Algebraic Variables Research Articles

The Jacobi Matrix for Functions in Noncommutative Algebras

We develop a general tool for constructing the exact Jacobi matrix for functions defined in noncommutative algebraic systems without using any partial derivative. The construction is applied to solving nonlinear problems of the form f(x) = 0 with the aid of Newton’s method in algebras defined in \({\mathbb{R}^N}\). We apply this to eight (commutative and noncommutative) algebras in \({\mathbb{R}^4}\). The Jacobi matrix is explicitly constructed for polynomials in x−a and for polynomials in the reciprocals (x−a)1 such that Jacobi matrices for functions defined by Taylor and Laurent expansions can be constructed in general algebras over \({\mathbb{R}^N}\). The Jacobi matrix for the algebraic Riccati equation with matrix elements from an algebra in \({\mathbb{R}^N}\) is presented, and one particular algebraic Riccati equation is numerically solved in all eight algebras over \({\mathbb{R}^4}\). Another case treated was the exponential function with algebraic variables including a numerical example. For cases where the computation of the exact Jacobi matrix for finding solutions of f(x) = 0 is time consuming, a hybrid method is recommended, namely to start with an approximation of the Jacobi matrix in low precision and only when \({\|f(x)\|}\) is sufficiently small, to switch to the exact Jacobi matrix.

Extended space method for parameter identifiability of DAE systems

Mathematical models of physical systems often have parameters that must be identified from physical data. This makes the analysis of the parameter identifiability of the given model system an essential prerequisite. Thus far, several methods have been proposed for analyzing the parameter identifiability of ordinary differential equation (ODE) systems. But, to the best of our knowledge, the parameter identifiability of differential algebraic equation (DAE) systems has scarcely been analyzed as a specific topic. Traditional differential algebraic (DA) methods developed for ODE systems are often applied directly on DAE systems. These methods, however, are not always applicable, e.g., when the prime ideal condition is not satisfied by a DAE system. In this paper, we propose a novel method to analyze the identifiability of DAE systems, based on the concept of space extension, through which the algebraic and differential variables can be decoupled. Furthermore, an inherent, low-dimensional, regular ODE system can be obtained, which is the external equivalent of the original DAE system. Subsequently, the differential algebraic (DA) method can then be used to analyze the identifiability of the low-dimension ODE system. Theoretical analysis is also presented for the proposed method. Two examples, including a simplified interaction model and an isothermal reactor system, are presented to illustrate the detailed steps and effectiveness of the proposed method.

Indirect Solution of Inequality Constrained and Singular Optimal Control Problems Via a Simple Continuation Method

This paper develops a simple continuation method for the approximate solution of optimal control problems. The class of optimal control problems considered include (i) problems with bounded controls, (ii) problems with state variable inequality constraints (SVIC), and (iii) singular control problems. The method used here is based on transforming the state variable inequality constraints into equality constraints using nonnegative slack variables. The resultant equality constraints are satisfied approximately using a quadratic loss penalty function. Similarly, singular control problems are made nonsingular using a quadratic loss penalty function based on the control. The solution of the original problem is obtained by solving the transformed problem with a sequence of penalty weights that tends to zero. The penalty weight is treated as the continuation parameter. The paper shows that the transformed problem yields necessary conditions for a minimum that can be written as a boundary value problem involving index-1 differential–algebraic equations (BVP-DAE). The BVP-DAE includes the complementarity conditions associated with the inequality constraints. It is also shown that the necessary conditions for optimality of the original problem and the transformed problem differ by a term that depends linearly on the algebraic variables in the DAE. Numerical examples are presented to illustrate the efficacy of the proposed technique.

Invariant Trapezoidal Kalman Filter for Application to Attitude Estimation

This paper incorporates formal concepts of invariance of numerical integration schemes into the design of Kalman filters. In general terms, invariant discretizations of dynamical systems can form the basis for the derivation of the covariance propagation during the prediction and update phases of a Kalman filter, and this paper presents a framework to that effect. Specifically, natural invariants of angular motion are introduced, as part of a symmetry-preserving trapezoidal integration rule, to form the basis for the derivation of a discrete-time invariant Kalman filter for an attitude estimation problem. The proposed filter is realized by expressing all zero-mean random variables in the Lie algebra, while the state manipulations are performed in special orthogonal group 3. Simulation and experimental results are obtained using a neutrally buoyant spherical blimp to validate the proposed method against state-of-the-art Kalman filters in a broad range of angular speeds and sampling rates. Furthermore, the results support claims of invariance of the estimator with respect to angular speed, as well as the preservation of the system’s symmetries through the covariance calculations.

A Module-Theoretic Interpretation of Schiffler's Expansion Formula

We give a module-theoretic interpretation of Schiffler's expansion formula which is defined combinatorially in terms of complete (Γ, γ)-paths in order to get the expansion of the cluster variables in the cluster algebra of a marked surface (S, M). Based on the geometric description of the indecomposable objects of the cluster category of the marked surface (S, M), we show the coincidence of Schiffler–Thomas’ expansion formula and the cluster character defined by Palu.

Friezes and a construction of the Euclidean cluster variables

Let Q be a Euclidean quiver. Using friezes in the sense of Assem–Reutenauer–Smith, we provide an algorithm for computing the (canonical) cluster character associated with any object in the cluster category of Q . In particular, this algorithm allows us to compute all the cluster variables in the cluster algebra associated with Q . It also allows us to compute the sum of the Euler characteristics of the quiver Grassmannians of any module M over the path algebra of Q .

Group Field Theory with Noncommutative Metric Variables

We introduce a dual formulation of group field theories as a type of noncommutative field theories, making their simplicial geometry manifest. For Ooguri-type models, the Feynman amplitudes are simplicial path integrals for BF theories. We give a new definition of the Barrett-Crane model for gravity by imposing the simplicity constraints directly at the level of the group field theory action.

Discrete Integrable Systems, Positivity, and Continued Fraction Rearrangements

In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the $Q$- and $T$-systems based on $A_r$. The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve manifest positivity. We also show how these techniques apply as well to non-commutative settings.

F-Polynomials in Quantum Cluster Algebras

F-polynomials and g-vectors were defined by Fomin and Zelevinsky to give a formula which expresses cluster variables in a cluster algebra in terms of the initial cluster data. A quantum cluster algebra is a certain noncommutative deformation of a cluster algebra. In this paper, we define and prove the existence of analogous quantum F-polynomials for quantum cluster algebras. We prove some properties of quantum F-polynomials. In particular, we give a recurrence relation which can be used to compute them. Finally, we compute quantum F-polynomials and g-vectors for a certain class of cluster variables, which includes all cluster variables in type \(\mbox{A}_{n}\) quantum cluster algebras.

Grade Transition Dynamic Optimization of the Living Nitroxide‐Mediated Radical Polymerization of Styrene in a Tubular Reactor

AbstractIn this work, the dynamic optimization of tubular reactors where nitroxide‐mediated radical polymerization (NMRP) of styrene takes place is performed. The effect of potential manipulated variables is addressed. The system is governed by partial differential equations (PDEs). The PDEs are discretized spatially giving rise to a differential‐algebraic equation (DAE) system. The DAE optimization problem is then solved using a simultaneous approach wherein the differential and the algebraic variables are fully discretized leading to a large‐scale nonlinear programming (NLP) problem. The resulting optimization problem is solved using an interior point algorithm capable of handling large‐scale NLPs. magnified image

The negative answer to Kameko's conjecture on the hit problem

The goal of this paper is to give a negative answer to Kameko's conjecture on the hit problem stating that the cardinal of a minimal set of generators for the polynomial algebra P k , considered as a module over the Steenrod algebra A , is dominated by an explicit quantity depending on the number of the polynomial algebra's variables k. The conjecture was shown by Kameko himself for k ⩽ 3 in his PhD thesis in the Johns Hopkins University in 1990, and recently proved by us and Kameko for k = 4 . However, we claim that it turns out to be wrong for any k > 4 . In order to deny Kameko's conjecture we study a minimal set of generators for A -module P k in some so-call generic degrees. What we mean by generic degrees is a bit different from that of other authors in the fields such as Crabb–Hubbuck, Nam, Repka–Selick, Wood …We prove an inductive formula for the cardinal of the minimal set of generators in these generic degrees when the number of the variables, k, increases. As an immediate consequence of this inductive formula, we recognize that Kameko's conjecture is no longer true for any k > 4 .

Encoding simplicial quantum geometry in group field theories

An extended group field theory formalism for quantum gravity, based on a field that is a function of both group variables, interpreted as discretized connection, and Lie algebra variables, interpreted as discretized triads, has been proposed recently as an attempt to define models with a clearer link with simplicial geometry. In the context of such a formalism, we introduce a new symmetry requirement on the field. This leads, in 3D, to Feynman amplitudes interpreted as simplicial path integrals based on the Regge action, to a proper relation between the discrete connection and the triad vectors appearing in the Regge action, and to a much more satisfactory and transparent encoding of simplicial geometry already at the level of the group field theory action.

Positivity of the T-System Cluster Algebra

We give the path model solution for the cluster algebra variables of the $T$-system of type $A_r$ with generic boundary conditions. The solutions are partition functions of (strongly) non-intersecting paths on weighted graphs. The graphs are the same as those constructed for the $Q$-system in our earlier work, and depend on the seed or initial data in terms of which the solutions are given. The weights are "time-dependent" where "time" is the extra parameter which distinguishes the $T$-system from the $Q$-system, usually identified as the spectral parameter in the context of representation theory. The path model is alternatively described on a graph with non-commutative weights, and cluster mutations are interpreted as non-commutative continued fraction rearrangements. As a consequence, the solution is a positive Laurent polynomial of the seed data.

Calculus Students' Use and Interpretation of Variables: Algebraic vs. Arithmetic Thinking

Abstract The ability to use and interpret algebraic variables as generalized numbers and changing quantities is fundamental to the learning of calculus. This study considers the use of variables in these advanced ways as a component of algebraic thinking. College introductory calculus students' (n = 174) written responses to algebra problems requiring the use and interpretation of variables as changing quantities were examined for evidence of algebraic and arithmetic thinking. A framework was developed to describe and categorize examples of algebraic, transitional, and arithmetic thinking reflected in these students' uses of variables. The extent to which students' responses showed evidence of algebraic or arithmetic thinking was quantified and related to their course grades. Only one third of the responses of these entering calculus students were identified as representative of algebraic thinking. This study extends previous research by showing that evidence of algebraic thinking in students' work was po...

Denominators of cluster variables

Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional indecomposable objects in the cluster category, inducing a correspondence between clusters and cluster-tilting objects. Fix a cluster-tilting object T and a corresponding initial cluster. By the Laurent phenomenon, every cluster variable can be written as a Laurent polynomial in the initial cluster. We give conditions on T that are equivalent to the fact that the denominator in the reduced form for every cluster variable in the cluster algebra has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T.

Mathematical Variables as Indigenous Concepts

This paper explores the semiotic status of algebraic variables. To do that we build on a structuralist and post‐structuralist train of thought going from Mauss and Lévi‐Strauss to Baudrillard and Derrida. We import these authors’ semiotic thinking from the register of indigenous concepts (such as mana), and apply it to the register of algebra via a concrete case study of generating functions. The purpose of this experiment is to provide a philosophical language that can explore the openness of mathematical signs to reinterpretation, and bridge some barriers between philosophy of mathematics and critical approaches to knowledge.

A comparison of curricular effects on the integration of arithmetic and algebraic schemata in pre-algebra students

This research examines students’ ability to integrate algebraic variables with arithmetic operations and symbols as a result of the type of instruction they received, and places their work on scales that illustrate its location on the continuum from arithmetic to algebraic reasoning. It presents data from pre and post instruction clinical interviews administered to a sample of middle school students experiencing their first exposure to formal pre-algebra. Roughly half of the sample (n = 15) was taught with a standards-based curriculum emphasizing representation skills, while a comparable group (n = 12) of students received traditional instruction. Analysis of the pre and post interviews indicated that participants receiving a standards-based curriculum demonstrated more frequent and sophisticated usage of variables when writing equations to model word problems of varying complexity. This advantage was attenuated on problems that provided more representational support in which a diagram with a variable was presented with the request that an expression be written to represent the perimeter and area. Differences in strategies used by the two groups suggest that the traditional curriculum encouraged students to continue using arithmetic conventions, such as focusing on finding specific values, when asked to model relations with algebraic notation.

On quantity calculus and units of measurement

‘Quantity calculus’ is a name sometimes given to algebra as it is applied to quantities. It is assumed that the rules of algebra apply equally well to quantities as they do to numbers, and that they apply to units of measurement because they too are, by definition, quantities in their own right. This paper examines that assumption and concludes that, while it may be true of abstract quantities, it is not universally applicable to measurable quantities, whose practical units are not algebraic variables.

Change of order for regular chains in positive dimension

We discuss changing the variable order for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations. We propose a modular method, reducing the problem to computations in dimension zero and one. The problems raised by the choice of the specialization points and the lack of the (crucial) information of what are the free and algebraic variables for the new order are discussed. Strong (but not unusual) hypotheses for the initial regular chain are required; the main required subroutines are change of order in dimension zero and a formal Newton iteration.

On the numerical damping of time integrators for coupled mechatronic systems

The generalized-α time integrator is considered for the simulation of mechatronic systems. In this context, the fundamental concept of numerical damping is analysed for coupled sets of first and second-order differential–algebraic equations. First, it appears that the algebraic variables do not influence the spectral properties of the dynamic variables. Second, we demonstrate that the coupling between the dynamic variables does not influence the high-frequency spectral response, so that the numerical damping can be determined as usual from elementary characteristic polynomials. Those results are exploited to assess the stability properties of the scheme and to select an algorithm with optimal damping properties.