Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

We prove the existence of a global heat flow u : Ω ×  \mathbbR⁺ ? \mathbbR^N {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR⁺ {\mathbb{R}^{+}}) ⊂  \mathbbRⁿ {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbR^N {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbR^N {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.     

Analogs of the Ikoma–Schwartz lemma and Liouville theorem for mappings with unbounded characteristic

We obtain results on the local behavior of open discrete mappings f:D ? \mathbbRⁿ f:D \to {\mathbb{R}^n} , n ≥ 2, that satisfy certain conditions related to the distortion of capacities of condensers. It is shown that, in an infinitesimal neighborhood of zero, the indicated mapping cannot grow faster than an integral of a special type that corresponds to the distortion of the capacity under this mapping, which is an analog of the well-known Ikoma growth estimate proved for quasiconformal mappings of the unit ball into itself and of the classic Schwartz lemma for analytic functions. For mappings of the indicated type, we also obtain an analog of the well-known Liouville theorem for analytic functions.     

On some properties of generalized quasiisometries with unbounded characteristic

We consider a family of open discrete mappings f:D ?[`(\mathbb R<sup>n</sup>)] f:D \to \overline {{{\mathbb R}^n}} that distort, in a special way, the p-modulus of a family of curves that connect the plates of a spherical condenser in a domain D in \mathbb R<sup>n</sup> {{\mathbb R}^n} ; p > n-1; p < n; and bypass a set of positive p-capacity. We establish that this family is normal if a certain real-valued function that controls the considered distortion of the family of curves has finite mean oscillation at every point or only logarithmic singularities of order not higher than n - 1: We show that, under these conditions, an isolated singularity x <sub>0</sub> ∈ D of a mapping f:D\{ x<sub>0</sub> } ?[`(\mathbb Rⁿ)] f:D\backslash \left\{ {{x_0}} \right\} \to \overline {{{\mathbb R}^n}} is removable, and, moreover, the extended mapping is open and discrete. As applications, we obtain analogs of the known Liouville and Sokhotskii–Weierstrass theorems.     

Approximations of the exponential of an operator with periodic coefficients

We consider the operator exponential e ^−tA , t > 0, where A is a selfadjoint positive definite operator corresponding to the diffusion equation in \mathbbRⁿ {\mathbb{R}^n} with measurable 1-periodic coefficients, and approximate it in the operator norm ||   ·   ||_{L²( \mathbbRⁿ ) ? L²( \mathbbRⁿ )} {\left\| {\; \cdot \;} \right\|_{{{L^2}\left( {{\mathbb{R}^n}} \right) \to {L^2}\left( {{\mathbb{R}^n}} \right)}}} with order O( t ^{- \fracm2} ) O\left( {{t^{{ - \frac{m}{2}}}}} \right) as t → ∞, where m is an arbitrary natural number. To construct approximations we use the homogenized parabolic equation with constant coefficients, the order of which depends on m and is greater than 2 if m > 2. We also use a collection of 1-periodic functions N _α (x), x ? \mathbbRⁿ x \in {\mathbb{R}^n} , with multi-indices α of length | a| \leqslant m \left| \alpha \right| \leqslant m , that are solutions to certain elliptic problems on the periodicity cell. These results are used to homogenize the diffusion equation with ε-periodic coefficients, where ε is a small parameter. In particular, under minimal regularity conditions, we construct approximations of order O(ε ^m ) in the L ²-norm as ε → 0. Bibliography: 14 titles.     

Pluripotential Energy

We discuss a notion of the energy of a compactly supported measure in \mathbbCⁿ \mathbb{C}^n for n > 1 which we show is equivalent to that defined by Berman, Boucksom, Guedj and Zeriahi. This generalizes the classical notion of logarithmic energy of a measure in the complex plane \mathbbC \mathbb{C} ; i.e., the case n = 1.     

An asymmetric affine Pólya–Szeg? principle

An affine rearrangement inequality is established which strengthens and implies the recently obtained affine Pólya–Szeg? symmetrization principle for functions on \mathbb Rⁿ{\mathbb R^n} . Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is established which is stronger than its classical Euclidean counterpart.     

Tartar’s conjecture and localization of the quasiconvex hull in $ \mathbb{R}^{{2 \times 2}} $

We give a concrete and surprisingly simple characterization of compact sets K ì \mathbbR^{2 ×2} K \subset \mathbb{R}^{{2 \times 2}} for which families of approximate solutions to the inclusion problem Du∈K are compact. In particular our condition is algebraic and can be tested algorithmically. We also prove that the quasiconvex hull of compact sets of 2 × 2 matrices can be localized. This is false for compact sets in higher dimensions in general.     

Hardy–Littlewood Inequalities for Fractional Derivatives of Invariant Harmonic Functions

We establish Hardy–Littlewood inequalities for fractional derivatives of M?bius invariant harmonic functions over the unit ball of \mathbb Rⁿ{\mathbb R^n} in mixed-norm spaces. In doing so we introduce a new criteria for the boundedness of operators in mixed-norm L ^p -spaces in terms of hyperbolic geometry of the real unit ball.     

On the sets of branch points of mappings more general than quasiregular

It is shown that if a point x ₀ ∊ ℝ ⁿ , n ≥ 3, is an essential isolated singularity of an open discrete Q-mapping f : D → [`(\mathbb R<sup>n</sup>)] \overline {\mathbb {R}^n} , B <sub>f</sub> is the set of branch points of f in D; and a point z <sub>0</sub> ∊ [`(\mathbb Rⁿ)] \overline {\mathbb {R}^n} is an asymptotic limit of f at the point x ₀; then, for any neighborhood U containing the point x ₀; the point z ₀ ∊ [`(f( B<sub>f</sub> ?U ))] \overline {f\left( {B_f \cap U} \right)} provided that the function Q has either a finite mean oscillation at the point x <sub>0</sub> or a logarithmic singularity whose order does not exceed n − 1: Moreover, for n ≥ 2; under the indicated conditions imposed on the function Q; every point of the set [`(\mathbb Rⁿ)] \overline {\mathbb {R}^n} \ f(D) is an asymptotic limit of f at the point x ₀. For n ≥ 3, the following relation is true: [`(\mathbbR<sup>n</sup> )] \f( D ) ì [`(f B_f )] \overline {\mathbb{R}^n } \backslash f\left( D \right) \subset \overline {f\,B_f } . In addition, if ￥ ? f( D ) \infty \notin f\left( D \right) , then the set f B _f is infinite and x₀ ? [`(B_f )] x_0 \in \overline {B_f } .     

On fractal measures and diophantine approximation

We study diophantine properties of a typical point with respect to measures on \mathbbRⁿ .\mathbb{R}^n . Namely, we identify geometric conditions on a measure μ on \mathbbRⁿ \mathbb{R}^n guaranteeing that μ-almost every y  ?  \mathbbRⁿ {\bf y}\,\in\,\mathbb{R}^n is not very well multiplicatively approximable by rationals. Measures satisfying our conditions are called ‘friendly’. Examples include smooth measures on nondegenerate manifolds; thus this paper generalizes the main result of [KM]. Another class of examples is given by measures supported on self-similar sets satisfying the open set condition, as well as their products and pushforwards by certain smooth maps.     

Criteria for monogenicity of Clifford algebra-valued functions on fractal domains

Suppose that Ω is a bounded domain with fractal boundary Γ in ${\mathbb R^{n+1}}Suppose that Ω is a bounded domain with fractal boundary Γ in \mathbb Rⁿ⁺¹{\mathbb R^{n+1}} and let \mathbb R_0,n{\mathbb R_{0,n}} be the real Clifford algebra constructed over the quadratic space \mathbb Rⁿ{\mathbb R^{n}}. Furthermore, let U be a \mathbb R_0,n{\mathbb R_{0,n}}-valued function harmonic in Ω and H?lder-continuous up to Γ. By using a new Clifford Cauchy transform for Jordan domains in \mathbb Rⁿ⁺¹{\mathbb R^{n+1}} with fractal boundaries, we give necessary and sufficient conditions for the monogenicity of U in terms of its boundary value u = U|_Γ. As a consequence, the results of Abreu Blaya et al. (Proceedings of the 6th International ISAAC Congress Ankara, 167–174, World Scientific) are extended, which require Γ to be Ahlfors-David regular.     

Numerical integration with polynomial exactness over a spherical cap

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \mathbbS²\mathbb{S}^2, we discuss tensor product rules with n ²/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n ³) nodes for numerical integration over spherical caps on \mathbbS²\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that have O(n ^d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n ^d ) nodes and possess a certain regularity property.     

Risk measures on the space of infinite sequences

Axiomatically based risk measures have been the object of numerous studies and generalizations in recent years. In the literature we find two main schools: coherent risk measures (Artzner, Coherent Measures of Risk. Risk Management: Value at Risk and Beyond, 1998) and insurance risk measures (Wang, Insur Math Econ 21:173–183, 1997). In this note, we set to study yet another extension motivated by a third axiomatically based risk measure that has been recently introduced. In Heyde et al. (Working Paper, Columbia University, 2007), the concept of natural risk statistic is discussed as a data-based risk measure, i.e. as an axiomatic risk measure defined in the space \mathbb Rⁿ{\mathbb R^n} . One drawback of these kind of risk measures is their dependence on the space dimension n. In order to circumvent this issue, we propose a way to define a family {ρ _n } _n=1,2,... of natural risk statistics whose members are defined on \mathbbRⁿ{\mathbb{R}^n} and related in an appropriate way. This construction requires the generalization of natural risk statistics to the space of infinite sequences l ^∞.     

Translation-Invariant Bilinear Operators with Positive Kernels

We study L ^r (or L ^{r, ∞}) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on \mathbb Rⁿ{\mathbb {R}^n}. We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on \mathbb R²ⁿ{\mathbb R^{2n}}, while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some sufficient conditions on nonnegative kernels yielding boundedness for the corresponding operators on products of Lebesgue spaces. We also prove that, unlike the linear case where boundedness from L ¹ to L ¹ and from L ¹ to L ^{1, ∞} are equivalent properties, boundedness from L ¹ × L ¹ to L ^1/2 and from L ¹ × L ¹ to L ^{1/2, ∞} may not be equivalent properties for bilinear translation-invariant operators with nonnegative kernels.     

Simultaneous inhomogeneous diophantine approximation on manifolds

In 1998, Kleinbock and Margulis proved Sprindzuk’s conjecture pertaining to metrical Diophantine approximation (and indeed the stronger Baker–Sprindzuk conjecture). In essence, the conjecture stated that the simultaneous homogeneous Diophantine exponent w ₀(x) = 1/n for almost every point x on a nondegenerate submanifold M \mathcal{M} of \mathbbRⁿ {\mathbb{R}^n} . In this paper, the simultaneous inhomogeneous analogue of Sprindzuk’s conjecture is established. More precisely, for any “inhomogeneous” vector θ ∈ \mathbbRⁿ {\mathbb{R}^n} we prove that the simultaneous inhomogeneous Diophantine exponent w ₀(x , θ) is 1/n for almost every point x on M \mathcal{M} . The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent w ₀(x) is 1/n for almost all x ∈ M \mathcal{M} if and only if, for any θ ∈ \mathbbRⁿ {\mathbb{R}^n} , the inhomogeneous exponent w ₀(x , θ) = 1/n for almost all x ∈ M \mathcal{M} . The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered by us. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory.     

On random surface area

Consider a random smooth Gaussian field G(x): F ? \mathbbR F \to \mathbb{R} , where F is a compact set in \mathbbR^d {\mathbb{R}^d} . We derive a formula for the average area of a surface determined by the equation G(x) = 0 and give some applications. As an auxiliary result, we obtain an integral expression for the area of a surface determined by zeros of a nonrandom smooth field. Bibliography: 13 titles.     

On the Representation of Compact Nilpotent Rings by Triangular Matrices

We prove that every compact nilpotent ring R of characteristic p > 0 can be embedded in a ring of upper triangular matrices over a compact commutative ring. Furthermore, we prove that every compact topologically nilpotent ring R of characteristic p > 0, is embedded in a ring of infinite triangular matrices over \mathbbF_p^w(R)\mathbb{F}_{p}^{w(R)}.     

Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces

We define nonnegative quasi-nearly subharmonic functions on so called locally uniformly homogeneous spaces. We point out that this function class is rather general. It includes quasi-nearly subharmonic (thus also subharmonic, quasisubharmonic and nearly subharmonic) functions on domains of Euclidean spaces \mathbbRⁿ{{\mathbb{R}}^n}, n ≥ 2. In addition, quasi-nearly subharmonic functions with respect to various measures on domains of \mathbbRⁿ{{\mathbb{R}}^n}, n ≥ 2, are included. As examples we list the cases of the hyperbolic measure on the unit ball B ⁿ of \mathbbRⁿ{{\mathbb{R}}^n}, the M{{\mathcal{M}}}-invariant measure on the unit ball B ²ⁿ of \mathbbCⁿ{{\mathbb{C}}^n}, n ≥ 1, and the quasihyperbolic measure on any domain D ì \mathbbRⁿ{D\subset {\mathbb{R}}^n}, D 1 \mathbbRⁿ{D\ne {\mathbb{R}}^n}. Moreover, we show that if u is a quasi-nearly subharmonic function on a locally uniformly homogeneous space and the space satisfies a mild additional condition, then also u ^p is quasi-nearly subharmonic for all p > 0.     

Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Polynomial n × n matrices A(x) and B(x) over a field \mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over \mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over \mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A _0x - A ₁, where A ₀ and A ₁ are n × n matrices over \mathbbF \mathbb{F} , and A ₀ is nonsingular.     

Covering and packing in {\mathbb Z^n} and {\mathbb R^n}. (II)

In the present part (II) we will deal with the group \mathbb G = \mathbb Zⁿ{\mathbb G = \mathbb Z^n} , and we will study the effect of linear transformations on minimal covering and maximal packing densities of finite sets A ì \mathbb Zⁿ{\mathcal A \subset {\mathbb Z}^n} . As a consequence, we will be able to show that the set of all densities for sets A{\mathcal A} of given cardinality is closed, and to characterize four-element sets A ì \mathbb Zⁿ{\mathcal A \subset {\mathbb Z}^n} which are “tiles”. The present work will be largely independent of the first part (I) presented in [4].