Compactness Assumption Research Articles

The generalized Łojasiewicz inequality for definable and subanalytic multifunctions

This paper is devoted to obtaining the Łojasiewicz inequality (version for two functions), in both the definable and subanalytic cases, under the most relaxed assumptions. It means that we drop the usual continuity and compactness assumptions. In the second part of the paper we concentrate on the Łojasiewicz inequality for multifunctions and apply it to the natural multifunctions related to the medial axis of a set (basic notion in pattern recognition).

Evolutionary equations are G-compact

We prove a compactness result related to G-convergence for autonomous evolutionary equations in the sense of Picard. Compared to previous work related to applications, we do not require any boundedness or regularity of the underlying spatial domain; nor do we assume any periodicity or ergodicity assumption on the potentially oscillatory part. In terms of abstract evolutionary equations, we remove any compactness assumptions of the resolvent modulo kernel of the spatial operator. To achieve the results, we introduced a slightly more general class of material laws. As a by-product, we also provide a criterion for G-convergence for time-dependent equations solely in terms of static equations.

Sections of Fell bundles over étale groupoids

We construct the reduced and essential C*-algebra of a Fell bundle over an étale groupoid (in full generality, without any second countability, local compactness or Hausdorff assumptions, even on the unit space) directly from sections under convolution. This eliminates the j-map commonly seen in the groupoid and Fell bundle C*-algebra literature. We further show how multiplier algebras and other Banach *-bimodules can again be constructed directly from sections. Finally, we extend Varela's morphisms from C*-bundles to Fell bundles, thus making the reduced C*-algebra construction functorial.

On equilibria in constrained generalized games with the weak continuous inclusion property

In this paper, we present results that use Himmelberg's fixed point theorem to highlight substantive trade-offs between compactness, continuity and convexity postulates in the setting of a constrained generalized game. The primary contribution is a focus on weakening the compactness assumption on the action sets and on two versions of the continuity assumption encapsulated as the continuous inclusion property (CIP). We show that the results are invariant to cardinality assumptions on the set of agents and on the dimension of the space in which the action sets are situated. This yields as special cases recent results in both mathematical economics and applied mathematics, and underscores the development of a rich and useful theory in bringing these two registers and two communities together.

Optimal Control of Probability on a Target Set for Continuous-Time Markov Chains

In this paper, a stochastic optimal control problem is considered for a continuous-time Markov chain taking values in a denumerable state space over a fixed finite horizon. The optimality criterion is the probability that the process remains in a target set before and at a certain time. The optimal value is a superadditive capacity of target sets. Under some minor assumptions for the controlled Markov process, we establish the dynamic programming principle, based on which we prove that the value function is a classical solution of the Hamilton-Jacobi-Bellman equation on a discrete lattice space. We then prove that there exists an optimal deterministic Markov control under the compactness assumption of control domain. We further prove that the value function is the unique solution of the HJB equation. We also consider the case starting from the outside of target set and give the corresponding results. Finally, we apply our results to two examples.

On entropy martingale optimal transport theory

In this paper, we give an overview of (nonlinear) pricing-hedging duality and of its connection with the theory of entropy martingale optimal transport (EMOT), recently developed, and that of convex risk measures. Similarly to Doldi and Frittelli (Finance Stoch 27(2):255–304, 2023), we here establish a duality result between a convex optimal transport and a utility maximization problem. Differently from Doldi and Frittelli (Finance Stoch 27(2):255–304, 2023), we provide here an alternative proof that is based on a compactness assumption. Subhedging and superhedging can be obtained as applications of the duality discussed above. Furthermore, we provide a dual representation of the generalized optimized certainty equivalent associated with indirect utility.

Spatially Resolved Modeling of the Nonlinear Dynamics of a Laminar Premixed Flame With a Multilayer Perceptron—Convolution Autoencoder Network

Abstract This work presents a multilayer perceptron-convolutional auto-encoder (MLP-CAE) neural network, which accurately predicts the two-dimensional flame dynamics of an acoustically excited premixed laminar flame. The architecture maps the acoustic perturbation time series into a heat release rate field, capturing flame lengths and shapes. This extends previous neural network models, which predicted only the field-integrated value. The MLP-CAE comprises two submodels: an MLP and a CAE. The idea behind the CAE network is to find a lower dimensional latent space of the heat release rate field. The MLP is responsible for modeling the flame dynamics by transforming the acoustic forcing signal into this latent space, enabling the decoder to produce the flow field distributions. To train the MLP-CAE, computational fluid dynamics (CFD) flame simulations with a broadband acoustic forcing were used. Its normalized amplitude was set to 0.5 and 1.0, ensuring a nonlinear flame response. The network was found to accurately predict the perturbed flame shapes. Additionally, it conserved the correct frequency response as verified by the global and local flame describing functions. The MLP-CAE provides a building block toward a potential shift away from a “0D” flame analysis with the acoustic compactness assumption. Combined with an acoustic network, the generated flame fields could provide more physical insight into the thermoacoustic dynamics. Those capabilities do not come at an additional significant computational cost, as even previous nonspatial flame models had to train on the CFD data, which included field distributions.

Weighted pseudo S-asymptotically Bloch type periodic solutions for a class of mean field stochastic fractional evolution equations

This paper concerns a class of mean-field stochastic fractional evolution equations. Initially, we establish some auxiliary results for weighted pseudo $S$-asymptotically Bloch type periodic stochastic processes. Without a compactness assumption on the resolvent operator and some additional conditions on forced terms, the existence and uniqueness of weighted pseudo $S$-asymptotically Bloch type periodic mild solutions on the real line of the referred equation are obtained. In addition, we show the existence of weighted pseudo $S$-asymptotically Bloch type periodic mild solutions with sublinear growth assumptions on the drift term and compactness conditions. Finally, an example is provided to verify the main outcomes.

The damped vibrating string equation on the positive half-line

In this paper, the existence of a solution to the problem describing the small vertical vibration of an elastic string on the positive half-line is investigated in the case when both viscous and material damping coefficients are present. The result is obtained by transforming the original partial differential equation into an appropriate abstract second-order ordinary differential equation in a suitable infinite dimensional space. The abstract problem is then studied using the combination of the Kakutani fixed point theorem together with the approximation solvability method and the weak topology. The applied procedure enables obtaining the existence result also for problems depending on the first derivative, without any strict compactness assumptions put on the right-hand side and on the fundamental system generated by the linear term. The paper ends by applying the obtained result to the studied mathematical model describing the small vertical vibration of an elastic string with a nonlinear Balakrishnan–Taylor-type damping term.

Optimal Controls for Fractional Backward Nonlocal Evolution Systems

This article analyzes a Caputo type fractional backward nonlocal evolution control system. By means of a joint combination of resolvent theory with the approximation solvability technique, we dispense with the compactness assumption on the semigroup and the Lipschitz restriction on the nonlinear term when we treat the existence of solutions. Furthermore, we launch a new method of formulating minimizing sequences twice and the weak topology technique to explore the optimal control problem (OP). Finally, the plausibility of our mentioned results is supported by a simple application.

Relative Morse Index Theory Without Compactness Assumption

We develop the relative Morse index theory for linear self-adjoint operator equation without compactness assumption and give the relationship with the index defined in [J. Differential Equations, 2016, 260(4): 3749–3784].

Singular cotangent models in fluids with dissipation

In this article we analyze several mathematical models with singularities where the classical cotangent model is replaced by a b-cotangent model. We provide physical interpretations of the singular symplectic geometry underlying in b-cotangent bundles featuring two models: the canonical (or non-twisted) model and the twisted one. The canonical one models systems on manifolds with boundary and the twisted one represents Hamiltonian systems with a singularity on the fiber. The twisted cotangent model includes (for linear potentials) the case of fluids with dissipation. We prove (non)-existence of cotangent lift dynamics and show the existence of an infinite number of escape orbits in this model. We also discuss more general physical interpretations of the twisted and non-twisted b-symplectic models. Twisted b-symplectic models yield in a natural way escape orbits that go to the critical set. Under compactness assumptions those escape orbits are continued as singular periodic orbits in the sense of Miranda and Oms (2021) and Miranda (2020). These models offer a Hamiltonian formulation for systems which are dissipative, extending the horizons of Hamiltonian dynamics and opening a new approach to study non-conservative systems.

Accelerated Distributed Projected Gradient Descent for Convex Optimization with Clique-wise Coupled Constraints

This paper addresses a distributed convex optimization problem with a class of coupled constraints, which arises in a multi-agent system composed of multiple communities modeled by cliques. First, we propose a fully distributed gradient-based algorithm with a novel operator inspired by the convex projection, called the clique-based projection. Next, we scrutinize the convergence properties for both diminishing and fixed step sizes. For diminishing ones, we show the convergence to an optimal solution under the assumptions of the smoothness of an objective function and the compactness of the constraint set. Additionally, when the objective function is strongly monotone, the exact convergence to the unique solution is proved without the assumption of compactness. For fixed step sizes, we prove the convergence rate of O(1/k) concerning an approximated objective residual under the assumption of the smoothness of the objective function. Furthermore, we apply Nesterov's acceleration method to the proposed algorithm and establish the convergence rate of O(1/k2). Numerical experiments illustrate the effectiveness of the proposed method.

Two-stage distributionally robust noncooperative games: Existence of Nash equilibrium and its application to Cournot–Nash competition

Two-stage distributionally robust stochastic noncooperative games with continuous decision variables are studied. In such games, each player solves a two-stage distributionally robust optimization problem depending on the decisions of the other players. Existing studies in this area have been limited with strict assumptions, such as linear decision rules, and supposed that each player solves a two-stage linear distributionally robust optimization with a specifically structured ambiguity set. This limitation motivated us to generalize and analyze the game in a nonlinear case. The contributions of this study are (ⅰ) demonstrating the conditions for the existence of two-stage Nash equilibria under convexity and compactness assumptions, and (ⅱ) consideration of a two-stage distributionally robust Cournot–Nash competition as an application, as well as an investigation into the conditions for the existence of market equilibria in an economic sense. We also report some results of numerical experiments to illustrate how distributional robustness affects the decision of each player in the Cournot–Nash competition.

Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups

Let k k be a number field and X X a smooth, geometrically integral quasi-projective variety over k k . For any linear algebraic group G G over k k and any G G -torsor g : Z → X g: Z \to X , we observe that if the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S S for all twists of Z Z by elements in H^1_{\text {\'{e}t}}(k,G), then the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S S for X X . As an application, we show that any homogeneous space of the form G / H G/H with G G a connected linear algebraic group over k k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some compactness assumptions when k k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G / H G/H with G G semisimple simply connected and H H finite, using the theory of torsors and descent.

Spectral properties of the inhomogeneous Drude-Lorentz model with dissipation

We establish spectral enclosures and spectral approximation results for the inhomogeneous lossy Drude-Lorentz system with purely imaginary poles, in a possibly unbounded Lipschitz domain of R3. Under the assumption that the coefficients θe, θm of the material are asymptotically constant at infinity, we prove that spectral pollution due to domain truncation can lie only in the essential numerical range of a curlcurl0−f(ω) pencil.As an application, we consider a conducting metamaterial at the interface with the vacuum; we prove that the complex eigenvalues with non-trivial real part lie outside the set of spectral pollution. We believe this is the first result of enclosure of spectral pollution for the Drude-Lorentz model without assumptions of compactness on the resolvent of the underlying Maxwell operator.

STRONG COMPACTNESS, SQUARE, GCH, AND WOODIN CARDINALS

Abstract We show the consistency, relative to the appropriate supercompactness or strong compactness assumptions, of the existence of a non-supercompact strongly compact cardinal $\kappa _0$ (the least measurable cardinal) exhibiting properties which are impossible when $\kappa _0$ is supercompact. In particular, we construct models in which $\square _{\kappa ^+}$ holds for every inaccessible cardinal $\kappa $ except $\kappa _0$ , GCH fails at every inaccessible cardinal except $\kappa _0$ , and $\kappa _0$ is less than the least Woodin cardinal.

SOLO FTRL ALGORITHM FOR PRODUCTION MANAGEMENT WITH TRANSFER PRICES

We consider a firm producing and selling $d$ commodities, and consisting from $n$ production and $m$ sales divisions. The firm manager tries to stimulate the best division performance by sequentially selecting internal commodity prices (transfer prices). In the static problem under general strong convexity and compactness assumptions we show that the SOLO FTRL algorithm of Orabona and Pal (2018), applied to the dual problem, gives the estimates of order $T^{-1/4}$ in the number $T$ of iterations for the optimality gap and feasibility residuals. This algorithm uses only the information on division reactions to current prices. It does not depend on any parameters and requires no information on the production and cost functions. The results of similar nature are obtained for the dynamic problem, where these functions depend on an i.i.d. sequence of random variables. We present two computer experiments with one and two commodities. In the static case the transfer prices and the difference between the supply and demand demonstrate fast stabilization. In the dynamic case the same quantities fluctuate around equilibrium values after a short transition phase.

A novel approach for solving simultaneously one-parameter nonexpansive semigroup, convex minimization and fixed point problems involving set-valued operators

"In this paper, we introduce a new iterative process for solving simultaneously one-parameter nonexpansive semigroup, convex minimization and fixed point problems involving set-valued operators in real Hilbert spaces and establish strong convergence theorems for the proposed iterative process. There is no compactness assumption. Our results improve important recent results."

Equilibrium non-existence in generalized games

A generalized game is a strategic situation in which agents' behavior restricts their opponents' available action choices, giving rise to interdependencies in terms of what strategy profiles remain mutually feasible. This paper proposes a novel example of a simple jointly convex generalized game in which the well-known convexity, compactness, continuity, and concavity assumptions are satisfied, but no Nash equilibrium exists. The essence of this contribution lies in answering a question left open by Banks and Duggan (2004): whether the supplemental condition of lower hemicontinuity of feasibility correspondences can be dropped from these authors' equilibrium-existence theorem. It cannot.