Equivalent Characterizations Research Articles

Asymptotic behavior and quasi-limiting distributions on time-fractional birth and death processes

AbstractIn this article we provide new results for the asymptotic behavior of a time-fractional birth and death process $N_{\alpha}(t)$ , whose transition probabilities $\mathbb{P}[N_{\alpha}(t)=\,j\mid N_{\alpha}(0)=i]$ are governed by a time-fractional system of differential equations, under the condition that it is not killed. More specifically, we prove that the concepts of quasi-limiting distribution and quasi-stationary distribution do not coincide, which is a consequence of the long-memory nature of the process. In addition, exact formulas for the quasi-limiting distribution and its rate convergence are presented. In the first sections, we revisit the two equivalent characterizations for this process: the first one is a time-changed classic birth and death process, whereas the second one is a Markov renewal process. Finally, we apply our main theorems to the linear model originally introduced by Orsingher and Polito [23].

Extension constructions of quasi-overlap functions and their derivative concepts on function spaces

Due to the successful applications in abundant practical application problems involving fuzzy sets and systems, overlap functions on unit closed interval have attracted continuous attentions of many scholars since they were proposed. In particular, recently, the author extended the concept of overlap functions on unit closed interval to the so-called quasi-overlap functions on bounded partially ordered sets. In this paper, we pay attention to the extension constructions of quasi-overlap functions and their derivative concepts on function spaces with bounded partially ordered sets as underlying sets. Specifically, first, based on quasi-overlap functions and their derivative concepts on any bounded partially ordered set, we give the way to construct quasi-overlap functions and their derivative concepts on function space composed of all fuzzy sets with that bounded partially ordered set as the truth values set along with the function space composed of all order-preserving functions from arbitrary sup semilattice to that bounded partially ordered set, respectively. Second, we introduce the concepts of representable quasi-overlap functions and representable derivative concepts of quasi-overlap functions on function spaces and obtain their equivalent characterizations. Third, we discuss some vital properties of representable quasi-overlap functions and representable derivative concepts of quasi-overlap functions on function spaces. Fourth, it should be mentioned that the obtained results cover the cases of quasi-overlap functions and their derivative concepts on function spaces composed of all interval-valued fuzzy sets and type-2 fuzzy sets when the underlying bounded partially ordered set is taken as the corresponding truth values set, respectively.

Equivalent characterization of pre-strained material properties and mechanical behavior prediction of steel/aluminum self-piercing riveted joints

Self-piercing riveting formation results in additional straining and property variations of the sheet material and rivet of the joint. As the riveted forming strain was concentrated within a limited area, it was difficult to characterize the properties of the pre-strained materials through conventional tensile tests. The present study dealt with the effects of considering riveted forming strain on the mechanical behavior prediction accuracy of the self-piercing riveted (SPR) joints of DP590 high-strength steel and AC43500-T7 cast aluminum alloy sheets. Main efforts were put on the evaluation of the failure parameters and behaviors of the pre-strained sheet materials. Equivalent specimens of the two sheet materials with different pre-strain levels were prepared via a flat rolling process for the characterization of the sheet deformation during the riveting process. Based on the material parameters obtained from the tensile tests of the rolled sheets and the compression tests of the rivet, a three-dimensional simulation model of the SPR joints was established. The mechanical behaviors of the joints were investigated under three loading conditions of 0° pull-out, 90° lap shear, and 45° tension–shear. The prediction accuracies of pull-off displacement under the three loading conditions reached 94.07%, 91.61% and 94.02%, and were 5.57%, 26.43% and 6.18% higher than those predicted by the model without considering the riveted forming history, respectively. The prediction accuracies of the peak force under the three loading conditions were 98.43%, 89.70% and 95.35%, respectively. The error difference of predicted peak force of the two models was less than 3%.

Time-temperature equivalence and unified characterization of asphalt mixture fatigue properties

This work presents a new method for evaluating the time-temperature equivalence and unified characterization of the fatigue performance for asphalt mixtures. For this purpose, direct tensile strength tests and direct tensile fatigue tests were carried out on asphalt mixtures at different temperatures and loading rates. A time-temperature-dependent stress ratio was established. By exploring the correlation between stress ratio and fatigue life, a new fatigue model considering the effect of temperature and loading time was built. The time-temperature equivalent of fatigue performance was confirmed by analysis of the fatigue dates under several temperatures and loading frequencies. The results showed that the direct tensile strength of the asphalt mixture changed non-linearly with the variation of the loading rate or temperature. The stress ratio was also related to temperature and loading rate. The time-temperature stress ratio was linearly related to fatigue life in a double logarithmic coordinate system, and a new SN fatigue curve was formed. The fatigue life under several temperatures and various loading frequencies can be described in one fatigue model and the time-temperature equivalent can be explored. The master curve of fatigue life or stress ratio was built. The time-temperature superposition principle was applicable for fatigue life estimation and stress ratio calculation. The fatigue life under different loading frequencies and various temperatures of the pavement in service can be completely estimated. The relative error between the test results and the calculated values is less than 30%.

Equilibrium characterization and shock propagation in conflict networks

We establish the general properties of equilibria (existence and uniqueness) in a very general model of interconnected multiplayer conflicts. In particular, under mild conditions on the cost function and the contest technology, we show that a pure-strategy Nash equilibrium always exists and the set of Nash equilibria is convex. Furthermore, under the strong monotonicity of the cost function, the equilibrium is unique, regardless of the conflict structure. To establish these properties of equilibria, we use variational inequality techniques to obtain equivalent equilibrium characterizations. After conducting comparative statics analysis with respect to the model primitives, we explore, in several simple examples of conflict structures, how shocks that originated locally propagate to the rest of the conflict network. Our general framework subsumes a number of models studied in the contest literature, as we are able to generalize many results obtained in these papers.

Equivalent constructions of nilpotent quadratic Lie algebras

The double extension and the T⁎-extension are classical methods for constructing finite dimensional quadratic Lie algebras. The first one gives an inductive classification in characteristic zero, while the latest produces quadratic non-associative algebras (not only Lie) out of arbitrary ones in characteristic different from 2. The classification of quadratic nilpotent Lie algebras can also be reduced to the study of free nilpotent Lie algebras and their invariant forms. In this work we will establish an equivalent characterization among these three construction methods. This equivalence reduces the classification of quadratic 2-step nilpotent to that of trivectors in a natural way. In addition, theoretical results will provide simple rules for switching among them.

Bubbles in discrete-time models

We introduce a new definition of bubbles in discrete-time models based on the discounted stock price losing mass under an equivalent martingale measure at some finite drawdown. We provide equivalent probabilistic characterisations of this definition and give examples of discrete-time martingales that are bubbles and others that are not. In the Markovian case, we provide sufficient analytic conditions for the presence of bubbles. We also show that the existence of bubbles is directly linked to the existence of a non-trivial solution to a linear Volterra integral equation of the second kind involving the Markov kernel. Finally, we show that our definition of bubbles in discrete time is consistent with the strict local martingale definition of bubbles in continuous time in the sense that a properly discretised strict local martingale in continuous time is a bubble in discrete time.

The many faces of the stochastic zeta function

We introduce a framework to study the random entire function $\zeta_\beta$ whose zeros are given by the Sine$_\beta$ process, the bulk limit of beta ensembles. We present several equivalent characterizations, including an explicit power series representation built from Brownian motion. We study related distributions using stochastic differential equations. Our function is a uniform limit of characteristic polynomials in the circular beta ensemble; we give upper bounds on the rate of convergence. Most of our results are new even for classical values of $\beta$. We provide explicit moment formulas for $\zeta$ and its variants, and we show that the Borodin-Strahov moment formulas hold for all $\beta$ both in the limit and for circular beta ensembles. We show a uniqueness theorem for $\zeta$ in the Cartwright class, and deduce some product identities between conjugate values of $\beta$. The proofs rely on the structure of the Sine$_\beta$ operator to express $\zeta$ in terms of a regularized determinant.

Connecting characterizations of equivalence of expressions: design research in Grade 5 by bridging graphical and symbolic representations

One typical challenge in algebra education is that many students justify the equivalence of expressions only by referring to transformation rules that they perceive as arbitrary without being able to justify these rules. A good algebraic understanding involves connecting the transformation rules to other characterizations of equivalence of expressions (e.g., description equivalence that both expressions describe the same situation or figure). In order to overcome this disconnection even before variables are introduced, a design research study was conducted in Grade 5 to design and investigate an early algebra learning environment to establish stronger connections between different mental models and representations of equivalence of expressions. The qualitative analysis of design experiments with 14 fifth graders revealed deep insights into complexities of connecting representations. It confirmed that many students first relate the representations in ways that are too superficial without establishing deep connections. Analyzing successful students’ processes helped to identify an additional characterization that can support students in bridging the connection between other characterizations, which we call restructuring equivalence. By including learning opportunities for restructuring equivalence, students can be supported to compare expressions in graphical and symbolic representation simultaneously and dynamically. The design research study disentangles the complex requirements for realizing the design principle of connecting multiple representations, which should be of relevance beyond the specific concept of equivalence and applicable to other mathematical topics.

Computing a Possibility Theory Repair for Partially Preordered Inconsistent Ontologies

We address the problem of handling inconsistency in uncertain knowledge bases that are specified in the lightweight fragments of description logics DL-Lite. More specifically, we assume that the TBox component is coherent, stable, and fully reliable. However, the ABox component may be inconsistent with respect to the TBox, partially preordered and uncertain. Uncertainty is encoded in the framework of possibility theory. In this context, we propose an extension of standard possibilistic DL-Lite. We represent the ABox as a symbolic weighted base, where the weights attached to the assertions are ordered according to a strict partial order. We define a tractable method for computing a single possibilistic repair for a partially preordered weighted ABox. The idea is to consider the possibilistic compatible bases of such an ABox, which intuitively encode all the possible extensions of a partial order, and compute the possibilistic repair of each compatible base. We then compute the intersection of all these possibilistic repairs to obtain a single repair for the initial ABox. We also provide an equivalent characterization by introducing the notion of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pi$</tex-math></inline-formula> -accepted assertions. This ensures that the computation of the partially preordered possibilistic repair can be achieved in polynomial time in DL-Lite.

Strong Equivalence of Logic Programs with Ordered Disjunction: A Logical Perspective

AbstractLogic Programs with Ordered Disjunction (LPODs) extend classical logic programs with the capability of expressing preferential disjunctions in the heads of program rules. The initial semantics of LPODs, although simple and quite intuitive, is not purely model-theoretic. As a result, certain properties of programs appear non-trivial to formalize in purely logical terms. For example, the current characterization of strong equivalence for LPODs, does not coincide with logical equivalence in some specific logic. This comes in sharp contrast with the well-known characterization of strong equivalence for classical logic programs, which coincides with logical equivalence in the logic of here-and-there. In this paper we obtain a purely logical characterization of strong equivalence for LPODs as logical equivalence in a four-valued logic. Moreover, we provide a new proof of the coNP-completeness of strong equivalence for LPODs, which has an interest in its own right since it relies on the special structure of such programs. Our results are based on the recent logical semantics of LPODs, a fact which we believe indicates that this new semantics may prove to be a useful tool in the further study of LPODs.

What Makes a Strong Monad?

Strong monads are important for several applications, in particular, in the denotational semantics of effectful languages, where strength is needed to sequence computations that have free variables. Strength is non-trivial: it can be difficult to determine whether a monad has any strength at all, and monads can be strong in multiple ways. We therefore review some of the most important known facts about strength and prove some new ones. In particular, we present a number of equivalent characterizations of strong functor and strong monad, and give some conditions that guarantee existence or uniqueness of strengths. We look at strength from three different perspectives: actions of a monoidal category V, enrichment over V, and powering over V. We are primarily motivated by semantics of effects, but the results are also useful in other contexts.

Incompatibility in General Probabilistic Theories, Generalized Spectrahedra, and Tensor Norms

In this work, we investigate measurement incompatibility in general probabilistic theories (GPTs). We show several equivalent characterizations of compatible measurements. The first is in terms of the positivity of associated maps. The second relates compatibility to the inclusion of certain generalized spectrahedra. For this, we extend the theory of free spectrahedra to ordered vector spaces. The third characterization connects the compatibility of dichotomic measurements to the ratio of tensor crossnorms of Banach spaces. We use these characterizations to study the amount of incompatibility present in different GPTs, i.e. their compatibility regions. For centrally symmetric GPTs, we show that the compatibility degree is given as the ratio of the injective and the projective norm of the tensor product of associated Banach spaces. This allows us to completely characterize the compatibility regions of several GPTs, and to obtain optimal universal bounds on the compatibility degree in terms of the 1-summing constants of the associated Banach spaces. Moreover, we find new bounds on the maximal incompatibility present in more than three qubit measurements.

Inner functions in QK spaces and multipliers

Abstract In this paper, under some conditions on weighted function K, we give some new equivalent characterizations of inner functions in Q K spaces. Meanwhile, we also studied inner functions in Q K spaces as multipliers of C 𝓑(Q K ) ∩ BMOA, where C 𝓑(Q K ) denoted the closure of Q K spaces in Bloch spaces.

A Novel Approach to the Fuzzification of Fields

There are many symmetries in L-fuzzy algebras. In this paper, a novel approach to the fuzzification of a field is introduced. We define a mapping F:LX→L from the family of all the L-fuzzy sets on a field X to L such that each L-fuzzy set is an L-fuzzy subfield to some extent. Some equivalent characterizations F(μ) are given by means of cut sets. It is proved that F is L-fuzzy convex structure on X, hence (X,F) forms an L-fuzzy convexity space. A homomorphism between fields is exactly an L-fuzzy convexity preserving mapping and an L-fuzzy convex-to-convex mapping. Finally, we discuss some operations of L-fuzzy subsets.

𝒲(ξ)-Gorenstein objects in extriangulated categories

We fix a proper class of [Formula: see text]-triangles [Formula: see text] in an extriangulated category [Formula: see text]. Let [Formula: see text] be a class of objects in [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text] and all [Formula: see text]. In this paper, we study [Formula: see text]-Gorenstein objects, and use the relative homological methods to show that the stability of the Gorenstein category [Formula: see text] in [Formula: see text]. Moreover, we give some equivalent characterizations of [Formula: see text]-Gorenstein objects and finite [Formula: see text]-(co)resolution dimensions for any object in [Formula: see text]. As an application, we consider the equivalent characterizations of [Formula: see text]-[Formula: see text]projective objects, [Formula: see text]-[Formula: see text]projective dimension and their duals.

On a new kind of ordered fuzzy group

This paper aims to further study the new kind of ordered fuzzy group named ordered L-group, which is put forward in literature [20]. Some algebraic properties of ordered L-groups, such as the relationship between elements, the equivalent characterizations and the products of these groups are discussed. Following that, the properties of substructures including characterization theorems, the convexity, the products of (normal) subgroups maintain the original substructure, along with the properties of ordered L-group homomorphisms are explored. The discussion of ordered fuzzy groups in this paper is from the perspective of fuzzy binary operation, which is different from the commonly method that just discuss the fuzzification of substructures in the research of fuzzy algebra. It can better reflect the essence of fuzzy groups logically just like that of classical groups.

Anisotropic variable Campanato-type spaces and their Carleson measure characterizations

Let $$p(\cdot ):\ {\mathbb {R}^n}\rightarrow (0,\infty )$$ be a variable exponent function satisfying the globally log-Hölder continuous condition and A a general expansive matrix on $${\mathbb {R}^n}$$ . In this article, the authors introduce the anisotropic variable Campanato-type spaces and give some applications. Especially, using the known atomic and finite atomic characterizations of anisotropic variable Hardy space $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ , the authors prove that this Campanato-type space is the appropriate dual space of $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ with full range $$p(\cdot )$$ . As applications, the authors first deduce several equivalent characterizations of these Campanato-type spaces. Furthermore, the authors also introduce the anisotropic variable tent spaces and show their atomic decomposition. Combining this and the obtained dual theorem, the Carleson measure characterizations of these anisotropic variable Campanato-type spaces are established.

Structure and Complexity of Bag Consistency

Since the early days of relational databases, it was realized that acyclic hypergraphs give rise to database schemas with desirable structural and algorithmic properties. In a bynow classical paper, Beeri, Fagin, Maier, and Yannakakis established several different equivalent characterizations of acyclicity; in particular, they showed that the sets of attributes of a schema form an acyclic hypergraph if and only if the local-to-global consistency property for relations over that schema holds, which means that every collection of pairwise consistent relations over the schema is globally consistent. Even though real-life databases consist of bags (multisets), there has not been a study of the interplay between local consistency and global consistency for bags. We embark on such a study here and we first show that the sets of attributes of a schema form an acyclic hypergraph if and only if the local-to-global consistency property for bags over that schema holds. After this, we explore algorithmic aspects of global consistency for bags by analyzing the computational complexity of the global consistency problem for bags: given a collection of bags, are these bags globally consistent? We show that this problem is in NP, even when the schema is part of the input. We then establish the following dichotomy theorem for fixed schemas: if the schema is acyclic, then the global consistency problem for bags is solvable in polynomial time, while if the schema is cyclic, then the global consistency problem for bags is NP-complete. The latter result contrasts sharply with the state of affairs for relations, where, for each fixed schema, the global consistency problem for relations is solvable in polynomial time.

A novel difference and derivative of linearly correlated fuzzy number-valued functions

In the present paper, the notion of the linearly correlated difference for linearly correlated fuzzy numbers is introduced. Especially, the linearly correlated difference and the generalized Hukuhara difference are coincident for interval numbers or even symmetric fuzzy numbers. Accordingly, an appropriate metric is induced by using the norm and the linearly correlated difference in the set of linearly correlated fuzzy numbers. Based on the symmetry of the basic fuzzy number, the linearly correlated derivative is proposed by the linearly correlated difference of linearly correlated fuzzy number-valued functions. In both non-symmetric and symmetric cases, the equivalent characterizations of the linearly correlated differentiability of a linearly correlated fuzzy number-valued function are established, respectively. Moreover, it is shown that the linearly correlated derivative is consistent with the generalized Hukuhara derivative for interval-valued functions.