Serdica Mathematical Journal

Volume 43, Number 1, 2017

C O N T E N T S

A B S T R A C T S

MATRIX COEFFICIENTS OF THE IRREDUCIBLE UNITARY REPRESENTATION OF SU(n,1)
Fouzia El Wassouli elwassouli@gmail.com

2010 Mathematics Subject Classification: 32M15, 32A45, 30E20, 33C60.
Key words: Hypergeometric function, matrix coefficients, weighted Bergman space.

This paper is devoted to presenting an explicit expression for the A-radial part of matrix coefficients of the irreducible unitary representations in terms of Gaussian hypergeometric series and some involved expressions of binomial coefficients

CENTER MANIFOLDS FOR EVOLUTION EQUATIONS ASSOCIATED WITH THE STEFAN PROBLEM
Martin Lukarevski martin.lukarevski@ugd.edu.mk

2010 Mathematics Subject Classification: Primary 35R35; Secondary 35B65, 35J70.
Key words: Center manifold, Evolution equation, Stefan problem, Free boundary problem.

Evolution equations can be used for solving the Stefan problem. We show the existence of a center manifold for an evolution equation that is associated with a quasilinear Stefan problem with variable surface tension and undercooling. This generalizes previous result for existence of center manifold for a Stefan problem where the relaxation coefficient is constant.

SYMPLECTIC DECOMPOSITION OF THE MASSIVE COADJOINT ORBITS OF A SEMIDIRECT PRODUCT
Benjamin Cahen benjamin.cahen@univ-lorraine.fr

2010 Mathematics Subject Classification: 81S10, 22E46, 22E45, 81R05.
Key words: Semidirect product, coadjoint orbit, unitary representation, symplectomorphism, Weyl quantization, Berezin quantization, Poincaré group.

Let G be the semidirect product V⋊ K where K is a connected semisimple non-compact Lie group acting linearily on a finite-dimensional real vector space V. Let 𝒪 be a coadjoint orbit of G whose little group K₀ is a maximal compact subgroup of K. We construct an explicit symplectomorphism between 𝒪 and the symplectic product ℝ²ⁿ×𝒪' where 𝒪' is a little group coadjoint orbit. We treat in details the case of the Poincaré group.

DIFFERENTIAL GEOMETRY OF CONCIRCULAR SUBMANIFOLDS OF EUCLIDEAN SPACES
Bang-Yen Chen bychen@math.msu.edu,
Shihshu Walter Wei wwei@ou.edu

2010 Mathematics Subject Classification: 53A07, 53C40, 53C42.
Key words: Euclidean submanifold, position vector field, concurrent vector field, concircular vector field, rectifying submanifold.

A Euclidean submanifold is called a rectifying submanifold if its position vector field x always lies in its rectifying subspace [B.-Y. Chen. Differential geometry of rectifying submanifolds. Int. Electron. J. Geom. 9, No 2 (2016), 1−8]. It was proved in [B.-Y. Chen. Differential geometry of rectifying submanifolds. Int. Electron. J. Geom. 9, No 2 (2016), 1−8] that a Euclidean submanifold M is rectifying if and only if the tangential component x^T of its position vector field is a concurrent vector field.
Since concircular vector fields are natural extension of concurrent vector fields, it is natural and fundamental to study a Euclidean submanifold M such that the tangential component x^T of the position vector field x of M is a concircular vector field. We simply call such a submanifold a concircular submanifold. The main purpose of this paper is to study concircular submanifolds in a Euclidean space. Our main result completely classifies concircular submanifolds in an arbitrary Euclidean space.

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SOLITARY PULSE SOLUTIONS OF A COUPLED NONLINEAR SCHRÖDINGER SYSTEM ARISING IN OPTICS
Sharad Silwal sdsilwal@jchs.edu

2010 Mathematics Subject Classification: 35A15, 35C08, 35Q51, 35Q55.
Key words: nonlinear Schrödinger, solitary pulse solutions, optics, concentration compactness, existence.

We prove the existence of travelling-wave solutions for a system of coupled nonlinear Schrödinger equations arising in nonlinear optics. Such a system describes second-harmonic generation in optical materials with χ⁽²⁾ nonlinearity. To prove the existence of travelling waves, we employ the method of concentration compactness to prove the relative compactness of minimizing sequences of the associated variational problem.

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EXTENDING THE CONVERGENCE DOMAIN OF~NEWTON'S METHOD FOR~GENERALIZED EQUATIONS
Ioannis K. Argyros iargyros@cameron.edu,
Santhosh George sgeorge@nitk.ac.in

2010 Mathematics Subject Classification: 65B05, 65G99, 65N35, 47H17, 49M15.
Key words: Hilbert space, generalized equation, Newton's method, Lipschitz conditions, Newton−Kantorovich hypothesis, local-semilocal convergence theorems, coercivity, multivalued maximal monotone operator, radius of convergence.

We present semi-local convergence results for Newton's method to solve generalized equations. Using a combination of Lipschitz and center-Lipschitz conditions on the operators involved instead of just Lipschitz conditions we show that our Newton−Kantorovich criteria are weaker than earlier sufficient conditions for the convergence of Newton's method. In particular, we provide finer error bounds and a better information on the location of the solution. Our results apply to solve generalized equations involving single as well as multivalued operators, which include variational inequalities, nonlinear complementarity problems and non smooth convex minimization problems. Numerical examples validate the theoretical results by showing that equations that could not be solved before can be solved using our new approach.

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HOMEOMORPHISMS OF FUNCTION SPACES AND~TOPOLOGICAL DIMENSION OF DOMAINS
Vadim Lazarev lazarev@math.tsu.ru

2010 Mathematics Subject Classification: 54C35.
Key words: pointwise convergence topology, topological dimension.

It is known since 1982, that dim X = dim Y whenever the function spaces C_p(X) and C_p(Y) are linearly homeomorphic. This statement was later extended to uniform homeomorphisms of the spaces C_p(X) and C_p(Y). We obtain, in the case of separable function spaces, a generalization of the first result to another direction.
We introduce, for each X, some subspace E(X) ⊂ C_pC_p(X), which is significantly wider, than the space L_p(X) of all linear continuous functionals on C_p(X). Our generalization includes homeomorphisms h : C_p(X) → C_p(Y), such that the image of Y under the dual mapping h^* of h is contained in E(X) and the image of X under (h⁻¹)^* is contained in E(Y).

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