Equivalent Characterizations Research Articles

Characterizing Nonuniform Hyperbolicity by Mather-Type Admissibility

AbstractWe consider linear cocycles acting on Banach spaces which satisfy the assumptions of the multiplicative ergodic theorem. A cocycle is nonuniformly hyperbolic if all Lyapunov exponents are non-zero, which is equivalent to the existence of a tempered exponential dichotomy. We provide an equivalent characterization of nonuniform hyperbolicity in terms of a Mather-type admissibility of a pair of weighted function spaces. As an application we give a short proof of the robustness of tempered exponential dichotomies under small linear perturbation.

Mixed orthogonality graphs for continuous-time stationary processes

In this paper, we introduce different concepts of Granger causality and contemporaneous correlation for multivariate stationary continuous-time processes to model different dependencies between the component processes. Several equivalent characterisations are given for the different definitions, in particular by orthogonal projections. We then define two mixed graphs based on different definitions of Granger causality and contemporaneous correlation, the (mixed) orthogonality graph and the local (mixed) orthogonality graph. In these graphs, the components of the process are represented by vertices, directed edges between the vertices visualise Granger causal influences and undirected edges visualise contemporaneous correlation between the component processes. Further, we introduce various notions of Markov properties in analogy to Eichler (2012), which relate paths in the graphs to different dependence structures of subprocesses, and we derive sufficient criteria for the (local) orthogonality graph to satisfy them. Finally, as an example, for the popular multivariate continuous-time AR (MCAR) processes, we explicitly characterise the edges in the (local) orthogonality graph by the model parameters.

Quasi-D-overlap functions: Construction and characterization

In this paper, we mainly consider the construction and characterization of so-called quasi-D-overlap functions, which are derived from D-overlap functions and irreducible quasi-D-overlap functions proposed by the author lately. The main contributions are: (i) it presents the concept of quasi-D-overlap functions and obtains construction manner of them by means of matrixes; (ii) it illustrates the relationship between quasi-D-overlap functions and 1-Lipschitz semi-quasi-overlap functions; (iii) it gives equivalent characterization of quasi-D-overlap functions and proper quasi-D-overlap functions by dint of matrixes, moreover, it obtains that the convex combination of quasi-D-overlap functions is also a quasi-D-overlap function.

Hypocoercivity in Hilbert spaces

The concept of hypocoercivity for linear evolution equations with dissipation is discussed and equivalent characterizations that were developed for the finite-dimensional case are extended to separable Hilbert spaces. Using the concept of a hypocoercivity index, quantitative estimates on the short-time and long-time decay behavior of a hypocoercive system are derived. As a useful tool for analyzing the structural properties, an infinite-dimensional staircase form is also derived and connections to linear systems and control theory are presented. Several examples illustrate the new concepts and the results are applied to the Lorentz kinetic equation.

The L-Fuzzy Prime Filter Degrees on Lattices and Its Induced L-Fuzzy Convex Structure

The aim of this paper is to examine the L-fuzzy prime filter degrees on lattices and their induced L-fuzzy convex structure. Firstly, the notion of L-fuzzy prime filter degrees on lattices is established using the implication operator when L is a completely distributive lattice. Secondly, an equivalent characterization of L-fuzzy prime filter degrees on lattices is provided. The equivalence relation, through the definitions of reflexivity, symmetry, and transitivity, provides a method for partitioning subsets within a lattice that possesses the prime filter property. Finally, the L-fuzzy convex structure induced by the L-fuzzy prime filter degrees on lattices is examined. Simultaneously, the properties of L-fuzzy prime filter degrees on lattices in relation to images and preimages under homomorphic mappings are discussed.

Intrinsic square function and wavelet characterizations of variable Hardy–Lorentz spaces

Let p(⋅):Rn→(0,∞] be a variable exponent function satisfying the globally log-Hölder continuous condition and q∈(0,1]. The goal of this article is to characterize the variable Hardy–Lorentz space Hp(⋅),q(Rn) in terms of various intrinsic square functions including the intrinsic g-function, the intrinsic Lusin area integral and the intrinsic gλ⁎-function. Using the intrinsic square function and the known atomic characterizations of Hp(⋅),q(Rn), and establishing some estimates on a discrete Littlewood–Paley g-function and a Peetre-type maximal function, the authors further give several equivalent characterizations of Hp(⋅),q(Rn) via wavelets. What is different from the existing works is that we merely assume p+=ess supx∈Rnp(x)∈(0,∞). One of the important tools that make it possible is to find a new dual space of Hp(⋅),q(Rn).

On the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature

We prove that an open noncollapsed manifold with nonnegative Ricci curvature and linear volume growth always splits off a line at infinity. This completes the final step to prove the existence of isoperimetric sets for given large volumes in the above setting. We also find that under our assumptions, the diameters of the level sets of any Busemann function are uniformly bounded as opposed to a classical result stating that they can have sublinear growth when the end is collapsing. Moreover, some equivalent characterizations of linear volume growth are given. Finally, we construct an example to show that for manifolds in our setting, although their limit spaces at infinity are always cylinders, the cross sections can be nonhomeomorphic.

Auslander‐type conditions and weakly Gorenstein algebras

AbstractLet be an Artin algebra. Under certain Auslander‐type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander‐type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if is left quasi‐Auslander, then is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if satisfies the Auslander condition, then is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander–Reiten's conjecture, which states that is Gorenstein if satisfies the Auslander condition.

Characterizations for convolution lattices based on non-distributive lattices

Recently, De Miguel, Bustince and De Baets have conducted a systematic study on convolution lattices based on distributive lattices. There have been few reports on applying non-distributive lattices to a domain of functions. As a complement to their work, in this paper, we carry out an in-depth investigation of convolution operations of the functions between a non-distributive lattice (domain) and a frame (co-domain). We first present an equivalence characterization between non-distributive lattices and idempotent functions and further show that a subset of the set of idempotent functions is closed under convolution operations. We demonstrate that this subset also is a bisemilattice and satisfies the Birkhoff equation under join- and meet-convolution operations. Finally, we analyze and study the lattice structure related to the obtained algebraic structure.

Gorenstein derived functors for extriangulated categories

Let ( C , E , s ) be an extriangulated category with a proper class ξ of E -triangles. In this paper, we study Gorenstein derived functors for extriangulated categories. More precisely, we first introduce the notion of the proper ξ ‐ G projective resolution for an object in C and define the functors ξ xt G P ( ξ ) and ξ xt G I ( ξ ) . Under some assumptions, we give some equivalent characterizations for ξ ‐ G projective and ξ ‐ G injective dimensions of objects in C by vanishing of such functors. As an application, our main results generalize Ren and Liu’s work. Moreover, our proof is not far from the usual module categories or triangulated categories.

Higher Tate traces of Chow motives

Abstract We establish the complete classification of Chow motives of projective homogeneous varieties for p-inner semi-simple algebraic groups, with coefficients in ℤ / p ⁢ ℤ {{\mathbb{Z}}/p{\mathbb{Z}}} . Our results involve a new motivic invariant, the Tate trace of a motive, defined as a pure Tate summand of maximal rank. They apply more generally to objects of the Tate subcategory generated by upper motives of irreducible, geometrically split varieties satisfying the nilpotence principle. Using Chernousov–Gille–Merkurjev decompositions and their interpretation through Bialynicki–Birula–Hesselink–Iversen filtrations due to Brosnan, we then generalize the characterization of the motivic equivalence of inner semi-simple groups through the higher Tits p-indexes. We also define the motivic splitting pattern and the motivic splitting towers of a summand of the motive of a projective homogeneous variety, which correspond for quadrics to the classical splitting pattern and Knebusch tower of the underlying quadratic form.

On small injective group rings

A ring [Formula: see text] is called right small injective if every homomorphism from a small right ideal of [Formula: see text] to [Formula: see text] can be extended to a homomorphism from [Formula: see text] to [Formula: see text]. Recall that a right ideal [Formula: see text] of [Formula: see text] is called small if for any proper right ideal [Formula: see text] of [Formula: see text], [Formula: see text]. It is proved that if [Formula: see text] is a locally finite group and the group ring [Formula: see text] is right small injective, then [Formula: see text] must be right small injective. The converse is not true even if [Formula: see text] is a finite group. An example is given to show a right small injective group ring [Formula: see text] whose coefficient ring [Formula: see text] is not right small injective. In the end, let [Formula: see text] be a finite group, an equivalent characterization is given for [Formula: see text] to be right small injective.

Stochastic orders and distortion risk contribution ratio measures

Relative spillover effects play a crucial role in the analysis and comparison of systemic risks. This paper introduces a novel approach, referred to as distortion risk contribution ratio measures, for quantifying such effects. Various types of contribution ratio measures are defined based on the newly proposed conditional distortion risk measures by Dhaene et al. (2022), and useful integral-based representations are provided as well. An interesting equivalent characterization result for the convex transform order is also presented, which is not only relevant to proving our main results but also has independent value in other research areas. We then establish comparison results between the distortion risk contribution ratio measures of two different bivariate random vectors with either the same or different copulas. Sufficient conditions are established in terms of stochastic orders, copula functions, distortion functions, and stress levels. Furthermore, we investigate the ordering behaviors of these measures in relation to the interaction between paired risks. Numerical examples are presented to illustrate the conditions and main findings.

The Intrinsic Characterization of a Fuzzy Consistently Connected Domain

The concepts of a fuzzy connected set (fc set) and a fuzzy consistently connected set (fcc set) are introduced on fuzzy posets, along with a discussion of their basic properties. Inspired by some equivalent conditions of crisp connected sets, the characterizations of the fc sets are given, and we also explore fuzzy completeness and fuzzy compactness in addition to defining a new fuzzy way-below relation based on fcc complete sets. Using this relationship as a basis, the fcc domain is also provided and studied, and its equivalent characterizations are obtained. In summary, we develop a method to establish fcc completeness from a continuous poset.

Dynamic Response and Damage Assessment of Partially Confined Metallic Cylinders Under Transverse Blast Loading

An analysis of the dynamic response and damage assessment of partially confined metallic cylinders under transverse air-blast loading is presented in this paper. The examination of the blast shock wave load distribution and the consequences of standoff distances on the plastic deformation of cylinders was conducted with meticulous attention in the experimental testing. The charges of TNT were determined to have masses of 16.3 kg and 29.5 kg, while the distances at which they were placed from the target varied between 1.5 m and 4.25 m. Three unique deformation modes were found to exist in the unilaterally supported cylinder, all of which were directly connected to the distribution of the blast load. In order to obtain an improved understanding of the dynamic behaviors exhibited in cylinders, a finite element simulation model was utilized and subsequently validated using an examination of the experimental results. The assessment of shock wave damage was conducted by utilizing the residual deformation of the metallic cylinder as an equivalent characterization under lateral blast shock wave loading. The duration of the positive pressure zone of the shock wave was determined to be between one-fourth and ten times the vibration period of a unilaterally supported cylinder, indicating that the P-I (overpressure-impulse) criterion could be utilized for damage evaluation. The incorporation of the P-I criterion was subsequently employed to evaluate the detrimental consequences, taking into consideration the radial distribution of residual deformation identified in metal cylindrical shell constructions subjected to lateral blast shock wave loading. This research contributed to a better comprehension of the dynamic behavior of partially confined metallic cylinders subjected to lateral blast loading and highlights the significance of the P-I criterion for evaluating and mitigating the damage effects in such scenarios.

N-Ary aggregation operators on function spaces: perspective of construction

For disposing numerous practical application problems involving expert systems, decision-making, image processing, classifications and etc, the investigations on the constructions and basic properties of n-ary aggregation operators (nAAOs) have always been a hot research topic with important research value and significance at theoretical investigations on aggregation operators (AOs). Herein, first, we propose a method for constructing nAAOs on function spaces via a family of known ones defined on a bounded poset, where those function spaces are composed by all fuzzy sets with that bounded poset as the truth values set. This method is different from the existing construction methods of nAAOs on bounded posets and provides a unified way of constructing usual nAAOs (like t-norms, uninorms, overlap functions, etc.) on function spaces via a family of known ones. Second, we present notion of representable nAAOs on function spaces and afford their equivalent characterization. Third, we discuss some vital properties of representable nAAOs on function spaces. Fourth, it is worth noticing that the obtained results cover the cases of nAAOs on function spaces composed of all interval-valued fuzzy sets and type-2 fuzzy sets when underlying bounded poset is taken as the corresponding truth values set, respectively. As a consequence, the theoretical results obtained herein have certain promotion and basic theoretical value for the mining of new potential applications of nAAOs in real problems, especially in expert systems, decision-making, image processing and etc.

Matrix-weighted Besov–Triebel–Lizorkin spaces with logarithmic smoothness

In this article, the authors study the matrix-weighted Besov–Triebel–Lizorkin spaces with logarithmic smoothness. Via first obtaining the Lp(Rn)-boundedness and the Fefferman–Stein type vector-valued inequality of matrix-weighted Peetre-type maximal functions with the exquisite ranges of indices in terms of the Ap dimension of matrix weights under consideration, the authors establish an equivalent characterization of these spaces in terms of the matrix-weighted Peetre-type maximal functions, which further implies that these spaces are well defined. As an application, the authors obtain the boundedness of some pointwise multipliers on these spaces and, even back to classical Besov–Triebel–Lizorkin spaces, some of them are also new.

Convergence order of one point large deviations rate functions for backward Euler method of stochastic delay differential equations with small noise

The present work focuses on the convergence order of one point large deviations rate functions for the backward Euler method of stochastic delay differential equations (SDDEs) with small noise. The drift and diffusion coefficients of SDDEs are allowed to grow super-linearly with respect to both the state variables and the delay variables. It is shown that the backward Euler method satisfies the one point large deviations principle (LDP) with good rate function. Further, the local uniform convergence orders of the one point large deviations rate function are derived by means of the equivalent characterizations for the original and numerical rate functions. These theoretical results are finally supported by a series of numerical experiments.

Equivalent characterization of initial crack size of blunt notched components based on test data and physical model

The precise definition of initial crack size applicable to fracture mechanics is crucial for predicting the total fatigue life of structural components. Given that the demarcation between crack formation and crack propagation depends not only on the material but also on the stress level, theoretical and experimental procedures for defining the crack boundary point are developed. Underlying the procedure is that the designed notched members have the same local stress distribution as the structure. Consequently, the fatigue life and initial crack size of S355J2 steel notched components with different scales but similar stress fields were predicted and compared with experimental data. Results show that fatigue life predictions fall within ±2 factor band, and the crack initiation size obtained by AM is closest to the experimental results, with a difference of less than 30.2%, which proves that the approach of treating the initial crack size as a function related to fatigue life and fatigue notch factor is reasonable.

A unified framework of fuzzy implications and coimplications

Fuzzy implications and coimplications play important roles in both theoretic and applied communities of fuzzy set theory. In this paper, we provide a unified framework for fuzzy implications and coimplications. Specifically, firstly, we introduce the concept of uni-implications, which is the unification of fuzzy implications and coimplications, and describe the structure of uni-implications. Secondly, we discuss the relationships among uni-implications, fuzzy boundary weak implications and fuzzy (co)implications. Thirdly, we present two constructions and equivalent characterizations of non-trivial uni-implications, respectively. Finally, we propose two binary operations ♠,♡ on some subset of the set of non-trivial uni-implications, denoted by FSNT, which make (FSNT,♠) a semigroup and (FSNT,♡) a non-commutative and non-idempotent monoid.