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An important problem in applications of quasiconformal analysis and in its numerical aspect is to establish algorithms for explicit or approximate determination of the basic quasiinvariant curvelinear and analytic functionals intrinsically…
Let A and B be normal matrices with coefficients that are continuous complex-valued functions on a topological space X that has the homotopy type of a CW complex, and suppose these matrices have the same distinct eigenvalues at each point…
The density of polynomials in a weighted space of infinitely differentiable functions in a multidimensional real space is proved under minimal conditions on weight functions and on differences between weight functions. We apply this result…
The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudoreflection groups over Dedekind domains. In the special case of a principal ideal domain in which the group…
The covariance matrix function is characterized in this paper for a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the compact two-point homogeneous space. Necessary and…
We prove a differential analog of a theorem of Chevalley on extending homomorphisms for rings with commuting derivations, generalizing a theorem of Kac. As a corollary, we establish that, under suitable hypotheses, the image of a…
Interpolation inequalities for $C^m$ functions allow to bound derivatives of intermediate order $0 < j<m$ by bounds for the derivatives of order $0$ and $m$. We review various interpolation inequalities for $L^p$-norms ($1 \le p \le…
We show that a differential version of the classical Chebyshev-Markov-Stieltjes inequalities holds for a broad family of weight functions. Such a differential version appears to be new. Our results apply to weight functions which are…
In this paper, we investigate the relationship between positive definite functions on the unit sphere $\sph$ and on the Euclidean space $\RR^d$. For the dimension $d$ to be odd, a new technique is developed to establish the inheritance of…
We study transnormal and isoparametric functions on closed Riemannian 4-manifolds and establish fundamental restrictions on their topology and geometry. In particular, we show that such manifolds cannot be endowed with negatively curved…
We obtain multiplicity results for a class of first-order superquadratic Hamiltonian systems and a class of indefinite superquadratic elliptic systems which lead to the study of strongly indefinite functionals. There is no assumption to the…
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…
We introduce a nonstandard extension of the category of diffeological spaces, and demonstrate its application to the study of generalized functions. Just as diffeological spaces are defined as concrete sheaves on the site of Euclidean open…
Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly…
For a finite group acting on a polynomial ring, the Chevalley-Shephard-Todd Theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo-reflections. In recent years, progress was…
We construct harmonic functions in the quarter plane for discrete Laplace operators. In particular, the functions are conditioned to vanish on the boundary and the Laplacians admit coefficients associated with transition probabilities of…
Continuous, dually epi-translation invariant valuations on the space of finite-valued convex functions on $\mathbb{C}^n$ that are invariant under the unitary group are investigated. It is shown that elements belonging to the dense subspace…
We study the behaviour of differential forms in a manifold having at least one of their maximal isotropic local distributions endowed with the special algebraic property of being decomposable. We show that they can be represented as the sum…
The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…
We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an infinitely transitive model. The latter is an algebraic variety with…