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On n class of capacities on complex manifolds endowed with an hermitian structure and their relation to elliptic and hyperbolic quasiconformal mappings

100%
Ławrynowicz J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction......................................................................................................................................... 5  1. An outline of results.................................................................................................................. 5  2. A fibre bundle model of elementary particles as a motivation  for the capacities in question..................................................................................................... 9  3. An example................................................................................................................................ 10  4. A potential-theoretical motivation for the capacities in question..................................... 12  5. Capacities and plurisubharmonic functions....................................................................... 14  6. A homology approach and the general definition of capacity........................................... 16  7. Finiteness and relations between capacities dependent on the chosen covering  and independent of it.................................................................................................................... 19  8. Behaviour under holomorphic and biholomorphic mappings......................................... 22  9. Some lemmas on Riemann surfaces................................................................................. 25  10. Comparison of the "complex" and "real" capacities in the case  of Riemann surfaces................................................................................................................... 30  11. Dependence on the universal covering manifold............................................................ 33  12. Relation to elliptic and hyperbolic quasiconformal mappings...................................... 36  13. Mathematical and physical conclusions............................................................................ 39 References......................................................................................................................................... 41
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Structure fractals and para-quaternionic geometry

61%
Ławrynowicz J., Vaccaro M.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2011
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tom 65
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nr 2
EN
It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions \(p = 2, 4\) and \(8\), respectively, but the procedure fails for \(p = 16\) in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are \(n = 2^7\). Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process \(p \to p + 2 \to p + 4 \to ...\), they have constructed \(2^4\)-dimensional “bipetals” for \(p = 9\) and \(2^7\)-dimensional “bisepals” for \(p = 13\). The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the “pistil” and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the “stamens”. The first named author, M. Nowak-Kepczyk, and S. Marchiafava (2006, 2009a, b) gave an effective, explicit determination of the periods and expressed them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. In contrast to earlier results, the fractal bundle flower structure, in particular petals, sepals, pistils, and stamens are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. Physical concepts of dual and conjugate objects as well as of antiparticles led us to extend the periodicity theorem to structure fractals in para-quaternionic formulation, applying some results in this direction by the second named author. The paper is concluded by outlining some applications.
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Foliations by complex manifolds involving the complex Hessian

54%
Ławrynowicz J., Kalina J., Okada M.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Summary...................................................................................................................................................................3 1. Introduction and an outline of results....................................................................................................................5 2. Capacities on complex manifolds and the generalized complex Monge-Ampère equations..................................8 3. Foliations............................................................................................................................................................10 4. Proof of the existence theorem in the holomorphically decomposable case.......................................................12 5. Proof of the existence theorem in the exterior product case...............................................................................14 6. Natural Markov processes connected with the foliation $ℒ_{k+p-1}$.................................................................16 7. Properties of canonical diffusions.......................................................................................................................18 8. Laplace-Beltrami operator on Riemannian manifolds.........................................................................................21 9. Harmonic theory on compact complex manifolds................................................................................................23 10. Laplace-Beltrami operator as the generator of a canonical diffusion................................................................27 11. Laplace-Beltrami operator in the case of the sphere and the hyperboloid........................................................29 12. Complex Hessian involving convex functions....................................................................................................33 13. Some examples of applications.........................................................................................................................36 14. Hypersurfaces in ℂ³ depending on two holomorphic functions.........................................................................41 References.............................................................................................................................................................43
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Differential and integral calculus for a Schauder basis on a fractal set (I) (Schauder basis 80 years after)

49%
Ławrynowicz J., Ogata T., Suzuki O.
Banach Center Publications
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2009
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tom 87
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nr 1
115-140
EN
In this paper we introduce a concept of Schauder basis on a self-similar fractal set and develop differential and integral calculus for them. We give the following results: (1) We introduce a Schauder/Haar basis on a self-similar fractal set (Theorems I and I'). (2) We obtain a wavelet expansion for the L²-space with respect to the Hausdorff measure on a self-similar fractal set (Theorems II and II'). (3) We introduce a product structure and derivation on a self-similar fractal set (Theorem III). (4) We give the Taylor expansion theorem on a fractal set (Theorem IV and IV'). (5) By use of the Taylor expansion for wavelet functions, we introduce basic functions, for example, exponential and trigonometrical functions, and discuss the relationship between the usual and introduced corresponding special functions (Theorem V). (6) Finally we discuss the relationship between the wavelet functions and the generating functions of the dynamical systems on a fractal set and show that our wavelet expansions can describe several fluctuations observed in nature.
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Mixed formulation for elastic problems - existence, approximation, and applications to Poisson structures

49%
Ławrynowicz J., Mignot A. L., Papaloucas L. C., Surry C.
Banach Center Publications
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1996
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tom 37
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nr 1
343-349
EN
A mixed formulation is given for elastic problems. Existence and uniqueness of the discretized problem are given for conformal continuous interpolations for the stress tensor components and for the components of the displacement vector. A counterpart of the problem is discussed in the case of an even-dimensional Euclidean space with an associated Hamiltonian vector field and the Poisson structure. For conformal interpolations of the same order the question remains open.
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Supercomplex structures, surface soliton equations, and quasiconformal mappings

49%
Ławrynowicz J., Kędzia K., Suzuki O.
Annales Polonici Mathematici
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1991
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tom 55
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nr 1
245-268
EN
Hurwitz pairs and triples are discussed in connection with algebra, complex analysis, and field theory. The following results are obtained: (i) A field operator of Dirac type, which is called a Hurwitz operator, is introduced by use of a Hurwitz pair and its characterization is given (Theorem 1). (ii) A field equation of the elliptic Neveu-Schwarz model of superstring theory is obtained from the Hurwitz pair (𝔼⁴,𝔼³) (Theorem 2), and its counterpart connected with the Hurwitz triple $(𝔼^{11},𝔼^{11},𝔼^{26})$ is mentioned. (iii) Isospectral deformations of the Hurwitz operator of the Hurwitz pair (𝔼²,𝔼²) induce various soliton equations (Theorem 3). (iv) A special complex structure, which is called a supercomplex structure, is introduced on separable Hilbert spaces (Definition 10). A correspondence between such structures and reduction solutions of Sato's version of Kadomtsev-Petviashvili system is established (Theorem 4). (v) The general class of quasiconformal mappings in the plane is obtained from generalized Hurwitz pairs (Theorem 5). From these results we conclude that Hurwitz pairs and triples give rise to several interesting applications.
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Generalized Hurwitz maps of the type S × V → W, anti-involutions, and quantum braided Clifford algebras

49%
Ławrynowicz J., Rembieliński J., Succi F.
Banach Center Publications
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1996
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tom 37
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nr 1
223-240
EN
The notion of a $J^3$-triple is studied in connection with a geometrical approach to the generalized Hurwitz problem for quadratic or bilinear forms. Some properties are obtained, generalizing those derived earlier by the present authors for the Hurwitz maps S × V → V. In particular, the dependence of each scalar product involved on the symmetry or antisymmetry is discussed as well as the configurations depending on various choices of the metric tensors of scalar products of the basis elements. Then the interrelation with quantum groups and related Clifford-type structures is indicated via anti-involutions which also play a central role in the theory of symmetric complex manifolds. Finally, the theory is linked with a natural generalization of general linear inhomogeneous groups as quantum braided groups. This generalization is in the spirit of the theory initiated and developed by S. Majid, however, our construction differs in the interrelation between the homogeneous and inhomogeneous parts of the group. In order to study the quantum braided orthogonal groups, we consider a kind of quantum geometry in the covector space. This enables us to investigate a quantum braided Clifford algebra structure related to the spinor representation of that group.
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On estimating a fifth order functional for bounded univalent functions

43%
Ławrynowicz J., Tammi O.
Colloquium Mathematicum
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1972
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tom 25
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nr 2
307-313
9
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Foliations and the generalized complex Monge-Ampère equations

43%
Kalina J., Ławrynowicz J.
Banach Center Publications
|
1983
|
tom 11
|
nr 1
111-119
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