Finite Set Of Equations Research Articles

Quantum ratchet with Lindblad rate equations.

A quantum random-walk model is established on a one-dimensional periodic lattice that fluctuates between two possible states. This model is defined by Lindblad rate equationsthat incorporate the transition rates between the two lattice states. Leveraging the system's symmetries, the particle velocity can be described using a finite set of equations, even though the state space is of infinite dimension. These equationsyield an analytical expression for the velocity in the long-time limit, which is employed to analyze the characteristics of directed motion. Notably, the velocity can exhibit multiple inversions, and to achieve directed motion, distinct, nonzero transition rates between lattice states are required.

Determinantal Representations and the Image of the Principal Minor Map

Abstract In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of $(\textrm{SL}_{2}(\mathbb{R}))^{n} \rtimes S_{n}$. We also study such representations over more general fields with quadratic extensions. Factorizations of Rayleigh differences prove an effective tool for capturing subtle behavior of the principal minor map. In contrast to the Hermitian case, we give examples to show for any field $\mathbb{F}$, there is no finite set of equations whose orbit under $(\textrm{SL}_{2}(\mathbb{F}))^{n} \rtimes S_{n}$ cuts out the image of $n\times n$ matrices over ${\mathbb{F}}$ under the principal minor map for every $n$.

Equations in Type-2 Fuzzy Sets

The main concern of this paper is with the equations satisfied by the algebra of truth values of type-2 fuzzy sets. That algebra has elements all mappings from the unit interval into itself with operations given by certain convolutions of operations on the unit interval. There are a number of positive results. Among them is a decision procedure, similar to the method of truth tables, to determine when an equation holds in this algebra. One particular equation that holds in this algebra implies that every subalgebra of it that is a lattice is a distributive lattice. It is also shown that this algebra is locally finite. Many questions are left unanswered. For example, we do not know whether or not this algebra has a finite equational basis, that is, whether or not there is a finite set of equations from which all equations satisfied by this algebra follow. This and various other topics about the equations satisfied by this algebra will be discussed.

Synchronization of a Josephson junction array in terms of global variables

We consider an array of Josephson junctions with a common LCR load. Application of the Watanabe-Strogatz approach [Physica D 74, 197 (1994)] allows us to formulate the dynamics of the array via the global variables only. For identical junctions this is a finite set of equations, analysis of which reveals the regions of bistability of the synchronous and asynchronous states. For disordered arrays with distributed parameters of the junctions, the problem is formulated as an integro-differential equation for the global variables; here stability of the asynchronous states and the properties of the transition synchrony-asynchrony are established numerically.

Observability at an initial state for polynomial systems

We consider observability at an initial state for polynomial systems. When testing for local observability for nonlinear systems, the observability rank condition based on the inverse function theorem is commonly used. However, the rank condition is a sufficient condition, and we cannot test for global observability using the rank condition. In this paper, first we derive necessary and sufficient conditions for global observability at an initial state for continuous-time polynomial systems. Then, necessary and sufficient conditions for local observability are derived from one of the global observability conditions using the localization of a polynomial ring. Using the same procedure, we derive both global and local observability conditions for discrete-time polynomial systems. Each condition is characterized by a finite set of equations, since polynomial rings are Noetherian. Finally, examples demonstrate the proposed criteria for testing for observability.

Support sets in exponential families and oriented matroid theory

The closure of a discrete exponential family is described by a finite set of equations corresponding to the circuits of an underlying oriented matroid. These equations are similar to the equations used in algebraic statistics, although they need not be polynomial in the general case. This description allows for a combinatorial study of the possible support sets in the closure of an exponential family. If two exponential families induce the same oriented matroid, then their closures have the same support sets. Furthermore, the positive cocircuits give a parameterization of the closure of the exponential family.

An Algebraic Approach to Local Observability at an Initial State for Discrete-Time Polynomial Systems

AbstractIn this paper, we consider local observability at an initial state for discrete-time autonomous polynomial systems. When testing for observability, for discrete-time nonlinear systems, a condition based on the inverse function theorem is commonly used. However, it is a sufficient condition. In this paper, first we derive a necessary and sufficient condition for global observability at an initial state for these polynomial systems. Then, a necessary and sufficient condition for local observability is derived from the global observability condition using the localization of a polynomial ring. Each condition is characterized by a finite set of equations, since polynomial rings are noetherian. Finally, an example demonstrates the proposed criteria for testing the observability.

Critical behavior of the particle mass spectra in a family of gelling systems

The critical and post-critical behavior of coagulating systems, where the coagulation efficiency grows with the masses of colliding particles g and l as K(g,l)=0.5(g(alpha)(l)(beta)+g(beta)(l)(alpha), lambda=alpha+beta>1, mu=/alpha - beta/ <1 is studied. The instantaneous sink that removes the particles with masses exceeding G is introduced which allows one to describe the coagulation kinetics by a finite set of equations and define the gel as the deposit of particles with masses between G+1 and 2G . This system displays the critical behavior (the sol-gel transition) in the limit G-->infinity if lambda=alpha+beta>1. The exact post-critical particle mass spectrum is shown to be an algebraic function of g times a growing exponent. All critical parameters of the systems are determined as the functions of alpha and beta .

Time saving techniques for electronic structure calculations of infinite and semi-infinite crystals, interfaces, and slabs of arbitrary thickness

Hard confined functions (HCF) are proposed as a basis set for electronic structure calculations. The basis functions have enough number of continuous derivatives to perform space integrations numerically with desired accuracy. Replacing unconfined basis functions in ADF-BAND package by HCF with cut-off radiuses of the order of the nearest neighbor distance leads to the reduction of the time of computations without losing the accuracy. For the crystals with surfaces, special techniques based on representation of the wave function as a linear combination of a finite number of HCF and Bloch waves, were elaborated. Exact finite sets of equations for coefficients of HCF and Bloch waves have been developed. The order of the sets depends only on the thickness of a perturbed region, but not on the size of the whole system. For integration of the density of states over perpendicular to the surface wave vector component and energy, the residue theorem and a shift of the energy path into the complex plane are used.

An extension of Willard?s Finite Basis Theorem: Congruence meet-semidistributive varieties of finite critical depth

Ross Willard proved that every congruence meet-semidistributive variety of algebras that has a finite residual bound and a finite signature can be axiomatized by some finite set of equations. We offer here a simplification of Willard’s proof, avoiding its use of Ramsey’s Theorem. This simplification also extends Willard’s theorem by replacing the finite residual bound with a weaker condition.

An axiomatization of graphs

The magmoid of hypergraphs labelled over a finite doubly ranked alphabet, is characterized as the quotient of the free magmoid generated by this alphabet divided by a finite set of equations. Thus a relevant open problem, posed by Engelfriet and Vereijken, is being solved.

On the Complexity of Reasoning in Kleene Algebra

We study the complexity of reasoning in Kleene algebra and *-continuous Kleene algebra in the presence of extra-equational assumptions E ; that is, the complexity of deciding the validity of universal Horn formulas E → s = t , where E is a finite set of equations. We obtain various levels of complexity based on the form of the assumptions E . Our main results are as follows: for *-continuous Kleene algebra, (i) if E contains only commutativity assumptions pq = qp , the problem is Π 1 0 -complete; (ii) if E contains only monoid equations, the problem is Π 2 0 -complete; and (iii) for arbitrary equations E , the problem is Π 1 1 -complete. The last problem is the universal Horn theory of the *-continuous Kleene algebras. This resolves an open question of the author [D. Kozen, 1994, Inform. and Comput. 110 , 366–390].

Regularization of L ∞ -Optimal Control Problems for Distributed Parameter Systems

We consider L∞-norm minimal controllability problems for vibrating systems. In the common method of modal truncation controllability constraints are first reformulated as an infinite sequence of moment equations, which is then truncated to a finite set of equations. Thus, feasible controls are represented as solutions of moment problems. In this paper, we propose a different approach, namely to replace the sequence of moment equations by a sequence of moment inequalities. In this way, the feasible set is enlarged. If a certain relaxation parameter tends to zero, the enlarged sets approach the original feasible set. Numerical examples illustrate the advantages of this new approach compared with the classical method of moments. The introduction of moment inequalities can be seen as a regularization method, that can be used to avoid oscillatory effects. This regularizing effect follows from the fact that for each relaxation parameter, the whole sequence of eigenfrequencies is taken into account, whereas in the method of modal truncation, only a finite number of frequencies is considered.

A FINITE AXIOMATIZATION OF LOCALLY SQUARE CYLINDRIC-RELATIVIZED SET ALGEBRAS

We give a finite set of equations axiomatizing the class Gn of locally square cylindricrelativized set algebras of dimension n, if n is finite. For infinite n, we give an axiomatization of the equational theory of Gn. HereGn denotes the class of all cylindric-relativized set algebras of dimension n with unit a union of Cartesian spaces.

Three Dimensional Axisymmetric Piezothermoelastic Solution of Clamped Circular Elastic Plate Bonded to Piezoceramic Plate

A three dimensional (3D) solution in terms of potential functions is presented for a finite clamped circular isotropic elastic plate bonded to a piezoceramic plate subjected to axisymmetric thermoelectromechanical load. The boundary and interface conditions are satisfied using Fourier-Bessel expansions yielding an infinite system of algebraic equations for the arbitrary constants. These are solved to any desired degree of accuracy by truncating to a finite set of equations. Results are presented to illustrate the effect of the thickness parameter. The present solution would help to ascertain the accuracy and range of validity of two dimensional (2D) theories of piezoelectric hybrid plates.

Decision problems for equational theories of relation algebras

The foundation of an algebraic theory of binary relations was laid by C. S. Peirce, building on earlier work of Boole and De Morgan. The basic universe of discourse of this theory is a collection of binary relations over some set, and the basic operations on these relations are those of forming unions, complements, relative products (i.e., compositions), and converses (i.e., inverses). There is also a distinguished relation, the identity relation. Other operations and distinguished relations studied by Peirce are definable in terms of the ones just mentioned. Such an algebra of relations is called a set relation algebra. A modern development of this theory as a theory of abstract relation algebras, axiomatized by a finite set of equations, was undertaken by Tarski and his students and colleagues, beginning around 1940. In 1942, Tarski proved that all of classical mathematics could be developed within the framework of the equational theory of relation algebras. Indeed, he created a general method for interpreting into the equational theory of relation algebras first-order theories that are strong enough to form a basis for the development of mathematics, in particular, set theories and number theories. As a consequence, he established that the equational theories of relation algebras and of set relation algebras are undecidable (see [10] and [11], and see [2] or [11], in particular Chapter 8, for unexplained terminology). They were the first known examples of undecidable equational theories. As was pointed out in [11], Tarski’s proof actually shows more. Namely, any class of relation algebras that contains the full set relation algebra on some infinite set (i.e., the set relation algebra whose universe consists of all binary relations on the infinite set) or, equivalently, that contains all set relation algebras on infinite sets must have an undecidable equational theory.

The structure of interlaced bilattices

Bilattices were introduced and applied by Ginsberg and Fitting for a diversity of applications, such as truth maintenance systems, default inferences and logic programming. In this paper we investigate the structure and properties of a particularly important class of bilattices called interlaced bilattices, which were introduced by Fitting. The main results are that every interlaced bilattice is isomorphic to the Ginsberg-Fitting product of two bounded lattices and that the variety of interlaced bilattices is equivalent to the variety of bounded lattices with two distinguishable distributive elements, which are complements of each other. This implies that interlaced bilattices can be characterized using a finite set of equations. Our results generalize to interlaced bilattices some results of Ginsberg, Fitting and Jónsson for distributive bilattices.

Triphasic finite element model for swelling porous media

AbstractThe equations describing the mechanical behaviour of intervertebral disc tissue and other swelling porous media are three coupled partial differential equations in which geometric and physical non‐linearities occur. The boundary conditions are deformation‐dependent. To solve the equations for an arbitrary geometry and arbitrary boundary conditions, we use the finite element (FE) method. The differential equations are rewritten in an integral form by means of the weighted residual method. The domain of the integral is defined via a set of shape functions (i.e. finite elements). By applying the Gauss theorem and rewriting with respect to the reference state (total Lagrange), non‐linear equations are obtained. These are solved by means of the Newton‐Raphson technique. In order to get a finite set of equations, the weighted residual equations are discretized. The shape functions are chose as weighting functions (Galerkin method). This discretization results in a non‐symmetric stiffness matrix. A general description is given for the elements implemented into the commercial FE package DIANA (DIANA Analysis B.V., Delft, Netherlands). The numerical results of unconfined compression of a schematic intervertebral disc with varying proteoglycan concentration are given.

RELATIONAL ALGEBRA AND EQUATIONAL PROOFS

A proposal is made for a formal framework in which equational provability can be represented and investigated. The basic formalism is an algebra of binary relations on ground terms of a first-order language. Two special-purpose mappings are introduced to deal with the structure of terms. Equational provability is characterized in terms of least fixpoints of some mappings. The use of the relational framework is illustrated by a few examples. It is shown that provability of an equation from a finite set of equations, where all equations are variable-free, is decidable. Then we discuss a method by Reeves ([3]) to deal with equations in semantic tableaux, that we can now prove to be complete in a very simple way. Finally, we discuss an equational proof format that is naturally induced by the relational formulation and serves as a guideline in finding proofs. The relation between Reeves' rules and the construction of such proofs is made explicit.

LEARNING FROM EXAMPLES IN UNIVERSE OF DISCOURSE DESCRIBED BY A SET OF EQUATIONS

Generalization is an important operation for programs that learn. Anti-unification guarantees the existence of a term which is the most specific generalization of a collection of terms. This provides a formal basis for learning from examples. However, such generalization may be "too" general and therefore counterexamples are required to restrict them. In this case, the learner has to check whether the resulting formula provides a concept to learn, i.e. whether a formula of the form t/{t1,…,tn} is a generalization, where t is viewed as a generalization of a set of examples and t1,…,tn are counterexample. The formula t/{t1,…,tn} is often called an implicit representation. The implicit representation t/{t1,…,tn} is a generalization iff there exist ground (variable-free) instances of t which are not instances of t1,…,tn. This is a non-trivial task even when there is no background knowledge (see [15]). We present here a method to check whether an implicit representation t/{t1,…,tn} is a generalization with respect to a finite set of equations which describes the background knowledge problem, i.e. whether there exists a ground instance of t which is not equivalent to any ground instance of t1,…,tn with respect to a set E of equations. Whereas this problem is in general undecidable since the equality is so, we show in this paper that in this case where the set E of equations is compiled into a ground convergent term rewriting system, we can discover concepts in Universe of Discourse with background knowledge described by a finite set of equations. We show how the method applies to induce concepts in theories of elementary arithmetic.