The Pompeiu problem and discrete groups

We formulate a version of the Pompeiu problem in the discrete group setting. Necessary and sufficient conditions are given for a finite collection of finite subsets of a discrete abelian group, whose torsion free rank is less than the cardinal of the continuum, to have the Pompeiu property. We also prove a similar result for nonabelian free groups. A sufficient condition is given that guarantees the harmonicity of a function on a nonabelian free group if it satisfies the mean-value property over two spheres.     

Convolution equations and the local Pompeiu property on symmetric spaces and on phase space associated to the Heisenberg group

In this paper, we present new uniqueness results related to geometric aspects of mean periodicity on various homogeneous spaces. Among these results, we point out the equivalence of the local and the global Pompeiu property for arbitrary families of compactly supported distributions, the solution of the local Pompeiu problem for the class of non real-analytic functions, and local versions of the two-radii theorem on symmetric spaces of arbitrary rank.     

Rellich type identities for eigenvalue problems and application to the Pompeiu problem

For classical Neumann eigenvalue, buckling eigenvalue and clamped plate eigenvalue, we give the corresponding Rellich type identities. As an application of these results, then, we obtain a new necessary and sufficient condition for a domain without the Pompeiu property.     

Stochastic Optimization over a Pareto Set Associated with a Stochastic Multi-Objective Optimization Problem

We deal with the problem of minimizing the expectation of a real valued random function over the weakly Pareto or Pareto set associated with a Stochastic Multi-objective Optimization Problem, whose objectives are expectations of random functions. Assuming that the closed form of these expectations is difficult to obtain, we apply the Sample Average Approximation method in order to approach this problem. We prove that the Hausdorff–Pompeiu distance between the weakly Pareto sets associated with the Sample Average Approximation problem and the true weakly Pareto set converges to zero almost surely as the sample size goes to infinity, assuming that our Stochastic Multi-objective Optimization Problem is strictly convex. Then we show that every cluster point of any sequence of optimal solutions of the Sample Average Approximation problems is almost surely a true optimal solution. To handle also the non-convex case, we assume that the real objective to be minimized over the Pareto set depends on the expectations of the objectives of the Stochastic Optimization Problem, i.e. we optimize over the image space of the Stochastic Optimization Problem. Then, without any convexity hypothesis, we obtain the same type of results for the Pareto sets in the image spaces. Thus we show that the sequence of optimal values of the Sample Average Approximation problems converges almost surely to the true optimal value as the sample size goes to infinity.     

A partial solution of the Pompeiu problem

A nonempty bounded open subset D of ⁿ is said to have the Pompeiu property if and only if for every continuous complex-valued function f on ⁿ which does not vanish identically there is a rigid motion of ⁿ onto itself — taking D onto (D) — such that the integral of f over (D) is not zero. This article gives a partial solution of the Pompeiu problem, the problem of finding all sets D with the Pompeiu property.In the special case that D is the interior of a homeomorphic image of an(n–1)-dimensional sphere, the main result states that if D has a portion of an(n–1)-dimensional real analytic surface on its boundary, then either D has the Pompeiu property or any connected real analytic extension of the surface also lies on the boundary of D. Thus, for example, any such region D having a portion of a hyperplane as part of its boundary must have the Pompeiu property, since the entire hyperplane cannot lie in the boundary of the bounded set D.The research for this paper was done in part while on sabbatical at the Courant Institute of Mathematical Sciences, New York University.     

On the Kneser property for reaction-diffusion systems on unbounded domains

We prove the Kneser property (i.e. the connectedness and compactness of the attainability set at any time) for reaction-diffusion systems on unbounded domains in which we do not know whether the property of uniqueness of the Cauchy problem holds or not.Using this property we obtain that the global attractor of such systems is connected.Finally, these results are applied to the complex Ginzburg-Landau equation.     

Inversion of the local Pompeiu transformation on Riemannian symmetric spaces of rank one

We study the problem of inversion of the local Pompeiu transform on Riemannian symmetric spaces of rank one. The explicit formula for the reconstruction of a function by its averages over balls and spheres with a single fixed radius is obtained.     

Extremal Versions of the Pompeiu Problem

We investigate the local Pompeiu problem of functions with zero integrals over balls and cubes and related problems.     

On one extremal problem of Pompeiu sets

We determine upper bounds for the least radius of a ball in which a given set is a Pompeiu set (the set considered is a half right circular cone). The obtained estimates significantly improve known results. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 61–72, January, 2009.     

Pompeiu sets in compact Heisenberg manifolds

A necessary and sufficient condition is presented for a set to be a Pompeiu subset of any compact homogeneous space with a finite invariant measure. The condition, which is expressed in terms of the intertwining operators of each primary summand of the quasi-regular representation, is then interpreted in the case of the compact Heisenberg manifolds. Examples are presented demonstrating that the condition to be Pompeiu in these manifolds is quite different from the corresponding condition for a torus of the same dimension. This provides a contrast with the existing comparison between the Heisenberg group itself and Euclidean space in terms of Pompeiu sets. In addition, the closed linear span of all translates of any square integrable function on any compact homogeneous space is determined.     

Pompeiu Problem on Product of Heisenberg Groups

In this paper, we address the Pompeiu problem for a product of Heisenberg groups. We consider this problem both for cases of a ball and for a bidisk. Furthermore, we address this problem for a product of the Heisenberg group with Euclidean space.     

Maximal linear transformation groups preserving asymptotic properties of linear differential systems: I

We consider the problem of describing sets of linear piecewise differentiable transformations that preserve some asymptotic property of linear differential systems. We present definitions needed for solving this problem, obtain preliminary results, and describe the set of linear transformations preserving the property of boundedness of the coefficients of linear differential systems on the time half-line.     

Branch points, Fourier integrals and Pompeiu problem

Leth be the square root of a polynomial and assume thath is univalent on the unitary disk of the complex plane. Then the set Ω=h(D) has the Pompeiu property.

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Sunto Si prova che sep è un polinomio e la mappa è univalente sul disco unitarioD del piano complesso, allora Ω=h(D) ha la proprietà di Pompeiu.

     

Truncated nonconvex state-dependent sweeping process: implicit and semi-implicit adapted Moreau’s catching-up algorithms

In this paper, we deal with the existence of solutions for perturbed state-dependent Moreau’s sweeping processes. Two ways are investigated to realize such a study, depending on the nature of the used scheme, namely implicit or semi-implicit. In both cases, our evolution problem is described in a general Hilbert space by a prox-regular moving set controlled through the truncated Hausdorff–Pompeiu distance. The normal cone involved is perturbed by a sum of a single-valued mapping and a multimapping.     

Pompeiu transforms on geodesic spheres in real analytic manifolds

We prove a support theorem for Pompeiu transforms integrating on geodesic spheres of fixed radiusr>0 on real analytic manifolds when the measures are real analytic and nowhere zero. To avoid pathologies, we assume thatr is less than the injectivity radius at the center of each sphere being integrated over. The proof of the main result is local and it involves the microlocal properties of the Pompeiu transform and a theorem of Hörmander, Kawai, and Kashiwara on microlocal singularities.     

Extremal cases of the Pompeiu problem

The Pompeiu problem is studied for functions defined on a ballB ⁿ and having zero integrals over all sets congruent to a given compact setK B. The problem of finding the least radiusr=r(K) ofB for whichK is a Pompeiu set is considered. The solution is obtained for the cases in whichK is a cube or a hemisphere.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 671–680, May, 1996.     

A Generic relation on Recursively Enumerable Sets

We introduce the concept of a generic relation for algorithmic problems, which preserves the property of being decidable for a problem for almost all inputs and possesses the transitive property. As distinct from the classical m-reducibility relation, the generic relation under consideration does not possess the reflexive property: we construct an example of a recursively enumerable set that is generically incomparable with itself. We also give an example of a set that is complete with respect to the generic relation in the class of recursively enumerable sets.     

Pointwise well-posedness in set optimization with cone proper sets

This paper deals with the well-posedness property in the setting of set optimization problems. By using a notion of well-posed set optimization problem due to Zhang et al. (2009) [18] and a scalarization process, we characterize this property through the well-posedness, in the Tykhonov sense, of a family of scalar optimization problems and we show that certain quasiconvex set optimization problems are well-posed. Our approach is based just on a weak boundedness assumption, called cone properness, that is unavoidable to obtain a meaningful set optimization problem.     

One-phase inverse Stefan problems with unknown nonlinear sources

We consider quasilinear models of inverse problems with phase transitions in a domain whose external boundary is a phase front with an unknown dependence on time. Additional information for finding the sources is given in the form of final overdetermination for the solution of the direct Stefan problem. On the basis of the duality principle, we obtain sufficient conditions for the uniqueness of the solution in Hölder classes for the considered inverse Stefan problems. We present examples in which the uniqueness property is lost when extending the set of admissible solutions.