Equivalent Characterizations Research Articles

Variable Besov Spaces Associated with Heat Kernels

Let $$({\mathcal {X}},\rho , \mu )$$ be a space of homogeneous type. Suppose that $$p(\cdot ),\ q(\cdot ):\ {\mathcal {X}}\rightarrow (0,\infty ]$$ are such that both $$1/p(\cdot )$$ and $$1/q(\cdot )$$ satisfy the globally log-Holder continuous condition, and $$s(\cdot ):\ {\mathcal {X}}\rightarrow \mathbb R$$ is a bounded function satisfying the locally log-Holder continuous condition. In this article, the authors introduce the variable Besov space $$B_{p(\cdot ),q(\cdot )}^{s(\cdot ),L}({\mathcal {X}})$$ , associated with a nonnegative self-adjoint operator L whose heat kernels satisfy small time Gaussian upper bound estimates, the Holder continuity, and the Markov property, which is new even on the sphere and the ball of $$\mathbb R^d$$ . Equivalent characterizations of this space, in terms of Peetre maximal functions and the heat semigroup, are established. Moreover, under the additional assumptions that $$\mu $$ satisfies the reverse doubling condition and the non-collapsing condition, its frame characterization is obtained. When L is the Laplacian operator on $$\mathbb R^d$$ , this space coincides with the existing variable Besov space.

Weighted composition operator on quaternionic Fock space

This paper is concerned with several important properties of weighted composition operator acting on the quaternionic Fock space $${\mathcal {F}}^2({\mathbb {H}})$$ . Complete equivalent characterizations for its boundedness and compactness are established. As corollaries, the descriptions for composition operator and multiplication operator on $${\mathcal {F}}^2({\mathbb {H}})$$ are presented, which can indicate some well-known existing theories in complex Fock space. Finally, as an appendix the closed expression for the kernel function of $${\mathcal {F}}^2({\mathbb {H}})$$ is exhibited, which can deepen the understanding of $${\mathcal {F}}^2({\mathbb {H}})$$ .

Vexillary signed permutations revisited

We study the combinatorial properties of vexillary signed permutations, which are signed analogues of the vexillary permutations first considered by Lascoux and Schützenberger. We give several equivalent characterizations of vexillary signed permutations, including descriptions in terms of essential sets and pattern avoidance, and we relate them to the vexillary elements introduced by Billey and Lam.

T-positive semidefiniteness of third-order symmetric tensors and T-semidefinite programming

The T-product for third-order tensors has been used extensively in the literature. In this paper, we first introduce first-order and second-order T-derivatives for the multi-variable real-valued function with the tensor T-product. Inspired by an equivalent characterization of a twice continuously T-differentiable multi-variable real-valued function being convex, we present a definition of the T-positive semidefiniteness of third-order symmetric tensors. After that, we extend many properties of positive semidefinite matrices to the case of third-order symmetric tensors. In particular, analogue to the widely used semidefinite programming (SDP for short), we introduce the semidefinite programming over the space of third-order symmetric tensors (T-semidefinite programming or TSDP for short), and provide a way to solve the TSDP problem by converting it into an SDP problem in the complex domain. Furthermore, we give several TSDP examples and especially some preliminary numerical results for two unconstrained polynomial optimization problems. Experiments show that finding the global minimums of polynomials via the TSDP relaxation outperforms the traditional SDP relaxation for the test examples.

Conformal vector fields on Finsler manifolds

In this paper, we give an equivalent characterization of conformal vector fields on a Finsler manifold [Formula: see text], whose metric [Formula: see text] is defined by a Riemannian metric [Formula: see text] and a 1-form [Formula: see text]. This characterization contains all related results in [Z. Shen and Q. Xia, On conformal vector fields on Randers manifolds, Sci. China Math. 55(9) (2012) 1869–1882; Z. Shen and M. Yuan, Conformal vector fields on some Finsler manifolds, Sci. China Math. 59(1) (2016) 107–114; X. Cheng, Y. Li and T. Li, The conformal vector fields on Kropina manifolds, Diff. Geom. Appl. 56 (2018) 344–354] as special cases. Further, we determine conformal fields on some Finsler manifolds [Formula: see text] when [Formula: see text] is of constant sectional curvature and [Formula: see text] is a conformal 1-form with respect to [Formula: see text].

Support condition for equivalent characterization of graph laws

Support condition for equivalent characterization of graph laws

Equivalent Characterizations of the Spectra of Graphs and Applications to Measures of Distance-regularity

The spectrum of a graph usually provides a lot of information about its combinatorial structure. Moreover, from the spectrum, the so-called predistance polynomials can be defined, as a generalization, for any graph, of the distance polynomials of a distance-regular graph. Going further, the preintersection numbers generalize the intersection numbers of a distance-regular graph. This paper describes, for any graph, the closed relationships between its spectrum, predistance polynomials, and preintersection numbers. Then, some applications to derive combinatorial properties of the given graph, most of them related to some fundamental characterizations of distance-regularity, are presented. In particular, the so-called `spectral excess theorem' is revisited. This result states that a connected regular graph is distance-regular if and only if its spectral excess, which is a value computed from the spectrum, equals the average excess, that is, the mean of the numbers of vertices at maximum distance from every vertex.

Fraïssé–Hintikka theorem in institutions

Abstract We generalize the characterization of elementary equivalence by Ehrenfeucht–Fraïssé games to arbitrary institutions whose sentences are finitary. These include many-sorted first-order logic, higher-order logic with types, as well as a number of other logics arising in connection to specification languages. The gain for the classical case is that the characterization is proved directly for all signatures, including infinite ones.

On the prime spectrum of an le-module

Here, we continue to characterize a recently introduced notion, le-modules [Formula: see text] over a commutative ring [Formula: see text] with unity [A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules, Acta Math. Hunga. 158(1) (2019) 202–215]. This paper introduces and characterizes Zariski topology on the set Spec[Formula: see text] of all prime submodule elements of [Formula: see text]. Thus, we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec[Formula: see text] is connected if and only if R/Ann(M) contains no idempotents other than [Formula: see text] and [Formula: see text]. Open sets in the Zariski topology for the quotient ring R/Ann(M) induces a base of quasi-compact open sets for the Zariski topology on Spec[Formula: see text]. Every irreducible closed subset of Spec[Formula: see text] has a generic point. Besides, we prove a number of different equivalent characterizations for Spec[Formula: see text] to be spectral.

A generalization of Strassen’s Positivstellensatz

Strassen’s Positivstellensatz is a powerful but little known theorem on preordered commutative semirings satisfying a boundedness condition similar to Archimedeanicity. It characterizes the relaxed preorder induced by all monotone homomorphisms to in terms of a condition involving large powers. Here, we generalize and strengthen Strassen’s result. As a generalization, we replace the boundedness condition by a polynomial growth condition; as a strengthening, we prove two further equivalent characterizations of the homomorphism-induced preorder in our generalized setting.

Stochastic comparisons of largest-order statistics for proportional reversed hazard rate model and applications

AbstractWe investigate stochastic comparisons of parallel systems (corresponding to the largest-order statistics) with respect to the reversed hazard rate and likelihood ratio orders for the proportional reversed hazard rate (PRHR) model. As applications of the main results, we obtain the equivalent characterizations of stochastic comparisons with respect to the reversed hazard rate and likelihood rate orders for the exponentiated generalized gamma and exponentiated Pareto distributions. Our results recover and strengthen some recent results in the literature.

Strongly stable C-stationary points for mathematical programs with complementarity constraints

In this paper we consider the class of mathematical programs with complementarity constraints (MPCC). Under an appropriate constraint qualification of Mangasarian–Fromovitz type we present a topological and an equivalent algebraic characterization of a strongly stable C-stationary point for MPCC. Strong stability refers to the local uniqueness, existence and continuous dependence of a solution for each sufficiently small perturbed problem where perturbations up to second order are allowed. This concept of strong stability was originally introduced by Kojima for standard nonlinear optimization; here, its generalization to MPCC demands a sophisticated technique which takes the disjunctive properties of the solution set of MPCC into account.

Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms

In this paper, we establish stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincar\'e inequalities. In particular, we establish the connection between parabolic Harnack inequalities and two-sided heat kernel estimates, as well as with the H\"older regularity of parabolic functions for symmetric non-local Dirichlet forms.

Characterizations for the n-strong Drazin invertibility in a ring

Recently, a new type of generalized inverse called the [Formula: see text]-strong Drazin inverse was introduced by Mosić in the setting of rings. Namely, let [Formula: see text] be a ring and [Formula: see text] be a positive integer, an element [Formula: see text] is called the [Formula: see text]-strong Drazin inverse of [Formula: see text] if it satisfies [Formula: see text], [Formula: see text] and [Formula: see text]. The main aim of this paper is to consider some equivalent characterizations for the [Formula: see text]-strong Drazin invertibility in a ring. Firstly, we give an equivalent definition of the [Formula: see text]-strong Drazin inverse, that is, [Formula: see text] is the [Formula: see text]-strong Drazin inverse of [Formula: see text] if and only if [Formula: see text], [Formula: see text] and [Formula: see text]. Also, we obtain some existence criteria for this inverse by means of idempotents. In particular, the [Formula: see text]-strong Drazin invertibility of the product [Formula: see text] are investigated, where [Formula: see text] is regular and [Formula: see text] are arbitrary elements in a ring.

Entanglement-breaking superchannels

In this paper we initiate the study of entanglement-breaking (EB) superchannels. These are processes that always yield separable maps when acting on one side of a bipartite completely positive (CP) map. EB superchannels are a generalization of the well-known EB channels. We give several equivalent characterizations of EB supermaps and superchannels. Unlike its channel counterpart, we find that not every EB superchannel can be implemented as a measure-and-prepare superchannel. We also demonstrate that many EB superchannels can be superactivated, in the sense that they can output non-separable channels when wired in series.We then introduce the notions of CPTP- and CP-complete images of a superchannel, which capture deterministic and probabilistic channel convertibility, respectively. This allows us to characterize the power of EB superchannels for generating CP maps in different scenarios, and it reveals some fundamental differences between channels and superchannels. Finally, we relax the definition of separable channels to include(p,q)-non-entangling channels, which are bipartite channels that cannot generate entanglement usingp- andq-dimensional ancillary systems. By introducing and investigatingk-EB maps, we construct examples of(p,q)-EB superchannels that are not fully entanglement breaking. Partial results on the characterization of(p,q)-EB superchannels are also provided.

Gorenstein objects in the category of N-complexes

For an integer [Formula: see text] and three subcategories [Formula: see text] of modules, we study the notion of [Formula: see text]-Gorenstein [Formula: see text]-complexes. We show that under certain hypotheses, an [Formula: see text]-complex [Formula: see text] is [Formula: see text]-Gorenstein if and only if [Formula: see text] is a [Formula: see text]-Gorenstein module for each [Formula: see text], [Formula: see text] and [Formula: see text] are [Formula: see text]-acyclic for any [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] denote the subcategory of [Formula: see text][Formula: see text]-complexes and [Formula: see text][Formula: see text]-complexes, respectively. Also, the stability of [Formula: see text]-Gorenstein [Formula: see text]-complexes is considered. As applications, not only some known results are encompassed, but also some new results are obtained, for instance, some equivalent characterizations for Ding injective [Formula: see text]-complexes and Gorenstein AC-injective [Formula: see text]-complexes are given.

Pointwise Multipliers of Zygmund Classes on"Equation missing"

It is well known that Lipschitz spaces on the torus are an algebra. It is no more the case in the non compact situation because of the behavior at infinity. This is a companion article to Bonami et al. (J Math Pures Appl (9) 131:130–170, 2019), where pointwise multipliers on Lipschitz spaces on $${\mathbb {R}}^n$$ are characterized for non-integer values of the parameter. In this article, the authors first establish two equivalent characterizations of a modified Zygmund space, and then characterize the pointwise multipliers on Lipschitz spaces on $${\mathbb {R}}^n$$ for the integer values of the parameter, in particular, for the Zygmund class, via the intersection of the Lebesgue space $$L^\infty ({\mathbb {R}}^n)$$ and the modified Zygmund space. This result can be used to show that the bilinear decomposition of the pointwise product of the Hardy space and its dual, in the integer values of the parameter, obtained in the aforementioned reference is sharp in the dual space sense.

Tosha-Degree Equivalence Signed Graphs

The Tosha-degree of an edge $\alpha $ in a graph $\Gamma$ without multiple edges, denoted by $T(\alpha)$, is the number of edges adjacent to $\alpha$ in $\Gamma$, with self-loops counted twice. A signed graph (marked graph) is an ordered pair $\Sigma=(\Gamma,\sigma)$ ($\Sigma =(\Gamma, \mu)$), where $\Gamma=(V,E)$ is a graph called the underlying graph of $\Sigma$ and $\sigma : E \rightarrow \{+,-\}$ ($\mu : V \rightarrow \{+,-\}$) is a function. In this paper, we define the Tosha-degree equivalence signed graph of a given signed graph and offer a switching equivalence characterization of signed graphs that are switching equivalent to Tosha-degree equivalence signed graphs and $ k^{th}$ iterated Tosha-degree equivalence signed graphs. It is shown that for any signed graph $\Sigma$, its Tosha-degree equivalence signed graph $T(\Sigma)$ is balanced and we offer a structural characterization of Tosha-degree equivalence signed graphs

A short note on the migrativity properties of overlap functions over uninorms

Recently, as an addendum to Qiao and Hu (2018) [21], Zhu and Hu (2020) [25] studied the α-migrativity of overlap and grouping functions over uninorms and nullnorms. Later on, Zhou and Yan (2019) [24] further investigated the α-migrativity properties of overlap functions over uninorms. They showed equivalent characterizations of the (α,U)-migrativity equation when the uninorm U belongs to one certain class (e.g., Umin, Umax, the family of idempotent uninorms, representable uninorms or uninorms continuous on ]0,1[2). However, we find that Propositions 3.5, 3.8 and Remark 3.5 (2) presented by Zhu and Hu (2020) [25] are incorrect. In this note, at first, we establish the updated results. And then, we study the α-migrativity of overlap functions over t-norms when t-norms are continuous and give their characterizations. In particular, we point out that the relationship between the α-migrativity of overlap functions over uninorms and the α-migrativity of uninorms over overlap functions. Finally, we discuss the (α,U)-migrativity equation for grouping functions using an analogous method.

Necessary/sufficient conditions for Pareto optimality in finite horizon mean-field type stochastic differential game

This paper is concerned with necessary and sufficient conditions for the existence of Pareto solutions in finite horizon mean-field type stochastic cooperative differential game. Based on the equivalent characterization of Pareto optimality, the problem is transformed into a set of constrained mean-field type stochastic optimal control problems with a special structure. Utilizing the mean-field type stochastic minimum principle, the necessary conditions are put forward. Under certain convex assumptions, it is shown that the necessary conditions are also sufficient ones. Next, the indefinite linear quadratic (LQ) case is studied. It is pointed out that the solvability of two related generalized differential Riccati equations (GDREs) provides a sufficient condition under which Pareto efficient strategies are equivalent to weighted sum optimal controls. In addition, all Pareto solutions are obtained based on the solutions of two generalized differential Lyapunov equations (GDLEs). At last, an example sheds light on the effectiveness of the theoretical results.