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## On one representation of analytic functions by harmonic functions

### Siberian Advances in Mathematics (2008-06-01) 18: 103-117 , June 01, 2008

Let *u*(*x*) be a function analytic in some neighborhood *D* about the origin,
$$
\mathcal{D}
$$
⊂ ℝ^{n}. We study the representation of this function in the form of a series *u*(*x*) = *u*_{0}(*x*) + |*x*|^{2}*u*_{1}(*x*) + |*x*|^{4}*u*_{2}(*x*) + …, where *u*_{k}(*x*) are functions harmonic in
$$
\mathcal{D}
$$
. This representation is a generalization of the well-known Almansi formula.

## Singular Arcs in the Generalized Goddard’s Problem

### Journal of Optimization Theory and Applications (2008-11-01) 139: 439-461 , November 01, 2008

We investigate variants of Goddard’s problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method that we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle the problem of nonsmoothness of the optimal control.

## Degrees of stretched Kostka coefficients

### Journal of Algebraic Combinatorics (2008-05-01) 27: 263-273 , May 01, 2008

Given a partition *λ* and a composition *β*, the *stretched Kostka coefficient*
$\mathcal {K}_{\lambda \beta}(n)$
is the map *n**↦**K*_{nλ,nβ} sending each positive integer *n* to the Kostka coefficient indexed by *n**λ* and *n**β*. Kirillov and Reshetikhin (J. Soviet Math. 41(2), 925–955, 1988) have shown that stretched Kostka coefficients are polynomial functions of *n*. King, Tollu, and Toumazet have conjectured that these polynomials always have nonnegative coefficients (CRM Proc. Lecture Notes 34, 99–112, 2004), and they have given a conjectural expression for their degrees (Séminaire Lotharingien de Combinatoire 54A, 2006).

We prove the values conjectured by King, Tollu, and Toumazet for the degrees of stretched Kostka coefficients. Our proof depends upon the polyhedral geometry of Gelfand–Tsetlin polytopes and uses tilings of GT-patterns, a combinatorial structure introduced in De Loera and McAllister, (Discret. Comput. Geom. 32(4), 459–470, 2004).

## Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization

### Journal of Optimization Theory and Applications (2008-11-01) 139: 243-261 , November 01, 2008

Higher-order variational sets are proposed for set-valued mappings, which are shown to be more convenient than generalized derivatives in approximating mappings at a considered point. Both higher-order necessary and sufficient conditions for local Henig-proper efficiency, local strong Henig-proper efficiency and local *λ*-proper efficiency in set-valued nonsmooth vector optimization are established using these sets. The technique is simple and the results help to unify first and higher-order conditions. As consequences, recent existing results are derived. Examples are provided to show some advantages of our notions and results.

## MINISYMPOSIUM 9: Subspace Correction Methods

### Domain Decomposition Methods in Science and Engineering XVII (2008-01-01) 60: 381 , January 01, 2008

Subspace correction methods are well established as a unifying framework for multigrid and domain decomposition. It also serves as a powerful tool both for the construction and the analysis of efficient iterative solvers for discretized partial differential equations. This minisymposium provides an overview on recent results in this field with special emphasis on linear and nonlinear systems.

## Separation of Variables for Systems of First-Order Partial Differential Equations and the Dirac Equation in Two-Dimensional Manifolds

### Symmetries and Overdetermined Systems of Partial Differential Equations (2008-01-01) 144: 471-496 , January 01, 2008

The problem of solving the Dirac equation on two-dimensional manifolds is approached from the point of separation of variables, with the aim of creating a foundation for analysis in higher dimensions. Beginning from a sound definition of multiplicative separation for systems of two first order linear partial differential equations of Dirac type and the characterization of those systems admitting multiplicatively separated solutions in some arbitrarily given coordinate system, more structure is step by step added to the problem by requiring the separation constants are associated with commuting differential operators. Finally, the requirement that the original system coincides with the Dirac equation on a two-dimensional manifold allows the characterization of the orthonormal frames and metrics admitting separation of variables for the equation and of the symmetries associated with the separated coordinates.

## Some natural equivalence relations in the Solovay model

### Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (2008-06-01) 78: 91-98 , June 01, 2008

We obtain some non-reducibility results concerning some natural equivalence relations on reals in the Solovay model. The proofs use the existence of reals *x* which are minimal with respect to the cardinals in *L*[*x*], in a certain sense.

## Double degeneracy in the problem on unbounded branches of forced oscillations

### Doklady Mathematics (2008-05-04) 77: 170-174 , May 04, 2008

## On maximal actions and w-maximal actions of finite hypergroups

### Journal of Algebraic Combinatorics (2008-04-01) 27: 127-141 , April 01, 2008

Sunder and Wildberger (*J. Algebr. Comb.**18*, 135–151, 2003) introduced the notion of actions of finite hypergroups, and studied maximal irreducible actions and *-actions. One of the main results of Sunder and Wildberger states that if a finite hypergroup *K* admits an irreducible action which is both a maximal action and a *-action, then *K* arises from an association scheme. In this paper we will first show that an irreducible maximal action must be a *-action, and hence improve Sunder and Wildberger’s result (Theorem 2.9). Another important type of actions is the so-called *w*-maximal actions. For a *w*-maximal action *π*:*K*→Aff (*X*), we will prove that *π* is faithful and |*X*|≥|*K*|, and |*K*| is the best possible lower bound of |*X*|. We will also discuss the strong connectivity of the digraphs induced by a *w*-maximal action.

## Forcing with quotients

### Archive for Mathematical Logic (2008-08-27) 47: 719-739 , August 27, 2008

We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a *σ*-ideal and quotient forcings of subsets of countable sets modulo an ideal.