Serdica Mathematical Journal

Volume 44, Number 3−4, 2018

C O N T E N T S

A B S T R A C T S

ROBUST OPTIMIZATION: STABILIZATION METHODS AND WELL-POSEDNESS IN MATHEMATICAL PROGRAMMING AND SADDLE POINT PROBLEMS
Driss Mentagui dri_mentagui@yahoo.fr, d_mentagui@hotmail.com

2010 Mathematics Subject Classification: 49K40, 49J45, 49L25, 49J52, 90C26, 90C31.
Key words: well-posed optimization problems, variational asymptotic developments, conjugacy, stability, α-convexity, epi-convergence, convex-concave functions, subdifferentiability, infimal-convolution, well-posed saddle point problems and variational sets, saddle functions.

In this paper, we provide various characterizations of several well-posedness concepts in mathematical programming and saddle point problems. We introduce a large class of generalized stabilization methods and display variational asymptotic developments of minimum and saddle values of regularization schemes under consideration. The convex and nonconvex cases are studied. A class of well-posed problems has been also studied using infimal-convolution, epigraphical analysis and subdifferentiability. Many examples and applications illustrated our investigation. Notably an application to Legendre-Fenchel transform in locally convex spaces is given. A detailed study of Levitin-Polyak well-posedness in mathematical programming as well as the one for saddle point problems have been displayed in metric and normed spaces.

ON INDEX-EXPONENT RELATIONS OVER HENSELIAN FIELDS WITH LOCAL RESIDUE FIELDS
Ivan D. Chipchakov chipchak@math.bas.bg

2010 Mathematics Subject Classification: 16K50, 12J10 (primary); 11S99, 12E15, 13F30.
Key words: Brauer group, Schur index, exponent, index-exponent pair, Brauer p-dimension, Henselian field, quasifinite field, maximally complete field.

Let p be a prime and (K, v) a Henselian valued field with a residue field K̂. This paper determines the Brauer p-dimension of K, in case p ≠ char(K̂) and K̂ is a p-quasilocal field properly included in its maximal p-extension. When K̂ is a local field, it describes index-exponent pairs of central division K-algebras of p-primary degrees. The same goal is achieved, if (K, v) is maximally complete, char(K) = p and K̂ is local.

2010 Mathematics Subject Classification: 14G15, 14R20.
Key words: Affine varieties, locally finite modules, Noether normalization.

The aim of this note is to show that if a finite field k with absolute Galois group 𝔊 acts on a set M with finite orbits and for some m there is a 𝔊-equivariant map ξ : M → k̅^m, whose fibres are of bounded cardinality, then M admits a 𝔊-equivariant embedding in an affine space k̅ⁿ of sufficiently large dimension n.

ISOTHERMIC SURFACES AND SOLUTIONS\\ OF THE CALAPSO EQUATION
Armando M. V. Corro avcorro@gmail.com,
Carlos M. C. Riveros carlos@mat.unb.br,
Karoline V. Fernandes karoline.victor.mat@hotmail.com

2010 Mathematics Subject Classification: 53A30, 32A10, 34M05.
Key words: lines of curvature, Weierstrass representation, isothermic surfaces, Calapso equation.

In this paper, we introduce a class of surfaces called radial inverse mean curvature surface (RIMC-surfaces) and we show that there is a correspondence between these surfaces and Bryant surfaces in the hyperbolic space ℍ³, therefore, the RIMC-surfaces are isothermic. We obtain a Weierstrass type representation for RIMC-surfaces which depends on a meromorphic function and a holomorphic function and we obtain a characterization so that these surfaces are parametrized by lines of curvature. In [3] (P. Calapso. Sulle superficie a linee di curvatura isoterme. Palermo Rend. 17 (1903), 275−286) it is shown that for each isothermic surface parametrized by lines of curvature in the Euclidean space a solution of the Calapso equation is associated, in this work we show that for these surfaces we can associate another solution of the Calapso equation. Moreover, we give explicit solutions of the Calapso equation that depend on holomorphic functions.

EXTREME REPRESENTATIONS OF SEMIRINGS
Chih-Whi Chen chihwhichen@xmu.edu.cn,
Brendan Frisk Dubsky Brendan.Frisk.Dubsky@math.uu.se,
Helena Jonsson Helena.Jonsson@math.uu.se,
Volodymyr Mazorchuk Volodymyr.Mazorchuk@math.uu.se,
Elin Persson Westin Elin.Persson.Westin@math.uu.se,
Xiaoting Zhang Xiaoting.Zhang@math.uu.se,
Jakob Zimmermann Jakob.Zimmermann@math.uu.se

2010 Mathematics Subject Classification: 16Y60, 16D10, 16D60.
Key words: semiring, semimodule, module, elementary semimodule, minimal semimodule, simple semimodule, cell, Kazhdan-Lusztig basis.

This is a write-up of the discussions during the meetings of the study group on representation theory of semirings which was organized at the Department of Mathematics, Uppsala University, during the academic year 2017−2018. The main emphasis is on classification of various classes of "irreducible" representations for various concrete semirings.

n-DIMENSIONAL COPULAS AND WEAK DERIVATIVES
Nikolay Chervenov nikolay.tchervenov@gmail.com,
Iordan Iordanov iiordanov@ibsedu.bg,
Boyan Kostadinov boyan.sv.kostadinov@gmail.com

2010 Mathematics Subject Classification: 35L05, 35L20, 46FXX, 46F10, 46F12, 60E05, 62F99.
Key words: copula, n-increasing function, weak solution, test function.

In the copula theory universe the number of multivariate copulas is very limited. This is caused by both of non-trivial tasks − to check the n-increasing property and to define the copula. We generalize the notion of n-increasing property in terms of weak derivatives which allows us to simplify the otherwise complex former method. Furthermore, we demonstrate the applicability of our approach to the class of n-dimensional Archimedean copulas. Finally, we present a method which allows us to obtain a class of copulas as a solution of a boundary value problem in appropriate Sobolev spaces.