On the Vervaat and Vervaat-Error Processes

It has been known since the pioneering works of Wim Vervaat in the early 70"s that the appropriately normalized Vervaat process asymptotically behaves like one half times the squared empirical process. In this paper we present a survey of the results concerning various distances between the Vervaat process and one half times the squared empirical process. In particular, the pointwise, L_p-, and sup-distances between these two processes are given full consideration.     

Simulating Perpetuities

A perpetuity is a random variable that can be represented as , where the W _i's are i.i.d. random variables. We study exact random variate generation for perpetuities and discuss the expected complexity. For the Vervaat family, in which 0,U$$ " align="middle" border="0"> uniform [0, 1], all the details of a novel rejection method are worked out. There exists an implementation of our algorithm that only uses uniform random numbers, additions, multiplications and comparisons.     

Deciding the existence of uniform interpolants over transitive models

We consider the problem of the existence of uniform interpolants in the modal logic K4. We first prove that all ${\square}$ -free formulas have uniform interpolants in this logic. In the general case, we shall prove that given a modal formula ${\phi}$ and a sublanguage L of the language of the formula, we can decide whether ${\phi}$ has a uniform interpolant with respect to L in K4. The ${\square}$ -free case is proved using a reduction to the G?del L?b Logic GL, while in the general case we prove that the question of whether a modal formula has uniform interpolants over transitive frames can be reduced to a decidable expressivity problem on the???-calculus.     

Uniform approximation of the heat kernel on a manifold

We approximate the heat kernel h(x, y, t) on a compact connected Riemannian manifold M without boundary uniformly in \((x,y,t)\in M\times M\times [a,b]\), \(a>0\), by n-fold integrals over \(M^n\) of the densities of Brownian bridges. Moreover, we provide an estimate for the uniform convergence rate. As an immediate corollary, we get a uniform approximation of solutions of the Cauchy problem for the heat equation on M.     

Morrey–Sobolev Extension Domains

We show that every uniform domain of \({{{\mathbb {R}}}^n}\) with \(n\ge 2\) is a Morrey–Sobolev \({\mathscr {W}}^{1,\,p}\)-extension domain for all \(p\in [1,\,n)\), and moreover, that this result is essentially the best possible for each \(p\in [1,\,n)\) in the sense that, given a simply connected planar domain or a domain of \({{{\mathbb {R}}}^n}\) with \(n\ge 3\) that is quasiconformal equivalent to a uniform domain, if it is a \({\mathscr {W}}^{1,\,p} \)-extension domain, then it must be uniform.     

Uniform Continuity, Uniform Convergence, and Shields

Let $\mathcal{B}$ be a bornology in a metric space $\langle X,d \rangle$ , that is, a cover of X by nonempty subsets that also forms an ideal. In Beer and Levi (J Math Anal Appl 350:568–589, 2009), the authors introduced the variational notions of strong uniform continuity of a function on $\mathcal{B}$ as an alternative to uniform continuity of the restriction of the function to each member of $\mathcal{B}$ , and the topology of strong uniform convergence on $\mathcal{B}$ as an alternative to the classical topology of uniform convergence on $\mathcal{B}$ . Here we continue this study, showing that shields as introduced in Beer, Costantini and Levi (Bornological Convergence and Shields, Mediterranean J. Math, submitted) play a pivotal role. For example, restricted to continuous functions, the topology of strong uniform convergence on $\mathcal{B}$ reduces to the classical topology if and only if the natural closure of the bornology is shielded from closed sets. The paper also further develops the theory of shields and their applications.     

In some symmetric spaces monotonicity properties can be reduced to the cone of rearrangements

Geometric properties being the rearrangement counterparts of strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some symmetric spaces are considered. The relationships between strict monotonicity, upper local uniform monotonicity restricted to rearrangements and classical monotonicity properties (sometimes under some additional assumptions) are showed. It is proved that order continuity and lower uniform monotonicity properties for rearrangements of symmetric spaces together are equivalent to the classical lower local uniform monotonicity for any symmetric space over a \({\sigma}\)-finite complete and non-atomic measure space. It is also showed that in the case of order continuous symmetric spaces over a \({\sigma}\)-finite and complete measure space, upper local uniform monotonicity and its rearrangement counterpart shortly called ULUM* coincide. As an application of this result, in the case of a non-atomic complete finite measure a new proof of the theorem which is already known in the literature, giving the characterization of upper local uniform monotonicity of Orlicz–Lorentz spaces, is presented. Finally, it is proved that every rotund and reflexive space X such that both X and X* have the Kadec-Klee property is locally uniformly rotund. Some other results are also given in the first part of Sect. 2.     

Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation

In the case of the boundary value problem for a singularly perturbed convection-diffusion parabolic equation, conditioning of an ε-uniformly convergent finite difference scheme on a piecewise uniform grid is examined. Conditioning of a finite difference scheme on a uniform grid is also examined provided that this scheme is convergent. For the condition number of the scheme on a piecewise uniform grid, an ε-uniform bound O(δ ₁ ^?2 lnδ ₁ ^?1 + δ ₀ ^?1 ) is obtained, where δ₁ and δ₀ are the error components due to the approximation of the derivatives with respect to x and t, respectively. Thus, this scheme is ε-uniformly well-conditioned. For the condition number of the scheme on a uniform grid, we have the estimate O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ); this scheme is not ε-uniformly well-conditioned. In the case of the difference scheme on a uniform grid, there is an additional error due to perturbations of the grid solution; this error grows unboundedly as ε → 0, which reduces the accuracy of the grid solution (the number of correct significant digits in the grid solution is reduced). The condition numbers of the matrices of the schemes under examination are the same; both have an order of O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ). Neither the matrix of the ε-uniformly convergent scheme nor the matrix of the scheme on a uniform grid is ε-uniformly well-conditioned.     

Right-Left Symmetry of Right Nonsingular Right Max-Min CS Prime Rings

In this article we show, among others, that if R is a prime ring which is not a domain, then R is right nonsingular, right max-min CS with uniform right ideal if and only if R is left nonsingular, left max-min CS with uniform left ideal. The above result gives, in particular, Huynh et al. (2000 Huynh , D. V. , Jain , S. K. , López-Permouth , S. R. ( 2000 ). On the symmetry of the goldie and CS conditions for prime rings . Proc. Amer. Math. Soc. 128 : 3153 – 3157 . [Google Scholar]) Theorem for prime rings of finite uniform dimension.     

On properties of infimal topology of a map space

We study properties of the infimal topology τ_inf which is the infimum of the family of all topologies of uniform convergence defined on the set C(X, Y) of continuous maps into a metrizable space Y. One of the main results of the research consists in obtaining necessary and sufficient condition for the topology τ_inf to have the Fréchet–Urysohn property. We also establish necessary and sufficient conditions for coincidence of the topology τinf and a topology of uniform convergence τ_μ (“attaining” the infimum). We prove that for this coincidence it is sufficient for the topology τ_inf to satisfy the first axiom of countability.     

Complex convexity and monotonicity in quasi-Banach lattices

In this paper we study the monotonicity and convexity properties in quasi-Banach lattices. We establish relationship between uniform monotonicity, uniform ?-convexity, H-and PL-convexity. We show that if the quasi-Banach lattice E has α-convexity constant one for some 0 < α < ∞, then the following are equivalent: (i) E is uniformly PL-convex; (ii) E is uniformly monotone; and (iii) E is uniformly ?-convex. In particular, it is shown that if E has α-convexity constant one for some 0 < α < ∞ and if E is uniformly ?-convex of power type then it is uniformly H-convex of power type. The relations between concavity, convexity and monotonicity are also shown so that the Maurey-Pisier type theorem in a quasi-Banach lattice is proved.Finally we study the lifting property of uniform PL-convexity: if E is a quasi-Köthe function space with α-convexity constant one and X is a continuously quasi-normed space, then it is shown that the quasi-normed Köthe-Bochner function space E(X) is uniformly PL-convex if and only if both E and X are uniformly PL-convex.     

Uniformization of Sierpiński carpets in the plane

Let S _i , i∈I, be a countable collection of Jordan curves in the extended complex plane \(\widehat{\mathbb{C}}\) that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map \(f\colon\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}\) such that f(S _i ) is a round circle for all i∈I. This implies that every Sierpiński carpet in \(\widehat{\mathbb{C}}\) whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiński carpet by a quasisymmetric map.     

Form preservation under approximation by local exponential splines of an arbitrary order

We continue the study of the properties of local L-splines with uniform knots (such splines were constructed in the authors’ earlier papers) corresponding to a linear differential operator L of order r with constant coefficients and real pairwise different roots of the characteristic polynomial. Sufficient conditions (which are also necessary) are established under which an L-spline locally inherits the property of the generalized k-monotonicity (k ≤ r ? 1) of the input data, which are the values of the approximated function at the nodes of a uniform grid shifted with respect to the grid of knots of the L-spline. The parameters of an L-spline that is exact on the kernel of the operator L are written explicitly.     

Uniform continuity in {\omega_\mu}-metric spaces and uc {\omega_\mu}-metric extendability

Generally, the term uc-ness means some continuity is uniform. A metric space X is uc when any continuous function fromX to [0, 1] is uniformly continuous and a metrizable space X is a Nagata space when it can be equipped with a uc metric. We consider natural forms of uc-ness for the \({\omega_\mu}\)-metric spaces, which fill a very large and interesting class of uniform spaces containing the usual metric ones, and extend to them various different formulations of the metric uc-ness, by additionaly proving their equivalence. Furthermore, since any \({\omega_\mu}\)-compact space is uc and any uc \({\omega_\mu}\)-metric space is complete, in the line of constructing dense extensions which preserve some structure, such as uniform completions, we focus on the existence for an \({\omega_\mu}\)-metrizable space of dense topological extensions carrying a uc \({\omega_\mu}\)-metric. In this paper we show that an \({\omega_\mu}\)-metrizable space X is uc-extendable if and only if there exists a compatible \({\omega_\mu}\)-metric d on X such that the set X′ of all accumulation points in X is crowded, i.e., any \({\omega_\mu}\)-sequence in X′ has a d-Cauchy \({\omega_\mu}\)-subsequence in X′.     

The Folded (2<Emphasis Type="Italic">D</Emphasis> + 1)-cube and Its Uniform Posets

Let Γ denote the folded (2D + 1)-cube with vertex set X and diameter D ≥ 3. Fix x ∈ X. We first define a partial order ≤ on X as follows. For y, z ∈ X let y ≤ z whenever ?(x, y) + ?(y, z) = ?(x, z). Let R (resp. L) denote the raising matrix (resp. lowering matrix) of Γ. Next we show that there exists a certain linear dependency among RL², LRL,L²R and L for each given Q-polynomial structure of Γ. Finally, we determine whether the above linear dependency structure gives this poset a uniform structure or strongly uniform structure.     

Strong approximations for long memory sequences based partial sums,counting and their Vervaat processes

We study the asymptotic behaviour of partial sums of long range dependent random variables and that of their counting process, together with an appropriately normalized integral process of the sum of these two processes, the so-called Vervaat process. The first two of these processes are approximated by an appropriately constructed fractional Brownian motion, while the Vervaat process in turn is approximated by the square of the same fractional Brownian motion.     

A sharp bound on the Hausdorff dimension of the singular set of a uniform measure

The study of the geometry of n-uniform measures in \(\mathbb {R}^{d}\) has been an important question in many fields of analysis since Preiss’ seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely 3-Hausdorff measure restricted to the Kowalski–Preiss cone. Using this cone one can construct an n-uniform measure whose singular set has Hausdorff dimension \(n-3\). In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any n-uniform measure is at most \(n-3\).     

Correctness of Cauchy-type problems for abstract differential equations with fractional derivatives

We prove the uniform correctness of a Cauchy-type problem with two fractional derivatives and a bounded operator A. We propose a criterion for the uniform correctness of unbounded operator A.     

Local circular law for random matrices

The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point \(z\) away from the unit circle. More precisely, if \( | |z| - 1 | \ge \tau \) for arbitrarily small \(\tau > 0\) , the circular law is valid around \(z\) up to scale \(N^{-1/2+ {\varepsilon }}\) for any \({\varepsilon }> 0\) under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.