Uniform Continuity Properties Research Articles

The Characterization of Infinite Divisibility of Compound Negative Binomial Distribution as the Sum of Exponential Distribution

The sum of random variables that are identical and independent from an exponential distribution creates the compound distribution. It is called compound negative binomial distribution as the sum of exponential distribution when the number of random variables added follows the negative binomial distribution. This compound distribution’s characteristic function is established by using mathematical analysis methods, included its uniform continuity property. The characteristic function's parametric curves never disappear from the complex plane, which means it is a positively defined function. Another characteristic function's property shows that this compound distribution is one of infinitely divisible distribution.

Continuous and holomorphic semicocycles in Banach spaces

We study some fundamental properties of semicocycles over semigroups of self-mappings of a domain in a Banach space. We prove that any semicocycle over a jointly continuous semigroup is itself jointly continuous. For semicocycles over semigroups which have generator, we establish a sufficient condition for differentiability with respect to the time variable, and hence for the semicocycle to satisfy a linear evolution problem, giving rise to the notion of ‘generator’ of a semicocycle. Bounds on the growth of a semicocycle with respect to the time variable are given in terms of this generator. Special consideration is given to the case of holomorphic semicocycles, for which we prove an exact correspondence between certain uniform continuity properties of a semicocyle and boundedness properties of its generator.

SOS specifications for uniformly continuous operators

Behavioral metric semantics provide formal notions to compare probabilistic systems, giving a notion of behavioral distance characterizing how far the behavior of two systems is apart. Compositional reasoning over probabilistic systems with respect to behavioral metric semantics requires the language operators to be uniformly continuous, which ensures that a limited change in the behavior of a subsystem implies a smooth and limited change in the behavior of the whole system. We consider a hierarchy of uniform continuity properties for process algebra operators and, for each of these properties, we propose a Structural Operational Semantics specification format (namely a set of syntactical constraints on the form of the SOS rules) ensuring that the property is satisfied by construction by all operators captured by the format.

On a Finite Range Decomposition of the Resolvent of a Fractional Power of the Laplacian II. The Torus

In previous papers, [M1, M2], [M3], we proved the existence as well as regularity of a finite range decomposition for the resolvent $G_{\alpha} (x-y,m^2) = ((-\Delta)^{\alpha\over 2} + m^{2})^{-1} (x-y) $, for $0<\alpha <2$ and all real $m$, in the lattice ${\bf Z}^{d}$ for dimension $d\ge 2$. In this paper, which is a continuation of the previous one, we extend those results by proving the existence as well as regularity of a finite range decomposition for the same resolvent but now on the lattice torus ${\bf Z}^{d}/L^{N+1}{\bf Z}^{d} $ for $d\ge 2$ provided $m\neq 0$ and $0<\alpha <2$. We also prove differentiability and uniform continuity properties with respect to the resolvent parameter $m^{2}$. Here $L$ is any odd positive integer and $N\ge 2$ is any positive integer.

On viscosity solutions to the Dirichlet problem for elliptic branches of inhomogeneous fully nonlinear equations

For scalar fully nonlinear partial differential equations F(x, D2u(x)) = 0 with x ∈ Ω b RN , we present a general theory for obtaining comparison principles and well posedness for the associated Dirichlet problem, where F(x, ·) need not be monotone on all of S(N), the space of symmetric N ×N matrices. We treat admissible viscosity solutions u of elliptic branches of the equation in the sense of Krylov [20]and extend the program initiated by Harvey and Lawson [11] in the homogeneous case when F does not depend on x. In particular, for the set valued map Θ defining the elliptic branch by way of the differential inclusion D2u(x) ∈ ∂Θ(x), we identify a uniform continuity property which ensures the validity of the comparison principle and the applicability of Perron’s method for the differential inclusion on suitably convex domains, where the needed boundary convexity is characterized by Θ. Structural conditions on F are then derived which ensure the existence of an elliptic map Θ with the needed regularity. Concrete applications are given in which standard structuralconditions on F may fail and without the request of convexity conditions in the equation. Examples include perturbed Monge–Ampere equations and equations prescribing eigenvalues of the Hessian.

Optimal stopping under nonlinear expectation

Let X:[0,T]×Ω⟶R be a bounded càdlàg process with positive jumps defined on the canonical space of continuous paths Ω. We consider the problem of optimal stopping the process X under a nonlinear expectation operator E defined as the supremum of expectations over a weakly compact but nondominated family of probability measures. We introduce the corresponding nonlinear Snell envelope. Our main objective is to extend the Snell envelope characterization to the present context. Namely, we prove that the nonlinear Snell envelope is an E-supermartingale, and an E-martingale up to its first hitting time of the obstacle X. This result is obtained under an additional uniform continuity property of X. We also extend the result in the context of a random horizon optimal stopping problem.This result is crucial for the newly developed theory of viscosity solutions of path-dependent PDEs as introduced in Ekren et al. (2014), in the semilinear case, and extended to the fully nonlinear case in the accompanying papers (Ekren et al. [6,7]).

Supervised pre-clustering for sparse regression

Kernel algorithms for large data-sets are now an active research area motivated by the many real world problems producing very large numbers of data points. However, standard kernel methods scale poorly with the size of the data-set. We propose a mathematically motivated approach to sparse function estimation that utilises the uniform continuity properties of functions in continuous reproducing kernel Hilbert spaces (RKHS) defined on compact domains. Using the uniform continuity properties of the function a similarity measure on data points is defined that allows data to be pre-clustered. Unlike previous methods for sparse function estimation using clustering the proposed approach is supervised and relies on well-defined mathematical concepts. The cluster centres are used to form a sparse RKHS subspace within which the function is estimated. The greedy pre-clustering algorithms and sparse model based on pre-clustering and committee machine techniques are presented. Results are presented to demonstrate the application of the proposed algorithms on function approximation problems.

Continuous mappings between spaces of arcs

A blow-analytic homeomorphism is an arc-analytic subanalytic homeomorphism, and therefore it induces a bijective mapping between spaces of analytic arcs. We tackle the question of the continuity of this induced mapping between the spaces of arcs, giving a positive and a negative answer depending of the topology involved. We generalise the result to spaces of definable arcs in the context of o-minimal structures, obtaining notably a uniform continuity property.

Abstraction and invariance for algebraically indexed types

Reynolds' relational parametricity provides a powerful way to reason about programs in terms of invariance under changes of data representation. A dazzling array of applications of Reynolds' theory exists, exploiting invariance to yield "free theorems", non-inhabitation results, and encodings of algebraic datatypes. Outside computer science, invariance is a common theme running through many areas of mathematics and physics. For example, the area of a triangle is unaltered by rotation or flipping. If we scale a triangle, then we scale its area, maintaining an invariant relationship between the two. The transformations under which properties are invariant are often organised into groups, with the algebraic structure reflecting the composability and invertibility of transformations. In this paper, we investigate programming languages whose types are indexed by algebraic structures such as groups of geometric transformations. Other examples include types indexed by principals--for information flow security--and types indexed by distances--for analysis of analytic uniform continuity properties. Following Reynolds, we prove a general Abstraction Theorem that covers all these instances. Consequences of our Abstraction Theorem include free theorems expressing invariance properties of programs, type isomorphisms based on invariance properties, and non-definability results indicating when certain algebraically indexed types are uninhabited or only inhabited by trivial programs. We have fully formalised our framework and most examples in Coq.

(Strong) weak exhaustiveness and (strong uniform) continuity

In this paper we continue, in the realm of metric spaces, the study of exhaustiveness and weak exhaustiveness at a point of a net of functions initiated by Gregoriades and Papanastassiou in 2008. We prove that exhaustiveness at every point of a net of pointwise convergent functions is equivalent to uniform convergence on compacta. We extend exhaustiveness-type properties to subsets. First, we introduce the notion of strong exhaustiveness at a subset B for sequences of functions and prove its equivalence with strong exhaustiveness at P0 (B) of the sequence of the direct image maps, where the hypersets are equipped with the Hausdorff metric. Furthermore, we show that the notion of strong-weak exhaustiveness at a subset is the proper tool to investigate when the limit of a pointwise convergent sequence of functions fulfills the strong uniform continuity property, a new pregnant form of uniform continuity discovered by Beer and Levi in 2009.

Continuity of triple I methods based on several implications

This paper focuses on the continuity problem of fully implicational triple I methods for fuzzy reasoning. Based on the residual implications generated by continuous triangular norms, the residual implication generated by nilpotent minimum with the standard negation and the Zadeh implication, respectively, continuity and uniform continuity properties of triple I methods are examined in Hamming and uniform metrics. We also investigate the continuity of the model of a fuzzy if-then rule corresponding to the triple I methods.