Regularity Properties Research Articles

Existence of solutions for stress-rate type strain-limiting viscoelasticity in Gevrey spaces.

In this work, we deal with a one-dimensional stress-rate type model for the response of viscoelastic materials, in relation to the strain-limiting theory. The model is based on a constitutive relation of stress-rate type. Unlike classical models in elasticity, the unknown of the model under consideration is uniquely the stress, avoiding the use of the deformation. Here, we treat the case of periodic boundary conditions for a linearized model. We determine an optimal function space that ensures the local existence of solutions to the linearized model around certain steady states. This optimal space is known as the Gevrey-class [Formula: see text], which characterizes the regularity properties of the solutions. The exponent [Formula: see text] in the Gevrey-class reflects the specific dispersion properties of the equation itself. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

Boundedness of Wolff-type potentials and applications to PDEs

We provide a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff–Havin–Maz’ya type. As a consequence, we prove a reduction principle for that integral operators, that is, a characterization of those rearrangement invariant spaces between which the potentials are bounded via a one-dimensional inequality of Hardy-type.Since the special case of the mentioned potential is known to control precisely very weak solutions to a broad class of quasilinear elliptic PDEs of non-standard growth, we infer the local regularity properties of the solutions in rearrangement invariant spaces for prescribed classes of data.

Destructive computing with winner-lose-all competition in multi-layered neural networks

The present paper aims to propose a new learning method based on destructive computing, contrary to the conventional progressive computing or the steady-step learning. In spite of the existence of a large amount of biased or distorted information in inputs, the conventional learning methods fundamentally aim to gradually acquire information that is as faithful as possible to inputs, which has prevented us from acquiring intrinsic information hidden in the deepest level of inputs. At this time, it is permitted to suppose a leap to that level by changing information at hand not gradually but drastically. In particular, for the really drastic change of information, we introduce the winner-lose-all (WLA) to drastically destroy the supposedly most important information for immediately reaching or leaping to intrinsic information, hidden in complicated inputs. The method was applied to a target-marketing problem. The experimental results show that, with the new method, multi-layered neural networks had an ability to disentangle complicated network configurations into the simplest ones with simple and independent correlation coefficients between inputs and targets. This was realized by drastically changing the information content in the course of learning and, correspondingly, by mixing regular and irregular properties over connection weights.

Secant-inexact projection algorithms for solving a new class of constrained mixed generalized equations problems

In this paper, a new version of a secant-type method for solving constrained mixed generalized equations is addressed. The method is a combination of the secant method applied to generalized equations with the conditional gradient method. We use the contraction mapping principle to establish the convergence results. Moreover, by assuming the Lipschitz condition on the gradient and the metric regularity property, we show that the sequence generated by the proposed algorithm is well-defined and locally convergent for a solution with linear or superlinear rate.

Sharp log-Sobolev inequalities in [formula omitted] spaces with applications

Given p,N>1, we prove the sharp Lp-log-Sobolev inequality on noncompact metric measure spaces satisfying the CD(0,N) condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in CD(0,N) spaces, symmetrization, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf–Lax semigroup in CD(0,N) spaces. The proof of this result uses Hamilton–Jacobi inequality and Sobolev regularity properties of the Hopf–Lax semigroup, which turn out to be essential in the present setting of nonsmooth and noncompact spaces. Moreover, a sharp Gaussian-type L2-log-Sobolev inequality and a hypercontractivity estimate are obtained in RCD(0,N) spaces. Our results are new, even in the smooth setting of Riemannian/Finsler manifolds. In particular, an extension of the celebrated rigidity result of Ni (2004) [55] on Riemannian manifolds will be a simple consequence of our sharp log-Sobolev inequality.

The (reflected) Eberlein convolution of measures

In this paper, we study the properties of the Eberlein convolution of measures and introduce a reflected version of it. For functions we show that the reflected Eberlein convolution can be seen as a translation invariant function-valued inner product. We study its regularity properties and show its existence on suitable sets of functions. For translation bounded measures we show that the (reflected) Eberlein convolution always exists along subsequences of the given sequence, and is a weakly almost periodic and Fourier transformable measure. We prove that if one of the two measures is mean almost periodic, then the (reflected) Eberlein convolution is strongly almost periodic. Moreover, if one of the measures is norm almost periodic, so is the (reflected) Eberlein convolution.

On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular problems

Abstract We establish existence results for a class of mixed anisotropic and nonlocal p-Laplace equations with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, the Harnack inequality for solutions and the weak Harnack inequality for supersolutions.

A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique

In this paper we would like to address the classical optimization problem of minimizing a proper, convex and lower semicontinuous function via the second order in time dynamics, combining viscous and Hessian-driven damping with a Tikhonov regularization term. In our analysis we heavily exploit the Moreau envelope of the objective function and its properties as well as Tikhonov regularization properties, which we extend to a nonsmooth case. We introduce the setting, which at the same time guarantees the fast convergence of the function (and Moreau envelope) values and strong convergence of the trajectories of the system to a minimal norm solution—the element of the minimal norm of all the minimizers of the objective. Moreover, we deduce the precise rates of convergence of the values for the particular choice of parameters. Various numerical examples are also included as an illustration of the theoretical results.

Infinite Order of The Solutions of Higher Order Homogeneous Linear Differential Equations with Entire Coefficients Having Completely Regular Growth Property and Bounded Components Fatou Set

The homogeneous higher order complex linear differential equations (n-thCLDEs) with entire functions is considered in this paper. We investigated some conditions that can be put on the coefficients which guarantee that any nonzero solution of such equations has infinite order. The conditions we stated are the completely regular growth (CRG), the characteristic function of some coefficients is approximately equals to the logarithm of its maximum modulus and the Denjoy’s conjecture (DC) property.

Internal aggregation models with multiple sources and obstacle problems on Sierpiński gaskets

We consider the doubly infinite Sierpiński gasket graph \mathsf{SG}_0 , rescale it by factor 2^{-n} , and on the rescaled graphs \mathsf{SG}_n=2^{-n}\mathsf{SG}_0 , for every n\in\mathbb{N} , we investigate the limit shape of three aggregation models with initial configuration \sigma_n of particles supported on multiple vertices. The models under consideration are: divisible sandpile in which the excess mass is distributed among the vertices until each vertex is stable and has mass less or equal to one, internal DLA in which particles do random walks until finding an empty site, and rotor aggregation in which particles perform deterministic counterparts of random walks until finding an empty site. We denote by \mathsf{SG}=\text{cl}(\bigcup_{n=0}^{\infty}\mathsf{SG}_n) the infinite Sierpiński gasket, which is a closed subset of \mathbb{R}^2 , for which \mathsf{SG}_n represents the level- n approximating graph, and we consider a continuous function \sigma\colon \mathsf{SG}\to\mathbb{N} . For \sigma we solve the obstacle problem, and we describe the noncoincidence set D\subset \mathsf{SG} as the solution of a free boundary problem on the fractal \mathsf{SG} . If the discrete particle configurations \sigma_n on the approximating graphs \mathsf{SG}_n converge pointwise to the continuous function \sigma on the limit set \mathsf{SG} , we prove that, as n\to\infty , the scaling limits of the three aforementioned models on \mathsf{SG}_n starting with initial particle configuration \sigma_n converge to the deterministic solution D of the free boundary problem on the limit set \mathsf{SG}\subset\mathbb{R}^2 . For D we also investigate boundary regularity properties.

Investigation of the glass forming regularity and thermal behavior of lead-free V2O5-TeO2-RO (R = Ca, Sr, Ba) low temperature sealing glass for electronics packaging

Lead-free V2O5-TeO2 glass systems are promising new sealant materials for electronics vacuum packaging, with low-melting temperature and unique light absorption. The glass forming regularity, structure and corresponding properties of the V2O5-TeO2-RO (R=Ca, Sr, Ba) glass system are systematically studied and analyzed in this paper. The results show that there is a wide glass forming region for the V2O5-TeO2-RO glass system, and with the increase of TeO2/V2O5 molar ratio, the thermal stability of V2O5-TeO2-SrO glass with the RO content of 20 mol% increases and the coefficient of thermal expansion increase from 11.2 to 15.86 × 10–6/°C due to transformation process of glass network structure from the [VO4] to transition state [TeO3 + 1] and [TeO3] group. The higher initial crystallization temperature of 40 mol%V2O5-40 mol%TeO2-20 mol%SrO glass is conducive to the flow of the glass slurry during the sealing process, obtaining a sealing device with good airtightness at extremely low temperatures of 350 °C.

The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.

Robustness of Stochastic Optimal Control to Approximate Diffusion Models Under Several Cost Evaluation Criteria

In control theory, typically a nominal model is assumed based on which an optimal control is designed and then applied to an actual (true) system. This gives rise to the problem of performance loss because of the mismatch between the true and assumed models. A robustness problem in this context is to show that the error because of the mismatch between a true and an assumed model decreases to zero as the assumed model approaches the true model. We study this problem when the state dynamics of the system are governed by controlled diffusion processes. In particular, we discuss continuity and robustness properties of finite and infinite horizon α-discounted/ergodic optimal control problems for a general class of nondegenerate controlled diffusion processes as well as for optimal control up to an exit time. Under a general set of assumptions and a convergence criterion on the models, we first establish that the optimal value of the approximate model converges to the optimal value of the true model. We then establish that the error because of the mismatch that occurs by application of a control policy, designed for an incorrectly estimated model, to a true model decreases to zero as the incorrect model approaches the true model. We see that, compared with related results in the discrete-time setup, the continuous-time theory lets us utilize the strong regularity properties of solutions to optimality (Hamilton–Jacobi–Bellman) equations, via the theory of uniformly elliptic partial differential equations, to arrive at strong continuity and robustness properties. Funding: The research of S. Yüksel was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Analytic regularity of global solutions for the [formula omitted]-equation

In this paper, we delve into the b-family of equations and explore regularity properties of its global solutions. Our findings reveal that, irrespective of the real choice of the constitutive parameter, when the initial datum is confined to an analytic Gevrey function the resulting global solution is analytic in both temporal and spatial variables.

Model Checking of Possibilistic Linear-Time Properties Based on Generalized Possibilistic Decision Processes

Model checking possibilistic linear-time properties was investigated by Li (2017). However, nondeterminism of the system is absent in previous studies. Therefore, in order to permit both possibilistic and nondeterministic choices, we use the generalized possibilistic decision process (GPDP) as a model of the system. First, the definition of GPDP describing the behavior of nondeterministic system is given in detail, the resolution of nondeterminism is performed by using the notion of schedulers, and the semantics of generalized possibilistic linear-temporal logic (GPoLTL) with schedulers are defined. Second, we study possibilistic model checking of some fuzzy linear-time properties under GPDP. Since there are many (infinite) schedulers satisfying a certain linear timing property in a given state of a GPDP, it is particularly critical to study the optimal strategy and its corresponding possible measure, which is called extremal possibility model checking. For some special fuzzy linear-time properties, such as constrained reachability, step-bounded constrained reachability, reachability, always reachability, repeated reachability, persistence reachability, we present complete solution to the optimal (including maximum and minimum cases) possibilistic model checking of the above reachability using the fixpoint techniques. We also introduce fuzzy !-regular properties in GPDP and show that their model checking can be simplified by repeated reachability. The algorithms for model checking are also provided. Additionally, an example is presented to illustrate the methods described in the paper.

The vanishing relaxation time behavior of multi-term nonlocal Jordan–Moore–Gibson–Thompson equations

The family of Jordan–Moore–Gibson–Thompson (JMGT) equations arises in nonlinear acoustics when a relaxed version of the heat flux law is employed within the system of governing equations of sound motion. Motivated by the propagation of sound waves in complex media with anomalous diffusion, we consider here a generalized class of such equations involving two (weakly) singular memory kernels in the principal and non-leading terms. To relate them to the second-order wave equations, we investigate their vanishing relaxation time behavior. The key component of this singular limit analysis are the uniform bounds for the solutions of these nonlinear equations of fractional type with respect to the relaxation time. Their availability turns out to depend not only on the regularity and coercivity properties of the two kernels, but also on their behavior relative to each other and the type of nonlinearity present in the equations.

Regularity of interval max-plus matrices

Max-plus algebra is an algebraic structure in which classical addition and multiplication are replaced by maximum and addition, respectively. We say that the columns of a real matrix A are strongly independent if the max-plus linear system A⊗x=b has a unique solution for at least one real vector b. A square matrix A with strongly independent columns is called strongly regular. The investigation of the properties of regularity is important for applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. The present paper studies three versions of the regularity of matrices and interval matrices, namely, strong regularity, von Neumann regularity and Gondran-Minoux regularity. For each concept of regularity we will present equivalent conditions which can be verified in polynomial time.

Solvability of the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with variable viscosity coefficient

We show the existence and the regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with density-dependent viscosity coefficients, by analyzing a fourth-order nonlinear elliptic equation for the stream function. For some stationary symmetric flows, we reformulate the Navier–Stokes equations as ordinary differential equations and give explicit examples of weak solutions. We present some further (ir-)regularity results in the case of piecewise-constant viscosity coefficients.

The statistical complexity of early-stopped mirror descent

Abstract Recently there has been a surge of interest in understanding implicit regularization properties of iterative gradient-based optimization algorithms. In this paper, we study the statistical guarantees on the excess risk achieved by early-stopped unconstrained mirror descent algorithms applied to the unregularized empirical risk. We consider the set-up of learning linear models and kernel methods for strongly convex and Lipschitz loss functions while imposing only boundedness conditions on the unknown data-generating mechanism. By completing an inequality that characterizes convexity for the squared loss, we identify an intrinsic link between offset Rademacher complexities and potential-based convergence analysis of mirror descent methods. Our observation immediately yields excess risk guarantees for the path traced by the iterates of mirror descent in terms of offset complexities of certain function classes depending only on the choice of the mirror map, initialization point, step size and the number of iterations. We apply our theory to recover, in a clean and elegant manner via rather short proofs, some of the recent results in the implicit regularization literature while also showing how to improve upon them in some settings.

Solving stochastic equations with unbounded nonlinear perturbations

This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of ‘admissible’ unbounded linear operators (not necessarily closed or closable). This is done using Yosida extensions of such unbounded linear operators. After proving the well-posedness of these equations, we also establish the Feller property for the corresponding transition semigroups. Several examples like heat equations and Schrödinger equations with nonlocal perturbations terms are given. Finally, we give an application to a general class of semilinear neutral stochastic equations.