnetic “lenses” and some phenomenological and approximate formulae for the length as a preparation for constructing the electromagnetic space. In the second part we have constructed a Randers-type electromagnetic space. In this ﬁnal, third part we present a variational approach and calculation of the focal length in the constructed here Randers-type geometry. We conclude in comparison of the potentials generating functions dependence of the focal length calculated and of its immersion electromagnetic “lenses” dependence. 9. Torsion-depending deformations within the electromagnetic spaces Following the general idea of Randers [R], from the relations (49) and (50) in [IL2] we deduce e2 2e (54) m2 c2 γμν + 2 Aμ Aν dxμ dxν − Aμ dxμ ds = 0. c c With the notation e2 Aμ Aν , c2 the relation (54) becomes Gμν = m2 c2 γμν + (55) e Gμ5 = G5μ = − Aμ , c GAB dxA dxB = 0 G55 = 1, dx5 = ds, for A, B = 1, 2, . . . , 5. This trick of Randers leads us to a sort of Kaluża-Klein theory [LR] in which we consider a ﬁve-dimensional pseudoriemannian metric with the additional condition (55). 138 R. S. Ingarden and J. L awrynowicz Randers did not observe that the condition (55) is of a non-holonomic character. This follows from looking at a non-holonomic geometry of Wagner [W] which implies that the motion of test bodies does not follow the shortest line in four and in ﬁve dimensions as well. By this fact the initial point of the geometrization programme is violated. On the other side the variational principle of the Kaluża-Klein theory in ﬁve dimensions cannot be regarded an intrinsic geometrical principle in the Randers-type space [R, I] related to (49). A more direct solution of the problem was proposed in [SK], where a variational principle (56) 1 μν √ −g dτ = 0 δ R + Fμν F 2 with Fμν = 2A[μ|ν] , dτ = dx1 dx2 dx3 dx4 was postulated. The disadvantage of that approach is caused by the fact that in the √ Randers and even Finsler space the expression −g dτ has no invariant meaning μ because, in general, it depends on x or dxμ and hence the integral in (56) is not a functional of ﬁeld functions. Let us begin with the static case: where all ﬁeld functions are independent of time. Then it is possible to diminish the number of co-ordinates by one: to eliminate the time co-ordinate by means of the energy conservation law: 1 2 (57) 1 − 2 δjk ẋj ẋk + eϕ = E with j, k = 1, 2, 3, mc c where E is a (constant) total energy of the electron. For eliminating the time, we rearrange the Lagrangian (43) in [IL2] in the form 1 e i j 2 L(x , x ) = −mc 1 − 2 δjk ẋj ẋk + Aj (xj )ẋj − eϕ(xj ) + E, c c (where an additive constant is unessential). Hence by (57) we get 2 2 2 L = (E − eϕ) − (mc ) (E − eϕ) + (e/c)Aj ẋj . With the notation d = δjk ẋj ẋk dt, the expression (57) becomes dt = (E − eϕ)d x̂j = (d/d) xj , c ẋj = x̂j (d/dt) (E − eϕ)2 − (mc2 )2 . In consequence the initial Hamilton principle yields Maupertuis’ principle or Fermat’s principle in the relativistic form ⎡ ⎤ 2 E − eϕ e δ Ldt = δ ⎣ − (mc)2 + Aj x̂j ⎦ d = 0. c c Model of magnetic electron microscope including the scanning microscope III 139 In an arbitrary parametrization we have 2 E − eϕ(xj ) e j j j = − (mc)2 δjk xj xk + Aj (xj )x . L x ,x c c With the notation j ψ(x ) = (58) E − eϕ(xj ) c 2 − (mc)2 we ﬁnally get e ds = ψ(xj ) δjk dxj dxk + Aj (xj )dxj c and in arbitrary curvilinear co-ordinates: e ds = ψ(xj ) γjk (xj )dxj dxk + Aj (xj )dxj c with γ as in (54). Now, coming back to the deﬁnition of the unit directional vector (Pμ ) in [IL2], Section 6, we have k (59) δkl x k x + (e/c)Aj Pj = ψδjk x or, with the notation pj = ψδjk x k δkl x k x , we get Pj = pj + (e/c) Aj . Yet, δ jk pj pk = ψ 2 , so with 1 2 μ μ ν g μν xμ , x = ∂ /∂x ∂x L2 2 we obtain (60) gjk pj pk Pj Pk − 2 , = M δjk + M ψ e where M = δ jk Pj pk = ψ 2 + δ jk Aj pk . c Therefore, by Deﬁnition 1 in [IL2], Section 6, in the static case it is natural to accept Deﬁnition 2. In the electromagnetic space (M, ds, N c ) in the static case the torsion tensor may be presented in the form 1 M M M (61) Tjkl = M Pj − 2 pj δkl + Pk − 2 pk δij + P − 2 p δjk 2 ψ ψ ψ 1 M − 2 (Pj pk p + pj Pk p + pj pk P ) + 3 4 pj pk p . ψ ψ 140 R. S. Ingarden and J. L awrynowicz We may still calculate matrices reciprocal to αjk = ajk + λbj bk if the reciprocal to ajk is known: aj ak = δkj . We try to ﬁnd a solution in the form αjk = ajk + κbj bk , (62) where bj = ajk bk and κ is the coeﬃcient to be determined. Clearly, αj αk = δ jk + κ + λ + κλb b bm bm = δkj , and hence κ = −λ/(1 + λb2 ), (63) where b2 = b b . According formulae (62) and (63) twice to (61), we conclude that 1 1 jm km N jk pm pn − Pm pn − pm Pn , g = (64) δjk + δ δ M M M where N = M + δ jk Pj Pk = δ jk Pj (pk + Pk ). With the help of (61) and (64) we can calculate the torsion vector 3 M (65) Tj = Pj − 2 pj . 2 ψ This is an elegant extension of the formula (10) in [IL1]. By suitable changes we can adapt the formulae (61) and (65) to the non-static case. We have ν ν Pμ = imcδμν x δρσ x ρ x σ + (e/c)Aμ , pμ = imcδμν x δρσ x ρ x σ , (66) (67) gμν Pμ = pμ + (e/c)Aμ , δ μν pμ pν = −(mc)2 , pμ pν Pμ Pν + 2 2 , M = δ μν Pμ pν = −(mc)2 + δ μν Aμ pν , = M δμν + M m c 1 1 μρ νσ N μν μν g = δ δ pρ pσ − Pρ pσ − pρ Pσ , δ + M M M where N = M + δ μν Pμ Pν = δ μν (pν + Pν ). Finally we arrive at Theorem 2. With the previous notation, as the natural extension of the torsion tensor in the electromagnetic space (M, ds, N c ) in the static case, the torsion tensor in the non-static case may be presented in the form (cf. [I]): 1 M M M Tμν = M Pμ + 2 2 pμ δνρ + Pν + 2 2 pν δρμ + Pρ + 2 2 pρ δμν 2 m c m c m c 1 3M + 2 2 (Pμ pν pρ + pμ Pν pρ + pμ pν Pρ ) + pμ pν pρ . (68) m c (m2 c2 )2 Model of magnetic electron microscope including the scanning microscope III 141 In consequence the non-static torsion vector reads: M (69) Tmu = 2 Pμ + 2 2 pμ . m c 10. Application to an electromagnetic microscope In order to characterize an electromagnetic microscope in terms of the corresponding electromagnetic space, we introduce the principal curvature tensor of Varga [V]: (70) ν ν ν ν = Rμ·ρσ − Tμ·σ R0·ρσ Hp·ρσ and the Berwald aﬃne curvature tensor [B]: τ τ τ (71) Kμνρσ = Rμνσ − Tμ·ν R0τ ρσ + Tμ·ρ Tντ σ|0 − Tμ·σ|0 Tντ ρ|0 + Tμνρ|0|σ − Tμνσ|0|ρ or, equivalently, ∂ ∂ ν τ τ Kμ·ρσ = Gν − G ν + Gμ·ρ Gτν·σ − Gμ·σ Gτν·ρ + Gτρ Gτν·μσ − Gτσ Gτν·μσ , ∂xσ μ·σ ∂xρ μ·σ where ∂ ∂ 2 Gν ∂ 3 Gν ν ν ν Gνμ = Gμ·ρ = Gμ·ρσ = . μG , μ ν, μ ∂x ∂x ∂x ∂x ∂x ρ ∂x σ Here Gν is as deﬁned at the beginning of Section 6 in [IL2] and Gμ ν ·ρ is as in the formula (35) in [IL2]. Formula (70) may be, by (70) and the symmetry of Tμνρ rewritten as τ τ Tντ σ|0 − Tμ·σ|0 + Tμνρ|0|σ − Tμνσ|0|ρ , Kμνρσ = Hμνρσ + Tμ·ρ|0 so the Berwald and Varga curvature tensors are closely related. μ The Berwald curvature measure in the line element (xμ , x ) with respect to two μ μ linearly independent vectors (x ) and (η ) is given by the formula μ μ μ ρ μ ρ (72) R(xμ , x , η μ ) = Kμνρσ (xμ , x )x x η ν η ρ (gμρ gνσ − gμσ gρν )x x η ν η σ . A Randers space is said to have the scalar curvature if (72) does not depend on η μ , μ i.e. on two linearly independent vectors (x ) and (η μ ). If (72) is constant – does not μ μ μ depend on x , x and η , the Randers space is said to have a constant curvature. It μ appears [B] that if (72) does not depend on η μ and x , then it is also independent of xμ , that is the Randers space possesses a constant curvature. Besides, for a Randers space with constant curvature (73) Kμνρσ = R (gμρ gνσ − gμσ gρν ) , where R is a constant. This form of Kμνρσ is characteristic for spaces of constant curvature. Since in the case of the general electromagnetic microscope (73) appears to be quite complicated we apply Berwald’s result [B] that in a Randers space of dimension n and scalar curvature we have 1 (n + 1)Rμ + RTμ + Tμ|0|0 = 0. 3 142 R. S. Ingarden and J. L awrynowicz If the space is of constant curvature, then (74) Rμ = 0 and, by (73), RTμ + Tμ|0|0 = 0. For and electromagnetic space, n = 3, we use Latin indices. By (65) with Pj and M given by (59) and (60), respectively, we obtain Theorem 3. With the previous notation, in the electromagnetic space (M, ds, N c ) corresponding to an electromagnetic microscope, the corresponding torsion vector reads k 3e (δjk x )(A x ) Aj − . Tj = 2c δmn x m x n Corollary 2. In the case of an electromagnetic microscope, the torsion vector (Tj ) is independent of ψ [as deﬁned by (58)] and thus also of the electric (scalar) potential ϕ. The torsion vector vanishes with the magnetic (vector) potential (Aj ). By the second equation in (74), we have m n m k s δmn x x δjk − δjm x x L2 RAk + Ak||s x x = 0 which is a diﬀerential equation with respect to xj and thus Aj (xj ), but algebraic j with respect to x . We can rearrange it as m n m k s (75) L2 RAk + Ak||s x x = 0. δmn x x δjk − δjm x x We may look at (75) as to a system of homogeneous linear equations for (76) qk = L2 RAk + Ak||s x x with the discriminant (x 2 )2 + (x 3 )3 2 1 D= −x x 3 1 −x x 1 2 −x x 1 3 2 (x ) + (x )2 3 2 −x x s 1 3 −x x 2 3 −x x 1 2 (x )2 + (x )2 = 0. Therefore the system (75) has inﬁnitely many solutions (76) which may be expressed in the form m n qk = aδkl x δmn x x , where a is an arbitrary constant. Taking into account the expression for ds2 in Section 9, we arrive at m n Rψ 2 Ak − aδk x δmn + R(e/c)2 Ak Am An + Ak|m|n x x (77) +2(e/c)RψAk A x δmn x m x n = 0. j j j Next we take in (77) subsequently x = δ j1 , x = δ j2 , x = δ j3 , add the equations obtained and apply the Maxwell equations in the form [IL2], (52) with γ ρσ = δ ρσ : Model of magnetic electron microscope including the scanning microscope III 3ψ 2 + (e/c)2 δ mn Am An + 2(e/c)ψ(A1 + A2 + A3 Ak = λ, 143 λ = 3a/R. With the gauge transformation Ak ⇒ Ak + (∂/∂xk )χ we may always assume that A1 + A2 + A3 = 0, so ﬁnally the condition discussed takes a simpler form 2 3ψ + (e/c)2 δ mn Am An Ak = λ. The expression in the square brackets is positive as the sum of squares of real quantities, of which ψ does not vanish by (58). Hence A1 = A2 = A3 , so the vector (Ak ) has constant direction in the space, and therefore the magnetic ﬁeld vanishes: Hj = εjk (∂/∂xk )A = 0, where εjk stands for the Ricci antisymmetric pseudo-tensor. In such a way the general problem is reduced to an analogous problem for the electrostatic microscope [I]. 11. Deformation of potentials with the help of generating functions Let us consider again an electromagnetic ﬁeld with the rotational symmetry along the axis x3 = z. We set x1 = ρ cos α, x2 = ρ sin α and express the electric (scalar) potential ϕ by a generating function f whose existence follows from the theory of harmonic functions [Sch, Lu]: 1 ϕ(ρ, z) = 2π (78) 2π f (z + iρ cos ψ) dψ, ϕ(0, z) = f (z). 0 In analogy we express the magnetic (vector) potential (Aj ) by a generation function g: (79) (Aj (x)) = −x2 A(ρ, z), x1 A(ρ, z), 0 1 A(ρ, z) = 2π (80) 2π g (z + iρ cos ψ) sin2 ψdψ, A(0, z) = 0 1 g (z). 2 Condition (79) has to assure that div(Aj ) = 0. In spite of the imaginary unit appearing in the integrals in (78) and (80) the potentials ϕ and A are real. Indeed, if we expand f and g into the Taylor series in z and integrate over ψ, all the integrals containing i, which correspond to odd powers of cos ψ, are zero: 1 2π 2π 2ν+1 cos 0 ψdψ = 0, 1 2π 2π 0 cos2ν+1 ψ sin2 ψdψ = 0, ν = 0, 1, . . . 144 R. S. Ingarden and J. L awrynowicz Since 1 2π then (81) 2π 2ν cos 0 (2ν)! ψdψ = 2ν , 2 (ν!)2 1 2π 2π cos2ν ψ sin2 ψdψ = 0 (2ν)! , 22ν+1 ν!(ν + 1)! 2ν ∞ (−1)ν 1 ϕ(ρ, z) = ρ f (2ν) (z) 2 (ν!) 2 ν=0 1 1 = f (z) − ρ2 f (z) + ρ4 f (4) (z) + . . . , 4 64 ∞ (82) A(ρ, z) = 1 (−1)ν 2 ν=0 ν!(ν + 1)! 1 ρ 2 2ν g(2ν) (z) 1 1 4 (4) 1 ρ g (z) + . . . = g (z) − ρ2 g (z) + 2 16 192 If we construct the potentials ϕ, A with the help of formulae (78), (80), it is natural to assume that f, g are arbitrary ρ-dependent real-analytic functions of one variable z. We notice that ϕ and A satisfy the diﬀerential equations 2π d i 1 1 2 (sin ψ)f (z + i cos ψ) dψ ≡ 0 ∇ ϕ ≡ ϕρρ + ϕρ + ϕzz = ρ 2π dψ ρ 0 (since the function in square brackets is periodic with period 2π) and Aρρ + (3/ρ) Aρ + Azz = 0. We also recall that ϕ, A are not directly observable quantities in contrast to the ﬁeld vectors E, H and the Lorentz force F acting on an electron; they are directly observable and unique. The choice of the constant 12 in the second formula in (80) is due to the relations H = ∇ × (Aj ) = −x1 Az , −x2 Az , 2A + ρAρ , H(0, z) = (0, 0, 2A(0, z)) = (0, 0, g(z)). Take now the principle of energy conservation ẋj ∂/∂ ẋj L − L = C, where C is a constant and the Lagrangian L does not depend explicitly on time. In order to eliminate time, let us consider the problem of variation 1 L xj , xj /ρ + p(s)(ρ − t + C) t ds = 0, δ 0 where p(s) is a Lagrangian multiplier. Variations with respect to ρ and s give (83) L(xj xj /ρ) − (xj /ρ)(∂/∂ ẋj )L(xj , xj /ρ) + p(s) = 0 Model of magnetic electron microscope including the scanning microscope III 145 and p (s) = 0, (84) p(0) = p(1) = C, respectively. Variation with respect to p(s) gives ρ = t and from (84) we deduce that p(s) = C for any s ∈ [0; 1]. Taking into account (83), we arrive at L xj , xj /ρ + C − xj /ρ ∂/∂ ẋj L xj , xj /ρ = 0. = (∂/∂ρ) ρ C + L xj , xj /ρ By ρ = t and the latter relation, we may reformulate our problem as the Fermat principle: 1 (85) δ F xj , xj ds = 0, 0 where F(xj , xj ) = G xj , xj , ρ(xj , xj ) = ρ C + L(xj , xj /ρ) and ρ = ρ(xj , xj ) is determined by the equation (∂/∂ρ)G(xj , xj , ρ) = 0. Hence we get (86) F = (1/ρ)gjk xj xk + (C − U ), −(1/ρ2 )gjk xj xk + (C − U ) = 0, U being the potential energy: U = U (xj ) = T − L, where T = gjk ẋj ẋk denotes the kinetic energy. Eliminating ρ from the system (85) we ﬁnally arrive at the Maupertuis principle: 1 (87) δ F (xj , xj )ds = 0, 0 where j F (x , x ) = 2 C − U (x ) gjk xj xk . j j We can see that now F contains the constant C assumed as given. Since U is ﬁxed up to an arbitrary constant, the latter constant is eliminated in (87), because it is contained both in C and in U . Thus F has a unique physical meaning. Yet, F does not contain derivatives with respect to time, but derivatives with respect to an arbitrary parameter s. Therefore all the conditions of the Fermat principle are satisﬁed and we can see that F has the dimension of energy as the Lagrangian L has, provided that s has the dimension of time; [AIM], pp. 173–177. The role of generating functions f, g and the integral representations of potentials (78) and (80) will appear when we take into account that the image of electron beams is rotated in the electric microscope by the magnetic ﬁeld around the optical axis by an angle ω expressible by f and g; [Lu], pp. 386–398. In other words, we need geometry with torsion ([IL1], Sect. 3) and torsion-depending deformations ([IL2], Sect. 8). We shall study this problem in the next two sections. 146 R. S. Ingarden and J. L awrynowicz 12. “Lens”-thickness depending deformations in relation with the scanning microscope Let us consider the Lagrangian (in the previous notation) in the form L(x, ẋ) = mc2 1 − 1 − ẋ2 /c2 − eϕ(x) + (e/c) Aj (x)ẋj . Then from (85) we deduce 1 G/mc = cρ 1 − 1 − x 2 /ρ2 c2 + a · x + ρcΦ, 2 where Φ = 2 (C − eϕ) /mc2 , (88) a = e/mc2 A are dimensionless potentials [Lu]. Extremals of the variational problem (85) are not inﬂuenced by a multiplicative constant of F, so we may modify it as F = (1/mc) G xj , xj , ρ , (89) (∂/∂ρ) (1/mc)G xj , xj , ρ = 0 and therefore 1 Φ + Φ2 x 2 + a · x . 4 In absence of the magnetic ﬁeld we have a dimensionless index of refraction 1 n = Φ + Φ2 4 j j F(x , x ) = and the expression for F is 1(p)-homogeneous in xj . Indeed, by (85) we have xj Gxj = G − ρGρ and so Fxj = (1/mc) Gxj + Gρ ∂/∂xj ρ = (1/mc)Gxj , xj Fxj = (1/mc) (xj Gxj ) = (1/mc) (G − ρGρ ). Yet, Gρ = 0 and G = mcF, and we obtain the Euler equation of the required homogeneity xj Fxj = F. Since from the latter relation it follows the initial condition for xj = 0, the equation provides also a suﬃcient condition for homogeneity. It is clear that F may be treated as a Randers metric [I]: 1 F(x, y) = Φ(x) + Φ(x)2 (y 1 )2 + (y 2 )2 + (y 3 )2 + aj (x)y j . (90) 4 Because of an analogy of F with the index of refraction, an electromagnetic ﬁeld can thus be interpreted as an optical medium whose properties vary from point to point. Model of magnetic electron microscope including the scanning microscope III 147 Then to every point there is an associated ray surface given by the equation 1 Φ + Φ2 (y 1 )2 + (y 2 )2 + (y 3 )2 + aj y i = 1. 4 For slow electrons Φ2 (x) ≈ 0, so the equation approximatively represents an ellipsoid symmetric with respect to the vector a with centre oﬀset in the same direction. If a = 0, the ellipsoid becomes a sphere. The expression discussed is the indicatrix of the Randers space constructed in the case of nonrelativistic electrons and in the general case as well. We conclude with Theorem 4. In the electromagnetic space corresponding to a rotationally symmetric electron microscope we can insert into the Randers metric function (90) the general solution for the ﬁelds (78), (80) or (81), (82) using (88). The case of a scanning microscope is included. Remark. The system of equations involved in Theorem 4 does not include time which is estimated. Precisely, we may rewrite (90) in the form 1 F(x, y, z, ẋ, ẏ) = Φ(x, y, z) + Φ(x, y, z)2 1 + ẋ2 + ẏ 2 + a1 ẋ + a2 ẏ + a3 , 4 where ẋ = (d/dz)x and ẏ = (d/dz)y. 13. Explicit formula for the focal length and some numerical results Following the Remark, we calculate the Hamilton function (91) H(z, u, v, w) = ẋp + ẏq − F(x, y) = − n2 (u, z) − a2 u + aw − v, where u = x2 + y 2 , v = p2 + q 2 , w = 2(xq − yp), p = (∂/∂ ẋ)F, q = (∂/∂ ẏ)F, and n(u, z) = n(x, y, z) is the index of refraction. In terms of (91) we have the Hamilton equations ẋ = Hp , ṗ = −Hx , q̇ = −Hy . As we have already mentioned at the end of Sect. 11, the image of electron beams is rotated in the electric microscope by the magnetic ﬁeld around the optical axis by an angle ω expressible by f and g: z 1 1 2 α(ζ) + α(ζ) dζ, β(ζ) ω(z) = − 2 4 z0 148 R. S. Ingarden and J. L awrynowicz with 2 e [C − ef(ζ)] , β(ζ) = g(ζ); mc2 mc2 where f and g the generating functions for the potentials (78)–(80), whereas ζ = z0 and ζ = z are optically conjugated planes corresponding to the object and image, respectively. Finally one may calculate ([Lu], p. 398) that the focal length of an electromagnetic “lens” of thickness d is given by the formula α(ζ) = 1 1 = f 4 (92) d 0 α + β 2 + 12 αα dζ. α + 14 α2 In order to have an idea how the curves (93) f = f (d) = f (d; f, g, C) look like, we take f1 f2 f3 f4 f5 f6 f7 f8 1√ = f d; (1 + z 2 ) g · cm, s 1√ = f d; (1 + z 2 ) g · cm, s 1√ = f d; (1 − z 2 ) g · cm, s 1√ = f d; (1 − z 2 ) g · cm, s 1√ = f d; (1 − z 2 ) g · cm, s 1√ = f d; (1 − z 2 ) g · cm, s 1√ = f d; (1 + z 2 ) g · cm, s 1√ = f d; (1 + z 2 ) g · cm, s 1√ 1 3 (1 − z ) g · cm, −2 g · cm , s s 1√ 1 (1 + z 2 ) g · cm, −4 g · cm3 , s s 1 2 1√ 3 (1 − z ) g · cm, 6 g · cm , s s 1√ 1 (1 + z 2 ) g · cm, 8 g · cm3 , s s 1√ 1 (1 + z 2 ) g · cm, 2 g · cm3 , s s 1 2 1√ 3 (1 − z ) g · cm, 4 g · cm , s s 1√ 1 (1 + z 2 ) g · cm, −6 g · cm3 , s s 1 2 1√ 3 (1 − z ) g · cm, −8 g · cm . s s 2 The curves f1 , . . . , f4 are shown on Fig. 7, accompanied by Tab. 1 with the corresponding focal length values. The curves f5 , . . . , f8 are shown on Fig. 8, accompanied by Tab. 2 correspondingly. Model of magnetic electron microscope including the scanning microscope III 149 Fig. 7: Four cases f1 , ..., f4 of the focal length f dependence on the “lens” thickness d. Table 1. Focal length f values corresponding to f1 , ..., f4 . d [cm] 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 f1 [cm] 16.305 8.403 5.663 4.273 3.433 2.871 2.468 2.166 1.931 1.742 1.588 1.460 1.351 1.258 1.178 1.107 1.045 0.989 0.940 0.895 0.854 0.818 0.784 0.753 f2 [cm] 18.778 9.534 6.389 4.805 3.850 3.212 2.756 2.413 2.147 1.933 1.758 1.613 1.489 1.383 1.292 1.212 1.141 1.078 1.022 0.971 0.925 0.884 0.846 0.811 f3 [cm] 19.318 9.759 6.528 4.905 3.928 3.276 2.809 2.459 2.187 1.969 1.790 1.641 1.515 1.408 1.314 1.232 1.160 1.096 1.038 0.986 0.940 0.897 0.858 0.823 f4 [cm] 19.538 9.845 6.580 4.942 3.956 3.299 2.829 2.476 2.201 1.982 1.802 1.652 1.525 1.416 1.322 1.239 1.167 1.102 1.044 0.992 0.945 0.902 0.863 0.827 150 R. S. Ingarden and J. L awrynowicz Fig. 8: Further four cases f5 , ..., f8 of the focal length f dependence on the “lens” thickness d. Table 2. Focal length f values corresponding to f5 , ..., f8 . d [cm] 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 f5 [cm] 16.298 8.393 5.648 4.254 3.409 2.842 2.435 2.127 1.887 1.693 1.534 1.400 1.285 1.185 1.096 1.016 0.943 0.872 0.797 f6 [cm] 18.778 9.533 6.388 4.803 3.848 3.209 2.752 2.409 2.142 1.928 1.752 1.606 1.482 1.376 1.284 1.203 1.132 1.068 1.011 0.960 0.913 0.871 0.832 0.796 f7 [cm] 19.318 9.758 6.528 4.904 3.927 3.275 2.808 2.458 2.185 1.967 1.788 1.640 1.513 1.405 1.312 1.230 1.157 1.093 1.035 0.983 0.936 0.893 0.854 0.819 f8 [cm] 19.538 9.845 6.580 4.941 3.956 3.298 2.828 2.475 2.201 1.981 1.801 1.651 1.524 1.415 1.321 1.238 1.166 1.101 1.043 0.991 0.943 0.900 0.861 0.825 Model of magnetic electron microscope including the scanning microscope III 151 14. Third conclusion: potentials generating functions dependence vs. immersion electromagnetic “lenses” dependence Potentials generating functions f and g dependence of the focal length has been characterized in Sect. 13 by the relation (92) of the form (93) and illustrated by examples f1 , f2 , . . . , f8 with 1√ (94) g · cm, fj (z) = fj (z; aj ) = 1 + aj z 2 s where 1 for j = 1, 2, 7, 8, aj = −1 for j = 3, 4, 5, 6; (95) where (96) 1√ g · cm, gj (z) = gj (z; bj ) = 1 + bj z 2 s 1 for j = 2, 4, 5, 7, bj = −1 for j = 1, 3, 6, 8; 1 g · cm3 = −2, C2 = −4, C3 = 6, C1 in s C4 = 8, C5 = 2, C6 = 4, C7 = −6, C8 = −8. For 0.05 cm < d < 1.20 cm, let fj− denote the domains consisting of the points (d, p) situated below the curves fj , j = 1, 2, . . . , 8. Then we can see that (97) f1− ⊂ f2− ⊂ f3− ⊂ f4− , f5− ⊂ f6− ⊂ f7− ⊂ f8− and that, at every d ∈ (0.05 cm; 1.20 cm), the diﬀerences (98) f2− \ f1− , f3− \ f2− , f4− \ f3− ; f6− \ f5− , f7− \ f6− , f8− \ f7− are relatively small. On Figs. 9 and 10 we visualize the growth of fj with the help of arrows joining points: (99) (cj , aj ) → (cj+1 , aj+1 ), ◦(cj , aj ) → ◦(cj+1 , aj+1 ), j = 1, 2, 3; j = 5, 6, 7, and (100) (cj , bj ) → (cj+1 , bj+1 ), j = 1, 2, 3; ◦(cj , bj ) → ◦(cj , bj ) → ◦(cj+1 , bj+1 ), j = 5, 6, 7, respectively. The relationship between the potentials generating functions dependence of the focal length and the immersion electromagnetic “lenses” dependence of that length is rather a problem in engineering and requires further study. The dependence of the focal length on various parameters of the immersion electromagnetic “lenses”: U1 , U2 , ζ(H1 ), ζ(H2 ), h1 , h2 , x1 , x2 , z0 , z, γ1 , γ2 , r1 (z), r2 (z) were discussed in [IL1], Sects. 4 and 5. 152 R. S. Ingarden and J. L awrynowicz 1 1 a 0 b 0 -1 -8 -6 -4 -2 0 C 2 4 6 8 Fig. 9: Growth of the focal length shown by arrows (99) in the plane (C, a) according to the formulae (92)-(96). -1 -8 -6 -4 -2 0 C 2 4 6 8 Fig. 10: Growth of the focal length shown by arrows (100) in the plane (C, b) according to the formulae (92)-(96). Acknowledgments The authors are deeply indebted to Mrs. Malgorzata Nowak-Kȩpczyk for drawing Figs. 7 and 8, and making the corresponding computer calculations. This work (J.L) was partially supported by the Ministry of Sciences and Higher Education grant PB1 P03A 001 26 (Section 9 of the paper) and partially by the grant of the University of L ódź no. 505/692 (Sections 10–14). Errata to [IL1] On p. 112 in the ﬁrst formula replace twice C by A. References [AIM] P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic, Dordrecht 1993. [B] L. Berwald, Über Finslersche und Cartansche Geometrie. 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Wagner, Sur la géometrie diﬀérentielle des multiplicités anholonomes, Trudy Seminara po Vektornomu i Tensornomu Analizu 2–3 (1935), 269–280. [W2] —, On the embedding of a ﬁeld of local surfaces in Xn in a constant ﬁeld of surfaces in aﬃne space [in Russian], Dokl. Akad. Nauk SSSR N.S. 66 (1949), 785–788. Institute of Physics Nicolaus Copernicus University Grudzia̧dzka 5, PL-87-100 Toruń Poland e-mail: [email protected] Institute of Physics Univeristy of L ódź Pomorska 149/153, PL-90-236 L ódź Institute of Mathematics Polish Academy of Sciences L ódź Branch, Banacha 22, PL-90-238 L ódź Poland e-mail: [email protected] Presented by Julian L awrynowicz at the Session of the Mathematical-Physical Commission of the L ódź Society of Sciences and Arts on June 25, 2009 MODEL MAGNETYCZNEGO MIKROSKOPU ELEKTRONOWEGO UWZGLȨDNIAJA̧CY MIKROSKOP SKANINGOWY III PROCEDURA WARIACYJNA I WYZNACZENIE DLUGOŚCI OGNISKOWEJ Streszczenie W pierwszej czȩści pracy przypomnieliśmy ogólne wlasności “soczewek” elektromagnetycznych oraz pewne wzory fenomenologiczne i przybliżone na dlugość ogniskowej jako przygotowanie do konstrukcji przestrzeni elektromagnetycznej. W drugiej czȩści skonstruowaliśmy przestrzeń elektromagnetyczna̧ typu Randersa. W obecnej, końcowej, trzeciej czȩści przedstawiamy podejście wariacyjne i wyznaczenie dlugości ogniskowej w skonstruowanej tu geometrii typu Randersa. W podsumowaniu porównujemy zależność wyznaczonej dlugości ogniskowej od funkcji generuja̧cych potencjaly z zależnościa̧ tejże dlugości od parametrów użytych imersyjnych “soczewek” elektromagnetycznych. PL ISSN 0459-6854 BULLETIN DE LA SOCIÉTÉ DES SCIENCES ET DES LETTRES DE L ÓDŹ 2010 Vol. LX Recherches sur les déformations no. 1 pp. 155–174 Roman Stanislaw Ingarden and Julian L awrynowicz FINSLER-GEOMETRICAL MODEL OF QUANTUM ELECTRODYNAMICS I EXTERNAL FIELD vs. FINSLER GEOMETRY Summary After summarizing the physical demands caused by the necessity of including open systems, we study quantum Dirac-Maxwell equations using a complex-analytical approach and convolution equations. Then we pass to a more general case of Yang-Mills equations in the presence of an external ﬁeld, including the cases of an arbitrary symmetry within SO(m) or SU(m), the global case, a non-abelian generalization, and a generalization of the Lagrangian and its embedding in the electroweak model. In the second part of the paper we shall deal with ferroelectric crystals in a Finsler geometry and the physical interpretation of solenoidal and nonsolenoidal connections on the canonical principal ﬁbre bundles. 1. Basic physical demands In analogy to a particular case of an electromagnetic microscope [IL3], and classical methods of electrodynamics, in the variational procedure involved, we deform in the ﬁrst step the electron trajectories, and in the second step the potentials. If we include thermodynamical eﬀects [L3] or electroweak eﬀects [KBG], we have an additional motivation of involving a ﬁve-dimensional, Kaluża-Klein-like geometry. Independently [L5], in connection with Kaluża-Klein theories (e.g. [L3]), Beil [Be1, 2] has shown that the U (1)-symmetry of the electromagnetic ﬁeld yields a gauge transformation of the form 1 Yνμ = δνμ − B −2 1 − (1 + kB 2 ) 2 Bμ Bν , Yν∗μ = δνμ − B −2 1 1 − (1 + kB 2 )− 2 Bμ Bν , 156 R. S. Ingarden and J. L awrynowicz where B2 = ηαβ B α B β , α,β ηαβ is the initial base space metric in the Lorentz form, an k is, in general, velocity dependent. The resulting metric gμν = ημν + kBμ Bν is, in general, Finslerian, even in the case where k is a universal constant related to the gravitational constant. In this direction Kerner [K] had proposed a nonlinear generalization of electrodynamics derived from Kaluża-Klein theory in ﬁve dimensions. It is based on investigation of a Gauss-Bonnet-type addend Rαβγδ Rαβγδ − 4Rαβ Rαβ + R2 , which is no more a topological invariant in ﬁve dimensions. When added to the Einstein-Hilbert Lagrangian, it leads to non-trivial equations of motion of the second order. As we already know [L6], the ﬁrst of us had proposed to consider a Randers space M with metric F and a Lorentz nonlinear connection N or a Cartan nonlinear connection N C : (1) (M, F, N ) or (M, F, N C ). Since we are interested in including thermodynamical and electroweak eﬀects, here we allow (M, F ) to be Finslerian. Now the concept consists in considering the twoelement sequence of mappings (c, sc, sc) (c, c, s) (s, sc, c) m c =⇒ =⇒ = (F0 , N or N c , V ) A = (F# , N or N c , V0 ) B = (F# , N# , V# ) (2) with s = F0 – a (relatively simple) Finsler metric, c = F# – a (relatively complicated) Finsler metric, sc = N – a Lorentz nonlinear connection or N C – a Cartan nonlinear connection, c = N# – a (relatively complicated) nonlinear connection, s = V# – a (relatively simple) potential corresponding to an external ﬁeld, sc = V0 – a (slightly complicated) potential corresponding to an external ﬁeld, c = V – a (relatively complicated) potential corresponding to an external ﬁeld. Moreover, as in the case of thermodynamics [L6], we are interested in open (nonisolated) systems. If we come over to quantum electrodynamics of the so-called second quantization, the wave function (spinor) ψ (say) becomes an operator (observable) in the Heisenberg picture (depending on time), while the wave function (state) ϕ (say) of the whole ﬁeld is independent of time. Such a treatment enables one to calculate (after a renormalization for avoiding inﬁnities) only mean values of Finsler-geometrical model of quantum electrodynamics I 157 these observables for a given time (also correlations for diﬀerent times) of the whole ﬁeld [I]. As it is well known (e.g. [LL], Sections 62–63), the number of particles N (say) becomes also an observable with integer nonnegative eigenvalues, as it is usual when following the second quantization (in the non-relativistic case). In this more ambitious theory, the concept of a particle as an independent and constant physical object actually disappears in contradiction to the ﬁrst quantization theory: particles are created and annihilated as excitations of the whole ﬁeld and the number of particles is, in general, not preserved. However, the total electric charge of the ﬁeld is preserved as the diﬀerence of the positive and negative charges of positrons and electrons. Moreover, special quantum particles and their quantum agglomerations, as atoms, molecules, gases, ﬂuids, and solid states, can sometimes exist, in special conditions, for billions of years. Atoms and molecules are examples of open systems, more complicated are maser and laser systems. 2. A generalized Dirac-Maxwell system Let (M, g) be a curved space-time: a C ∞ -diﬀerentiable paracompact connected fourdimensional manifold endowed with a pseudoriemannian metric g: a symmetric C ∞ tensor ﬁeld of type (0, 2) which is nondegenerate and has at each point the index 1. (It is not diﬃcult to generalize the staﬀ of Sections 2–4 to the case of locally Minkowskian Ingarden spaces; cf. [L6], pp. 121–123 and papers quoted therein. It seems to us better to generalize ﬁrst the staﬀ to the Yang-Mills system.) Let ψ : M → C4 be a spinor, i.e. a C ∞ -diﬀerentiable function which is supposed to satisfy the generalized Dirac equation in the sense of [Sc] and [L1, 3], in presence of its C ∞ -diﬀerentiable self-electromagnetic ﬁeld A : M → R4 : (3) A = j, Dψ = 0, 1 2 3 Div A = 0. 0 In a local co-ordinate system (x , x , x , x ) we can express the generalized Dirac equation as (4) D = γ̂ k (∂/∂xk ) + ieAk + Γk + m, where m = m0 c/ and e = e0 /c, m0 and e0 denoting the rest mass and electric charge of the particle in question, respectively. The symbols γ̂ k denote the Dirac matrices obtained by the commutation rules γ̂ j γ̂ k + γ̂ k γ̂ j = 2g jk I4 , and Γk are the spinor connections (the generalized Christoﬀel symbols), determined by 1 1 Γk = ψ j (∂/∂xk )γ̂j − γ̂ Γjk − Tr γ̂γ̂ j (∂/∂xk )γ̂j γ̂ 4 32 with 1 εjkrs γ̂ j γ̂ k γ̂ r γ̂ s , γ̂k = gjk γ̂ k and γ̂ = 24 158 R. S. Ingarden and J. L awrynowicz where [εjkrs ] denotes the totally antisymmetric Levi-Civita tensor and Γjk denote the usual Christoﬀel symbols ∂ 1 ∂ ∂ g + g − g (5) grs Γrjk = sk js jk . 2 ∂xj ∂xk ∂xs For a study of the operator D we refer to [LW2, S]; in case of the Minkowski space-time we may take the representation (cf. [L3]): γ̂ μ = −iαμ , μ = 1, 2, 3, γ̂ 0 = −iγ0 ; 0 σμ I2 0 αμ = , , γ0 = σμ 0 0 −I2 σμ denoting the familiar Pauli matrices. The symbol j in (3) denotes the current generated by ψ: 1 2 3 0 , γ̂jk , γ̂jk , γ̂jk , j = eψ̄ j γ̂jk ψ k , γ̂jk = γ̂jk where ψ̄ j stands for the complex conjugate of ψ j . The divergence Div and the Laplace-Beltrami operator can be expressed in a local co-ordinate system as Div A = (∂/∂xj )uj + uk Γjjk and Ak = Div(Grad Ak ), where Ajk = g j (∂/∂x )Ak . The system (3) satisﬁes a compatibility condition being the continuity equation Div j = 0 which can be checked directly. The choice (4) for the generalized Dirac operator is not the only reasonable possibility. For instance one may take [LW1]: (6) D = gjk γ̂ j {g k [(∂/∂x ) + ieAk + Γk ] + δ k0 m}, where δk denotes the Kronecker symbol. 3. A complex-analytical approach Let pk = i[(∂/∂xk ) + ieAk ]. In case of the usual Minkowski space-time, following [GLW1]. we write the Dirac equation in (3) as the system (p1 − ip2 )ψ 0 + p3 ψ 3 + (p0 − m)ψ 1 = 0, (p1 + ip2 )ψ 3 − p3 ψ 0 + (p0 − m)ψ 2 = 0, (p1 − ip2 )ψ 2 + p3 ψ 1 − (p0 + m)ψ 3 = 0, (p1 + ip2 )ψ 1 − p3 ψ 2 − (p0 + m)ψ 0 = 0, which is equivalent to (p1 + ip2 ) (ψ 1 + ψ 3 ) = (p3 + m) (ψ 2 + ψ 0 ) − p0 (ψ 2 − ψ 0 ), (p1 − ip2 ) (ψ 2 + ψ 0 ) = −(p3 − m) (ψ 1 + ψ 3 ) − p0 (ψ 1 − ψ 3 ), (p1 + ip2 ) (ψ 1 − ψ 3 ) = (p3 − m) (ψ 2 − ψ 0 ) + p0 (ψ 2 + ψ 0 ), (p1 − ip2 ) (ψ 2 − ψ 0 ) = −(p3 + m) (ψ 1 − ψ 3 ) + p0 (ψ 1 + ψ 3 ). Finsler-geometrical model of quantum electrodynamics I 159 In the case of an arbitrary curved space-time with metric g we replace the above system by a system of the form 1 a1 (∂1 + ieA01 ) + ia12 ∂2 + ieA02 ϕ1 = a13 ∂3 + a10 ∂0 + c1 (e, m) ϕ2 +b1 (∂ 0 + ieA00 )ϕ0 , (7) a21 (∂1 + ieA01 ) + ia22 ∂2 + ieA02 ϕ2 = a23 ∂3 + a20 ∂0 + c2 (e, m) ϕ1 +b1 (∂ 0 + ieA00 )ϕ3 , a31 (∂1 + ieA01 ) + ia32 ∂2 + ieA02 ϕ3 = a33 ∂3 + a30 ∂0 + c3 (e, m) ϕ0 +b2 (∂ 0 + ieA00 )ϕ2 , (8) a01 (∂1 + ieA01 ) + ia02 ∂2 + ieA02 ϕ0 = a03 ∂3 + a00 ∂0 + c0 (e, m) ϕ3 +b2 (∂ 0 + ieA00 )ϕ1 , where the real-valued functionals ajk , bj , and cj (e, m) depend, in general, on g, but do not depend on the operators ∂ k = ∂/∂y k and, moreover, ajk and bj do not depend on the normalized electric charge e and the normalized mass m. We require that (9) ajk [g] = 1 and bj [g] = 0 for k = 1, 2, for a suitable representation of the Dirac matrices γ̂ k , together with (10) ΔA0 = j0 , div A0 = 0, div j0 = 0, where div and Δ are the two-dimensional analogues of Div and ; j0 standing for the two-dimensional current generated by the spinor (ϕ1 , ϕ2 )T (T – transposed) satisfying the generalized Dirac equation (7) with restrictions (9), (10), and (11) ajk [g] = 0 for k = 3, 0 and cj (e, m) = 0. Therefore we suppose that in each co-ordinate neighbourhood on M the fourteen real-valued functions gjk , 0 ≤ j ≤ k ≤ 3; A01 , A02 , f10 , f20 satisfy the fourteen real diﬀerential equations (9) and (10). We call (9) and (10) the saparability conditions for the Dirac-Maxwell system (3), equivalent to the system of equations (7), (8), and (12) A = j, Div A = 0. The further procedure with equations (7), (8), and (12) may start with considering the Fourier transform with respect to the variable e. Denote by ε the conjugate variable and set Z = ∂ 1 − i∂ 2 + 2 A01 − iA02 (∂/∂ε) , Z̄ = ∂ 1 + i∂ 2 + 2 A01 + iA02 (∂/∂ε). 160 R. S. Ingarden and J. L awrynowicz Then the system of (7) and (8), subjected to conditions (9) and (11), implies (13) Z̄ ϕ̂1 = 0, Z ϕ̂2 = 0, Z̄ ϕ̂3 = 0, Z ϕ̂0 = 0, whereˆstands for the partial Fourier transform with respect to ε. Now, for every co-ordinate neighbourhood on M , let us introduce the space of the two complex variables (y 1 + iy 2 , ε + iϑ), where ϑ is the complex conjugate of ε. Consider the hypersurface Z = y 1 + iy 2 , ε + ϑ ∈ C2 : ϑ = F (y 1 , y 2 ), F independent of ε . By (10), div A0 = 0, so we can choose a real-valued function F such that A01 = ∂/∂y 2 F, A02 = −(∂/∂y 1)F (14) and hence 1 0 A + iA02 = −i(∂/∂ z̄)F, where z = y 1 + iy 2 . 2 1 ¯ In consequence, Z̄ and Z can be indentiﬁed with the ∂-tangential and ∂-tangential on S, respectively. Choose now ϕ̂k so that (15) ϕ̂k = Φk |S, k = 1, 2, 3, 0, Φk : C → C being holomorphic for k odd and antiholomorphic for k even. Then we get Lemma 1. With the previous notation, the mapping (15) provides a solution to the system (13) for any choice of the potentials A01 and A02 . 4. Convolution equations Before investigating the general system of equations (7), (8), and (12) subjected to conditions (9) and (10), we concentrate on the equations (10). Let G denote the Green operator in two dimensions and the asterisk * the convolution. Then we can express the potentials A01 and A02 as (16) A01 = G ∗ f10 , A02 ∗ f20 . The procedure, originally due to [GLW1] and generalizing that of [GL], follows the same main idea as the method of generating functions used in [IL3] for a model of magnetic electron microscope. On the other hand, by (10), we have div j0 = 0. Consequently, relations (14) and (16) yield dF = −A02 dy 1 + A01 dy 2 = G ∗ −f20 dy 1 + f10 dy 2 . In turn we consider four functions Ψk : C2 → C, holomorphic for k odd: (17) Ψk = Ψk y 1 + iy 2 , ε + iϑ , k = 1, 3, and antiholomorphic for k even: Ψk = Ψk y 1 − iy 2 , ε − ϑ , (18) k = 2, 0, Finsler-geometrical model of quantum electrodynamics I 161 and calculate their Fourier transforms in the variable ε: Φk = Φk y 1 + iy 2 , e, ϑ , k = 1, 3, (19) Φk = Φk (y 1 − iy 2 , ε, ϑ), k = 2, 0, respectively. This yields the diﬀerential form 1 2 2 1 dJ(y 1 , y 2 , e, ϑ) = Φ Φ2 + Φ Φ1 dy 2 − i Φ Φ1 − Φ Φ2 dy 1 (20) and we are led to solve the equation dF (y 1 , y 2 ) = (G ∗ dJ) (y 1 , y 2 , F (y 1 , y 2 )). (21) Equation (21) implies (22) F (y01 , y02 ) = 2 1 2 2 1 K y 1 , y 2 , y01 , y02 Φ Φ + Φ Φ1 − i Φ Φ1 − Φ Φ2 dy 1 dy 2 C with [GL]: 1 K(y , y 2 , y01 , y02 ) = 1 0 (23) − z0 − z z0 z̄ + z z̄0 z0 − z Ln|tz0 − z|dt = −1 + ln ln − z 4|z0 |2 z 4|z0 |2 |z|2 + (z0 z̄ + z z̄0 ) |z0 |2 [4|z0 |2 |z|2 − (z0 z̄ + z z̄0 )]1/2 |z0 |2 ×Arctan . 2 2 2 2|z0 | |z| − |z0 | (z0 z̄ + z z̄0 ) − (z0 z̄ + z z̄0 )2 Therefore the solution to the system of equations (7), (8), and (10), subjected to conditions (9) and (11), is given by the formulae (14) and for k = 1, 3, ϕk0 = Φk y 1 + iy 2 , e, F (y 1 , y 2 ) (24) ϕk0 = Φk y 1 − iy 2 , e, F (y 1 , y 2 ) for k = 2, 0, where F and K are calculated in (22) and (23), respectively. Finally we come back to the general system of equations (7), (8), and (12) with conditions (9) and (10). We express any solution of this system in the form (25) ϕ1 − ϕ10 = Y1 ϕ2 − ϕ20 , ϕ2 − ϕ20 = Y1−1 ϕ1 − ϕ10 , (26) where (27) ϕ3 − ϕ30 = Y2 ϕ0 − ϕ00 , ϕ0 − ϕ00 = Y2−1 ϕ2 − ϕ20 , Yj [Ψ] = Yj y 1 , y 2 , y 3 , y 0 , e, m, ψ(y 1 , y 2 , y 3 , y 0 , e, m) , j = 1, 2. 162 R. S. Ingarden and J. L awrynowicz The transformation Y1 is determined by inserting the expressions (25) into equations (7) and, similarly, Y2 is determined by inserting (26) into (8). Hence we have proved Theorem 1. Let (M, g) be a C ∞ -diﬀerentiable four-dimensional pseudoriemannian manifold with metric g of index 1. Let further ψ : M → C1 be a C ∞ -diﬀerentiable solution to the system of equations (3) with the operator D given in a local co-ordinate system (x1 , x2 , x3 , x0 ), by (4), where the vector ﬁeld (potential) A is supposed to be C ∞ -diﬀerentiable. (i) Then this system is equivalent to the system of the form (7), (8), and (12) under the conditions (9) and (10), where the real-valued functionals ajk , bj , cj (e, m) depend, in general, on g, but do not depend on the operators ∂ k = ∂/∂y k and, in addition, ajk and bj do not depend on the parameters e and m. (ii) Moreover, any solution of the system can be expressed in the form (25) and (26) with Yj as in (27), where ϕk0 , F , and K are given by (24), (22), and (23), respectively, the transform (19) are four partial Fourier transforms with respect to the variable ε of arbitrary four corresponding functions (17) and (18), holomorphic for k odd (k = 1, 3) and antiholomorphic for k even (k = 1, 0). The transformation Y1 is determined by inserting the expressions (23) into the equations (7) and, similarly, Y2 is determined by inserting (24) into (8). The above method has an advantage of being applicable for certain curved spacetimes because its crucial point of applying the global Fourier transform together with suitable convolution equation concerns the electric charge e treated as a real variable whose conjugate variable ε is considered as the real part of the complex variable ε + iϑ. In fact, this means that the space-time in question is treated as the typical ﬁbre in the ﬁbre bundle [St] generated by the spectrum of all possible electric charges. For further generalization we refer to 5. A generalization: Yang-Mills system in the presence of an external field We are going to derive rigorously the system of local Yang-Mills equations in the case of an arbitrary (pseudo)riemannian metric or, diﬀerently speaking, the Yang-Mills equations in the presence of external ﬁelds. Let su(2) be the Lie algebra corresponding to SU(2). By a Yang-Mills ﬁeld we mean any vector foeld A = (Ak ), k = 1, 2, . . . , n, in an open set U in Rn , with values in su(2); [AL], pp. 12–13. Let further T be a representation of SU(2) in a vector space M and let Ψ : U → M be a C ∞ -mapping. Then the covariant derivative of the vector ﬁeld Ψ is given by ∇k Ψ = (∂/∂xk )Ψ + t(Ak )Ψ, t being the representation of su(2) corresponding to T and x = (xk ), k = 1, 2, . . . , n, denoting a co-ordinate system in U . It satisﬁes the identity (Vj ◦ Vk − Vk ◦ Vj )Ψ = t(Fjk )Ψ, where (28) Fjk = ∂/∂xj Ak − ∂/∂xk Aj + [Aj , Ak ]. Finsler-geometrical model of quantum electrodynamics I 163 If SU(2) is replaced by SU(1), the [Aj , Ak ] = 0 and [Fjk ] is the electromagnetic tensor. Next, let us consider a C ∞ -function ρ : U → SU(2) and the following gauge transformation of the pair (Ψ(x), Ak (x)), x ∈ U : (29) (30) Ψ(x) → Ψ (x) = T (ρ(x))Ψ(x), Ak (x) → Ak (x) = ρ(x)Ak (x)ρ−1 (x) − (∂/∂xk ρ(x) ρ−1 (x). With each functional of the form (29)–(30) we may associate the functional [GH]: L∼ (Ψ, A) = L [Ψ(x), (∇j Ψ(x))] dx + L[A], (31) where L[A] is a volume integral involving the Killing form acting on [Fjk ]: 1 L[A] = − (32) g js g rk Tr (Fjr Fsk ) dV ; 4 here g = [gjk ] is a pseudoriemannian (in particular, Riemannian) tensor on U , and dV is the volume element in the manifold (U, g). We observe that (31) is invariant with respect to (29)–(30). Therefore L∼ expresses the action of the vector ﬁeld Ψ on the Yang-Mills ﬁeld A, which may be interpreted as a compensation ﬁeld, whereas L expresses the action of a free compensation ﬁeld. In the case of Minkowski space-time we have g jj = −1 for j = 1, 2, 3, g 00 = 1, g jk = 0 for j = k; j, k = 1, 2, 3, 0, and the extremals for L satisfy the Yang-Mills equations [1], pp. 12–13: (33) ∇k Fjk = 0, with (34) ∇k = g k ∇ , j = 1, 2, 3, 0, ∇ Fjk = ∂/∂x Fjk + [A , Fjk ]. We have [GKL]: Lemma 2. Suppose that M# is an n-dimensional compact orientable (pseudo)riemannian manifold with metric g of index 0 or 1. Let su(2) be the Lie algebra corresponding to SU(2) and A = (Ak ) a C ∞ Yang-Mills ﬁeld in a co-ordinate neighbourhood U of M# . Further, in a co-ordinate system (xk ) in U , let Fjk be given by (28). Consider the functional (32), where dV is the volume element in U . If the functional attains its stationary value for some Yang-Mills ﬁeld A, then this ﬁeld satisﬁes the generalized Yang-Mills equations (35) (DivF )j + Ak , F jk = 0, j = 1, 2, . . . , n, where (36) (37) (DivF )j = g sk (∂/∂xs )Fkj − Γrsk Frj , F jk = g jr g sk Frs , and Γrsk are the usual Christoﬀel symbols. Fkj = g j Fk , 164 R. S. Ingarden and J. L awrynowicz Proof. Denote by Fjr , Fsk the Killing form −Tr(Fjr Fsk ) in (32) and calculate the ﬁrst local Gâteaux variation δL[A]; this means that the supports of δAj , j = 1, 2, . . . , n, are compact: 1 1 g js g rk (δFjr , Fsk + Fjr , δFsk ) dV = − g js g rk δFjr , Fsk dV. δL[A] = − 4 2 Yet, ∂ ∂ δFjr = δAr − r δAj + [δAj , Ar ] + [Aj , δAr ] . j ∂x ∂x Thus, integrating the integrand by parts under the assumption that the supports of δAj are compact, we obtain (38) δL[A] = 10 δk , k=1 where, with the notation |j = ∂/∂xj and dVeucl = (±detg)−1/2 dV , 1 δ1 = g js g rk δAr , (∂/∂xj )Fsk (±detg)1/2 dVeucl , 2 1 js rk δ2 = g δAr , Fsk (±detg)1/2 dVeucl , g|j 2 1 rk δ3 = δAr , Fsk (±detg)1/2 dVeucl , g js g|j 2 1 g js g rk δAr , Fsk (∂/∂xj )(±detg)1/2 dVeucl , δ4 = 2 1 δ5 = − g js g rk δAj , (∂/∂xr )Fsk (±detg)1/2 dVeucl , 2 1 δ6 = − g js g rk δAj , Fsk (±detg)1/2 dVeucl , 2 1 δ7 = − g js g rk δAj , Fsk (±detg)1/2 dVeucl , 2 1 δ8 = − g js g rk δAj , Fsk (∂/∂xr )(±detg)1/2 dVeucl , 2 1 δ9 = − g js g rk [Aj , δAr ]Fsk (±detg)1/2 dVeucl , 2 1 δ10 = − g js g rk [δAj , Ar ]Fsk (±detg)1/2 dVeucl . 2 By the properties of the Killing form, we have 1 δ10 = g js g rk δAj , [Ar , Fsk ](±detg)1/2 dVeucl 2 1 g js g rk δAr , [Fsk , Aj ](±detg)1/2 dVeucl =− 2 1 = g js g rk [δAr , Aj ], Fsk (±detg)1/2 dVeucl = δ9 . 2 Finsler-geometrical model of quantum electrodynamics I 165 Besides, δ5 = δ1 , δ7 = δ2 , δ6 = δ3 , δ8 = δ4 . Yet, with the notation Δjs = g js det g, we get ∂ ∂ ∂ 1 1 (±detg)1/2 = (±detg)1/2 g js r gjm = − (±detg)1/2 gjs r g js , ∂xr 2 ∂x 2 ∂x where the latter equality follows from the identity g js gjs = 4. Next we express the derivatives of g js in terms of g and the related Christoﬀel symbols: js = −Γjqr g qs − Γsqr g jq . g|r (39) Hence (40) 1/2 (∂/∂xr ) (±detg) 1/2 = (±detg) Γqqr , so the corresponding addends δk become: 1 g js Γrqr g qk − Γkqr g rq δAj , Fsk (±detg)1/2 dVeucl δ7 = δ2 = 2 1 js qk r = g g Γqr + g js g rq Γkqr δAj , Fsk (±detg)1/2 dVeucl , 2 1 rk qs j δ6 = δ3 = g g Γqr + g rk g jq Γsqr δAj , Fsk (±detg)1/2 dVeucl , 2 1 δ8 = δ4 = − g js g rk Γqqr δAj , Fsk (±detg)1/2 dVeucl . 2 Consequently, we obtain δ1 + δ5 + δ9 + δ10 = − g js g rk δAj , Δr Fsk (±detg)1/2 dVeucl , δ2 + δ3 + δ4 + δ6 + δ7 + δ8 = 2(δ2 + δ3 + δ4 ) = Rjsk δAj , Fsk (±detg)1/2 dVeucl with ∇Fsk given by (34) and Rjsk = g js g rq Γkqr + g rk g qs Γjqr + g rk g jq Γsqr . Since δAj , j = 1, 2, . . . , n, are arbitrary and our Killing form is nondegenerate, by (36) and (37) the relation δL[A] = 0 implies (41) g js g rk ∇r Fsk = Rjsk , j = 1, 2, . . . , n. We need to prove that the systems (41) and (35) are equivalent. For this we introduce the notation (38) and apply again formulae (39). In this direction from (41) we deduce that g rk (∂/∂xr )Fkj + Ar , Fkj = g rq Γkqr Fkj + g rk Γjqr Fkq − g jq Γsqr Fsr + g rk [(∂/∂xr )g rs ]Fsk = g rq Γkqr Fkj + g rk Γjqr Fkq − g jq Γsqr Fsr + g qs Γjqr Fsr − g jq Γsqr Fsr 166 R. S. Ingarden and J. L awrynowicz and distinguish in a natural way ﬁve addends a1 , . . . , a5 in the expression obtained. Evidently, a5 = −a3 . Since F rq = g sq Fsr equals −F qr = g rk Fkq , we also get a4 = −a2 . Hence g rk (∂/∂xr )Fkj + Ar , Fkj = g lq Γkqr Fkj , j = 1, 2, . . . , n. and, consequently, g rk (∂/∂xr )Fkj − Γqrk Fqj + g rk Ar , Fkj (42) = 0, j = 1, 2, . . . , n. Therefore we arrive indeed at (35) with the notation (36) and (37). Of course we can also proceed in the opposite way, and this completes the proof. 6. The case of an arbitrary symmetry within SO(m) or SU(m) We proceed to generalize our staﬀ to any symmetry within SO(m) or SU(m), m = 2, 3, . . . . Take an arbitrary compact subgroup G of SO(m) or SU(m) and consider a G-vector bundle E = (E, π, M# ) over the base space M# – an n-dimensional compact orientable (pseudo)riemannian manifold with metric g, E standing for the bundle space and π : E → M# for the projection. We include the case of complex, in particular, holomorphic vector bundles. It is well known that it exists a covering π = {Uj , j ∈ I} of M# with local frames over Uj such that the corresponding transition matrices have their values in SU(2). Then a connection N# on E is called a G-connection if for every local frame in question the connection matrix ω has its values in the Lie algebra G corresponding to G. It is well known (e.g. [W]) that, with the help of a partition of unity, for a given G-vector bundle we can show the existence of a G-connection. For applications in quantum electrodynamics it is worth-while to distinguish the notion of a G-vector ﬁeld deﬁned as any vector ﬁeld A on M# with values in G. Hence a Yang-Mills ﬁeld is an SU(2)-vector ﬁeld. A G-connection N# is called the connection corresponding to a G-vector ﬁeld if in any local frame belonging to the G-structure the curvature form corresponding to A satisﬁes the diﬀerential equation (43) δF + 2Trg (A ⊗G F) = 0, where ⊗G denotes the G-dependent tensor product operator and, locally, (44) F = Fjk dxj ∧ dxk , A = A dx , with Fjk and A given by (28) and A = (Aj ), respectively, and ∧ denoting the wedge product operator. These deﬁnitions are motivated by Lemma 3. Suppose M# , g, G, G and E are as before. Then the system of diﬀerential equations (35) with the notation (36) and (37) is well-posed and equivalent to (43). Proof. By the relations (42) and (45) (gjs )|r = Γqrs gjq + Γrj gqs , Finsler-geometrical model of quantum electrodynamics I 167 the latter being analogous to (39), the equations (35) with the notation (36) and (37) become g rk [(∂/∂xr )Fsk − Γqrk Fsq ] + g rk [Ar , Fsk ] = 0, s = 1, 2, . . . , n. Therefore g rk (∂/∂xr )Fsk − Γqrk Fsq − Γqrs gjq Fkj − Γqrj gqs Fkj + g rk [Ar , Fsk ] = 0. Thus g rk (∂/∂xr )Fsk − Γqrk Fsq − Γqrs Fqk − Γqrj gqs Fkj + g rk [Ar , Fsk ] = 0. Yet, by the anitisymmetry of F rj , g rk Γqrj gqs Fkj = Γqrj gqs F rj = Γqrj F rj gqs = 0, and hence −g rk (∂/∂xr )Fsk − Γqrk Fsq − Γqrs Fqk + g rk [Ar , Fks ] = 0, s = 1, 2, . . . , n. By the deﬁnitions of the ﬁrst Gâteaux variation δ and the G-structure-dependent tensor product ⊗G , the above system is identical with (43), as desired. 7. The global Yang-Mills system We are now prepared to derive the global Yang-Mills system which depends on an arbitrary (pseudo)riemannian metric g and an arbitrary non-abelian compact Lie group G [GKL]: Theorem 2. Take an arbitrary compact subgroup of SO(m) or SU(m) and consider a real or complex G-vector bundle E = (E, π, M# ) over the base space M# – an n-dimensional compact orientable (pseudo)riemannian manifold with metric g of index 0 or 1, E standing for the bundle space and π : E → M# for the projection. Suppose further that, in a local frame, (46) F = dA + A ∧ A, A = Ak dxk , where (X k ) is a co-ordinate system in a co-ordinate neighbourhood of M# and ∧ denotes the wedge product operator. Consider the functional 1 L[A] = − (47) Tr(F ∧ ∗g F), 4 M# where A is the G-vector ﬁeld of class C ∞ on M# such that, locally, A = (Ak ) with A = Ak dxk , and ∗g stands for the g-dependent Hodge ∗-operator. Finally, suppose that the functional (47) attains its stationary value for some G-vector ﬁeld A. Let P (M# , G) be the bundle of orthonormal frames of E, endowed with the connection N0 induced by a given G-connection N# of E, corresponding to A. Then the ﬁeld A satisﬁes on M# the system of generalized Yang-Mills equation (48) D(G Ω2 ) = 0, 168 R. S. Ingarden and J. L awrynowicz where D is the covariant derivative operator related to N0 , G : Ap (P ) → An−p (P ), (49) Aν (P ) being the modulus of horizontal ν-forms on P , of the type adG, and where Ω2 stands for the curvature form corresponding to A. Proof. By Lemmas 2 and 3, the G-vector ﬁeld A satisﬁes the diﬀerential equation (42). Denote by ω the connection matrix corresponding to A. Then for any cross-section σ of P (M# , G) we have σ ∗ Ω2 = F, (50) σ ∗ ω = A, where F and A are locally given by (46), and also (51) D(G Ω2 ) = d(G Ω2 ) + ω ∧ G Ω2 . Therefore σ ∗ D(G Ω2 ) = σ ∗ d(G Ω2 ) + (σ ∗ ω) ∧ (σ ∗ G Ω2 ) = dσ ∗ (G Ω2 ) + (σ ∗ ω) ∧ (σ ∗ G Ω2 ). It can be easily seen that σ ∗ G = ∗g σ ∗ , so relation (51) gives (52) ∗g σ ∗ D(G Ω2 ) = (∗g d∗g )σ ∗ Ω2 + ∗g (δ ∗ ω) ∧ ∗g (δ ∗ Ω2 ) . Consequently, by the deﬁnition of the codiﬀerential operator: δg = ∗g d∗g and relations (50), the condition (52) takes the form (53) ∗g σ ∗ D G Ω2 = δg F + ∗g (A ∧ ∗g F). In order to prove that the global diﬀerential equation (43) is equivalent to ∗g σ ∗ D G Ω2 = 0, (54) it suﬃces to verify the relation ∗g (A ∧ ∗g F) = 2Trg (A ⊗G F). (55) For this, let us consider an orthonormal system (eq ), q = 1, 2, . . . , m, of vector ﬁelds in any co-ordinate neighborhood u of M# , mentioned in Lemma 2. Consider also the system (e∗j ), j = 1, 2, . . . , n, of one-forms on U such that e∗j [ek ] = δjk for each (j, k). Then from (44) we infer 1 (56) F = f jk e∗j , A = a e∗ 2 and, furthermore, 1 jk 1 ∗ ∗ f ∗g (e∗j ∧ e∗k ) = f jk εrs jk er ∧ es , 2 4 where εrs jm stands for the totally antisymmetric Levi-Civita tensor, and 1 q jk rs ∗ 1 q jk rs q ∗ a ,f a ,f εjk er ∧ e∗s ∧ e∗q , ∗g (∗g F ∧ A) = εjk εrs e , A ∧ ∗g F = 4 4 where q q q εrs jk εrs = 2 δj δk − δj δk ∗g F = Finsler-geometrical model of quantum electrodynamics I 169 and δjk denotes the Kronecker symbol. Therefore 1 · 2 · 2 ar , f q δrq e∗ = 2Trg (A ⊗G F), 4 δjk being again the Kronecker symbol. Consequently, the global diﬀerential equation (43) is indeed equivalent to (54). The system of diﬀerential equations (54) holds, in particular, for any local crosssection σ on P (M# , G). Yet, the Hodge operator ∗g is an isomorphism, so the Gvector ﬁeld A satisﬁes on M# the system of diﬀerential equations (48), as desired. ∗g (A ∧ ∗g F) = 8. An SU(2)-based non-abelian generalization In Sect. 7 our non-abelian generalization was related with consideration of an arbitrary non-abelian compact Lie group G. Now let us concentrate, following [BI1, 2], on ﬁnding non-abelian solutions of ﬁnite mass to the Yang-Mills equations. Let us introduce the following generalization of the U(1)-action: 1 α μν 1 α μν 2 1 2 Fβγ Fα β (57) S = Fα − , (1 − )d4 x with = 1 + 2 Fβγ 4π 2β 16β 4 where the constant β appears for dimensional reasons, meaning essentially the limiting value of the electric ﬁeld in Mie’s nonlinear electrodynamics [Mi]. The nonlinearity breaks the conformal symmetry and the stress-energy tensor [Tνμ ] has the non-zero trace: α Fαμν = 0. Tμμ = −1 4β 2 (1 − ) − Fμν The trace vanishes as β → 0, and the approach reduces to the standard one. We assume that Aα 0 = 0, k Aα j = εαjk (n /r)[1 − w(r)], where nk = xk /r, r= x2 + y 2 + z 2 , and the function w is real-valued. Integration in (57) over the sphere gives a twodimensional action from which we eliminate the constant β by the co-ordinate rescal√ √ ing βt → t, βr → r. Then we obtain w 2 (1 − w2 )2 S = r2 (1 − )dr with = 1 + 2 2 + , r r4 where stands for the diﬀerentiation with respect to r. The nonlinearity appears α ] on the potentials Ajμ . here because of the nonlinear dependence of the tensor [Fμν The corresponding equation of motion reads [KBG]: (58) (w /) = (1/r2 )w(w2 − 1). 170 R. S. Ingarden and J. L awrynowicz For time-dependent conﬁgurations the energy density is equal to minus the Lagrangian, so the total energy (mass) is given by ∞ M [w] = (2 − 1)r2 dr. 0 In particular, 2 ∞ 3 1 √ 3 2 4 M [0] = r + 1 − r dr = π ≈ 1.236 Γ 3 4 0 corresponds to the point-like magnetic monopole with the unit magnetic charge being an embedded U(1)-solution. In order to assure the convergence of M [w], the ratio of its integrand to r has to tend to zero as r → ∞. Then the equation (58) should reduce to the ordinary Yang-Mills equation, equivalent to the autonomous system (59) ẇ = u, u̇ = u + (w2 − 1)w, where the dot stands for diﬀerentiation with respect to τ = ln r. We have thus constructed a dynamical system with three non-degenerate stationary points (u, w) = (0, −1), (0, 0), (0, 1), where (0, 0) is a focus and two others are saddle points with eigenvalues λ = −1, 2. The separatrices along the directions λ = −1 go from ∞ to the focus with eigenvalues √ √ 1 1 λ = (1 − 3i), (1 + 3i) 2 2 passing through the saddle points. Following [GK], let us consider ﬁnite-energy conﬁgurations with nonvanishing magnetic charge. Such solutions have w = 0 asymptotically which does not correspond to bounded solutions unless w ≡ 0, or w = −1, 0, 1 asymptotically, and this corresponds to zero magnetic charge. Consequently, the only ﬁnite-energy conﬁgurations with nonvanishing magnetic charge are the embedded U(1)-monopoles. In terms of the variable r, (58) implies w = −1 + cr−1 + O(r−2 ) or w = 1 + cr−1 + O(r−2 ), where c is a constant. Hence the corresponding integral M [w] converges as r → ∞. The values w = −1, 1 correspond to two neighbouring topologically distinct YangMills vacua. It seems important to consider the local solutions of (58) near r = 0. If M [w] converges, then w tends to a ﬁnite limit as r → 0. By (58) the only allowed limiting values are w = −1, 1. By the symmetry of (59) under reﬂection w → ±w, without any loss of generality we may take w(0) = 1, and then from (59) we get w = 1 − br2 + b2 (44b2 + 3) 4 r + O(r6 ), 10(4b2 + 1) 171 Finsler-geometrical model of quantum electrodynamics I where b is a constant. At r → 0, → 1 + 12b2 + O(r2 ), so this time we have not a solution of the initial system (58). It remains to ﬁnd proper values of b to smooth ﬁnite-energy solutions by gluing together the two asymptotic solutions between 0 and ∞. This is eﬀectively done in [GK, KBG]. 9. A generalization of the Lagrangian and its embedding in the electroweak model It seems instructive to compare the pure electromagnetic (abelian) Lagrangian of Born and Infeld, used in Sect. 8, with what can be extracted from its non-abelian version based on the symmetry group G = SU(2)× U(1) after deﬁning physical ﬁelds as linear combinations of the U(1) and SU(2) gauge ﬁelds; see [KBG] for details. Write the non-abelian generalizations P ans S of Maxwell’s tensor invariants P and S as 1 P = Fμν F μν , S = Fμν F̃ μν = εμνρσ Fμν Fρσ , 2 where a Fμν = Fμν Ja ; a = 0 for U(1), a = 1, 2, 3 for SU(2). We get 1 a aμν 0 LG = 2aFμν F 0μν + aFμν F 2 1 a aμν c cρσ 0 0 +β −2 M 2 b 2Fμν F 0μν Fρσ F 0ρσ + Fμν F Fρσ F 8 0 a 0 0 0 0 F̃ 0μν Fρσ F̃ 0ρσ + Fμν F 0μν Fρσ F aρσ + 2Fμν F aμν Fρσ F aρσ + c 2Fμν 1 a aμν c cρσ 0 a 0 0 F̃ F̃ 0μν Fρσ F̃ aρσ + 2Fμν F̃ aμν Fρσ F̃ aρσ + Fμν Fρσ F̃ + Fμν 8 + ... 2 1 1 1 2 LBI = − Fμν F μν + β −2 (Fμν F μν ) + β −2 Fμν F̃ μν 4 32 32 − 2 1 −4 1 −4 5 −6 3 4 β (Fμν F μν ) − β Fμν F μν Fμν F̃ μν + β (Fμν F μν ) 128 128 2048 + 2 4 3 −6 1 −6 Fμν F̃ μν + . . . , β (Fμν F μν )2 Fμν F̃ μν + β 1024 2048 where a, b, c, . . . are representation-dependent coeﬃcients coming from the traces. Next we introduce physical ﬁelds with linear combinations of the U(1) and SU(2) gauge ﬁelds and compare the two series term by term, ﬁxing the coeﬃcients. In particular, for the pure electromagnetic sector LEM in LBI , we arrive at the formula 172 R. S. Ingarden and J. L awrynowicz (related to rotation with the so-called Weinberg angle θ within the linear combinations of the U(1) and SU(2) gauge ﬁelds): 1 LG − LEM = a 2 cos2 θ + sin2 θ Fμν F μν 2 1 −2 2 4 4 2 2 +β M b 2 cos θ + sin θ + 3 sin θ cos θ Fμν F μν Fρσ F ρσ 8 (60) 1 +c 2 cos2 θ + sin4 θ + 3 sin2 θ cos2 θ Fμν F̃ μν Fρσ F̃ρσ 8 1 15 15 6 4 2 −4 3 2 4 6 sin θ + cos θ sin θ + cos θ sin θ + 2 cos θ + . . . . +β M g 32 8 2 For other instructive examples we recommend [To, Ma1, 2]. Examples strictly related with the Finsler geometry will be given in the second part of this paper [IL5]. Acknowledgments This work (J. L ) was partially supported by the Ministry of Scinces and Higher Education grant PB1 P03A 001 26 (Sections 1–2 of the paper) and partially by the grant of the University of L ódź no. 505/692 (Sections 3–9). References M. F. Atiyah, Geometry of Yang-Mills Fields (Lezioni Fermiane), Accademia Nazionale dei lincei – Scuola Normale Superiore, Pisa 1979. [BI1] M. Born and L. Infeld, Electromagnetic mass, Nature 132 (1932), 970–970. [BI2] —, —, Foundations of the new ﬁeld theory, Proc. Roy. Soc. London A 144 (1934), 425–451. [Be1] R. G. Beil, Finsler and Kaluza-Klein gauge theories, Internat. J. Theor. Phys. 32 (1933), 1021–1031. [Be2] —, Moving frame transport and gauge transformations, Found. Phys. 25 (1995), 717–742. [GH] S. G. Gindikin and G. M. Henkin, Penrose’s transformation and complex integral geometry [in Russian], in: Itogi nauki i tekhniki. Seriya: Sovremennye problemy matematiki 17, Moskva 1981, pp. 57–111. [GK] D. V. Gal’tsov and R. Kerner, Classical glueballs in non-abelian Born-Infeld theory, Phys. Rev. Lett. 84, no. 26 (2000), 5955–5958. [GKL] B. Gaveau, J. Kalina, J. L awrynowicz, and L. Wojtczak. The global Yang-Mills equations depending on an arbitrary metric, Ann. Polon. Math. 46 (1985), 105–114. [GL] — and G. Laville, On the Maxwell-Dirac system in two dimensions, Bull. Soc. Sci. Lettres L ódź 32 (1982), 1–8. 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Mie, Grundlagen einer Theorie der Materie. Erste Mitteilung, Annalen der Physik 37 (1912), 511–534; Ditto. Zweite Mitteilung, ibid. 39 (1912), 1–40; Ditto. Dritte Mitteilung, ibid. 40 (1913), 1–66. [S] N. Salingaros, Electromagnetism and the holomorphic properties of spacetime, J. Math. Phys. 22 (1981), 1919–1925. [Sc] E. Schmutzer, Relativistische Physik, 2. Auﬂ., B. G. Teubner, Leipzig 1968. [St] N. E. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, N. J. 1951. [To] I. T. Todorov, Conformal covariant instanton solutions of Euclidean Yang-Mills equations (An elementary introduction), in: III School of Elementary Particles and High Energy Physics, Primorsko, Bulgaria 1977, Bulgarian Acad. of Sci., Soﬁa 1978, pp. 131–182. [W] R. O. Wells, Jr., Diﬀerential Analysis on Complex Manifolds, Prentice Hall, Englewood Cliﬀs, N. J. 1973. Institute of Physics Nicolaus Copernicus University Grudzia̧dzka 5, PL-87-100 Toruń Poland e-mail: [email protected] Institute of Physics Univeristy of L ódź Pomorska 149/153, PL-90-236 L ódź Institute of Mathematics Polish Academy of Sciences L ódź Branch, Banacha 22, PL-90-238 L ódź Poland e-mail: [email protected] Presented by Julian L awrynowicz at the Session of the Mathematical-Physical Commission of the L ódź Society of Sciences and Arts on March 2, 2010 MODEL FINSLEROWSKO-GEOMETRYCZNY ELEKTRODYNAMIKI KWANTOWEJ I POLE ZEWNȨTRZNE A GEOMETRIA FINSLERA Streszczenie Po podsumowaniu ﬁzycznych oczekiwań spowodowanych koniecznościa̧ uwzglȩdnienia ukladów otwartych (nie izolowanych), rozważamy kwantowe równania Diraca-Maxwella przy użyciu podejścia analitycznego zespolonego oraz równań splotowych. Z kolei przechodzimy do ogólniejszego przypadku równań Yanga-Millsa w obecności pola zewnȩtrznego, z wla̧czeniem przypadków dowolnej symetrii w zakresie grup SO(m) lub SU(m), przypadku globalnego, uogólnienia nie-abelowego oraz uogólnienia lagranżianu i jego wlożenia w model elektro-slaby. W drugiej czȩści pracy bȩdziemy zajmowali siȩ krysztalami ferrelektrycznymi w geometrii Finslera oraz interpretacja̧ ﬁzyczna̧ koneksji solenoidalnych i niesolenoidalnych w kanonicznych glównych wia̧zkach wlóknistych. !" #$ %# & ' (% )%*!$ %# % (+ ,-. /*! ($0*!$ ,-. 1% 23$ ,-. )%% 2!% 10# 3 2% 0* % 2 2 40 * '0 %+ ,0%$ ,-. !$5$ ,-. 6 6 %+ $ 0* $"+%$ ,-. $ ,-. 70 '8 96 20 ': 0$0$ 10 ; 1$ %$!% ,-. 2%%6 $ ,-. 7 ) 0 2 9$ 0%$$ %# <0 ($0* *#$ ,-. "! !$ ,-. (% 0$!$ ,-. 25= $ ;$ 9 33 (-%6 $ 2$-! + $!%#$ >%! ;0 0%0$ 1$ (-%6 %0 10# 0 0 1 ?% @*$520%$ ,-. >"%% >%!$ ,-. )%% >!$ ,-. >""! >%#$ ,-. 9 /0 0 ! / 5) 70 /$AB2 ' /AB2 CONTENU DU VOLUME LX, no. 2 1. Yu. Zelinskiı̌, Continuous mappings between domains of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ca. 4 pp. 2. A. Touzaline, On the solvability of a quasistatic contact problem for elastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ca. 20 pp. 3. J. Rutkowski and C. Surry, Melting and related phenomena in thin lead ﬁlms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ca. 8 pp. 4. J. Zaja̧c and B. Falda, Inﬂuence of Professor Julian L awrynowicz and his Lublin colleagues during 20 years of PolishMexican collaboration in generalized complex analysis . . . . . . . . ca. 16 pp. 5. V. S. Shpakivskyi and S. A. Plaksa, Integral theorems and a Cauchy formula in a commutative three-dimensional harmonic algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ca. 10 pp. 6. D. Mierzejewski, The dimensions of sections of the sets of the solutions of some quadratic quaternionic equations . . . . . . . . . . . . ca. 12 pp. 7. A. K. Kwaśniewski, Some Cobweb posets digraphs elementary properties and questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ca. 7 pp. 8. M. Nowak-Kȩpczyk, Binary alloy thin ﬁlms vs. LennardJones and Morse potentials. A note on binary alloys with arbitrary atoms concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ca. 16 pp. 9. R. S. Ingarden and J. L awrynowicz, Finsler-geometrical model of quantum electrodynamics II. Physical interpretation of solenoidal and nonsolenoidal connections on the cannonical principal ﬁbre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ca. 16 pp. 10. R. S. Ingarden and J. L awrynowicz, Finsler geometry and physics. Physical overwiev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ca. 16 pp.