On the fastest moving off from a vertex in directed regular graphs

Let Γ be a directed regular locally finite graph, and let $\bar \Gamma $ be the undirected graph obtained by forgetting the orientation of Γ. Let x be a vertex of Γ and let n be a nonnegative integer. We study the length of the shortest directed path in Γ starting at x and ending outside of the ball of radius n centered at x in $\bar \Gamma $ .     

Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities

We show that if ${{\mathcal A} \subset \mathbb{R}^N}$ is an annulus or a ball centered at zero, the homogeneous Neumann problem on ${{\mathcal A}}$ for the equation with continuous data $$\nabla \cdot \left(\frac{\nabla v}{\sqrt{1 - |\nabla v|^2}} \right) = g(|x|,v) + h(|x|)$$ has at least one radial solution when g(|x|,·) has a periodic indefinite integral and ${\int_{\mathcal A} h(|x|)\,{\rm{d}}x = 0.}$ The proof is based upon the direct method of the calculus of variations, variational inequalities and degree theory.     

A Weighted Isoperimetric Inequality in an Orthant

Let c, k ₁ , ..., k _N be non-negative numbers, and define a measure μ in the orthant W :?=?{x?∈??by \(d\mu = e^{c|x|^2 } x_1 ^{k_1 }\cdots x_N ^{k_N } \, dx \) . It is shown that among all measurable subsets of W with fixed μ-measure, the intersection of W with a ball centered at the origin renders the weighted perimeter relative to W a minimum. Norm inequalities in weighted Sobolev spaces and a comparison result for solutions to boundary value problems for degenerate elliptic equations are also given.     

The rank and generating set for inverse limits of wreath products of Abelian groups

We derive an exact formula for the topological rank d(W) of the inverse limit ${W = \ldots \wr A_2 \wr A_1}$ of iterated wreath products of arbitrary nontrivial finite Abelian groups. By using the language of automorphisms of a spherically homogeneous rooted tree, we construct and study a topological generating set for W with cardinality ${d(A_1) + \rho'}$ , where ${\rho'}$ is the topological rank of the profinite Abelian group ${A_2 \times A_3 \times \cdots}$ . In particular, if the group A ₁ is cyclic, this approach gives a minimal generating set for W.     

Lyashko–Looijenga morphisms and submaximal factorizations of a Coxeter element

When W is a finite reflection group, the noncrossing partition lattice $\operatorname{NC}(W)$ of type W is a rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $\operatorname{NC}(W)$ as a generalized Fu?–Catalan number, depending on the invariant degrees of W. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $\operatorname{NC}(W)$ as fibers of a Lyashko–Looijenga covering ( $\operatorname{LL}$ ), constructed from the geometry of the discriminant hypersurface of W. We study algebraically the map $\operatorname{LL}$ , describing the factorizations of its discriminant and its Jacobian. As byproducts, we generalize a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorizations of a Coxeter element of W.     

Stability of approximation under the action of singular integral operators

Let T be a singular integral operator, and let 0 < α < 1. If t > 0 and the functions f and Tf are both integrable, then there exists a function $g \in B_{Lip_\alpha } (ct)$ such that $\left\| {f - g} \right\|_{L^1 } \leqslant Cdist_{L^1 } (f,B_{Lip_\alpha } (t))$ and $\left\| {Tf - Tg} \right\|_{L^1 } \leqslant C\left\| {f - g} \right\|_{L^1 } + dist_{L^1 } (Tf,B_{Lip_\alpha } (t)).$ . (Here B _X (τ) is the ball of radius τ and centered at zero in the space X; the constants C and c do not depend on t and f.) The function g is independent of T and is constructed starting with f by a nearly algorithmic procedure resembling the classical Calderón-Zygmund decomposition.     

Relative average widths of Sobolev spaces in L 2(ℝ d )

In this paper, in order to consider the problems of relative width on ? ^d , we proposed definitions of relative average width which combine the ideas of the relative width and the average width. We established the smallest number M which make the following equality $$ \overline K _\sigma (U(W_2^\alpha ),M(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) $$ hold, where U(W ₂ ^α ) is the Riesz potential or Bessel potential of the unit ball in L ²(? ^k ) and the notations $ \overline K _\sigma $ (·, ·,L ₂(? ^d )) and $ \overline d _\sigma $ (·, L ₂(? ^d )) denote respectively the relative average width in the sense of Kolmogorov and the average width in the sense of Kolmogorov in their given order. In 2001, Subbotin and Telyakovskii got similar results on the relative width of Kolmogorov type. We also proved that $$ \overline K _\sigma (U(W_2^\alpha ) \cap B(L_2 (\mathbb{R}^d )),U(W_2^\beta ) \cap B(L_2 (\mathbb{R}^d ))L_2 (\mathbb{R}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 (\mathbb{R}^d )), $$ where 0 × β × α.     

On Rationality of W-algebras

We study the problem of classification of triples ( $ \mathfrak{g} $ ; f; k), where g is a simple Lie algebra, f its nilpotent element and k ∈ $ \mathbb{C} $ , for which the simple W-algebra W _k ( $ \mathfrak{g} $ ; f) is rational.     

On a problem concerning the Banach algebra generated by the maps n\mapsto\lambda^{\binom{n}{k}}

In Jabbari and Namioka (Milan J. Math. 78:503?C522, 2010), the authors characterized the spectrum M(W) of the Weyl algebra W, i.e. the norm closure of the algebra generated by the family of functions $\{n\mapsto x^{n^{k}}; x\in\mathbb{T}, k\in\mathbb{N}\}$ , ( $\mathbb{T}$ the unit circle), with a closed subgroup of $E(\mathbb{T})^{\mathbb{N}}$ where $E(\mathbb{T})$ denotes the family of the endomorphisms of the multiplicative group $\mathbb{T}$ . But the size of M(W) in $E(\mathbb{T})^{\mathbb{N}}$ as well as the induced group operation were left as a problem. In this paper, we will give a solution to this problem.     

Radial functions and regularity of solutions to the Schrödinger equation

Letf be a radial function and setT ^* f(x)=sup_0<t<1 |T _t f(x)|, x ∈ ?ⁿ, n≥2, where(T_t f)^ (ξ)=e ^it|ξ|a \(\hat f\) (ξ),a > 1. We show that, ifB is the ball centered at the origin, of radius 100, then \(\int\limits_B {|T^ * f(x)|} dx \leqslant c(\int {|\hat f(\xi )|^2 (l + |\xi |^s )ds} )^{1/2} \) if and only ifs≥1/4.     

Some stability estimates for the symmetrized first eigenfunction of certain elliptic operators

We prove two stability-type estimates involving the Schwarz rearrangement of the normalized first eigenfunction u ₁?>?0 of certain linear elliptic operators whose first eigenvalue λ₁ is close to the lowest possible one (i.e., ${\lambda_1^\star}$ , the first eigenvalue of the Dirichlet Laplacian in a suitable ball). In particular, we prove that if ${\lambda_1\approx \lambda_1^\star}$ then the L ^∞-distance between the rearrangement ${u_1^\star}$ and the normalized first eigenfunction of the Dirichlet Laplacian corresponding to ${\lambda_1^\star}$ is less than a suitable power of the difference ${\lambda_1-\lambda_1^\star}$ times a universal constant. We also show that the L ^∞-distance between the first eigenfunction of the Dirichlet Laplacian in a ball whose first eigenvalue equals λ₁ and the rearrangement ${u_1^\star}$ can be controlled with a power of the value assumed by ${u_1^\star}$ on the boundary of that ball.     

Spectral norm of products of random and deterministic matrices

We study the spectral norm of matrices W that can be factored as W?=?BA, where A is a random matrix with independent mean zero entries and B is a fixed matrix. Under the (4?+???)th moment assumption on the entries of A, we show that the spectral norm of such an m × n matrix W is bounded by ${\sqrt{m} + \sqrt{n}}$ , which is sharp. In other words, in regard to the spectral norm, products of random and deterministic matrices behave similarly to random matrices with independent entries. This result along with the previous work of Rudelson and the author implies that the smallest singular value of a random m × n matrix with i.i.d. mean zero entries and bounded (4?+???)th moment is bounded below by ${\sqrt{m} - \sqrt{n-1}}$ with high probability.     

A gradient-projective basis of compactly supported wavelets in dimension n > 1

A given set W = {W _X } of n-variable class C ¹ functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum\limits_\chi {(\int_{\mathbb{R}^n } {\nabla f \cdot } \nabla W_\chi ^* )} W_\chi $ converges to f with respect to the norm $\left\| {\nabla ( \cdot )} \right\|_{L^2 (\mathbb{R}^n )} $ . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = {W _x } of compactly supported class C ^2?? functions on ? ⁿ such that where X has the structure $\chi = (\tilde \chi ,\nu )$ , ν ∈ ?. W is a wavelet set in the sense that the functions indexed by $\tilde \chi $ are generated by an averaging of lattice translations with wave propagations, and there are two additional discrete parameters associated with the latter. These functions indexed by $\tilde \chi $ are the unit-scale wavelets. The support volumes of our unit-scale wavelets are not uniformly bounded, however. While the practical value of this construction is doubtful, our motivation is theoretical. The point is that a gradient-orthonormal basis of compactly supported wavelets has never been constructed in dimension n > 1. (In one dimension the construction of such a basis is easy — just anti-differentiate the Haar functions.)     

Cubature Formulae Associated with the Dunkl Laplacian

In this paper, we study the integration of functions of the form $$u=\sum_{j=0}^{m-1}u_{j}(x) \Phi_{j}(|x|^{2}),$$ where (u _j ) are in ${\mathcal{C}^{1}(\overline{B(r)})\cap\mathcal{C}^{2}(B(r))}$ and harmonic in the open ball B(r) centered at the origin and with radius r > 0, with respect to the Dunkl Laplacian Δ _k and ${\{\Phi_{0},\ldots,\Phi_{m-1}\}}$ is a given system of linearly independent integrable functions on [0, r ²]. In particular, we construct cubature formulae having highest order of precision with respect to the class of k-polyharmonic functions of degree m, i.e. ${\Delta_{k}^{m}u=0,m\in\mathbb{N}\setminus\{0\}}$ and we give an extension of the Pizzetti formula type for functions in ${\mathcal{C}^{2m-1}(\overline{B(r)}) \cap\mathcal{C}^{2m}(B(r))}$ and k-polyharmonic of order m.     

Traces of monotone Sobolev functions

In this paper we prove that ifu: ${\mathbb{B}}^n \to {\mathbb{R}}$ , where ${\mathbb{B}}^n $ is the unit ball in ? ⁿ , is a monotone function in the Sobolev space W^1·p ( ${\mathbb{B}}^n $ ), andn ? 1 <p ≤n, thenu has nontangential limits at all the points of $\partial {\mathbb{B}}^n $ except possibly on a set ofp-capacity zero. The key ingredient in the proof is an extension of a classical theorem of Lindelöf to monotone functions in W^1·p ( ${\mathbb{B}}^n $ ),n ? 1 <p ≤n.     

Mass Transport and Uniform Rectifiability

In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance W ₂ from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance W ₂ which asserts that if??? and ?? are probability measures in ${{\mathbb{R}^n}}$ , ${{\varphi}}$ is a radial bump function smooth enough so that ${{\int \varphi d \mu \gtrsim 1}}$ , and??? has a density bounded from above and from below on supp( ${{\varphi}}$ ), then ${{W_2(\varphi \mu, a\varphi \nu) \leq cW_2(\mu, \nu)}}$ , where ${{a = \int \varphi d\mu/ \int \varphi d\nu}}$ .     

Sharp Logarithmic Bounds for Beurling-Ahlfors Operator Restricted to the Class of Radial Functions

We establish a class of sharp logarithmic estimates for the Beurling-Ahlfors transform B on the complex plane. For any K > 0 we determine the optimal constant ${L = L(K) \in (0, \infty]}$ such that the following holds. If ${F : \mathbb{C} \rightarrow \mathbb{C}}$ is a radial function, then for any R > 0, $$\frac{1}{|\mathcal{B}(0, R)|} \int_{\mathcal{B}(0, R)} |BF(z)| dz \leq \frac{K}{|\mathcal{B}(0, R)|} \int_{\mathcal{B}(0, R)} \Psi(|F(z)|) dz + L(K),$$ where Ψ(t) = (t + 1) log(t + 1) – t and ${\mathcal{B}(0, R) \subset \mathbb{C}}$ denotes the ball of center 0 and radius R. A related result in higher dimensions is also established. The proof rests on probabilistic methods and exploits a certain sharp inequality for martingales.     

On Estimating the Asymptotic Variance of Stationary Point Processes

We investigate a class of kernel estimators $\widehat{\sigma}^2_n$ of the asymptotic variance σ ² of a d-dimensional stationary point process $\Psi = \sum_{i\ge 1}\delta_{X_i}$ which can be observed in a cubic sampling window $W_n = [-n,n]^d\,$ . σ ² is defined by the asymptotic relation $Var(\Psi(W_n)) \sim \sigma^2 \,(2n)^d$ (as n →? ∞) and its existence is guaranteed whenever the corresponding reduced covariance measure $\gamma^{(2)}_{red}(\cdot)$ has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of $\gamma^{(2)}_{red}(\cdot)$ outside of an expanding ball centered at the origin, we determine optimal bandwidths b _n (up to a constant) minimizing the mean squared error of $\widehat{\sigma}^2_n$ . The case when $\gamma^{(2)}_{red}(\cdot)$ has bounded support is of particular interest. Further we suggest an isotropised estimator $\widetilde{\sigma}^2_n$ suitable for motion-invariant point processes and compare its properties with $\widehat{\sigma}^2_n$ . Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of $\widehat{\sigma}^2_n$ for planar Poisson, Poisson cluster, and hard-core point processes and for various values of n b _n .     

Curvature estimates for submanifolds with prescribed Gauss image and mean curvature

We study that the n-graph defined by a smooth map ${f:\Omega\subset\mathbb R^{n}\to \mathbb R^{m}, m\ge 2,}$ in ${\mathbb R^{m+n}}$ of the prescribed mean curvature and the Gauss image. Under the condition $$\Delta_f=\left[\text{det}\left(\delta_{ij}+\sum_\alpha{\frac {\partial {f^\alpha}}{\partial {x^i}}}{\frac {\partial {f^\alpha}}{\partial {x^j}}}\right)\right]^{\frac{1}{2}} < 2,$$ we derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le{\frac{C}{R^2}}$$ when 2 ≤ n ≤ 5 with constant C depending on the given geometric data. If there is no dimension limitation we obtain $$\sup_{D_R(x)}|B|^2\le CR^{-a}\sup_{D_{2R}(x)}(2-\Delta_f)^{-\left({\frac{3}{2}}+{\frac{1}{s}}\right)},\quad s=\min(m, n)$$ with a < 1. If the image under the Gauss map is contained in a geodesic ball of the radius ${{\frac{\sqrt{2}}{4}}\pi}$ in G _n,m we also derive corresponding estimates.