On the representation of localized functions in ℝ2 by the Maslov canonical operator

We prove that localized functions can be represented in the form of an integral over a parameter, the integrand being the Maslov canonical operator applied to an amplitude obtained from the Fourier transform of the function to be represented. This representation generalizes an earlier one obtained by Dobrokhotov, Tirozzi, and Shafarevich and permits representing localized initial data for wave type equations with the use of an invariant Lagrangian manifold, which simplifies the asymptotic solution formulas dramatically in many cases.     

Multipliers with singularities along a curve in ℝ n

We consider in ⁿ ,n2, the curve = (t,t ² ,...,t ⁿ ), ₀t₀,₀>0 a small number. We study the boundedness of operatorsT ,>0, defined by multipliers which present singularities along . Our results are derived from a sharp estimate on a suitable maximal function. In the casen=2 theT 's are Bochner-Riesz operators and our results coincide with the known ones.     

On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

Set of ambiguous points for functions in ℝ n

It is known that an arbitrary function in the open unit disk can have at most countable set of ambiguous points. Point ζ on the unit circle is an ambiguous point of a function if there exist two Jordan arcs, lying in the unit ball, except the endpoint ζ, such that cluster sets of function along these arcs are disjoint. We investigate whether it is possible to modify the notion of ambiguous point to keep the analogous result true for functions defined in the n-dimensional Euclidean unit ball.     

On Young hulls of convex curves in ℝ2n

For a convex curve in an even-dimensional affine space we introduce a series of convex domains (called Young hulls) describe their structure and give a formulas for the volume of the biggest of these domains.To our Teacher Vladimir Igorevich Arnold on the occasion of his 60-th birthday     

On the Agmon-Miranda maximum principle for solutions of strongly elliptic equations in domains of ℝn with conical points

In this paper the Agmon-Miranda maximum principle for solutions of strongly elliptic differential equations Lu = 0 in a bounded domain G with a conical point is considered. Necessary and sufficient conditions for the validity of this principle are given both for smooth solutions of the equation Lu = 0 in G and for the generalized solution of the problem Lu = 0 in G, D _k ^v u = gk on G (k = 0,...,m-1). It will be shown that for every elliptic operator L of order 2m > 2 there exists such a cone in ⁿ (n4) that the Agmon-Miranda maximum principle fails in this cone.     

A quadratic finite element method for solving biharmonic problems in ℝ n

Summary A family of simplicial finite element methods having the simplest possible structure, is introduced to solve biharmonic problems in ⁿ ,n3, using the primal variable. The family is inspired in the MORLEY triangle for the two dimensional case, and in some sense this element can be viewed as its member corresponding to the valuen=2.     

On the Morse index of minimal surfaces in ℝp with polygonal boundaries

The minimal surfaces spanning a polygon in ^p (p2) correspond to the critical points of an analytic function in finitely many variables, namely Shiffman's function. We shall prove that the Morse index of the minimal surface coincides with the Morse index of at the corresponding critical point. Alternatively expressed, the Schwarz operator of the minimal surface and the Hessian of have the same number of negative eigenvalues. Finally we control the degeneration of the critical points.     

On the representation of nonlinear systems with gaussian inputst †

An arbitrary nonlinear system with input a Gaussian process, which is such that its output process has finite second moments, admits two kinds of representations: the first in terms of a sequence of deterministic kernels and the second in terms of a single stochastic kernel. We consider here the identification of the sequence of deterministic kernels from the input and output processes, the representation of the system output when its input is a sample function of the Gaussian process or another equivalent Gaussian process, and the relationship of the sequence of kernels mentioned above to the Volterra expansion kernels when the system has a Volterra representation.     

On the Lévy–Khintchine representation of q-Bessel negative definite functions

We study q-Bessel negative definite functions and use their properties to give a proof of Lévy–Khintchine representation of q-infinitely divisible probability measures.     

On the structure of incoming and outgoing subspaces for a wave equations in ℝ n

We investigate the structure of incoming and outgoing subspaces in the Lax-Phillips scheme for the classic wave equation in ℝ ⁿ . Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 708–712, May, 1999.     

On the representation theory of Möbius groups inR n*

We will solve several fundamental problems of Möbius groupsM(R ⁿ) which have been matters of interest such as the conjugate classification, the establishment of a standard form without finding the fixed points and a simple discrimination method. Let \(g = \left[ {\begin{array}{*{20}c} a &; b \\ c &; d \\ \end{array} } \right]\) be a Clifford matrix of dimensionn, c ≠ 0. We give a complete conjugate classification and prove the following necessary and sufficient conditions:g is f.p.f. (fixed points free) iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; 0 \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α|<1 and |E?AE ¹| ≠ 0;g is elliptic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; \beta \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α| <1 and |E?AE ¹|=0;g is parabolic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; 0 \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α|=1; andg is loxodromic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; \beta \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α| >1 or rank (E?AE ¹) ≠ rank (E?AE ¹,ac ^?1+c ^?1 d), where α is represented by the solutions of certain linear algebraic equations and satisfies $\left| {c^{ - 1} \alpha '} \right| = \left| {\left( {E - AE^1 } \right)^{ - 1} \left( {\alpha c^{ - 1} + c^{ - 1} \alpha '} \right)} \right|.$      

On the extension of Hölder maps with values in spaces of continuous functions

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc ₀ or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc ₀ can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.     

Hamiltonization and Integrability of the Chaplygin Sphere in ℝ n

This paper studies a natural n-dimensional generalization of the classical nonholonomic Chaplygin sphere problem. We prove that for a specific choice of the inertia operator, the restriction of the generalized problem onto a zero value of the SO(n−1)-momentum mapping becomes an integrable Hamiltonian system after an appropriate time reparametrization.     

Darboux property of Gâteaux derivatives of functions on ℝ n

D. Preiss proved that the graph of the derivative of a continuous Gateaux-differentiable function f : ℝ² → ℝ is always connected. We show that this is no longer true in higher dimensions: we construct a continuous, Gateaux-differentiable function f : ℝ³ → ℝ for which the range of its gradient mapping {∇ f(x) : x ∈ ℝ³} is disconnected. We also give an example of an approximately differentiable continuous function on ℝ² such that the range of its gradient mapping is disconnected. The work is a part of the research project MSM 0021620839 financed by MSMT and it was also partly supported by GAČR 201/06/0198 and GAČR 201/06/0018.