Sufficient Conditions For Equivalence Research Articles

Soft Weakly Quasi-Continuous Functions Between Soft Topological Spaces

As an extension of quasi-continuity in general topology, we define soft quasi-continuity. We show that this notion is equivalent to the known notion of soft semi-continuity. Next, we define soft weak quasi-continuity. With the help of examples, we prove that soft weak quasi-continuity is strictly weaker than both soft semi-continuity and soft weak continuity. We introduce many characterizations of soft weak quasi-continuity. Moreover, we study the relationship between soft quasi-continuity and weak quasi-continuity with their analogous notions in general topology. Furthermore, we show that soft regularity of the co-domain of a soft function is a sufficient condition for equivalence between soft semi-continuity and soft weakly quasi-continuity. Furthermore, we provide several results of soft composition, restrictions, preservation, and soft graph theorems in terms of soft weak quasi-continuity.

State Bounding Description and Reachable Set Estimation for Discrete-Time Genetic Regulatory Networks With Time-Varying Delays and Bounded Disturbances

This article investigates the problems of state bounding description and reachable set estimation for discrete-time delayed genetic regulatory networks with bounded disturbances. A novel delay-dependent sufficient condition composing several simple linear matrix inequalities is first given to guarantee that the state trajectories converge globally exponentially into a Cartesian product of two polytopes. Furthermore, equivalent sufficient conditions directly represented by the system parameters are derived. As applications, it is shown that these sufficient conditions are exactly global exponential stability criteria of the zero equilibrium when the disturbances vanish, and the Cartesian product of two polytopes can be viewed as a reachable set estimation of the states when the initial functions are limited into a certain range. The effectiveness of the proposed approach is illustrated by two numerical examples. Compared with the existing results, it is worthy to stress that the presented method does not need to construct any Lyapunov–Krasovskii functional and can be easily realized.

State bounding and controller design for genetic regulatory networks with multiple delays and bounded disturbances

AbstractThis article addresses the problems of state bounding and state‐feedback controller design for genetic regulatory networks with multiple delays and bounded disturbances. Multiple time‐varying discrete delays and multiple constant distributed delays are involved. A delay‐dependent sufficient condition containing several simple inequalities is first given to guarantee that the state trajectories remain inside or converge exponentially into a Cartesian product of two polytopes. Furthermore, equivalent sufficient conditions directly represented by the system parameters are derived. Based on the state bounding results, one can obtain reachable set estimation and global exponential stability criteria. By designing state‐feedback controllers, the state trajectories of the close‐loop system are limited in a given Cartesian product of two polytopes. It is worthy to stress that the proposed method does not need to construct any Lyapunov–Krasovskii functional, and it can be easily realized. The effectiveness of the obtained theoretical results is illustrated by a numerical example.

Sufficient conditions for equivalence between safety factor-based and reliability-based design requirements

Although there is a perceptible trend of developing and adopting reliability-based design (RBD) in geotechnical engineering, geotechnical practitioners are used to safety factor-based design (SFBD) due to its simplicity. At the stage of transition from SFBD to RBD, geotechnical practitioners frequently compare RBD results with SFBD results that they get used to for cross-checking. Nevertheless, SFBD and RBD, in theory, may provide different design outcomes because they use different design methodologies and requirements. The equivalence between SFBD and RBD (i.e., with the same feasible designs) can only be achieved under some conditions. This study proposes sufficient conditions for achieving the equivalence between SFBD and RBD based on the probabilistic sufficiency factor (i.e., a fractile of the performance function at the target failure probability level PTE). Simplified verification methods of the proposed sufficient conditions in three simple cases are provided. The proposed sufficient conditions and their verification are illustrated using a gravity retaining wall example with a separable performance function and a deep excavation example with an inseparable performance function. Results demonstrate the validity of the proposed sufficient conditions, under which a unique mapping from the allowable factor of safety to PTE for a given design situation can be established.

Weaker Forms of Soft Regular and Soft T2 Soft Topological Spaces

Soft ω-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness is introduced. Then soft ω-regularity as a weaker form of both soft regularity and soft ω-local indiscreetness is defined and investigated. Additionally, soft ω-T2 as a new soft topological property that lies strictly between soft T2 and soft T1 is defined and investigated. It is proved that soft anti-local countability is a sufficient condition for equivalence between soft ω-locally indiscreetness (resp. soft ω-regularity) and soft locally indiscreetness (resp. soft ω-regularity). Additionally, it is proved that the induced topological spaces of a soft ω-locally indiscrete (resp. soft ω-regular, soft ω-T2) soft topological space are (resp. ω-regular, ω-T2) topological spaces. Additionally, it is proved that the generated soft topological space of a family of ω-locally indiscrete (resp. ω-regular, ω-T2) topological spaces is soft ω-locally indiscrete and vice versa. In addition to these, soft product theorems regarding soft ω-regular and soft ω-T2 soft topological spaces are obtained. Moreover, it is proved that soft ω-regular and soft ω-T2 are hereditarily under soft subspaces.

A study on association in time of a finite semi markov process

We derive three equivalent sufficient conditions for association in time of a finite state semi Markov process in terms of transition probabilities and crude hazard rates. This result generalizes the earlier results of Esary and Proschan (1970)for a binary Markov process and Hjort, Natvig and Funnemark (1985) for a multistate Markov process.

On Isometric Error-Correcting Codes over Finite Fields of Prime Order

Error-correcting codes obtained from each other via a Hamming distance preserving transformation, called isometry, are equivalent. In this paper, we consider three different isometries which yield permutationally equivalent linear codes, monomially equivalent linear codes, and equivalent nonlinear codes, respectively. For each case, we derive some new necessary and sufficient conditions for equivalence using the concept of weight vector of a code or distance matrix of a code. Results hold for error-correcting codes over a finite field of prime order.

Event‐triggered control for continuous‐time switched linear systems

The event-triggered control problem for switched linear system is addressed in this study. The periodical sampling scheme and event-triggering condition are incorporated in the closed-loop. The feedback control updates its value only at sampling instants as long as event-triggering condition is satisfied as well. In addition, the switchings are only allowed to occur at sampling instants and meanwhile the switching condition is satisfied. Three equivalent sufficient conditions are proposed to ensure the asymptotic stability of switched systems. In particular, one condition has a promising feature of affineness in system matrices, and as a consequence, it is extended to robust sampling case and ℒ 2 -gain analysis. Several examples are provided to illustrate the authors' results.

Passivity-Based Asynchronous Control for Markov Jump Systems

The issue of asynchronous passive control is addressed for Markov jump systems in this technical note. The asynchronization phenomenon appears between the system modes and controller modes, which is described by a hidden Markov model. Accordingly, a hidden Markov jump model is used to name the resultant closed-loop system. By utilizing the matrix inequality technique, three equivalent sufficient conditions are obtained, which can guarantee the hidden Markov jump systems to be stochastically passive. Based on the established conditions, the design of asynchronous controller, which covers the well-known mode-independent controller and synchronous controller as special cases, is addressed. The DC motor device is applied to demonstrate the practicability of the derived asynchronous synthesis scheme.

On the equivalence of systems of different sizes, with applications to system comparisons

Abstract The signature of a coherent system is a useful tool in the study and comparison of lifetimes of engineered systems. In order to compare two systems of different sizes with respect to their signatures, the smaller system needs to be represented by an equivalent system of the same size as the larger system. In the paper we show how to construct equivalent systems by adding irrelevant components to the smaller system. This leads to simpler proofs of some current key results, and throws new light on the interpretation of mixed systems. We also present a sufficient condition for equivalence of systems of different sizes when restricting to coherent systems. In cases where for a given system there is no equivalent system of smaller size, we characterize the class of lower-sized systems with a signature vector which stochastically dominates the signature of the larger system. This setup is applied to an optimization problem in reliability economics.

On α 2 − ν 2-products of automata

Two equivalent sufficient conditions are given for the completeness of classes of finite automata with respect to the isomorphic simulation under the α 2 − ν 2-product. It is conjectured that these conditions are also necessary with respect to the isomorphic or homomorphic simulation too.

Equivalence of rational representations of behaviors

This article deals with the equivalence of representations of behaviors of linear differential systems. In general, the behavior of a given linear differential system has many different representations. In this paper we restrict ourselves to kernel and image representations. Two kernel representations are called equivalent if they represent one and the same behavior. For kernel representations defined by polynomial matrices, necessary and sufficient conditions for equivalence are well known. In this paper, we deal with the equivalence of rational representations, i. e. kernel and image representations that are defined in terms of rational matrices. As the first main result of this paper, we will derive a new condition for the equivalence of rational kernel representations of possibly noncontrollable behaviors. Secondly we will derive conditions for the equivalence of rational representations of a given behavior in terms of the polynomial modules generated by the rows of the rational matrices. We will also establish conditions for the equivalence of rational image representations. Finally, we will derive conditions under which a given rational kernel representation is equivalent to a given rational image representation.

New results about the relationship between optimally weighted least squares estimate and linear minimum variance estimate

The optimally weighted least squares estimate and the linear minimum variance estimate are two of the most popular estimation methods for a linear model. In this paper, the authors make a comprehensive discussion about the relationship between the two estimates. Firstly, the authors consider the classical linear model in which the coefficient matrix of the linear model is deterministic, and the necessary and sufficient condition for equivalence of the two estimates is derived. Moreover, under certain conditions on variance matrix invertibility, the two estimates can be identical provided that they use the same a priori information of the parameter being estimated. Secondly, the authors consider the linear model with random coefficient matrix which is called the extended linear model; under certain conditions on variance matrix invertibility, it is proved that the former outperforms the latter when using the same a priori information of the parameter.

Equivalence of queries that are sensitive to multiplicities

The query equivalence problem has been studied extensively for set-semantics and, more recently, for bag and bag-set semantics. However, SQL queries often combine set, bag and bag-set semantics. For example, an SQL query that returns a multiset of elements may call a subquery or view that returns a set of elements. Queries may access both relations that do not contain duplicates, as well as relations with duplicates. As another example, in SQL one can compute a multiset-union of queries, each of which returns a set of answers. This paper presents combined semantics, which formally models query evaluation combining set, bag and bag-set semantics. The equivalence problem for queries evaluated under combined semantics is studied. A sufficient condition for equivalence is presented. For several important common classes of queries necessary and sufficient conditions for equivalence are presented.

Differential inequality conditions for dominance between continuous Archimedean t-norms

Dominance between triangular norms (t-norms) is a versatile relationship. For continuous Archimedean t-norms, dominance can be verified by checking one of many sufficient conditions derived from a generalization of the Mulholland inequality. These conditions pertain to various convexity properties of compositions of additive generators and their inverses. In this paper, assuming differentiability of these additive generators, we propose equivalent sufficient conditions that can be expressed as inequalities involving derivatives of the additive generators, avoiding the need of composing them. We demonstrate the powerfulness of the results by the straightforward rediscovery of dominance relationships in the Schweizer-Sklar t-norm family, as well as by unveiling some formerly unknown dominance relationships in the Sugeno-Weber t-norm family. Finally, we illustrate that the results can also be applied to members of different parametric families of t-norm.

Indicators and indexes of directional output loss and input allocative inefficiency

AbstractWe extend Grosskopf et al.'s method and create an output loss indicator using directional output distance functions that allows non‐radial efficiency gains in output. We compare our new output loss indicator with an indicator of input allocative inefficiency and derive the necessary and sufficient condition for equivalence between the two indicators. We also present an output loss index and corresponding input allocative inefficiency index and consider how indexes are related. Then we extend our analysis to productivity change and derive the necessary and sufficient condition for an output loss change indicator to be equivalent to an indicator of input allocative inefficiency change. Copyright © 2008 John Wiley & Sons, Ltd.

Deciding equivalences among conjunctive aggregate queries

Equivalence of aggregate queries is investigated for the class of conjunctive queries with comparisons and the aggregate operators count, count-distinct, min, max, and sum. Essentially, this class contains unnested SQL queries with the above aggregate operators, with a where clause consisting of a conjunction of comparisons, and without a having clause. The comparisons are either interpreted over a domain with a dense order (like the rationals) or with a discrete order (like the integers). Characterizations of equivalence differ for the two cases. For queries with either max or min, equivalence is characterized in terms of dominance mappings, which can be viewed as a generalization of containment mappings. For queries with the count-distinct operator, a sufficient condition for equivalence is given in terms of equivalence of conjunctive queries under set semantics. For some special cases, it is shown that this condition is also necessary. For conjunctive queries with comparisons but without aggregation, equivalence under bag-set semantics is characterized in terms of isomorphism. This characterization essentially remains the same also for queries with the count operator. Moreover, this characterization also applies to queries with the sum operator if the queries have either constants or comparisons, but not both. In the general case (i.e., both comparisons and constants), the characterization of the equivalence of queries with the sum operator is more elaborate. All the characterizations given in the paper are decidable in polynomial space.

Equivalence problem for annuli and Bell representations in the plane

In this paper we solve the problem of equivalence between annuli and Bell representations, which are canonical planar domains of connectivity 2. We give a necessary and sufficient condition for equivalence as an explicit formula in parameters of the domains.

H-infinity output feedback control for descriptor systems with delayed states

This paper considers the H-infinity dynamic output feedback control for descriptor systems with delay in states. The controller is a descriptor system without delay. Several equivalent sufficient conditions for the existence of one descriptor dynamic controller without impulsive models are given. Furthermore the explicit expression of the desired controller is obtained. The detailed design of the controller is presented using the cone complementarity linearization iterative algorithm and the LMI method. A numerical example is shown to illustrate the designed method.

ON THE EXISTENCE OF EQUIVALENT TARIFF VECTORS — WHEN THE STATUS QUO MATTERS

It has long been recognized that, in a restricted world of just two countries and two commodities, any feasible non-distorting international transfer can be replaced by a pair of individually distorting but collectively non-distorting equivalent import duties. In recent years it has sometimes been suggested that equivalence persists in broader contexts and there have been two attempts to provide interesting sufficient conditions for equivalence. Our purpose is three-fold. We first note that the sufficient conditions so far provided relate to the equilibrium values of endogenous variables (like prices) rather than to the given values of exogenous variables (like endowments, technologies and preferences) and that conditions of the former kind may be difficult to empirically justify. It is then shown, by means of a robust example with conditions imposed on exogenous variables only, that equivalence cannot be relied upon when there are more than two countries, even when the countries have no extraordinary features. Finally, we emphasize some of the difficulties in implementing equivalent tariffs even when they do exist.