Equivalent Characterizations Research Articles

Restricted modules and associated vertex algebras of extended Heisenberg-Virasoro algebra

In this paper, a family of infinite dimensional Lie algebras L˜ is introduced and investigated, called the extended Heisenberg-Virasoro algebra, denoted by L˜. These Lie algebras are related to the N=2 superconformal algebra and the Bershadsky-Polyakov algebra. We study restricted modules and associated vertex algebras of the Lie algebra L˜. More precisely, we construct its associated vertex (operator) algebras VL˜(ℓ123,0), and show that the category of vertex algebra VL˜(ℓ123,0)-modules is equivalent to the category of restricted L˜-modules of level ℓ123. Then we give uniform constructions of simple restricted L˜-modules. Also, we present several equivalent characterizations of simple restricted modules over L˜.

Irreducible quasi-D-overlap functions: Matrix representation and diagonal generation

As an emerging mathematical model of information aggregation successfully applied in image processing, pattern recognition, decision-making and etc, theoretical research of overlap functions has also continued to develop, especially the research on their related derivative concepts that have emerged in recent years. This work considers so-called irreducible quasi-D-overlap functions defined on grid of unit square, which are one class of discrete aggregation operators originated from overlap functions. The core contents are: (i) it presents the concept of irreducible quasi-D-overlap functions and shows basic properties of them; (ii) it provides equivalent characterization of irreducible quasi-D-overlap functions by means of strong permutation matrixes; (iii) it gives a construction manner for irreducible quasi-D-overlap functions by means of so-called diagonal operators and the greatest and smallest irreducible quasi-D-overlap functions with given diagonal operator are obtained.

A note on CSP rings

A ring $R$ is called right CSP if the sum of any two closed right ideals of $R$ is also a closed right ideal of $R$. Left CSP rings can be defined similarly. An example is given to show that a left CSP ring may not be right CSP. It is shown that a matrix ring over a right CSP ring may not be right CSP. It is proved that $\mathbb{M}_{2}(R)$ is right CSP if and only if $R$ is right self-injective and von Neumann regular. The equivalent characterization is given for the trivial extension $R\propto R$ of $R$ to be right CSP.

A tree-based algorithm for the integration of monomials in the Chow ring of the moduli space of stable marked curves of genus zero

The Chow ring of the moduli space of stable marked curves is generated by Keel's divisor classes. The top graded part of this Chow ring is isomorphic to the integers, generated by the class of a single point. In this paper, we give an equivalent graphical characterization on the monomials in this Chow ring, as well as the characterization on the algebraic reduction on such monomials. Moreover, we provide an algorithm for computing the intersection degree of tuples of Keel's divisor classes — we call it the forest algorithm; the complexity of which is O(n3) in the worst case, where n refers to the number of marks on the stable curves in the ambient moduli space. Last but not least, we introduce three identities on multinomial coefficients which naturally came into play, showing that they are all equivalent to the correctness of the base case of the forest algorithm.

On classification of singular matrix difference equations of mixed order

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl's method by selecting a suitable quasi-difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off-diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation.

The dual index and dual core generalized inverse

Abstract In this article, we introduce the dual index and dual core generalized inverse (DCGI). By applying rank equation, generalized inverse, and matrix decomposition, we give several characterizations of the dual index when it is equal to 1. We realize that if DCGI exists, then it is unique. We derive a compact formula for DCGI and a series of equivalent characterizations of the existence of the inverse. It is worth noting that the dual index of A ^ \widehat{A} is equal to 1 if and only if its DCGI exists. When the dual index of A ^ \widehat{A} is equal to 1, we study dual Moore-Penrose generalized inverse (DMPGI) and dual group generalized inverse (DGGI) and consider the relationships among DCGI, DMPGI, DGGI, Moore-Penrose dual generalized inverse, and other dual generalized inverses. In addition, we consider symmetric dual matrix and its dual generalized inverses. Finally, two examples are given to illustrate the application of DCGI in linear dual equations.

Stochastic linear quadratic optimal control problems with expectation-type linear equality constraints on the terminal states

A stochastic linear quadratic (LQ) optimal control problem with an expectation-type linear equality constraint on the terminal state is considered. Under the solvability condition on a stochastic Riccati equation and a surjectivity condition on the linear constraint mapping, the constrained stochastic LQ problem is solved completely by the Lagrangian duality theory. Some equivalent characterizations of the surjectivity condition are discussed by the controllability theory of linear control systems. Especially, the equivalence between the surjectivity condition and a Kalman-type rank condition is proved under proper conditions.

Periodic Gabor frames on positive half line

In this paper, we introduce the concept of periodic Gabor frames on positive half line. Firstly, we establish a necessary and sufficient condition for a periodic Gabor system to be a Gabor frame. Then, we present some equivalent characterizations of Parseval Gabor frames on positive half line by means of some fundamental equations in time domain.

A unified framework of 0-overlap functions and 1-grouping functions

Overlap and grouping functions have been studied by many scholars since they were proposed because of their successful application in various practical problems such as in image processing, decision-making, classifications etc.. At the same time, 0-overlap functions and 1-grouping functions as their generalization are not only necessary to characterize interval overlap and grouping functions, but also the proposed generalized overlap functions based on 0-overlap functions have been successfully applied to classification problem. In this paper, we focus on the unification of 0-overlap functions and 1-grouping functions. Specifically, firstly, we introduce the concept of Θ−Ξ functions, which is the unification of 0-overlap functions and 1-grouping functions, and give some basic properties of Θ−Ξ functions. Secondly, we investigate the construction methods of Θ−Ξ functions on the basis of two classes of peculiar binary functions, so-called (e,k)-pseudo-automorphisms and (e,k)-aggregation functions with e,k belonging to unit closed interval, respectively. Finally, we obtain two equivalent characterizations of Θ−Ξ functions.

Perron-Frobenius type theorem for nonnegative tubal matrices in the sense of t-product

This paper is focused on the study of Perron-Frobenius (PF) type theorem for nonnegative tubal matrices, where a tubal matrix is a matrix with every element being a tubal scalar (a vector in the usual sense). Such a matrix is often recognized as a third-order tensor. The product between tubal scalars, tubal vectors, and tubal matrices can be done by the powerful t-product. In this paper, we define nonnegative/positive/strongly positive tubal scalars/vectors/matrices, and establish several properties that are analogous to their matrix counterparts. In particular, we introduce irreducible tubal matrices and provide two equivalent characterizations. Then, the celebrated PF theorem is established on the nonnegative irreducible tubal matrices. We show that some conclusions of the PF theorem for nonnegative irreducible matrices, such as the existence of a positive eigenvector, can be generalized to the tubal matrix setting, while some are not. For those conclusions that can not be extended, weaker conclusions are proved. We also show that, if the nonnegative irreducible tubal matrix contains a strongly positive tubal scalar, then most conclusions of the matrix PF theorem hold.

Solution set bounds for LCPs over tensor spaces

The goal of this paper is to investigate a class of linear complementarity problems over tensor-spaces, denoted by TLCP, which is an extension of the classical linear complementarity problem. First, two classes of structured tensors over tensor-spaces (i.e., T-R tensor and T-RO tensor) are introduced and some equivalent characterizations are discussed. Then, the lower bound of the solutions in the sense of the infinity norm of the TLCP is obtained when the problem has a solution. Moreover, by introducing a new quantity, the upper bound of the solutions in the sense of the infinity norm of the TLCP is given when the tensor involved in the problem is a T-RO tensor.

Gorenstein (ℒ,𝒜)-flat dimension of complexes and relative singularity categories

Assume that the class of all Gorenstein [Formula: see text]-flat modules is closed under extensions. We define a notion of Gorenstein [Formula: see text]-flat dimension for complexes and consider equivalent characterizations of the finiteness of Gorenstein [Formula: see text]-flat dimension of complexes. Furthermore, in addition, assume that [Formula: see text] is a right coherent ring. We construct the relative singularity category with respect to Gorenstein [Formula: see text]-flat modules, as the triangulated quotient of the triangulated subcategory of [Formula: see text] consisting of all complexes with both finite Gorenstein [Formula: see text]-flat dimension and cotorsion dimension by the bounded homotopy category of flat-cotorsion modules, and show that the relative singularity category is triangulated equivalent to the stable category of the Frobenius category consisting of all Gorenstein [Formula: see text]-flat and cotorsion modules.

Chaos and weak mixing on uniform spaces

Let f0,∞={fn}n=0∞ be a sequence of uniformly continuous self-maps on a uniform space X. We prove that under some natural conditions, topological weak mixing implies dense distributional η-chaos in a sequence, and generic η-chaos is equivalent to dense η′-chaos for (X,f0,∞). We give several equivalent characterizations of sensitivity for (X,f0,∞), and as applications, we get the relationships between sensitivity and Li-Yorke sensitivity, and further obtain that topological weak mixing implies Li-Yorke sensitivity for a class of abelian (X,f0,∞). We show that (X×Y,f0,∞×g0,∞) is sensitive (resp. Li-Yorke sensitive and multi-sensitive) if and only if (X,f0,∞) or (Y,g0,∞) is sensitive (resp. Li-Yorke sensitive and multi-sensitive). We also prove that distributional chaos (resp. Li-Yorke chaos, sensitivity and Li-Yorke sensitivity) is equivalent between (X,f0,∞) and its N-th iteration system. Finally, we confirm that distributional chaos in a sequence, Li-Yorke chaos, sensitivity and Li-Yorke sensitivity are all preserved under topological equi-conjugacy.

On difunctions

The notion of a difunction was introduced by Jacques Riguet in 1948. Since then it has played a prominent role in database theory, type theory, program specification and process theory. The theory of difunctions is, however, less known in computing than it perhaps should be. The main purpose of the current paper is to give an account of difunction theory in relation algebra, with the aim of making the topic more mainstream.As is common with many important concepts, there are several different but equivalent characterisations of difunctionality, each with its own strength and practical significance. This paper compares different proofs of the equivalence of the characterisations.A well-known property is that a difunction is a set of completely disjoint rectangles. This property suggests the introduction of the (general) notion of the “core” of a relation; we use this notion to give a novel and, we believe, illuminating characterisation of difunctionality as a bijection between the classes of certain partial equivalence relations.

Rigid subcategories and related subcategories in extriangulated categories

In this paper, we firstly obtain the analogue of a Bongartz completion for functorially finite rigid subcategories in an extriangulated category and then we investigate the properties of n-cluster tilting subcategories in an extriangulated category for any positive integer n and show some equivalent characterization of them. Particularly, when the extriangulated category is 2-Calabi-Yau, we show any n-cluster tilting subcategories are in fact cluster tilting for any integer and they coincide with strongly contravariantly finite maximal n-rigid subcategories.

Improved equivalent circuit characterization of an ultracapacitor for power electronic applications

Ultracapacitor-based energy storage systems are becoming increasingly popular as a secondary power source in Renewable energy and Electric Vehicle applications. The design of a well-tuned ultracapacitor-based energy storage system requires accurate characterization of its simplified electrical equivalent circuit. The existing approaches for estimating parameters of the simplified equivalent circuit rely on the initial slope of the ultracapacitor response, which can lead to inaccurate estimation. This work focuses on developing a simple and accurate linear regression-based approach for estimation of equivalent circuit parameters. We highlight some critical observations related to the equivalent circuit parameters of a simplified equivalent circuit of an ultracapacitor. In particular, this work indicates the non-uniqueness of capacitance during the charging and discharging modes. The characterization and analysis reported in this study are based on a 24 V laboratory prototype comprising 10 series-connected ultracapacitors.

Construction and approximation for a class of feedforward neural networks with sigmoidal function

As we know, feedforward neural networks (FNNs) with sigmoidal activation function are universal approximators. Theoretically, for any continuous function defined on a compact set, there exists an FNN such that the FNN can approximate the function with arbitrary accuracy, which constitutes a theoretical guarantee that FNN can be used as an efficient learning machine. This paper addresses the construction and approximation for FNNs. We construct an FNN with sigmoidal activation function and estimate its approximation error. In particular, an inverse theorem of the approximation is established, which implies the equivalence characterization theorem of the approximation and reveals the relationship between the topological structure of the FNN and its approximation ability. As keys in this study, the concepts of modulus of continuity of function, K-functional, and their relationship are utilized, and two Bernstein-type inequalities are established.

Quasi-2-Segal sets

We show that the 2-Segal spaces (also called decomposition spaces) of Dyckerhoff-Kapranov and G\'alvez-Kock-Tonks have a natural analogue within simplicial sets, which we call quasi-2-Segal sets, and that the two ideas enjoy a similar relationship as the one Segal spaces have with quasi-categories. In particular, we construct a model structure on the category of simplicial sets whose fibrant objects are the quasi-2-Segal sets which is Quillen equivalent to a model structure for complete 2-Segal spaces (where our notion of completeness comes from one of the equivalent characterizations of completeness for Segal spaces). We also prove a path space criterion, which says that a simplicial set is a quasi-2-Segal set if and only if its path spaces (also called d\'ecalage) are quasi-categories, as well as an edgewise subdivision criterion.

Equivalent characterizations of martingale Hardy–Lorentz spaces with variable exponents

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Outdoor I-V characterization of tilted and vertical bifacial PV modules

This work proposes an experimental method to characterize bifacial photovoltaic modules through outdoor current-voltage (I-V) measurements. This characterization is an extension of methods developed for monofacial modules by considering the IEC TS 60904-1-2 standard for bifacial modules, particularly the introduction of the equivalent irradiance that accounts for albedo and bifaciality. We extract the short-circuit current, open-circuit voltage, and maximum power from the module I-V characteristic curve under outdoor conditions. We extrapolate these parameters to standard test conditions using the measured equivalent irradiance and module temperature. This method is applied on bifacial modules in two mounting orientations: 15° tilt north-oriented and vertical east-west orientation. Characterizing the bifacial modules’ electrical parameters using the proposed approach yields results with similar accuracy as applying the equivalent characterization of a tilted monofacial module.