Continuity Properties Research Articles

Continuity and sensitivity analysis of parameterized Nash games

In this paper we consider continuity of the set of Nash equilibria and approximate Nash equilibria for parameterized games. For parameterized games with unique Nash equilibria, the continuity of this equilibrium mapping is well-known. However, when the equilibria need not be unique, there may exist discontinuities in the equilibrium mapping. Because the parameters of a game need to be estimated in practice, these discontinuities can result in an incorrect understanding of the Nash equilibria and, thus also, the decision-making of the players. The focus of this work is to summarize continuity properties for parameterized Nash equilibria and prove continuity via the approximate Nash game with uniformly continuous objective functions over potentially non-compact strategy spaces. That is, we consider a robust representation for the set of Nash equilibria so that small perturbations in the parameters lead to small changes in the set of Nash equilibria.

Incentive design and profit sharing in multi-modal transportation networks

We consider the situation where multiple transportation service providers cooperate to offer an integrated multi-modal platform to enhance the convenience to the passengers through ease in multi-modal journey planning, payment, and first and last mile connectivity. This market structure allows the multi-modal platform to coordinate profits across modes and also provide incentives to the passengers. Accordingly, in this paper, we use cooperative game theory coupled with the hyperpath-based stochastic user equilibrium framework to study such a market. We assume that the platform sets incentives (price discount or excess charge on passengers) along every edge in the transportation network. We derive the continuity and monotonicity properties of the equilibrium flow with respect to the incentives along every edge. The optimal incentives that maximize the profit of the platform are obtained through a two time-scale stochastic approximation algorithm. We use the asymmetric Nash bargaining solution to design a fair profit sharing scheme among the service providers. We show that the profit for each service provider increases after cooperation on such a platform. Finally, we complement the theoretical results through two numerical simulations.

Continuity properties of Lyapunov exponents for surface diffeomorphisms

We study the entropy and Lyapunov exponents of invariant measures $\mu$ for smooth surface diffeomorphisms $f$, as functions of $(f,\mu)$. The main result is an inequality relating the discontinuities of these functions. One consequence is that for a $C^\infty$ surface diffeomorphisms, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semi-continuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion for the existence of SRB measures with positive entropy.

Sub-linear convergence of a stochastic proximal iteration method in Hilbert space

We consider a stochastic version of the proximal point algorithm for convex optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in this form. Indeed, most related results are confined to the finite-dimensional setting, where error bounds could depend on the dimension of the space. On the other hand, the few existing results in the infinite-dimensional setting only prove very weak types of convergence, owing to weak assumptions on the problem. In particular, there are no results that show strong convergence with a rate. In this article, we bridge these two worlds by assuming more regularity of the optimization problem, which allows us to prove convergence with an (optimal) sub-linear rate also in an infinite-dimensional setting. In particular, we assume that the objective function is the expected value of a family of convex differentiable functions. While we require that the full objective function is strongly convex, we do not assume that its constituent parts are so. Further, we require that the gradient satisfies a weak local Lipschitz continuity property, where the Lipschitz constant may grow polynomially given certain guarantees on the variance and higher moments near the minimum. We illustrate these results by discretizing a concrete infinite-dimensional classification problem with varying degrees of accuracy.

The augmented Lagrangian method can approximately solve convex optimization with least constraint violation

There are many important practical optimization problems whose feasible regions are not known to be nonempty or not, and optimizers of the objective function with the least constraint violation prefer to be found. A natural way for dealing with these problems is to extend the nonlinear optimization problem as the one optimizing the objective function over the set of points with the least constraint violation. This leads to the study of the shifted problem. This paper focuses on the constrained convex optimization problem. The sufficient condition for the closedness of the set of feasible shifts is presented and the continuity properties of the optimal value function and the solution mapping for the shifted problem are studied. Properties of the conjugate dual of the shifted problem are discussed through the relations between the dual function and the optimal value function. The solvability of the dual of the optimization problem with the least constraint violation is investigated. It is shown that, if the least violated shift is in the domain of the subdifferential of the optimal value function, then this dual problem has an unbounded solution set. Under this condition, the optimality conditions for the problem with the least constraint violation are established in term of the augmented Lagrangian. It is shown that the augmented Lagrangian method has the properties that the sequence of shifts converges to the least violated shift and the sequence of multipliers is unbounded. Moreover, it is proved that the augmented Lagrangian method is able to find an approximate solution to the problem with the least constraint violation and it has linear rate of convergence under an error bound condition. The augmented Lagrangian method is applied to an illustrative convex second-order cone constrained optimization problem with least constraint violation and numerical results verify our theoretical results.

Langevin dynamic for the 2D Yang–Mills measure

We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions.To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric.To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations.Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.

The Infinite Divisibility of Compound Negative Binomial Distribution as the Sum of Laplace Distribution

The infinite divisibility of compound negative binomial distribution especially as the sum of Laplace distribution has important roles in governing the mathematical model based on its characteristic function. In order to show the property of characteristic function of this compound negative binomial distribution, it is used Fourier-Stieltjes transform to have characteristic function and then governed the property of continuity and quadratic form by using analytical approaches. The infinite divisibility property is obtained by introducing a function satisfied the criteria to be a characteristic function such that its convolution has the characteristic function of compound negative binomial distribution. Then it is concluded that the characteristic function of compound negative binomial distribution as the sum of Laplace distribution satisfies the property of continuity, quadratic form and infinite divisibility.

Coexistence of ferroelectric phases in barium titanate single-domain thin films

In about twenty years, the temperature-misfit strain phase diagrams have become valuable tools to describe ferroelectric thin film properties. The background theoretical approach consists in searching for the position of the global minimum of the potential landscape and deducing the thermodynamically stable phase. In temperature-misfit strain phase diagrams, lines separate the stable phase regions, and the analysis of the polarization continuity properties across these lines allows us to determine the order of phase transition. This work revealed additional important information relative to the properties of first order phase transitions; we determined the limit of stability of the metastable phases and plotted the corresponding lines in the traditional temperature-misfit strain diagrams. These one dimensional boundaries are the analog to the superheating and supercooling temperature points in the context of bulk ferroelectric crystal. We highlight two dimensional coexistence regions adjacent to first order thermodynamic transition lines. The location and the shape of these regions are strongly dependent on the set of parameters used to describe the bulk crystal. We illustrate the interest in the revelation of coexistence regions, showing that the corresponding multi-stability properties could be used to design non-binary memory cells by using barium titanate thin films.

The Calderón Problem for the Fractional Wave Equation: Uniqueness and Optimal Stability

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension $n\in \mathbb{N}$.

Green’s Functions for First-Order Systems of Ordinary Differential Equations without the Unique Continuation Property

This paper is a contribution to the spectral theory associated with the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory. Despite this fact we are able to describe symmetric restrictions of the maximal relation associated with $Ju'+qu=wf$ and show the existence of Green's functions for self-adjoint relations even if unique continuation of solutions fails.

On the continuity of topological entropy of certain partially hyperbolic diffeomorphisms

In this paper, we consider certain partially hyperbolic diffeomorphisms (PHDs) with centre of arbitrary dimension and obtain continuity properties of the topological entropy under C 1 perturbations. The systems considered have subexponential growth in the centre direction and uniform exponential growth along the unstable foliation. Our result applies to PHDs which are Lyapunov stable in the centre direction. It applies to another important class of systems which do have subexponential growth in the centre direction, for which we develop a technique to use exponential mixing property of the systems to get uniform distribution of unstable manifolds. A primary example are the translations on homogeneous spaces which may have centre of arbitrary dimension and of polynomial orbit growth.

Markovian imprecise jump processes: Extension to measurable variables, convergence theorems and algorithms

The existing framework of Markovian imprecise jump processes, also known as imprecise continuous-time Markov chains, is limited to bounded real variables that depend on the state of the system at a finite number of (future) time points. This is an issue in many applications, because typically the variables of interest depend on the state of the system at all time points in some – possibly unbounded – (future) interval, and they can be unbounded or even extended real valued; examples of such variables are temporal averages, the number of (selected) jumps in some interval and hitting times. To eliminate this shortcoming, we assume that the sample paths are càdlàg and use measure theory to extend the domain of Markovian imprecise jump processes to extended real-valued variables that may depend on the state of the system at all (future) time points – that is, the extended real variables that are bounded below or above and are measurable with respect to the σ-algebra generated by the cylinder events. We investigate the continuity properties of the extended lower and upper expectations with respect to point-wise convergent sequences, and this yields generalisations of the Monotone Convergence Theorem and Lebesgue's Dominated Convergence Theorem. For two particular classes of variables, we strengthen these convergence theorems and present an iterative scheme to approximate their lower and upper expectations. The first class is the number of selected jumps in some interval, and the second class are real variables that take the form of a Riemann integral over some interval; this second class includes temporal averages and occupancy times.

Robust treatment of cross-points in optimized Schwarz methods

In the field of domain decomposition, the optimized Schwarz method (OSM) appears to be one of the prominent techniques to solve large scale time-harmonic wave propagation problems. It is based on appropriate transmission conditions using carefully designed impedance operators to exchange information between subdomains. The efficiency of such methods is however hindered by the presence of cross-points, where more than two subdomains abut, if no appropriate treatment is provided. In this work, we propose a new treatment of the cross-point issue for the Helmholtz equation that remains valid in any geometrical interface configuration. We exploit the multi-trace formalism to define a new exchange operator with suitable continuity and isometry properties. We then develop a complete theoretical framework that generalizes classical OSM to partitions with cross-points and contains a rigorous proof of geometric convergence, uniform with respect to the mesh discretization, for appropriate positive impedance operators. Extensive numerical results in 2D and 3D are provided as an illustration of the proposed method.

The fractional Schrödinger equation on compact manifolds: global controllability results

The goal of this work is to prove global controllability and stabilization properties for the fractional Schrödinger equation on d-dimensional compact Riemannian manifolds without boundary (M, g). To prove our main results we use techniques of pseudo-differential calculus on manifolds. More precisely, by using microlocal analysis, we are able to prove propagation of regularity which together with the so-called Geometric Control Condition and Unique Continuation Property help us to prove global control results for the system under consideration. As a main novelty this manuscript presents the relation between the geometric control condition and the controllability for the fractional Schrödinger equation.

On Some Properties of Dyadic Operators

In this paper, the objects of our investigation are some dyadic operators, including dyadic shifts, multilinear paraproducts and multilinear Haar multipliers. We mainly focus on the continuity and compactness of these operators. First, we consider the continuity properties of these operators. Then, by the Frechet–Kolmogorov–Riesz–Tsuji theorem, the non-compactness properties of these dyadic operators will be studied. Moreover, we show that their commutators are compact with $\operatorname{CMO}$ functions, which is quite different from the non-compactness properties of these dyadic operators. These results are similar to those for Calderon–Zygmund singular integral operators.

On δ-Sequential Spaces

Sequence converges is an important research object in topology and analysis, since it is closely related to continuity, compactness and other related properties. In this article, we use the notion of regularly convergence, which is a generalization of the convergence notion, to define the regularly sequentially closed sets and the operator of regularly sequential closure; then we consider thier charactarizations and prove that the convergence and regularly convergence are coincide in regular spaces. Finally we introduce new axioms by involving δ-open sets and regular open sets with the concept of regularly convergence; namely δ-sequential space and r-sequential space, when we show that there are no general relations between these new spaces and the sequential space, in addition we prove some statements as; r-first countable space is δ-sequential, r-sequential space is stronger than δ-sequential space and the spaces δ-sequential and sequential are coincide both in regular spaces and in compact r-T_2 spaces.

Strict convergence with equibounded area and minimal completely vertical liftings

Minimal lifting measures of vector-valued functions of bounded variation were introduced by Jerrard–Jung. They satisfy strong continuity properties with respect to the strict convergence in BV. Moreover, they can be described in terms of the action of the optimal Cartesian currents enclosing the graph of u. We deal with a good notion of completely vertical lifting for maps with values into the two dimensional Euclidean space. We then prove lack of uniqueness in the high codimension case. Relationship with the relaxed area functional in the strict convergence is also discussed.

Linear operators associated with p-Dunford, p-Pettis and p-Bochner integrable functions with values in a Banach space

In the present paper, we study several aspects of linear operators associated with p-Bochner, p-Dunford and p-Pettis integrable functions defined on a finite measure space with values in a Banach space. Continuity and compactness properties of said operators are discussed in details. We investigate the operators in the light of p-summingness and order boundedness. We study p′-Pettis representability and p′-Bochner representability of an operator T∈L(Lp(μ),X), p∈(1,∞). Equicompactness and collective compactness of a family of linear operators associated with a family of integrable functions are also studied. Some results on equicompactness and collective compactness are obtained in the sequel.

Exponential decay for the KdV equation on ℝ with new localized dampings

In this paper, we prove several results on the exponential decay in $L^{2}$ norm of the KdV equation on the real line with localized dampings. First, for the linear KdV equation, the exponential decay holds if and only if the averages of the damping coefficient on all intervals of a fixed length have a positive lower bound. Moreover, under the same damping condition, the exponential decay holds for the (nonlinear) KdV equation with small initial data. Finally, with the aid of certain properties of propagation of regularity in Bourgain spaces for solutions of the associated linear system and the unique continuation property, the exponential decay for the KdV equation with large data holds if the damping coefficient has a positive lower bound on $E$, where $E$ is equidistributed over the real line and the complement $E^{c}$ has a finite Lebesgue measure.

Uniqueness of inverse source problems for general evolution equations

In this paper, we investigate inverse source problems for a wide range of PDEs of parabolic and hyperbolic types as well as time-fractional evolution equations by partial interior observation. Restricting the source terms to the form of separated variables, we establish uniqueness results for simultaneously determining both temporal and spatial components without non-vanishing assumptions at [Formula: see text], which seems novel to the best of our knowledge. Remarkably, mostly we allow a rather flexible choice of the observation time not necessarily starting from [Formula: see text], which fits into various situations in practice. Our main approach is based on the combination of the Titchmarsh convolution theorem with unique continuation properties and time-analyticity of the PDEs under consideration.