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In this paper, we consider an operator for refinement of variables to b e used in the design of hybrid system. Variables on a given level of abstraction are replaced by more complicated processes on a lower level just like the function are called in the program. Then we established the equivalence, bisimulation equivalence and approximate bisimulation equivalence which are by polynomial flow event structures. These equivalence, bisimulation equivalence a nd approximate bisimulation equivalence are based on the common forms of their zeros. The example show that the equivalence, bisimulation equivalence and approximate bisimulation equivalence are preserved or not under the variables refinement, if the equivalence is not preserved precisely, then we can use the approximate methods to make them approximate equivalence. Lastly we show that our refineme nt has some nice properties.

The article discusses the set of square n × n binary matrices with the same number of 1’s in each row and each column. An equivalence relation on this set is introduced. Each binary matrix is represented using ordered n-tuples of natural numbers. We are looking for a formula which calculates the number of elements of each factor-set by the introduced equivalence relation. We show a relationship between some particular values of the parameters and the Fibonacci sequence.

We consider left-invariant control affine systems, evolving on three-dimensional matrix Lie groups. Equivalence and controllability are investigated. All full-rank systems are classified, under detached feedback equivalence. A representative is identified for each equivalence class. The controllability nature of these representatives is determined.

Let be the algebra of all complex matrices. We characterize linear maps satisfyingfor all , .

We concern with the classification problem of generalized linear control systems, which might be useful for some engineering applications. Inspired by the work of Shayman and Zhou in 1987, we give the definition of linear, differentiable and topological equivalence for a special class of generalized linear systems in a unified way. Then we derive some properties on these three kinds of equivalences and show the canonical forms through a concrete example. In this paper, we obtain natural generalizationsof the Brunovsky's feedback equivalence theorem and the Willems' topological classification theorem for usual linear control systems.

For a matrix pencil A 0 x − A 1 , where A 0 and A 1 are (n × n) -matrices over an arbitrary field F , and A0 is a nonsingular matrix, we establish the normal form with respect to semiscalar equivalence. We also describe the structure of nonsingular polynomial matrices over the field F , which can be reduced to the established form by the transformations of semiscalar equivalence.

In this paper using the second order formula, namely the ∀∃(∀)-identities, we characterize some subclasses of the invertible algebras that are linear over an Abelian group and have restrictions on the use of the automorphisms of the corresponding group.

In this article, the invertible matrices over commutative semirings are studied. Some properties and equivalent descriptions of the invertible matrices are given and the inverse matrix of an invertible matrix is presented by analogues of the classic adjoint matrix. Also, Cramer's rule over a commutative semiring is established. The main results obtained in this article generalize the corresponding results for matrices over commutative rings, for lattice matrices, for incline matrices, for matrices over zerosumfree semirings and for matrices over additively regular semirings.

We study the notion of polynomial-time relation reducibility among computable equivalence relations. We identify some benchmark equivalence relations and show that the reducibility hierarchy has a rich structure. Specifically, we embed the partial order of all polynomial-time computable sets into the polynomial-time relation reducibility hierarchy between two benchmark equivalence relations $\mathsf{E}_{\lambda}$ and $\mathsf{id}$ . In addition, we consider equivalence relations with finitely many nontrivial equivalence classes and those whose equivalence classes are all finite.

A new method for solving the problem of equivalence of linear unary recursive programs is proposed. The main idea behind this method is to reduce the equivalence problem to known problems on graphs and to problems in group theory. A class of program semantics is singled out for which the problem of the equivalence of the programs in question is solvable in polynomial time by the proposed method.

We present a constructive solution to the problem of full output feedback equivalence, of linear, minimal, time-invariant systems. The equivalence relation on the set of systems is transformed to another on the set of invertible block Bezout/Hankel matrices using the isotropy subgroups of the full state feedback group and the full output injection group. The transformation achieving equivalence is calculated solving linear systems of equations. We give a polynomial version of the results proving that two systems are full output feedback equivalent, if and only if they have the same family of generalized Bezoutians. We present a new set of output feedback invariant polynomials that generalize the breakaway polynomial of scalar systems.

We show that the equivalence of deterministic linear top-down tree-to-word transducers is decidable in polynomial time. Linear tree-to-word transducers are non-copying but not necessarily order-preserving and can be used to express XML and other document transformations. The result is based on a partial normal form that provides a basic characterization of the languages produced by linear tree-to-word transducers., short version of this paper will be published in the proceedings of the 20th Conference on Developments in Language Theory (DLT 2016), Montreal, Canada

In this paper some numerical algorithms generating the full invertible matrices and full matrices with a given spectrum are presented.

The paper is devoted to the notion of typical equivalence introduced by B. I. Plotkin. We give some examples of elementarily equivalent objects that are not typically equivalent and show two ways to construct nonisomorphic typically equivalent algebras. We also prove A. I. Maltsev’s theorem on elementary equivalence of linear groups over fields for the case of typical equivalence.

Let U k be the general Boolean algebra and T a linear operator on M m,n (U k ). If for any A in M m,n (U k ) (M n (U k ), respectively), A is regular (invertible, respectively) if and only if T(A) is regular (invertible, respectively), then T is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over U k . Meanwhile, noting that a general Boolean algebra U k is isomorphic to a finite direct product of binary Boolean algebras, we also give some characterizations of linear operators that strongly preserve regular (invertible, respectively) matrices over 169-7 k from another point of view.

In this paper we introduce a new class, called F , of linear transformations defined from the space of real n × n symmetric matrices into itself. Within this new class, we show the equivalence between Q - and Q b -transformations. We also provide conditions under which a linear transformation belongs to F . Moreover, this class, when specialized to square matrices of size n , turns out to be the largest class of matrices for which such equivalence holds true in the context of standard linear complementary problems.

Group classification of classes of mKdV-like equations with time-dependent coefficients is carried out. The usage of equivalence transformations appears a crucial point for the exhaustive solution of the problem. We prove that all the classes under consideration are normalized. This allows us to formulate the classification results in three ways: up to two kinds of equivalence (which are generated by transformations from the corresponding equivalence groups and all admissible point transformations) and using no equivalence. A simple way of the construction of exact solutions of mKdV-like equations using equivalence transformations is described., 10 pages, minor changes

While a pair of N × N matrices can almost always be exactly and simultaneously diagonalized by a generalized eigendecomposition, no exact solution exists in the case of a set with more than two matrices. This problem, termed approximate joint diagonalization (AJD), is instrumental in blind signal processing. When the set of matrices to be jointly diagonalized includes at least N linearly independent matrices, we propose a suboptimal but closed-form solution for AJD in the direct least-squares sense. The corresponding non-iterative algorithm is given the acronym DIEM (DIagonalization using Equivalent Matrices). Extensive numerical simulations show that DIEM is both fast and accurate compared to the state-of-the-art iterative AJD algorithms.

Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser ("graph") equivalence are inequivalent under the finer ("bundle") equivalence. Here we relate graphs to transversals and splitting relative difference sets (RDSs) and introduce an intermediate relation, canonical equivalence, to aid in distinguishing the classes. We identify very precisely the conditions under which a graph equivalence determines a bundle equivalence, using transversals and extensions. We derive a new and easily computed algebraic measure of nonlinearity for a function f, calculated from the image of its coboundary ?f. This measure is preserved by bundle equivalence but not by the coarser equivalences. It takes its minimum value if f is a homomorphism, and takes its maximum value if the graph of f contains a splitting RDS.

In this paper we explore equivalence conditions and invariants for behaviors given in kernel representations. In case the kernel representation is given in terms of a linear matrix pencil, the invariants for strict equivalence are given by the Kronecker canonical form which, in turn, we interpret in geometric control terms. If the behavior is given in a kernel representation by a higher order rectangular polynomial matrix, the natural equivalence concept is behavior equivalence. These notions are closely related to the Morse group that incorporates state space similarity transformations, state feedback, and output injection. A simple canonical form for behavioral equivalence is given that clearly exhibits the reachable and autonomous parts of the behavior. Using polynomial models we also present a unified approach to pencil equivalence that elucidates the close connections between classification problems from linear algebra, geometric control theory, and behavior theory. We also indicate how to derive the invariants under behavior equivalence from the Kronecker invariants.

We show (among other things) that Brauer's k(B)-conjecture holds for defect groups with are central extensions of metacyclic 2-groups by cyclic groups. The same holds for defect groups which contain a central cyclic subgroup of index at most 9. In particular the k(B)-conjecture holds for 2-blocks of defect at most 4 and 3-blocks of defect at most 3. As a byproduct we obtain the block invariants of 2-blocks with minimal nonmetacyclic defect groups., Comment: 26 pages

Let = max(a, b), =a+b, a, . In max-plus algebra we work on the linear algebra structure for the pair of operations () extended to matrices and vectors over . In this paper our main aim is to reproduce the work of R. A. Cuninghame-Green [3] on the linear systems over a max-plus semi-field .

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.

We investigate the asymptotic equivalence for linear differen- tial systems by means of the notions of t1 -similarity and strong stability.

We prove that orbit equivalence of measure preserving ergodic a.e. free actions of a countable group with the relative property (T) is a complete analytic equivalence relation.

We give a new and elementary proof that simultaneous similarity and simultaneous equivalence of families of matrices are invariant under extension of the ground field, a result which is non-trivial for finite fields and first appeared in a paper of Klinger and Levy., Comment: 10 pages (minor corrections)

A new method to describe the structures of zeolites and related open-framework crystalline solids was developed on the basis of the atom–atom distance matrix (DM). It has been mathematically proved that each such distance matrix corresponds to no more than two different (racemic) structures. For a given structure, the distance matrices for different atom-labeling schemes within a unit cell can be transformed to each other through a matrix transformation, and all these equivalent matrices correspond to the same (or the mirror) structure. A procedure was developed to standardize such representing matrices for each structure, which can be represented as a heat map, the 2-dimensional (2D) barcode or finger print of a structure. With this method, information on the connectivity of the framework atoms in the structure of zeolites or related open-framework crystalline solids was included in the heat map. The structural features of rings, channels, and cages of zeolites and related open-framework crystalline materials can also be recognized in the heat map. The inherent correlation between the structures and their heat maps is discussed. With this new method, the structure of zeolites or related open-framework crystalline solids can be studied by the well developed matrix theory, graph theory, or other mathematic tools.

In this paper, we consider equivalence relations of C*-algebra extensions and describe the relationship between the isomorphism equivalence and the unitary equivalence. We also show that a certain group homomorphism is the obstruction for these equivalence relations to be the same.

We examine the effective categoricity of equivalence structures via Ershov's difference hierarchy. We explore various kinds of categoricity available by distinguishing three different notions of isomorphism available in this hierarchy. We prove several results relating our notions of categoricity to computable equivalence relations: for example, we show that, for such relations, computable categoricity is equivalent to our notion of weak ω-c.e. categoricity, and that -categoricity is equivalent to our notion of graph-ω-c.e. categoricity.

We show that the L2-Betti numbers of equivalence relations defined by R. Sauer coincide with those defined by D. Gaboriau.

In this paper, a generalization to arbitrary fields of the usual Wiener–Hopf equivalence of complex valued rational matrix functions is given and the left local Wiener–Hopf factorization indices defined in our previous work [A. Amparan, S. Marcaida, I. Zaballa, Local realizations and local polynomial matrix representations of systems, Linear Algebra Appl. 425 (2007) 757–775] are proved to form a complete system of invariants for this equivalence relation. For the case when the field is algebraically closed a reduced form of a controllable matrix pair under the feedback equivalence is presented for which the controllability indices can be written as sums of the local controllability indices [A. Amparan, S. Marcaida, I. Zaballa, On the existence of linear systems with prescribed invariants for system similarity, Linear Algebra Appl. 413 (2006) 510–533].

Let be the set of all subspaces of . The following three groups act on : (i) the symmetric group on the coordinate positions of ; (ii) the group of monomial transformations of ; (iii) the group of semi-linear monomial transformations of . The orbits of under these actions are the equivalence classes of linear codes in under three notions of equivalence: permutation equivalence, monomial equivalence and equivalence. Let , , N n,q Γ denote the numbers of the orbits under the three actions on respectively. It was recently proved that as n → ∞, and . In this article, we show that , where q = pr and p is a prime.

We consider the set of quadruples of matrices defining regular singular linear time-invariant dynamical systems under restricted system equivalence and derive miniversal deformations from a basis of the normal space to orbits under the Lie group action related to this equivalence relation. A lower bound and an upper bound for the dimension of the orbits are obtained. We conclude with examples and further comments about genericity.

Skew-Hadamard matrices are of special interest due to their use, among others, in constructing orthogonal designs. In this paper, we give a survey on the existence and equivalence of skew-Hadamard matrices. In addition, we present some new skew-Hadamard matrices of order 52 and improve the known lower bound on the number of the skew-Hadamard matrices of this order.

We introduce an equivalence between central simple strongly G-graded algebras. Such classes are associated in a natural way to absolutely irreducible characters of semisimple G-graded algebras. We study invariants of this equivalence relation, and also the structure of certain representatives of the equivalence classes.

Linear Map of MatricesThe paper is concerned with a generalization of concepts introduced in [13], i.e. introduced are matrices of linear transformations over a finitedimensional vector space. Introduced are linear transformations over a finitedimensional vector space depending on a given matrix of the transformation. Finally, I prove that the rank of linear transformations over a finite-dimensional vector space is the same as the rank of the matrix of that transformation.

Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also show that if there is a Baire measurable assignment of means to the equivalence classes of E, then E is generically smooth.

We present criteria for a pair of maps to constitute a quaternion-symbol equivalence (or a Hilbert-symbol equivalence if we deal with global function fields) expressed in terms of vanishing of the Clifford invariant. In principle, we prove that a local condition of a quaternion-symbol equivalence can be transcribed from the Brauer group to the Brauer-Wall group.

Throughout this paper we consider data structures as sorted algebras endowed with a designated subset of their visible part, which represents the set of truth values. The originality of our approach is the application of the standard abstract algebraic logic theory of deductive systems to the hidden heterogeneous case. We generalize the well-known equivalence relation between finite automata, which relies on the Nerode equivalence relation between states, to k-data structures. This is obtained via the Leibniz congruence, which can be viewed as a generalization of the Nerode equivalence in automata theory.

The equivalence between the classical moment matching and the Laguerre-based reduction approaches is shown. In addition, this equivalence is generalized to include a family of coefficients known as generalized Markov parameters.

In this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice. We give certain characterizations of fuzzy semi-partitions and fuzzy partitions over a complete residuated lattice, as well as over a linearly ordered complete Heyting algebra. In the latter case, for a fuzzy equivalence relation over a linearly ordered complete Heyting algebra, we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it. Most of the presented results are new, but some of them are generalizations of known results given in a way which simplifies and clarifies them.

Each additive cut in the nonstandard natural numbers *ℕ induces the equivalence relation MU on *ℕ defined as xMU y if |x − y| e U. Such equivalence relations are said to be monadic. Reducibility between monadic equivalence relations is studied. The main result (Theorem 3.1) is that reducibility can be defined in terms of cofinality (or coinitiality) and a special parameter of a cut, called its width. Smoothness and the existence of transversals are also considered. The results obtained are similar to theorems of modern descriptive set theory on the reducibility of Borel equivalence relations.

The equivalence problem for a group G is the problem of deciding which equations hold in G. It is known that for finite nilpotent groups and certain other solvable groups, the equivalence problem has polynomial-time complexity. We prove that the equivalence problem for a finite nonsolvable group G is co-NP-complete by reducing the k-coloring problem for graphs to the equivalence problem, where k is the cardinality of G.

We give a complete derived equivalence classification of all nonstandard representation-infinite domestic selfinjective algebras over an algebraically closed field. As a consequence, also a complete stable equivalence classification of these algebras is obtained., Comment: 13 pages, 7 figures

We derive a computable set of necessary and sufficient conditions for the existence of a homomorphism from one shift of finite type to another. Also we consider an equivalence relation on subshifts, called weak equivalence, which was introduced and studied by Beal and Perrin. We classify arbitrary shifts of finite type up to weak equivalence.

There exist several models for writing the equations of a multidimensional (nD) linear system and equivalence transformations can be used to pass from one representation to another. Within the constructive algebraic analysis approach to nD linear systems theory, this equivalence issue is studied by means of isomorphisms of finitely presented modules. The present paper illustrates this general algebraic analysis approach by focussing on the equivalence problem for two frequently used 2D models, namely the generalized Fornasini-Marchesini models and the Roesser models.

Two matrices $A$ and $B$ are called unitary (resp. orthogonal) equivalent if $AU=VB$ for two unitary (resp. orthogonal) matrices $U$ and $V$. Using trace identities, criteria are given for simultaneous unitary, orthogonal or complex orthogonal equivalence between two sets of matrices., 8 pages, 13 refs

A hyperoperation ◦ on a nonempty set H is a function from H×H into P ∗(H) where P ∗(H) is the set of all nonempty subset of H and (H, ◦) is call a hypergroupoid. A hypergroupoid (H, ◦) is called a semihypergroup if the hyperoperation ◦ is associative. Thus, semihypergroups generalize semigroups. Moreover, if S is a semigroup; we can define a hyperoperation ◦ on S in order to make (S, ◦) a semihypergroup. In 2013, R.I. Sararnrakskul defined a hyperoperation ◦ on the partial transformation semigroup P (X) to make a semihypergroup. In this paper, we define a regular equivalence relation ρ on (P (X), ◦) so that P (X)/ρ is a semihypergroup and then we studies some subsemihypergroup of P (X)/ρ. AMS Subject Classification: 20M20, 20N20 Received: November 10, 2014 c © 2015 Academic Publications, Ltd. url: www.acadpubl.eu §Correspondence author 22 R.I. Sararnrakskul, S. Pianskool

The max algebra consists of the nonnegative real numbers equipped with two binary operations, maximization and multiplication. We characterize the invertible linear operators that preserve the set of commuting pairs of matrices over a subalgebra of max algebra.