Sufficient Conditions For Regularity Research Articles

Reduced‐order observer‐based finite‐time control for one‐sided Lipschitz nonlinear switched singular systems

SummaryThe reduced‐order observer‐based finite‐time control problem for one‐sided Lipschitz nonlinear switched singular systems is addressed in this paper. First, the design method of the reduced‐order observer is given via state transformation. Then, based on the average dwell time (ADT) approach, some new sufficient conditions for regularity, impulse‐freeness, have a unique solution and finite‐time boundedness (FTB) of the dynamic augmented systems are obtained by exploring the reduced‐order observer‐based controller. Further, the lower finite‐time bound can be obtained by using singular value decomposition method. And the state feedback gain and the observer gain are computed by solving linear matrix inequalities (LMIs). Finally, the validity of the obtained method is illustrated by means of a numerical example and a DC motor system.

Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz-Sobolev-Slobodeckij and total variation spaces

We describe higher-order chain rules for multivariate functions and tensor fields. We estimate Sobolev-Slobodeckij norms, Musielak-Orlicz norms, and the total variation seminorms of the higher derivatives of tensor fields after a change of variables and determine sufficient regularity conditions for the coordinate change. We also introduce a novel higher-order chain rule for composition chains of multivariate functions that is described via nested set partitions and generalized Bell polynomials; it is a natural extension of the Faà di Bruno formula. Our discussion uses the coordinate-free language of tensor calculus and includes Fréchet-differentiable mappings between Banach spaces.

Regular integral transformations on time scales and generalized statistical convergence

In this work, using regular integral transformations on time scales, we generalize the concept of statistical convergence. This enables us not only to unify discrete and continuous cases known in the literature but also to derive new convergence methods with choices of appropriate transformations and time scales. This is a continuation of our earlier work and includes many new methods. We obtain sufficient conditions for regularity of kernel functions on time scales and also we prove a characterization theorem for the generalized statistical convergence. At the end of the paper we display some applications and special cases of our results.

Sufficient conditions for regularity, positive recurrence, and absorption in level‐dependent QBD processes and related block‐structured Markov chains

This paper is concerned with level‐dependent quasi‐birth‐death (LD‐QBD) processes, i.e., multi‐variate Markov chains with a block‐tridiagonal ‐matrix, and a more general class of block‐structured Markov chains, which can be seen as LD‐QBD processes with total catastrophes. Arguments from univariate birth‐death processes are combined with existing matrix‐analytic formulations to obtain sufficient conditions for these block‐structured processes to be regular, positive recurrent, and absorbed with certainty in a finite mean time. Specifically, it is our purpose to show that, as is the case for competition processes, these sufficient conditions are inherently linked to a suitably defined birth‐death process. Our results are exemplified with two Markov chain models: a study of target cells and viral dynamics and one of kinetic proof‐reading in T cell receptor signal transduction.

An Improved Regularity Criterion for the 3D Magnetic Bénard System in Besov Spaces

This article notably targets the more general (extended) function spaces by investigating the regularity of the weak solutions or turbulent solutions to the Cauchy problem of the 3D magnetic Bénard system by converting it into mathematical symmetric form, in the absence of thermal diffusion, in terms of pressure. In that regard, we successfully improved the results by obtaining sufficient integrable regularity conditions for the pressure and gradient pressure in the homogeneous Besov spaces.

Functional group bridge for simultaneous regression and support estimation.

This paper is motivated by studying differential brain activities to multiple experimental condition presentations in intracranial electroencephalography (iEEG) experiments. Contrasting effects of experimental conditions are often zero in most regions and nonzero in some local regions, yielding locally sparse functions. Such studies are essentially a function-on-scalar regression problem, with interest being focused not only on estimating nonparametric functions but also on recovering the function supports. We propose a weighted group bridge approach for simultaneous function estimation and support recovery in function-on-scalar mixed effect models, while accounting for heterogeneity present in functional data. We use B-splines to transform sparsity of functions to its sparse vector counterpart of increasing dimension, and propose a fast nonconvex optimization algorithm using nested alternative direction method of multipliers (ADMM) for estimation. Large sample properties are established. In particular, we show that the estimated coefficient functions are rate optimal in the minimax sense under the L2 norm and resemble a phase transition phenomenon. For support estimation, we derive a convergence rate under the norm that leads to a selection consistency property under δ-sparsity, and obtain a result under strict sparsity using a simple sufficient regularity condition. An adjusted extended Bayesian information criterion is proposed for parameter tuning. The developed method is illustrated through simulations and an application to a novel iEEG data set to study multisensory integration.

Mixing properties of non-stationary INGARCH(1,1) processes

We derive mixing properties for a broad class of Poisson count time series satisfying a certain contraction condition. Using specific coupling techniques, we prove absolute regularity at a geometric rate not only for stationary Poisson-GARCH processes but also for models with an explosive trend. We provide easily verifiable sufficient conditions for absolute regularity for a variety of models including classical (log-)linear models. Finally, we illustrate the practical use of our results for hypothesis testing.

Regularity of Lie groups

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the $C^0$-topological context, and provide necessary and sufficient regularity conditions for the (standard) $C^k$-topological setting. We prove that the evolution map is $C^0$-continuous on its domain $\textit{iff}\hspace{1pt}$ the Lie group $G$ is locally $\mu$-convex. We furthermore show that if the evolution map is defined on all smooth curves, then $G$ is Mackey complete. Under the assumption that $G$ is locally $\mu$-convex, we show that each $C^k$-curve for $k\in \mathbb{N}_{\geq 1}\sqcup\{\mathrm{lip},\infty\}$ is integrable (contained in the domain of the evolution map) $\textit{iff}\hspace{1pt}$ $G$ is Mackey complete and $\mathrm{k}$-confined. The latter condition states that each $C^k$-curve in the Lie algebra $\mathfrak{g}$ of $G$ can be uniformly approximated by a special type of sequence that consists of piecewise integrable curves. A similar result is proven for the case $k\equiv 0$; and, we provide several mild conditions that ensure that $G$ is $\mathrm{k}$-confined for each $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$. We finally discuss the differentiation of parameter-dependent integrals in the (standard) $C^k$-topological context. In particular, we show that if the evolution map is defined and continuous on $C^k([0,1],\mathfrak{g})$ for $k\in \mathbb{N}\sqcup\{\infty\}$, then it is smooth thereon $\textit{iff}\hspace{1pt}$ it is differentiable at zero $\textit{iff}\hspace{1pt}$ $\mathfrak{g}$ is $\hspace{0.2pt}$ Mackey$\hspace{1pt}/ \hspace{1pt}$integral$\hspace{1pt}$ complete for $k\in \mathbb{N}_{\geq 1}\sqcup\{\infty\}\hspace{1pt}/\hspace{1pt}k\equiv 0$. This result is obtained by calculating the directional derivatives explicitly, recovering the standard formulas that hold, e.g., in the Banach case.

On the local in time well-posedness of an elliptic–parabolic ferroelectric phase-field model

We consider a state-of-the-art ferroelectric phase-field model arising from the engineering area in recent years, which is mathematically formulated as a coupled elliptic–parabolic differential system. We utilize the maximal parabolic regularity theory to show the local in time well-posedness of the ferroelectric problem in both 2D and 3D spaces, which is sharp in the sense that the local solution is unique and a blow-up criterion is present. The well-posedness result will firstly be proved under some general assumptions. Afterwards we give sufficient geometric and regularity conditions which will guarantee the fulfillment of the imposed assumptions.

A monotone frequentist measure of evidence for testing variance components in linear mixed models

This paper studies the s-value, a monotone frequentist measure of evidence, in a linear mixed model with a general parameter space that includes null variance components for the random effects. Under this nonregular parameter space, an asymptotic version of the s-value based on likelihood ratio statistics is defined and discussed under null variance components. Three sufficient regularity conditions to obtain the asymptotic distributions of the likelihood statistics are provided and some properties of the asymptotic s-value discussed. It is shown that the s-value satisfies logical consequences which are not satisfied by typical p-values and also that the test based on the s-value has significance level α, i.e., the maximum probability of the s-value not exceeding a fixed threshold α does not surpass α for each fixed α∈(0,1). Finally, the theoretical results are illustrated through Monte Carlo simulations and applied to a famous Orthodontic growth data.

A new sufficient condition for local regularity of a suitable weak solution to the MHD equations

We assume that Ω is either the whole space R3 or a half-space or a smooth bounded or exterior domain in R3, T>0 and (u,b,p) is a suitable weak solution of the MHD equations in Ω×(0,T). We show that (x0,t0)∈Ω×(0,T) is a regular point of the solution (u,b,p) if the limit inferior (for t→t0−) of the sum of the L3–norms of u and b over an arbitrarily small ball Bρ(x0) is less than infinity.

On regularity of functions of Markov chains

We consider processes which are functions of finite-state Markov chains. It is well known that such processes are rarely Markov. However, such processes are often regular in the following sense: the distant past values of the process have diminishing influence on the distribution of the present value. In the present paper, we present novel sufficient conditions for regularity of functions of Markov chains.

On Mixed Pressure-Velocity Regularity Criteria to the Navier-Stokes Equations in Lorentz Spaces

In this paper we derive regular criteria in Lorentz spaces for Leray-Hopf weak solutions $v$ of the three-dimensional Navier-Stokes equations based on the formal equivalence relation $\pi\cong|v|^2$, where $\pi$ denotes the fluid pressure and $v$ the fluid velocity. It is called the mixed pressure-velocity problem (the P-V problem). It is shown that if $\f{\pi}{(e^{-|x|^2}+|v|)^{\theta}}\in L^p(0,T;L^{q,\infty})\,,$ where $0\leq\theta\leq1$ and $\f2p+\f3q=2-\theta$, then $v$ is regular on $(0,T]$. Note that, if $\Om$ is periodic, we may replace $\,e^{-|x|^2} \,$ by a positive constant. This result improves a 2018 statement obtained by one of the authors. Furthermore, as an integral part of our contribution, we give an overview on the known results on the P-V problem, and also on two main techniques used by many authors to establish sufficient conditions for regularity of the so-called Ladyzhenskaya-Prodi-Serrin (L-P-S) type.

The stability of a generalized variational inequality problem with the nuclear norm function

This paper is to study the stability of the generalized variational inequality (GVI) problem whose convex function is the nuclear norm of a linear matrix mapping. The strict Robinson constraint qualification and the second-order sufficient optimality conditions for the GVI problem are proposedand are demonstrated as sufficient conditions for the isolated calmness of the inverse of the KKT (Karush-Kuhn-Tucker) mapping. The constraint non-degeneracy condition and the strong second-order sufficient optimality conditions for the GVI problem are proposed and are proven to be sufficient conditions for the strong regularity of the KKT system.

Equilibrium analysis of CH4, CO, CO2, H2O, H2, C mixtures in C–H–O atom space using Gibbs free energy global minimization

AbstractGibbs free energy minimization is employed to carry out thermodynamic equilibrium analysis studies of mixtures containing methane, carbon monoxide, carbon dioxide, water, and hydrogen ideal gases, and possibly solid carbon. The employed global minimization approach represents a general, unifying, conceptual framework that allows reaction and phase equilibrium analysis to be simultaneously carried out in the atom‐mol fraction space (aH, aO), thus capturing in a comprehensive manner the equilibrium behavior of a number of industrially important processes, such as methane reforming (steam, dry, energetically enhanced), and methanation. Two theorems are presented, establishing necessary and sufficient (necessary) feasibility and regularity (optimality) conditions for the aforementioned minimization problem. The equilibrium results obtained through application of these theorems are guaranteed to be globally optimal. They quantify in (aH, aO) space, the feasible region, the carbon formation region, and all species mole over total atom‐mol normalized ratios, for a range of temperatures and pressures.

Strong solutions and trajectory attractors to the thin-film equation with absorption

We prove local and global in time existence of non-negative weak solutions to the thin-film equation with absorption and obtain sufficient conditions for extra regularity of these solutions. Moreover, for the class of global strong solutions, we show existence of a trajectory attractor.

Note on global regular solution to the 3D liquid crystal equations

In this paper we provide a sufficient condition for regularity of solutions to the 3D nematic liquid crystal flows. More precisely, we prove that if the horizontal velocity component uh=(u1,u2) satisfies ∇uh∈Lq(0,T;Lp(R3)),3p+2q≤32,2≤p≤∞, then the solution (u,d) is regular.

Urban Courier: Operational Innovation and Data-driven Coverage-and-Pricing

We study a novel practice in the urban logistics sector, the transshipment-based urban courier (TUC) system. In the TUC system, large-capacity vehicles are utilized as moving warehouses to flexibly consolidate packages from individual couriers. The goal is to achieve a better trade-off between transportation costs and delivery efficiency. Because the performance of the TUC system depends significantly on the coordination of the transportation services and the pricing system for consumers, we consider the joint transportation coverage and pricing problem. In particular, we propose a data-driven robust pricing model in coordination with a transportation coverage model. Theoretical performance guarantees and numerical results verify the robustness of the pricing decision. The coverage-and-pricing model can be solved by a decomposition-based heuristic. This heuristic is a valid approximation algorithm under sufficient regularity conditions. After decomposition, the coverage-and-pricing problem can be reformulated as a maximum weighted coverage problem and solved via column generation, where the column-generating subproblems can be solved efficiently by a dynamic programming algorithm. Numerical experiments and a case study based on operational data from DML-Express demonstrate the efficiency of our method and provide managerial implications for the pricing policy in a TUC system.

Consequences of Model Misspecification for Maximum Likelihood Estimation with Missing Data

Researchers are often faced with the challenge of developing statistical models with incomplete data. Exacerbating this situation is the possibility that either the researcher’s complete-data model or the model of the missing-data mechanism is misspecified. In this article, we create a formal theoretical framework for developing statistical models and detecting model misspecification in the presence of incomplete data where maximum likelihood estimates are obtained by maximizing the observable-data likelihood function when the missing-data mechanism is assumed ignorable. First, we provide sufficient regularity conditions on the researcher’s complete-data model to characterize the asymptotic behavior of maximum likelihood estimates in the simultaneous presence of both missing data and model misspecification. These results are then used to derive robust hypothesis testing methods for possibly misspecified models in the presence of Missing at Random (MAR) or Missing Not at Random (MNAR) missing data. Second, we introduce a method for the detection of model misspecification in missing data problems using recently developed Generalized Information Matrix Tests (GIMT). Third, we identify regularity conditions for the Missing Information Principle (MIP) to hold in the presence of model misspecification so as to provide useful computational covariance matrix estimation formulas. Fourth, we provide regularity conditions that ensure the observable-data expected negative log-likelihood function is convex in the presence of partially observable data when the amount of missingness is sufficiently small and the complete-data likelihood is convex. Fifth, we show that when the researcher has correctly specified a complete-data model with a convex negative likelihood function and an ignorable missing-data mechanism, then its strict local minimizer is the true parameter value for the complete-data model when the amount of missingness is sufficiently small. Our results thus provide new robust estimation, inference, and specification analysis methods for developing statistical models with incomplete data.

Metric Regularity of Quasidifferentiable Mappings and Optimality Conditions for Nonsmooth Mathematical Programming Problems

This article is devoted to the analysis of necessary and/or sufficient conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova quasidifferentials. We obtain new necessary and sufficient conditions for the local metric regularity of a multifunction in terms of quasidifferentials of the distance function to this multifunction. We also propose a new MFCQ-type constraint qualification for a parametric system of quasidifferentiable equality and inequality constraints and prove that it ensures the metric regularity of a multifunction associated with this system. As an application, we utilize our constraint qualification to strengthen existing optimality conditions for quasidifferentiable programming problems with equality and inequality constraints. We also prove the independence of the optimality conditions of the choice of quasidifferentials and present a simple example in which the optimality conditions in terms of quasidifferentials detect the non-optimality of a given point, while optimality conditions in terms of various subdifferentials fail to disqualify this point as non-optimal.