Sufficient Conditions For Continuity Research Articles

Necessary and sufficient conditions for continuity of hypercontractive processes and fields

Sample path properties of random processes are an interesting and extensively studied topic, especially in the case of Gaussian processes. In this article, we study the continuity properties of hypercontractive fields, providing natural extensions for some known Gaussian results beyond Gaussianity. Our results apply to both random processes and random fields alike.

Stochastic integration with respect to fractional processes in Banach spaces

In the article, integration of temporal functions in (possibly non-UMD) Banach spaces with respect to (possibly non-Gaussian) fractional processes from a finite sum of Wiener chaoses is treated. The family of fractional processes that is considered includes, for example, fractional Brownian motions of any Hurst parameter or, more generally, fractionally filtered generalized Hermite processes. The class of Banach spaces that is considered includes a large variety of the most commonly used function spaces such as the Lebesgue spaces, Sobolev spaces, or, more generally, the Besov and Lizorkin-Triebel spaces. In the article, a characterization of the domains of the Wiener integrals on both bounded and unbounded intervals is given for both scalar and cylindrical fractional processes. In general, the integrand takes values in the space of γ-radonifying operators from a certain homogeneous Sobolev-Slobodeckii space into the considered Banach space. Moreover, an equivalent characterization in terms of a pointwise kernel of the integrand is also given if the considered Banach space is isomorphic with a subspace of a cartesian product of mixed Lebesgue spaces. The results are subsequently applied to stochastic convolution for which both necessary and sufficient conditions for measurability and sufficient conditions for continuity are found. As an application, space-time continuity of the solution to a parabolic equation of order 2m with distributed noise of low time regularity is shown as well as measurability of the solution to the heat equation with Neumann boundary noise of higher regularity.

Family of Continuous Strain-Consistent Convective Tensor Rates and Its Application in Hooke-Like Isotropic Hypoelasticity

This paper presents a new family of objective (Lagrangian and Eulerian) continuous strain-consistent convective (corotational and non-corotational) tensor rates. Objective strain-consistent convective tensor rates are defined as having the property that there exist objective (Lagrangian and Eulerian) strain tensors from the Hill family such that the considered rates of these strain tensors are equal to the rotated/standard (Lagrangian/Eulerian) stretching (strain rate) tensors. The family of such Eulerian strain-consistent convective tensor rates was introduced by Bruhns et al. (Proc. R. Soc. Lond. A 460:909–928, 2004). On the one hand, the family of continuous strain-consistent convective tensor rates presented in this paper extends the Bruhns et al. family by including Lagrangian tensor rates and, on the other hand, it is narrower than the Bruhns et al. family due to the continuity requirement imposed on tensor rates, which is necessary for applications. The theorem that any strain tensors only from the Doyle–Ericksen family (which is a subfamily of the Hill family) provide sufficient continuity conditions for strain-consistent convective tensor rates was formulated and proved. The expressions obtained by proving this theorem for convector tensors show that each tensor from the Doyle–Ericksen family is associated with a single strain-consistent convective tensor rate (in the Bruhns et al. family, any strain tensor from the Hill family generates infinitely many convector tensors associated with infinitely many strain-consistent convective tensor rates). In addition, a new family of Hooke-like isotropic hypoelastic material models based on objective continuous strain-consistent convective rates of the rotated/standard (Lagrangian and Eulerian) Kirchhoff stress tensors was constructed. The theorem that any material model from this family is self-consistent provided that the Lame parameter $\lambda =0$ and/or the deformation of the body is isochoric was formulated and proved. By self-consistent hypoelastic material models are meant those models for which constitutive hypoelastic relations are counterparts of constitutive relations for Cauchy/Green elasticity. Some material models from the new family were tested by solving the simple shear problem. Both new and well-known solutions of this problem for material models were obtained using an approach that takes into account the self-consistency property of material models from the new family.

Universal Feedback Controllers and Inverse Optimality for Nonlinear Stochastic Systems

Abstract In this paper, we develop a constructive finite time stabilizing feedback control law for stochastic dynamical systems driven by Wiener processes based on the existence of a stochastic control Lyapunov function. In addition, we present necessary and sufficient conditions for continuity of such controllers. Moreover, using stochastic control Lyapunov functions, we construct a universal inverse optimal feedback control law for nonlinear stochastic dynamical systems that possess guaranteed gain and sector margins. An illustrative numerical example involving the control of thermoacoustic instabilities in combustion processes is presented to demonstrate the efficacy of the proposed framework.

The Weyl product on quasi-Banach modulation spaces

We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.

The Weyl Product on Quasi-Banach Modulation Spaces

We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.

Complex Monge-Ampère Equation in Strictly Pseudoconvex Domains

We study the complex Monge-Amp\`ere equation $(dd^c u)^n=\mu$ in a strictly pseudoconvex domain $\Omega$ with the boundary condition $u=\varphi$, where $\varphi\in C(\partial\Omega)$. We provide a non-trivial sufficient condition for continuity of the solution $u$ outside "small sets".

Analytic aspects of evolution algebras

We prove that every evolution algebra A is a normed algebra, for an l 1 -norm defined in terms of a fixed natural basis. We further show that a normed evolution algebra A is a Banach algebra if and only if A = A 1 ⊕ A 0 , where A 1 is finite-dimensional and A 0 is a zero-product algebra. In particular, every nondegenerate Banach evolution algebra must be finite-dimensional and the completion of a normed evolution algebra is therefore not, in general, an evolution algebra. We establish a sufficient condition for continuity of the evolution operator L B of A with respect to a natural basis B , and we show that L B need not be continuous. Moreover, if A is finite-dimensional and B = { e 1 , … , e n } , then L B is given by L e , where e = ∑ i e i and L a is the multiplication operator L a ( b ) = a b , for b ∈ A . We establish necessary and sufficient conditions for convergence of ( L a n ( b ) ) n , for all b ∈ A , in terms of the multiplicative spectrum σ m ( a ) of a . Namely, ( L a n ( b ) ) n converges, for all b ∈ A , if and only if σ m ( a ) ⊆ Δ ∪ { 1 } and ν ( 1 , a ) ≤ 1 , where ν ( 1 , a ) denotes the index of 1 in the spectrum of L a .

Weighted Exponential Random Graph Models: Scope and Large Network Limits

We study models of weighted exponential random graphs in the large network limit. These models have recently been proposed to model weighted network data arising from a host of applications including socio-econometric data such as migration flows and neuroscience. Analogous to fundamental results derived for standard (unweighted) exponential random graph models in the work of Chatterjee and Diaconis, we derive limiting results for the structure of these models as the number of nodes goes to infinity. Our results are applicable for a wide variety of base measures including measures with unbounded support. We also derive sufficient conditions for continuity of functionals in the specification of the model including conditions on nodal covariates. Finally we include a number of open problems to spur further understanding of this model especially in the context of applications.

The Weyl product on quasi-Banach modulation spaces

We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.

Systems Parabolic in Petrovskii’s Sense in Hörmander Spaces

We study a general initial-boundary-value problem for systems parabolic in Petrovskii’s sense with trivial initial Cauchy data in some anisotropic Hormander inner-product spaces. It is proved that the operators corresponding to this problem are isomorphisms between the appropriate Hormander spaces. As an application of this result, we establish a theorem on the local increase in regularity of the solutions of the problem. We also establish new sufficient conditions of continuity for the generalized partial derivatives of a given order for the chosen component of the solution.

Weak Continuity of Risk Functionals with Applications to Stochastic Programming

Measuring and managing risk has become crucial in modern decision making under stochastic uncertainty. In two-stage stochastic programming, mean risk models are essentially defined by a parametric recourse problem and a quantification of risk. From the perspective of qualitative robustness theory, we discuss sufficient conditions for continuity of the resulting objective functions with respect to perturbation of the underlying probability measure. Our approach covers a fairly comprehensive class of both stochastic programming related risk measures and relevant recourse models. Not only does this unify previous approaches but also extends known stability results for two-stage stochastic programs to models with mixed-integer quadratic recourse and mixed-integer convex recourse, respectively.

A Note on Continuity of Solution Set for Parametric Weak Vector Equilibrium Problems

We consider the parametric weak vector equilibrium problem. By using a weaker assumption of Peng and Chang (2014), the sufficient conditions for continuity of the solution mappings to a parametric weak vector equilibrium problem are established. Examples are provided to illustrate the essentialness of imposed assumptions. As advantages of the results, we derive the continuity of solution mappings for vector optimization problems.

A sufficient condition for the continuity of permanental processes with applications to local times of Markov processes

We provide a sufficient condition for the continuity of real valued permanental processes. When applied to the subclass of permanental processes which consists of squares of Gaussian processes, we obtain the sufficient condition for continuity which is also known to be necessary. Using an isomorphism theorem of Eisenbaum and Kaspi which relates Markov local times and permanental processes, we obtain a general sufficient condition for the joint continuity of local times.

On classical capacities of infinite-dimensional quantum channels

A coding theorem for entanglement-assisted communication via an infinite-dimensional quantum channel with linear constraints is extended to a natural degree of generality. Relations between the entanglement-assisted classical capacity and ?-capacity of constrained channels are obtained, and conditions for their coincidence are given. Sufficient conditions for continuity of the entanglement-assisted classical capacity as a function of a channel are obtained. Some applications of the obtained results to analysis of Gaussian channels are considered. A general (continuous) version of the fundamental relation between coherent information and the measure of privacy of classical information transmission via an infinite-dimensional quantum channel is proved.

On a continuity of characteristic exponents of linear discrete time-varying systems

On a continuity of characteristic exponents of linear discrete time-varying systemsIn this paper we present a sufficient condition for continuity of Lyapunov exponents of discrete time-varying linear system. Basing on this result we show that Lyapunov exponents of time-invariant systems depend continuously on the time-varying perturbations.

Continuity Of The Selectorial Maps

In this paper the selectorial map notion is introduced and the continuity property of selectorial map is studied. The necessary and sufficient conditions for continuity of selectorial maps are obtained.

Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds

In this paper we continue the investigation of the regularity of optimal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma-Trudinger-Wang and the geometry of the cut locus. We derive some sufficient and some necessary conditions to ensure that the optimal transport map is always continuous. In dimension two, we can sharpen our result into a necessary and sufficient condition. We also provide some sufficient conditions for regularity, and review existing results.

Formula omitted]-approximative compactness and continuity of the generalized projection operator in Banach spaces

We introduce the notion of W -approximative (weak) compactness for a nonempty subset of a smooth Banach space, and present its characterizations. With this notion, we give the sufficient conditions for continuity and the upper semi-continuity of the generalized projection operator.

Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation

The behavior of solutions to the biharmonic equation is well-understood in smooth domains. In the past two decades substantial progress has also been made for the polyhedral domains and domains with Lipschitz boundaries. However, very little is known about higher order elliptic equations in the general setting. In this paper we introduce new integral identities that allow to investigate the solutions to the biharmonic equation in an arbitrary domain. We establish: (1) boundedness of the gradient of a solution in any three-dimensional domain; (2) pointwise estimates on the derivatives of the biharmonic Green function; (3) Wiener-type necessary and sufficient conditions for continuity of the gradient of a solution.