#### On normal Hankel matrices of low orders.

by Ikramov, Kh. D., Chugunov, V. N. [2008-07-01]

In the previous work of the authors, the problem of describing complex n × n matrices that are simultaneously normal and Hankel was reduced to a system of n − 1 real equations with respect to 2 n unknowns. These equations are quadratic, and it is not at all clear whether they have real solutions. It is shown here that the systems corresponding to n = 3 and n = 4 are solvable and have infinitely many real solutions. [ABSTRACT FROM AUTHOR]

#### Report Finds Advanced Math Students Faltering at Algebra.

No author [2008-10-20]

The article focuses on the study of Math students in the U.S. released by the Brookings Institution. The study found that while eight-graders are doing better on national math tests, students in advanced classes are faring worse. Algebra is becoming widely accepted as a must-have for eight-graders. The study also revealed that enrollment of low achievers has more than doubled in eight-grade algebra and teachers of low achievers have less experience and fewer formal credentials.

#### QUOTIENT HYPER PSEUDO BCK-ALGEBRAS.

by HARIZAVI, HABIB, KOOCHAKPOOR, TAYEBEH, BOORZOEI, RAJAB ALI [2013-12-01]

In this paper, we first investigate some properties of the hyper pseudo BCK-algebras. Then we define the concepts of strong and reflexive hyper pseudo BCK- ideals and establish some relationships among them and the other types of hyper pseudo BCK- ideals. Also, we introduce the notion of regular congruence relation on hyper pseudo BCK-algebras and investigate some related properties. By using this relation, we construct the quotient hyper pseudo BCK-algebra and give some related results. [ABSTRACT FROM AUTHOR]

#### Equivalence of Pepin's and the Lucas-Lehmer Tests.

by Jaroma, John H. [2009-07-01]

Pepin's test provides a necessary and sufficient condition for a Fermat number to be prime. The Lucas-Lehmer test does similarly for a Mersenne number. These tests share a common nature. However, this is evident neither by their usual statements nor their usual treatment in the literature. Furthermore, it is unusual to even find a proof of the latter result in elementary textbooks. The intent of this paper is to bring to light the equivalent structure of these two primality tests [ABSTRACT FROM AUTHOR]

#### DERIVATION FROM QUEUEING THEORY OF AN IDENTITY RELATED TO ABEL'S GENERALIZATION OF THE BINOMIAL THEOREM, WHICH IS USEFUL IN GRAPH THEORY.

by Cooper, Robert B. [1973-01-01]

A well-known theorem of graph theory gives a simple formula for the calculation of the number of spanning trees of a complete graph with n labeled vertices. A well-known proof of this theorem uses a combinatorial identity, related to Abel's generalization of the binomial theorem, that is difficult to prove from first principles. It is the purpose of this note to observe that this identity is an easy consequence of an analysis of the busy period for the single-server queue with Poisson input and constant service times. [ABSTRACT FROM AUTHOR]

#### ON THE INVERSION THEOREM OF WEI-LIANG CHOW.

by Mitreva, Mariya I. [2013-09-01]

#### BINARY RELATIONS AND ALGEBRAS ON MULTISETS.

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

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]

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]

#### A Partial Supply Simulation Relation and its Proof System in PADS.

by Xinghua Yao, Hengyang Wu [2017-12-01]

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]

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]

#### On degenerate Carlitz-type (h, q)-tangent numbers and polynomials.

by Ryoo, C. S. [2018-09-01]

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]

#### A note on fast approximate minimum degree orderings for symmetric matrices with some dense rows.

by Dollar, H. S., Scott, J. A. [2010-01-01]

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]

#### SIMULTANEOUS BINARY COLLISIONS IN THE EQUAL-MASS COLLINEAR FOUR-BODY PROBLEM.

by TIANCHENG OUYANG, DUOKUI YAN [2015-06-01]

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]

#### On the Definitional Embeddability of Some Elementary Algebraic Theories into the First-Order Predicate Calculus.

by SHALACK, V. I. [2016-04-01]

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 

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 

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 

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]

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]

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]

#### Método versus Cálculo en las críticas de Newton a Descartes y Leibniz.

by Guicciardini, Niccolò [2009-01-01]

#### Asymptotic Properties of the Factors of Words Over a Finite Alphabet.

by Tomescu, Ioan [2005-01-01]

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]

#### A Temporal Logic Approach to Binding-Time Analysis.

by DAVIES, ROWAN [2017-03-01]

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]

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]

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 , 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]

#### A Teacher’s Mediation of a Thinking-Aloud Discussion in a 6th Grade Mathematics Classroom.

by Zolkower, Betina, Shreyar, Sam [2007-06-01]

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]