scholarly journals Ordered Structures of Polynomials over Max-Plus Algebra

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1137
Author(s):  
Cailu Wang ◽  
Yuanqing Xia ◽  
Yuegang Tao
Keyword(s):  
Equivalence Class ◽  
Minimal Element ◽  
Algebraic Method ◽  
Upper And Lower Bounds ◽  
Order Structure ◽  
Single Element ◽  
Polynomial Functions ◽  
Ordered Structures ◽  
Finite Set ◽  
The Continuum

The ordered structures of polynomial idempotent algebras over max-plus algebra are investigated in this paper. Based on the antisymmetry, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs). Based on the symmetry, the quotient POIA of formal polynomials is then obtained. The order structure relationships among these three POIAs are described: the POIA of polynomial functions and the POIA of formal polynomials are orderly homomorphic; the POIA of polynomial functions and the quotient POIA of formal polynomials are orderly isomorphic. By using the partial order on formal polynomials, an algebraic method is provided to determine the upper and lower bounds of an equivalence class in the quotient POIA of formal polynomials. The criterion for a formal polynomial to be the minimal element of an equivalence class is derived. Furthermore, it is proven that any equivalence class is either an uncountable set with cardinality of the continuum or a finite set with a single element.

scholarly journals The binomial equivalence classes of finite words

2020 ◽  
Vol 30 (07) ◽  
pp. 1375-1397
Author(s):  
Marie Lejeune ◽  
Michel Rigo ◽  
Matthieu Rosenfeld
Keyword(s):  
Nilpotent Group ◽  
Equivalence Class ◽  
Free Monoid ◽  
Equivalence Classes ◽  
Single Element ◽  
Free Nilpotent Group ◽  
Context Free ◽  
Abelian Equivalence

Two finite words [Formula: see text] and [Formula: see text] are [Formula: see text]-binomially equivalent if, for each word [Formula: see text] of length at most [Formula: see text], [Formula: see text] appears the same number of times as a subsequence (i.e., as a scattered subword) of both [Formula: see text] and [Formula: see text]. This notion generalizes abelian equivalence. In this paper, we study the equivalence classes induced by the [Formula: see text]-binomial equivalence. We provide an algorithm generating the [Formula: see text]-binomial equivalence class of a word. For [Formula: see text] and alphabet of [Formula: see text] or more symbols, the language made of lexicographically least elements of every [Formula: see text]-binomial equivalence class and the language of singletons, i.e., the words whose [Formula: see text]-binomial equivalence class is restricted to a single element, are shown to be non-context-free. As a consequence of our discussions, we also prove that the submonoid generated by the generators of the free nil-[Formula: see text] group (also called free nilpotent group of class [Formula: see text]) on [Formula: see text] generators is isomorphic to the quotient of the free monoid [Formula: see text] by the [Formula: see text]-binomial equivalence.

A general treatment of equivalent modalities

1989 ◽  
Vol 54 (4) ◽  
pp. 1460-1471
Author(s):  
Fabio Bellissima ◽  
Massimo Mirolli
Keyword(s):  
Modal Logic ◽  
Minimal Element ◽  
Continuum Hypothesis ◽  
Classical Problem ◽  
Point Of View ◽  
General Treatment ◽  
General Point ◽  
Whole Class ◽  
The Continuum

The problem of the nonequivalent modalities available in certain systems is a classical problem of modal logic. In this paper we deal with this problem without referring to particular logics, but considering the whole class of normal propositional logics. Given a logic L let P(L) (the m-partition generated by L) denote the set of the classes of L-equivalent modalities. Obviously, different logics may generate the same m-partition; the first problem arising from this general point of view is therefore to determine the cardinality of the set of all m-partitions. Since, as is well known, there exist normal logics, and since one immediately realizes that there are infinitely many m-partitions, the problem consists in choosing (assuming the continuum hypothesis) between ℵ0 and . In Theorem 1.2 we show that there are m-partitions, as many as the logics.The next problem which naturally arises consists in determining, given an m-partition P(L), the number of logics generating P(L) (in symbols, μ(P(L))). In Theorem 2.1(ii) we show that ∣{P(L): μ(P(L)) = }∣ = . Now, the set {L′) = P(L)} has a natural minimal element; that is, the logic L* axiomatized by K ∪ {φ(p) ↔ ψ(p): φ, ψ are L-equivalent modalities}; P(L) and L* can be, in some sense, identified, thus making the set of m-partitions a subset of the set of logics.

A New Dynamic Basis Algorithm for Solving Linear Programming Problems for Engineering Design

1990 ◽  
Vol 112 (2) ◽  
pp. 208-214 ◽  
Author(s):  
Y. Wang ◽  
E. Sandgren
Keyword(s):  
Linear Programming ◽  
Simplex Method ◽  
Inequality Constraints ◽  
Simplex Algorithm ◽  
Upper And Lower Bounds ◽  
Programming Algorithm ◽  
Single Element ◽  
Basis Matrix ◽  
Active Constraint ◽  
Design Variables

A new linear programming algorithm is proposed which has significant advantages compared to a traditional simplex method. A search direction is generated along a common edge of the active constraint set. This direction is followed in order to identify candidate constraints and to modify the current basis. The dimension of the basis matrix begins with a single element and dynamically increases but remains less than or equal to the number of design variables. This is true regardless of the number of inequality constraints present including upper and lower bounds. The proposed method can operate equally well from a feasible or infeasible point. The pivot operation and artificial variable strategy of the simplex method are not used. Examples are presented and results are compared to those generated by a traditional revised simplex algorithm. Extensions are presented for both exterior and interior versions of the approach.

The first-order structure of weakly Dedekind-finite set

2005 ◽  
Vol 70 (4) ◽  
pp. 1161-1170 ◽  
Author(s):  
A. C. Walczak-Typke
Keyword(s):  
Order Structure ◽  
Finite Sets ◽  
First Order ◽  
Finite Set ◽  
Infinite Sets ◽  
Order Structures

AbstractWe show that infinite sets whose power-sets are Dedekind-finite can only carry ℵ0-categorical first order structures. We identify other subclasses of this class of Dedekind-finite sets, and discuss their possible first order structures.

COVERING A SET OF POINTS WITH A MINIMUM NUMBER OF TURNS

2004 ◽  
Vol 14 (01n02) ◽  
pp. 105-114 ◽  
Author(s):  
MICHAEL J. COLLINS
Keyword(s):  
Euclidean Space ◽  
Lower Bounds ◽  
Piecewise Linear ◽  
Upper And Lower Bounds ◽  
Linear Path ◽  
Change Direction ◽  
Finite Set ◽  
Minimum Number ◽  
Pass Through ◽  
Set Of Points

Given a finite set of points in Euclidean space, we can ask what is the minimum number of times a piecewise-linear path must change direction in order to pass through all of them. We prove some new upper and lower bounds for the rectilinear version of this problem in which all motion is orthogonal to the coordinate axes. We also consider the more general case of arbitrary directions.

scholarly journals Discrete anatomical coordinates for speech production and synthesis

2017 ◽  
Author(s):  
M Florencia Assaneo ◽  
Daniela Ramirez Butavand ◽  
Marcos A Trevisan ◽  
Gabriel B Mindlin
Keyword(s):  
Vocal Tract ◽  
Electronic Device ◽  
Threshold Strategy ◽  
Experimental Phonology ◽  
Finite Set ◽  
Intelligible Speech ◽  
Low Dimensional ◽  
Open Question ◽  
Continuous Movements ◽  
The Continuum

The sounds of all languages are described by a finite set of symbols, which are extracted from the continuum of sounds produced by the vocal organ. How the discrete phonemic identity is encoded in the continuous movements producing speech remains an open question for the experimental phonology. In this work, this question is assessed by using Hall-effect transducers and magnets -mounted on the tongue, lips and jaw- to track the kinematics of the oral tract during the vocalization ofvowel-consonant-vowelstructures. Using athreshold strategy, the time traces of the transducers were converted into discrete motor coordinates unambiguously associated with the vocalized phonemes. Furthermore, the signals of the transducers combined with the discretization strategy were used to drive a low-dimensional vocal model capable of synthesizing intelligible speech. The current work not only addressed a relevant inquiry of the biology of language, but also shows the performance of the experimental technique to monitor the displacement of the main articulators of the vocal tract while speaking. This novel electronic device represents an economic and portable option to the standard system used to study the vocal tract movements.

Grain boundaries as heterogeneous systems: atomic and continuum elastic properties

1992 ◽  
Vol 339 (1655) ◽  
pp. 555-586 ◽  
Keyword(s):  
Grain Boundary ◽  
Grain Boundaries ◽  
Elastic Properties ◽  
Discrete Model ◽  
Continuum Model ◽  
Elastic Moduli ◽  
Atomic Level ◽  
Upper And Lower Bounds ◽  
Effective Moduli ◽  
The Continuum

The relation between atomic structure and elastic properties of grain boundaries is investigated theoretically from both atomistic and continuum points of view. A heterogeneous continuum model of the boundary is introduced where distinct phases are associated with individual atoms and possess their atomic level elastic moduli determined from the discrete model. The effective elastic moduli for sub-blocks from an infinite bicrystal are then calculated for a relatively small number of atom layers above and below the grain boundary. These effective moduli can be determined exactly for the discrete atomistic model, while estimates from upper and lower bounds are evaluated in the framework of the continuum model. The complete fourth-order elastic modulus tensor is calculated for both the local and the effective properties. Comparison between the discrete atomistic results and those for the continuum model establishes the validity of this model and leads to criteria to assess the stability of a given grain boundary structure. For stable structures the continuum estimates of the effective moduli agree well with the exact effective moduli for the discrete model. Metastable and unstable structures are associated with a significant fraction of atoms (phases) for which the atomic-level moduli lose positive definiteness or even strong ellipticity. In those cases, the agreement between the effective moduli of the discrete and continuum systems breaks down.

scholarly journals Equivalence classes of functions on finite sets

1982 ◽  
Vol 5 (4) ◽  
pp. 745-762
Author(s):  
Chong-Yun Chao ◽  
Caroline I. Deisher
Keyword(s):  
Equivalence Class ◽  
Equivalence Classes ◽  
Permutation Groups ◽  
Weak Equivalence ◽  
Finite Sets ◽  
The Family ◽  
Polya's Theorem ◽  
Finite Set ◽  
Pólya’S Theorem

By using Pólya's theorem of enumeration and de Bruijn's generalization of Pólya's theorem, we obtain the numbers of various weak equivalence classes of functions inRDrelative to permutation groupsGandHwhereRDis the set of all functions from a finite setDto a finite setR,Gacts onDandHacts onR. We present an algorithm for obtaining the equivalence classes of functions counted in de Bruijn's theorem, i.e., to determine which functions belong to the same equivalence class. We also use our algorithm to construct the family of non-isomorphicfm-graphs relative to a given group.

Dual Neural Network Method for Solving Multiple Definite Integrals

2019 ◽  
Vol 31 (1) ◽  
pp. 208-232 ◽  
Author(s):  
Haibin Li ◽  
Yangtian Li ◽  
Shangjie Li
Keyword(s):  
Neural Network ◽  
Calculation Method ◽  
Upper And Lower Bounds ◽  
Primitive Function ◽  
Neural Network Method ◽  
Definite Integrals ◽  
Network Method ◽  
Finite Set ◽  
Data Points ◽  
Repeated Applications

This study, which examines a calculation method on the basis of a dual neural network for solving multiple definite integrals, addresses the problems of inefficiency, inaccuracy, and difficulty in finding solutions. First, the method offers a dual neural network method to construct a primitive function of the integral problem; it can approximate the primitive function of any given integrand with any precision. On this basis, a neural network calculation method that can solve multiple definite integrals whose upper and lower bounds are arbitrarily given is obtained with repeated applications of the dual neural network to construction of the primitive function. Example simulations indicate that compared with traditional methods, the proposed algorithm is more efficient and precise in obtaining solutions for multiple integrals with unknown integrand, except for the finite input-output data points. The advantages of the proposed method include the following: (1) integral multiplicity shows no influence and restriction on the employment of the method; (2) only a finite set of known sample points is required without the need to know the exact analytical expression of the integrand; and (3) high calculation accuracy is obtained for multiple definite integrals whose integrand is expressed by sample data points.

ACCELERATING MONTE CARLO MARKOV PROCESSES

COSMOS ◽  
2005 ◽  
Vol 01 (01) ◽  
pp. 87-94 ◽  
Author(s):  
CHII-RUEY HWANG
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Process ◽  
Markov Processes ◽  
Transition Matrix ◽  
Asymptotic Variance ◽  
Suitable Condition ◽  
Transition Matrices ◽  
Finite Set ◽  
The Continuum

Let π be a probability density proportional to exp - U(x) in S. A convergent Markov process to π(x) may be regarded as a "conceptual" algorithm. Assume that S is a finite set. Let X0,X1,…,Xn,… be a Markov chain with transition matrix P and invariant probability π. Under suitable condition on P, it is known that [Formula: see text] converges to π(f) and the corresponding asymptotic variance v(f, P) depends only on f and P. It is natural to consider criteria vw(P) and va(P), defined respectively by maximizing and averaging v(f, P) over f. Two families of transition matrices are considered. There are four problems to be investigated. Some results and conjectures are given. As for the continuum case, to accelerate the convergence a family of diffusions with drift ∇U(x) + C(x) with div(C(x)exp - U(x)) = 0 is considered.

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