Subset Of Rd Research Articles

A discrete probabilistic model yielding multidimensional q − Gaussians in the thermodynamic limit for specific parameter values

We show that the N → ∞ limiting probability distributions (with N being the number of random variables to be summed) of a particular case belonging to a family of d − dimensional scale-invariant probabilistic models based on Leibniz-like (d + 1) − dimensional hyperpyramids (introduced in Rodríguez and Tsallis, J. Math. Phys 2012) are given, after an appropriate change o variables, by d − dimensional q − Gaussian distributions ( fq,β(x)∝eq−βxTΣ−1x , with x taking values in a subset of Rd , q and β real parameters, Σ a positive definite matrix, and eqx=[1+1(1−q)x]+11−q , with [u]+=max{u,0} ), for the particular parameter values q = 3 and β < 0. Some features of these distributions, which arise in the context of Nonextensive Statistical Mechanics, as well as a connection between Dirichlet distributions and q − Gaussian distributions, which generalizes the one given for dimension 1 by Rodríguez et al, are also exposed.

Validity and failure of the integral representation of Γ-limits of convex non-local functionals

We prove an integral-representation result for limits of non-local quadratic forms on H01(Ω), with Ω a bounded open subset of Rd, extending the representation on Cc∞(Ω) given by the Beurling-Deny formula in the theory of Dirichlet forms. We give a counterexample showing that a corresponding representation may not hold if we consider analogous functionals in W01,p(Ω), with p≠2 and 1<p⩽d.

Quasi-uniform designs with optimal and near-optimal uniformity constant

A design is a collection of distinct points in a given set X, which is assumed to be a compact subset of Rd, and the mesh-ratio of a design is the ratio of its fill distance to its separation radius. The uniformity constant of a sequence of nested designs is the smallest upper bound for the mesh-ratios of the designs. We derive a lower bound on this uniformity constant and show that a simple greedy construction achieves this lower bound. We then extend this scheme to allow more flexibility in the design construction.

Dissipation for a non-convex gradient flow problem of a Patlack–Keller–Segel type for densities on [formula omitted], [formula omitted

We study an evolution equation that is the gradient flow in the 2-Wasserstein metric of a non-convex functional for densities in Rd with d≥3. Like the Patlack–Keller–Segel system on R2, this evolution equation features a competition between the dispersive effects of diffusion, and the accretive effects of a concentrating drift. We determine a parameter range in which the diffusion dominates, and all mass leaves any fixed compact subset of Rd at an explicit polynomial rate.

Infinitely many solutions for problems in fractional Orlicz–Sobolev spaces

We use a symmetric mountain pass lemma of Kajikiya to prove the existence of infinitely many weak solutions for the Schrödinger Φ-Laplace equation ( − Δ ) Φ u + V ( x ) φ ( u ) = ξ ( x ) f ( u ) in ℝ d , where Φ(t)= ∫ 0tφ(s)ds is an N-function, ΔΦ is the Φ-Laplacian operator, V:ℝd→ℝ is a continuous function, ξ is a function with sign-changing on ℝd and the nonlinearity f is sublinear as |u|→∞. During the study of our problem, we deal with a new compact embedding theorem for the Orlicz–Sobolev spaces. We also study the existence and multiplicity of solutions to the general fractional Φ-Laplacian equations of Kirchhoff type M ( ∫ ℝ 2 d Φ ( u ( x ) − u ( y ) K ( | x − y | ) ) d x d y N ( | x − y | ) ) ( − Δ ) Φ K , N u = f ( x , u ) in Ω , u = 0 in ℝ d ∖ Ω , where Ω is an open bounded subset of ℝd with smooth boundary ∂Ω, d>2, and M:ℝ0+→ℝ+ is a continuous function and f:Ω×ℝ→ℝ is a Carathéodory function. The proofs rely essentially on the fountain theorem and the genus theory.

Stochastic invariance of closed sets with non-Lipschitz coefficients

This paper provides a new characterization of the stochastic invariance of a closed subset of Rd with respect to a diffusion. We extend the well-known inward pointing Stratonovich drift condition to the case where the diffusion matrix can fail to be differentiable: we only assume that the covariance matrix is. In particular, our result can be applied to construct affine and polynomial diffusions on any arbitrary closed set.

Dirichlet problem and subclasses of Baire-one functions

Let U be a bounded open subset of ℝd, d ≥ 2 and f ∈ C(∂U). The Dirichlet solution fCU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure Ū of U in general if U is not regular but it is always Baire-one. Let H(U) be the space of all functions continuous on the closure Ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that fCU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus fCU belongs to the subclass B1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure Ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10]. A generalization to the abstract context of simplicial function space on a metrizable compact space is provided. We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].

Global uniqueness in an inverse problem for time fractional diffusion equations

Given (M,g), a compact connected Riemannian manifold of dimension d⩾2, with boundary ∂M, we consider an initial boundary value problem for a fractional diffusion equation on (0,T)×M, T>0, with time-fractional Caputo derivative of order α∈(0,1)∪(1,2). We prove uniqueness in the inverse problem of determining the smooth manifold (M,g) (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solutions on a subset of ∂M at fixed time. In the “flat” case where M is a compact subset of Rd, two out the three coefficients ρ (density), a (conductivity) and q (potential) appearing in the equation ρ∂tαu−div(a∇u)+qu=0 on (0,T)×M are recovered simultaneously.

Product of Simplices and sets of positive upper density in ℝd

AbstractWe establish that any subset of ℝd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d ⩾ 4.We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Δk1 and Δk2 are two fixed non-degenerate simplices of k1 + 1 and k2 + 1 points respectively, then any subset of ℝd of positive upper Banach density with d ⩾ k1 + k2 + 6 will necessarily contain an isometric copy of all sufficiently large dilates of Δk1 × Δk2.A new direct proof of the fact that any subset of ℝd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of k + 1 points provided d ⩾ k + 1, a result originally due to Bourgain, is also presented.

A characterization of 𝜇-equicontinuity for topological dynamical systems

Two different notions of measure theoretical equicontinuity ( μ − \mu - equicontinuity) for topological dynamical systems with respect to Borel probability measures appeared in works by Gilman (1987) and Huang, Lee and Ye (2011). We show that if the probability space satisfies Lebesgue’s density theorem and Vitali’s covering theorem (for example a Cantor set or a subset of R d \mathbb {R}^{d} ), then both notions are equivalent. To show this we characterize Lusin measurable maps using μ − \mu - continuity points. As a corollary we also obtain a new characterization of μ − \mu - mean equicontinuity.

Transitive and series transitive maps on [formula omitted

Motivated by the behavior of topologically transitive homomorphisms of Polish abelian groups, we say a continuous map f:Rd→Rd is ‘series transitive’ if for any two nonempty open sets U,V⊂Rd, there exist x∈U and n∈N such that ∑j=0n−1fj(x)∈V. We show that any map on a discrete and closed subset of Rd can be extended to a mixing map of Rd, and use this result to produce a mixing map f:Rd→Rd (for each d∈N) which is also series transitive. We have examples to say that transitivity and series transitivity are independent properties for continuous self-maps of Rd. We also construct a chaotic map (i.e., a transitive map with a dense set of periodic points) f:Rd→Rd such that f is arbitrarily close to and asymptotic to the identity map. Finally, we make a few observations about topological transitivity of continuous homomorphisms of Polish abelian groups.

Approximating a diffusion by a finite-state hidden Markov model

For a wide class of continuous-time Markov processes evolving on an open, connected subset of Rd, the following are shown to be equivalent: (i)The process satisfies (a slightly weaker version of) the classical Donsker–Varadhan conditions;(ii)The transition semigroup of the process can be approximated by a finite-state hidden Markov model, in a strong sense in terms of an associated operator norm;(iii)The resolvent kernel of the process is ‘v-separable’, that is, it can be approximated arbitrarily well in operator norm by finite-rank kernels. Under any (hence all) of the above conditions, the Markov process is shown to have a purely discrete spectrum on a naturally associated weighted L∞ space.

Conservativeness criteria for generalized Dirichlet forms

We develop sufficient analytic conditions for conservativeness of non-sectorial perturbations of symmetric Dirichlet forms which can be represented through a carré du champ on a locally compact separable metric space. These form an important subclass of generalized Dirichlet forms which were introduced in [21]. In case there exists an associated strong Feller process, the analytic conditions imply conservativeness, i.e. non-explosion of the associated process in the classical probabilistic sense. As an application of our general results on locally compact separable metric state spaces, we consider a generalized Dirichlet form given on a closed or open subset of Rd which is given as a divergence free first order perturbation of a symmetric energy form. Then using volume growth conditions of the carré du champ and the non-sectorial first order part, we derive an explicit criterion for conservativeness. We present several concrete examples which relate our results to previous ones obtained by different authors. In particular, we show that conservativeness can hold for a large variance if the anti-symmetric part of the drift is strong enough to compensate it. This work continues our previous work on transience and recurrence of generalized Dirichlet forms.

Radial mollifiers, mean value operators and harmonic functions in Dunkl theory

In this paper we show how to use mollifiers to regularize functions relative to a set of Dunkl operators in Rd with Coxeter–Weyl group W, multiplicity function k and weight function ωk. In particular for Ω a W-invariant open subset of Rd, for ϕ∈D(Rd) a radial function and u∈Lloc1(Ω,ωk(x)dx), we study the Dunkl-convolution product u⁎kϕ and the action of the Dunkl-Laplacian and the volume mean operators on these functions. The results are then applied to obtain an analog of the Weyl lemma for Dunkl-harmonic functions and to characterize them by invariance properties relative to mean value and convolution operators.

Heat content for convolution semigroups

Let X={Xt}t≥0 be a Lévy process in Rd and Ω be an open subset of Rd with finite Lebesgue measure. In this article we consider the quantity H(t)=∫ΩPx(Xt∈Ωc)dx related to X which is called the heat content. We study its asymptotic behaviour as t goes to zero for isotropic Lévy processes under some mild assumptions on the characteristic exponent. We also treat the class of Lévy processes with finite variation in full generality.

Optimal polynomial admissible meshes on some classes of compact subsets of [formula omitted

We show that any compact subset of Rd which is the closure of a bounded star-shaped Lipschitz domain Ω, such that ∁Ω has positive reach in the sense of Federer, admits an optimal AM (admissible mesh), that is a sequence of polynomial norming sets with optimal cardinality. This extends a recent result of A. Kroó on C2 star-shaped domains.Moreover, we prove constructively the existence of an optimal AM for any K:=Ω¯⊂Rd where Ω is a bounded C1,1 domain. This is done by a particular multivariate sharp version of the Bernstein Inequality via the distance function.

Estimates of s-numbers of a Sobolev embedding involving spaces of variable exponent

Let Ω be a bounded open subset of Rd, suppose that p(⋅):Ω→(1,∞) is a bounded, log-Hölder continuous function, and let Lp(⋅)(Ω), W∘(Ω)p(⋅)1 be the usual variable exponent Lebesgue space and the corresponding Sobolev space. The natural embedding id:W∘(Ω)p(⋅)1→Lp(⋅)(Ω) is compact; when Ω is a bounded domain it is shown that there are positive constants K1,K2 such that for all n∈N,K1≤n1/dsn(id)≤K2, where sn(id) is the nth approximation, Bernstein, Gelfand or Kolmogorov number of id. When p is constant this result is familiar; for variable p and d>1 it appears to be the first result available for s-numbers of Sobolev embeddings. The paper also contains a sharp estimate of the norm of embeddings between Lp(⋅)(Ω) spaces which is interesting in its own right.

On Schur’s conjecture in ℝ4

A diameter graph in ℝd is a graph in which vertices are points of a finite subset of ℝd and two vertices are joined by an edge if the distance between them is equal to the diameter of the vertex set. This paper is devoted to Schur’s conjecture, which asserts that any diameter graph on n vertices in ℝd contains at most n complete subgraphs of size d. It is known that Schur’s conjecture is true in dimensions d ≤ 3. We prove this conjecture for d = 4 and give a simple proof for d = 3.

Ergodic BSDEs and related PDEs with Neumann boundary conditions under weak dissipative assumptions

We study a class of ergodic BSDEs related to PDEs with Neumann boundary conditions. The randomness of the driver is given by a forward process under weakly dissipative assumptions with an invertible and bounded diffusion matrix. Furthermore, this forward process is reflected in a convex subset of Rd not necessarily bounded. We study the link of such EBSDEs with PDEs and we apply our results to an ergodic optimal control problem.

Lifshitz tails for a percolation model in the continuum

Abstract. In this work we study Lifshitz tails for Laplacians in a percolation model on ℝ d ${\mathbb {R}}^d$ . At any lattice point i in ℝ d ${\mathbb {R}}^d$ we remove a set S + i ${S+i}$ with a certain probability p. We consider the Laplacian on the remaining subset of ℝ d ${\mathbb {R}}^d$ with either Dirichlet or Neumann boundary conditions. We prove that the integrated density of states exhibits Lifshitz behavior at the bottom of the spectrum when we consider Dirichlet boundary conditions, while when we consider Neumann boundary conditions, it exhibits a van Hove behavior.