Context Of Algebras Research Articles

Null–space construction using cofactors from a screw–algebra context

This paper develops an approach for constructing the null space N ( A ) of a linear system of homogeneous equations using the cofactors of an augmented coefficient matrix A . The relationship between the row space R ( A T ) and null space is exploited by introducing an augmenting vector which is linearly independent of the row space and dependent on the null space. The resultant null space is shown to be a vector of cofactors of the augmenting row of the coefficient matrix and is invariant. This provides a straightforward solution to a linear system of homogeneous equations without going through Gauss-Seidel elimination. The approach is derived from a onedimensional null space and is extended to a multidimensional one by partitioning the coefficient matrix and consequently constructing a set of ( n − m ) null–space vectors based on cofactors. Examples are given and accuracy is compared with Gauss–Seidel elimination. The approach is further used in a screw–algebra context with a simple procedure to obtain a system of reciprocal screws representing a set of constraint wrenches from a set of twists of freedom, in the form of a linear system of homogeneous equations in R 6 . The paper provides rigorous proofs and applications in both linear algebra and advanced kinematics.

Discrete series of fusion algebras

AbstractWe show that the left regular representation of a countably infinite (discrete) group admits no finite-dimensional invariant subspaces. We also discuss a consequence of this fact, and the reason for our interest in this statement.We then formally state, as a ‘conjecture’, a possible generalisation of the above statement to the context of fusion algebras. We prove the validity of this conjecture in the case of the fusion algebra arising from the dual of a compact Lie group.We finally show, by example, that our conjecture is false as stated, and raise the question of whether there is a ‘good’ class of fusion algebras, which contains (a) the two ‘good classes’ discussed above, namely, discrete groups and compact group duals, and (b) only contains fusion algebras for which the conjecture is valid.

Finite-dimensional duality on the generalized Lie algebra 𝔰𝔩(2)q

We show a local duality between the left and right finite-dimensional representations on the enveloping algebra of (2)q, the generalized Lie algebra introduced by Lyubashenko and Sudbery. This duality works in a general context of algebras satisfying certain algebraic properties; our goal in this paper is to prove that the enveloping algebra of (2)q satisfies these abstract properties.

The Topological Version of Free Entropy

Notions of topological free entropy and of free capacity are introduced in the C*-algebra context. Basic properties, basic problems and connections to potential theory and random matrix theory are discussed.

Completeness of Timed μCRL

In [25] a straightforward extension of the process algebra μCRL was proposed to explicitly deal with time. The process algebra μCRL has been especially designed to deal with data in a process algebraic context. Using the features for data, only a minor extension of the language was needed to obtain a very expressive variant of time. But [25] contains syntax, operational semantics and axioms characterising timed μCRL. It did not contain an in depth analysis of theory of timed μCRL. This paper fills this gap, by providing soundness and completeness results. The main tool to establish these is a mapping of timed to untimed μCRL and employing the completeness results obtained for untimed μCRL.

Sharing Teaching Ideas: Mathematical Modeling: Compound Functions and the IRS Tax Rate Schedules

When a teacher finds a real-life application of mathematics, especially one to which every student in a class can relate, expounding on that application is certainly worthwhile. This article focuses on the concept of compound functions, as presented in a second-year algebra context, and on an application of this concept using the tax rate schedules from 1999. Teachers can adapt this activity to reflect current tax information.

Recursion and Petri nets

This paper shows how to define Petri nets through recursive equations. It specifically addresses this problem within the context of the box algebra, a model of concurrent computation which combines Petri nets and standard process algebras. The paper presents a detailed investigation of the solvability of recursive equations on nets in what is arguably the most general setting. For it allows an infinite number of possibly unguarded equations, each equation possibly involving infinitely many recursion variables. The main result is that by using a suitable partially ordered domain of nets, it is always possible to solve a system of equations by constructing the limit of a chain of successive approximations.

Unitized Jordan algebras and dispersionless KdV equations

Multicomponent KdV systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having higher-order symmetries, recursion operators and hierarchies of conservation laws. In this paper the dispersionless limits of these Jordan KdV equations are studied. Recursion laws for conserved densities are given under the assumption that the algebra possesses a unity element. Sufficient conditions are given for the unitized counterpart of a diagonalizable non-unital system to be diagonalizable. Hamiltonian structure is discussed within the context of DN Jordan algebras and N scattering problems.

Linking Algebraic Concepts and Contexts: Every Picture Tells a Story

Current trends in middle school curricula highlight the importance of integrating algebraic reasoning throughout the middle grades. The use of innovative technologies offers one exciting way to engage students in algebraic problem-solving activities. In this article, we report on our efforts to design for seventh graders a computer-based instructional unit that would not only address state standards but, we hoped, also lay the groundwork for the students' subsequent formal study of algebra. The software we chose, SimCalc MathWorlds, is available free of charge via the World Wide Web at www.simcalc. umassd.edu. Look on the first page of this site. With this program, students can animate characters on the screen by creating graphs of position and velocity.

Subbehaviors and interconnections for delay differential systems

The goal of this paper is to investigate linear delay differential systems with constant coefficients and commensurate point delays from a behavioral point of view. For the appropriate algebraic context, a larger ring of generalized delay differential operators is introduced. The classical systems theoretic notions of controllability, i/o-systems. nonanticipation. defined in behavioral terminology, are algebraically characterized in terms of kernel representations. Achievability of subsystems via regular interconnec-tions is put into relation with controllability in the sense of trajectory steering.

Computational experiments in arithmetic geometry over finite fields and their computer support

Some selected findings of an investigation into the following two problems of arithmetic geometry of algebraic curves over finite prime fields are presented: (a) the existence of rational points and (b) the distribution of angles of Kloosterman sums. A brief description of computer experiments supporting the investigation is presented, and some algebraic and analytical context of the problems considered is given.

Factorization of Formal Exponentials and Uniformization

Let g be a Lie algebra over a field of characteristic zero equipped with a vector space decomposition g=g−⊕g+, and let s and t be commuting formal variables commuting with g. We prove that the map C: sg−[[s,t]]×tg+[[s,t]]→sg−[[s,t]]⊕tg+[[s,t]] defined by the Campbell–Baker–Hausdorff formula and given by esg−etg+=eC(sg−,tg+) for g±∈g±[[s,t]] is a bijection, as is well known when g is finite-dimensional over R or C, by geometry. It follows that there exist unique Ψ±∈g±[[s,t]] such that etg+esg−=esΨ−etΨ+ (also well known in the finite-dimensional geometric setting). We apply this to a Lie algebra g consisting of certain formal infinite series with coefficients in a Z-graded Lie algebra p, for instance, an affine Lie algebra, the Virasoro algebra, or a Grassmann envelope of the N=1 Neveu–Schwarz superalgebra. For p the Virasoro algebra, the result was first proved by Huang as a step in the construction of a geometric formulation of the notion of vertex operator algebra, and for p a Grassmann envelope of the Neveu–Schwarz superalgebra, it was first proved by Barron as a corresponding step in the construction of a supergeometric formulation of the notion of vertex operator superalgebra. In the special case of the Virasoro (resp., N=1 Neveu–Schwarz) algebra with zero central charge the result gives the precise expansion of the uniformizing function for a sphere (resp., supersphere) with tubes resulting from the sewing of two spheres (resp., superspheres) with tubes in two-dimensional genus-zero holomorphic conformal (resp., N=1 superconformal) field theory. The general result places such uniformization problems into a broad formal algebraic context.

Quantum groups with invariant integrals.

Quantum groups have been studied intensively for the last two decades from various points of view. The underlying mathematical structure is that of an algebra with a coproduct. Compact quantum groups admit Haar measures. However, if we want to have a Haar measure also in the noncompact case, we are forced to work with algebras without identity, and the notion of a coproduct has to be adapted. These considerations lead to the theory of multiplier Hopf algebras, which provides the mathematical tool for studying noncompact quantum groups with Haar measures. I will concentrate on the *-algebra case and assume positivity of the invariant integral. Doing so, I create an algebraic framework that serves as a model for the operator algebra approach to quantum groups. Indeed, the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context.

Ariadne's Thread: The Life and Times of Oughtred's Clavis

William Oughtred's Clavis mathematicae, first published in 1631, was regarded for the remainder of the seventeenth century as a classic text in algebra, reprinted, translated, and explained for over 70 years. Yet its content was limited, its style obscure and its notation old-fashioned even when it was first written, and it now seems extraordinary that it should have had such a long life. The early success of the book, and the great esteem in which Oughtred was held by contemporary and later mathematicians, can only be understood in the light of the general state of algebra teaching in England during the first half of the seventeenth century. In later years the book was kept alive by the efforts of one of its greatest admirers, John Wallis. The aim of this paper is to set the story of the Clavis into the wider context of English algebra, and the political background of the time.

A Refinement of the Lecture Hall Theorem

Forn⩾1, let Lnbe the set of lecture hall partitions of lengthn, that is, the set ofn-tuples of integersλ=(λ1,…,λn) satisfying 0⩽λ1/1⩽λ2/2⩽…⩽λn/n. Let ⌈λ⌉ be the partition (⌈λ1/1⌉,…,⌈λn/n⌉), and leto(⌈λ⌉) denote the number of its odd parts. We show that the identity∑λ∈Lnq|λ|u|⌈λ⌉|vo(⌈λ⌉)=(1+uvq)(1+uvq2)…(1+uvqn)(1−u2qn+1)(1−u2qn+2)…(1−u2q2n)is equivalent to a refinement of Bott's formula for the affine Coxeter groupCn, obtained by I. G. Macdonald (Math. Ann.199(1972), 161–174) and V. Reiner (Electron. J. Combin.2(1995), R25). The caseu=v=1 of the above identity, called the lecture hall theorem, was proved by us in (Ramanujan J.1(1997), 101–111), and then by Andrews (“Mathematical Essays in Honor of G.-C. Rota,” pp. 1–22, Birkhäuser, Cambridge, MA, 1998). In the present paper, we give two direct proofs of the above identity. The first one is rather short, but requires a bit ofq-calculus; the second one is the first truly bijective proof ever found in the domain of lecture hall partitions. Although we describe our bijection in completely combinatorial terms, it finds its origin in the algebraic context of Coxeter groups. Both proofs are completely independent of all earlier proofs of the Lecture Hall Theorem.

Structure Sense: The Relationship between Algebraic and Numerical Contexts.

Several researchers suggest that students' difficulties with the algebraic structure are in part due to their lack of understanding of structural notions in arithmetic. They assume that the algebraic system used by students inherits structural properties associated with the number system with which students are familiar. This study explored this assumption. In an attempt to discover whether wrong interpretations of the algebraic structure found in an algebraic context occur in a purely numerical one, we interviewed 53 sixth-graders individually. The assessment confirms the assumption: students' difficulties with the algebraic structure were found in purely numerical contexts. However, the study also confirms two, seemingly, contradictory observations. On the one hand, the students' interpretations of the structures of the expressions were very consistent; that is, the same tendencies were found in many students' answers. In this sense the students' behaviour was consistent. On the other hand, it was clearly observed that the same student may give an incorrect answer in one context and a correct answer in another. In this sense, it often seemed that the students were inconsistent.

On the problem of feedback linearization

The paper studies and solves in a geometric framework the problem of partial feedback linearization for discrete-time dynamics. An algorithm for computing the largest linearizable subsystem is proposed. This approach can be considered as dual to the one already proposed in literature in an algebraic context.

On Polyakov's basic variational formula for loop spaces

We use the homological algebra context to give a more rigorous proof of Polyakov's basic variational formula for loop spaces.

CONNECTEDNESS, DISCONNECTEDNESS AND CLOSURE OPERATORS: FURTHER RESULTS

Let X be an arbitrary category with an (E, M)-factorization structure for sinks. A notion of constant morphism that depends on a chosen class of monomorphisms was previously used to provide a generalization of the connectedness-disconnectedness Galois connection (also called torsion-torsion free in algebraic contexts). This Galois connection was shown to factor through the class of all closure operators on X with respect to M. Here, properties and implications of this factorization are investigated. In particular, it is shown that this factorization can be further factored. Examples are provided.

Closed-Loop Pole Design for Vibration Suppression

A technique is discussed for selecting the closed-loop pole locations in the state or output feedback problem. The damping is the only parameter that is allowed to vary. The geometric interpretation of this is that each closed-loop pole is constrained to lie on a circular arc whose radius corresponds to that pole’s open-loop undamped natural frequency. An analytic approach is then proposed to compute the required state and output feedback gain. The method is based on a sensitivity analysis of the closed-loop eigenvalues to each gain element. An Euler–Bernoulli pinnedbeamexample is used to demonstrate the procedure. Thisnew formulationoffers insight into the uniqueness issue in state design, the possibility of state eigenstructure assignability, and the limited pole placement in output design from a linear algebra context.