BINARY RELATIONS AND ALGEBRAS ON MULTISETS.

by Ghilezan, Silvia, Pantović, Jovanka, Vojvodić, Gradimir [2014-01-01]

Academic Journal

pages 7

Contrary to the notion of a set or a tuple, a multiset is an unordered collection of elements which do not need to be different. As multisets are already widely used in combinatorics and computer science, the aim of this paper is to get on track to algebraic multiset theory. We consider generalizations of known results that hold for equivalence and order relations on sets and get several properties that are specific to multisets. Furthermore, we exemplify the novelty that brings this concept by showing that multisets are suitable to represent partial orders. Finally, after introducing the notion of an algebra on multisets, we prove that two algebras on multisets, whose root algebras are isomorphic, are in general not isomorphic. [ABSTRACT FROM AUTHOR]


Existence and uniqueness of fuzzy solutions for the nonlinear second-order fuzzy Volterra integrodifferential equations.

by Momani, Shaher, Arqub, Omar Abu, Al-Mezel, Saleh, Kutbi, Marwan [2016-08-01]

Academic Journal

pages 15

Formulation of uncertainty Volterra integrodifferential equations (VIDEs) is very important issue in applied sciences and engineering; whilst the natural way to model such dynamical systems is to use the fuzzy approach. In this work, we present and prove the existence and uniqueness of four solutions of fuzzy VIDEs based on the Hausdorff distance under the assumption of strongly generalized differentiability for the fuzzy-valued mappings of a real variable whose values are normal, convex, upper semicontinuous, and compactly supported fuzzy sets in . In addition to that, we utilize and prove the characterization theorem for solutions of fuzzy VIDEs which allow us to translate a fuzzy VIDE into a system of crisp equations. The proof methodology is based on the assumption of the generalized Lipchitz property for each nonlinear term appears in the fuzzy equation subject to the specific metric used, while the main tools employed in the analysis are founded on the applications of the Banach fixed point theorem and a certain integral inequality with explicit estimate. An efficient computational algorithm is provided to guarantee the procedure and to confirm the performance of the proposed approach. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 15

PADS (Process Algebra for Demand and Supply) is a formal framework to analyze hierarchical scheduling in real-time embedded systems. Inspired by the supply simulation relation in PADS, we introduce a partial supply simulation relation in order to describe the fact that an unschedulable task may finish on time. It is more general than the supply simulation relation. Then, we explore some properties of partial supply simulation relation. Furthermore, we establish a proof system for the partial supply simulation relation in a decomposingcomposing way, which helps to infer tasks' partial schedulabilities. Finally, it is proved that the proof system is sound and complete with respect to the semantic definition of partial supply simulation relation. [ABSTRACT FROM AUTHOR]


On the Downhill Method.

by Timlake, W. T., Bach, H. [1969-12-01]

Academic Journal

pages 4

The downhill method is a numerical method for solving complex equations f(z) = 0 on which the only restriction is that the function w = f(z) must be analytical. An introduction to this method is given and a critical review of relating literature is presented. Although in theory the method always converges, it is shown that a fundamental dilemma exists which may cause o breakdown in practical applications. To avoid this difficulty and to improve the rate of convergence toward a root, some modifications of the original method are proposed and a program (FORTRAN) based on the modified method is given in Algorithm 365. Some numerical examples are included. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 12

In this paper, we construct the degenerate Carlitz-type (h, q)-tangent numbers and polynomials associated with the p-adic q-integral on Zp. We also give some explicit formulas for degenerate Carlitz-type (h, q)-tangent numbers and polynomials. Moreover, we show some relations with the Stirling numbers of the first kind and Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 13

Recently, a number of variants of the approximate minimum degree algorithm have been proposed that aim to efficiently order symmetric matrices containing some dense rows. We compare the performance of these variants on a range of problems and highlight their potential limitations. This leads us to propose a new variant that offers both speed and robustness. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 34

In the four-body problem, it is not clear what initial conditions can lead to simultaneous binary collision (SBC), even in the collinear case. In this paper, we study SBC in the equal-mass collinear four-body problem and have a partial answer for initial conditions leading to SBC. After introducing a Levi-Civita type transformation, we analyze the new transformed differential system of SBC and solve for all possible solutions. The problem is studied in two cases: decoupled case and coupled case. In the decoupled case where SBC is treated as two separated binary collisions, the initial conditions leading to SBC satisfy several simple algebraic identities. This result gives insights to the coupled case, which is SBC in the equal-mass collinear four-body problem. Furthermore, we show from a different perspective that solutions passing through SBC must be analytic in the transformed system and the initial condition set leading to SBC has a measure 0. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 6

In this article we prove a theorem on the definitional embeddability into first-order predicate logic without equality of such well-known mathematical theories as group theory and the theory of Abelian groups. This result may seem surprising, since it is generally believed that these theories have a non-logical content. It turns out that the central theory of general algebra are purely logical. Could this be the reason that we find them in many branches of mathematics? This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics. [ABSTRACT FROM AUTHOR]


College Algebra

by Jay Abramson [2015]

Book

pages unknown

College Algebra provides a comprehensive and multi-layered exploration of algebraic principles. The text is suitable for a typical introductory Algebra course, and was developed to be used flexibly. The modular approach and the richness of content ensures that the book meets the needs of a variety of programs. College Algebra guides and supports students with differing levels of preparation and experience with mathematics. Ideas are presented as clearly as possible, and progress to more complex understandings with considerable reinforcement along the way. A wealth of examples - usually several dozen per chapter - offer detailed, conceptual explanations, in order to build in students a strong, cumulative foundation in the material before asking them to apply what they've learned


Algebra and Trigonometry

by Jay Abramson [2015]

Book

pages unknown

Algebra and Trigonometry provides a comprehensive and multi-layered exploration of algebraic principles. The text is suitable for a typical introductory Algebra & Trigonometry course, and was developed to be used flexibly. The modular approach and the richness of content ensures that the book meets the needs of a variety of programs. Algebra and Trigonometry guides and supports students with differing levels of preparation and experience with mathematics. Ideas are presented as clearly as possible, and progress to more complex understandings with considerable reinforcement along the way. A wealth of examples - usually several dozen per chapter - offer detailed, conceptual explanations, in order to build in students a strong, cumulative foundation in the material before asking them to apply what they've learned. This is a full-color textbook


Abstract Algebra - Theory and Applications

by Thomas W. Judson [2016]

Book

pages unknown

Abstract Algebra: Theory and Applications is an open-source textbook that is designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many non-trivial applications. The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galois Theory


PLANAR 2-HOMOGENEOUS COMMUTATIVE RATIONAL VECTOR FIELDS.

by ALKAUSKAS, GIEDRIUS [2018-06-01]

Academic Journal

pages 21

In this article we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In the latter case, orbits of each flow are given in terms of 1-homogeneous rational functions W as curves W (x, y) = const. An exhaustive method to construct such commuting algebraic flows is presented. The degree of the so-obtained algebraic functions in two variables can be arbitrarily high. [ABSTRACT FROM AUTHOR]


The natural brackets on couples of vector fields and 1-forms.

by DOUPOVEC, Miroslav, KUREK, Jan, MIKULSKI, Włodzimierz M. [2018-08-01]

Academic Journal

pages 10

All natural bilinear operators transforming pairs of couples of vector fields and 1-forms into couples of vector fields and 1-forms are found. All natural bilinear operators as above satisfying the Leibniz rule are extracted. All natural Lie algebra brackets on couples of vector fields and 1-forms are collected. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 8

Let A be an alphabet of cardinality m, k_n be a sequence of positive integers and ω∈ A* (|ω|=k_n). In this paper it is shown that if lim sup_{n→∞}k_n/ln n1/ln m, then this property is not true. Also, if lim inf_{n→∞}k_n/ln n>1/ln m, then almost all words of length n over A do not contain the factor ω. Moreover, if lim_{n→∞}(ln n-k_nln m)=α∈ R, then lim sup_{n→∞}|W(n,k_n,ω,A)| /m^m≤1-exp (-exp(α)) and lim inf_{n→∞}|W (n, k_n,ω,A)|/m^n≥1-exp (-(1-1/m) exp(α)), where W(n, k_n, ω, A) denotes the set of words of length n over A containing the factor ω of length k_n. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 45

This article demonstrates that there is a fundamental relationship between temporal logic and languages that involve multiple stages, such as those used to analyze binding times in the context of partial evaluation. This relationship is based on an extension of the Curry-Howard isomorphism, which identifies proofs with programs, and propositions with types. Our extension involves the "next time" (□) operator from linear-time temporal logic and yields a λ-calculus λ° with types of the form □ A for expressions in the subsequent stage, with appropriate introduction and elimination forms. We demonstrate that λ° is equivalent to the core of a previously studied multilevel binding-time analysis. This is similar to work by Davies and Pfenning on staged computation based on the necessity (□) operator of modal logic, but □ only allows closed code, and naturally supports a code evaluation construct, whereas □ captures open code, thus is more flexible, but is incompatible with such a construct. Instead, code evaluation is an external global operation that is validated by the proof theory regarding closed proofs of □ formulas. We demonstrate the relevance of λ° to staged computation directly by showing that that normalization can be done in an order strictly following the times of the logic. We also extend λ° to small functional language and show that it would serve as a suitable basis for directly programming with multiple stages by presenting some example programs. [ABSTRACT FROM AUTHOR]


On Entropy of a Logical System.

by BORIČIĆ, MARIJA [2013-12-01]

Academic Journal

pages 14

Logical system associated with the partition induced by the corresponding Lindenbaum-Tarski algebra makes possible to define its entropy. We consider three approaches to define the entropy of a logical system, metaphorically called algebraic, probabilistic and philosophical, and give some reasons to discard or accept some of them, resulting with a proposal to found our definition on geometric distribution of measures over matching partition of set of formulae. This definition enables to classify finite-valued propositional logics regarding their entropies. Asymptotic approximations for some infinite-valued logics are proposed as well. The considered examples include Lukasiewicz's, Kleene's and Priest's three-valued logics, Belnap's four-valued logic, Gödel's and McKay's m-valued logics, and Heyting's and Dummett's infinite-valued logics. [ABSTRACT FROM AUTHOR]


Metric Spaces With Distances in Representable Autometrized Algebras.

by Subba Rao, B. V., Yedlapalli, Phani [2018-05-01]

Academic Journal

pages 10

In this paper, we introduce A-metric spaces (X, A, d), where X is a non empty set, A = (A, +, ≤, *) is a Representable Autometrized Algebra in [6], d : X × X → A is a function satisfying the formal properties of a distance function. We proved that any Ai-metric space (X, A, d) is congruent to a subdirect product of Aimetric spaces where each Ai is a totally ordered Representable autometrized algebra. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 26

This article presents a Vygotsky-inspired analysis of how a teacher mediated a “thinking aloud” whole-group discussion in a 6th grade mathematics classroom. This discussion centered on finding patterns in a triangular array of consecutive numbers as a phase towards building recursive and direct algebraic formulas. By a “thinking aloud” discussion we mean a conversation wherein students exchange and further develop ideas-in-the-making with their peers under the teacher’s guidance. Drawing upon Halliday’s systemic functional linguistics (SFL), we treated the selected discussion as a text. We then analyzed how the teacher mediated the conjoined making of this text so that it served as an interpersonal gateway for students to practice searching for patterns and signifying these patterns in propositional form. This analysis was guided by the following questions: How did the discussion as a text-in-the-making mean what it did? What was the role of the teacher in the conjoined making of this text? Our study illustrates the power of SFL for capturing the inner grammar of instructional conversations thus illuminating the complexities and subtleties of the teacher’s role in mediating semiotic mediation in mathematics classrooms. [ABSTRACT FROM AUTHOR]


Algebraizable logics and a functorial encoding of its morphisms.

by Pinto, Darllan Conceição, Mariano, Hugo Luiz [2017-08-01]

Academic Journal

pages 38

This work presents some results about the categorial relation between logics and their associated categories of structures. A (propositional, finitary) logic is a pair given by a signature and Tarskian consequence relation on its formula algebra. The logics are the objects in our categories of logics; the morphisms are certain signature morphisms that are translations between logics. Morphisms between algebraizable logics are translations that preserve algebraizing pairs: they can be completely encoded by certain functors defined on the quasivarieties canonically associated to the algebraizable logics. This kind of results will be useful in the development of a categorial approach to the representation theory of general logics. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 33

We introduce a network model that allows us to capture the time-varying service delivered to a traffic stream due to the presence of random perturbations (e.g., cross-traffic in a communication network). We first present the model for a single network element and then describe how such elements may be interconnected using the operations of fork and join. Such networks may be seen as a generalization of stochastic event graphs and of the class of fork-join networks. The departure processes in such a network satisfy a system of equations along with the exogenous arrival processes and the service processes of the various network elements. This system can be seen as a general time-varying linear system in the (min, plus) semi-field. We obtain an explicit representation of the departure process in terms of the exogenous arrival processes and the service processes. Sufficient conditions are derived for this system to have a unique solution. We also study liveness and absence of explosion in this class of networks. Under appropriate stationarity and ergodicity assumptions, we establish stability theorems for such networks. To this end we first obtain a rate result for the departure process in terms of the rates of the exogenous arrival process and the throughput of a saturated system. We then use this result to show that in a network with a single exogenous arrival, all queue lengths are finite with probability one if the arrival rate is less than the throughput of the saturated system, and to give a representation of the queue-length process. This class of networks allows for a detailed description of controlled sessions in integrated service networks. We also show that it contains several earlier discrete event models of the literature pertaining to stochastic Petri nets, service curves, and fork-join networks and show how the present model unifies them in a single algebraic structure. This structure is that of a semi-ring of functions of two real variables, where ... [ABSTRACT FROM AUTHOR]


Collected Papers Of Wei-liang Chow, The

by Chow, Wei-Liang, Chern, Shiing-Shen, Shokurov, Vyacheslav V. [2002-01-01]

eBook

pages unknown

This invaluable book contains the collected papers of Prof Wei-Liang Chow, an original and versatile mathematician of the 20th Century. Prof Chow's name has become a household word in mathematics because of the Chow ring, Chow coordinates, and Chow's theorem on analytic sets in projective spaces. The Chow ring has many advantages and is widely used in intersection theory of algebraic geometry. Chow coordinates have been a very versatile tool in many aspects of algebraic geometry. Chow's theorem — that a compact analytic variety in a projective space is algebraic — is justly famous; it shows the close analogy between algebraic geometry and algebraic number theory.About Professor Wei-Liang ChowThe long and distinguished career of Prof Wei-Liang Chow (1911-95) as a mathematician began in China with professorships at the National Central University in Nanking (1936-37) and the National Tung-Chi University in Shanghai (1946-47), and ultimately led him to the United States, where he joined the mathematics faculty of Johns Hopkins University in Baltimore, Maryland, first as an associate professor from 1948 to 1950, then as a full professor from 1950 until his retirement in 1977.In addition to serving as chairman of the mathematics department at Johns Hopkins from 1955 to 1965, he was Editor-in-Chief of the American Journal of Mathematics from 1953 to 1977.


eBook

pages unknown

Professor Peter Hilton is one of the best known mathematicians of his generation. He has published almost 300 books and papers on various aspects of topology and algebra. The present volume is to celebrate the occasion of his sixtieth birthday. It begins with a bibliography of his work, followed by reviews of his contributions to topology and algebra. These are followed by eleven research papers concerned with various topics of current interest in algebra and topology. The articles are contributed by some of the many mathematicians with whom he has worked at one time or another. This book will be of interest to both topologists and algebraists, particularly those concerned with homotopy theory.


eBook

pages unknown

The algebraic theory of corner subrings introduced by Lam (as an abstraction of the properties of Peirce corners eRe of a ring R associated with an idempotent e in R) is investigated here in the context of Banach and C•-algebras. We propose a general algebraic approach which includes the notion of ranges of (completely) contractive conditional expectations on C•-algebras and on ternary rings of operators, and we investigate when topological properties are consequences of the algebraic assumptions. For commutative C•-algebras we show that dense corners cannot be proper and that self-adjoint corners must be closed and always have closed complements (and may also have non-closed complements). For C•-algebras we show that Peirce corners and some more general corners are similar to self-adjoint corners. We show uniqueness of complements for certain classes of corners in general C•-algebras, and establish that a primitive C•-algebra must be prime if it has a prime Peirce corner. Further we consider corners in ternary rings of operators (TROs) and characterise corners of Hilbertian TROs as closed subspaces.