Algebraic Treatment Research Articles

Common Misconceptions About Absolute Value and Related Thinking Strategies

This research paper seeks to identify the widespread misconceptions held by pre-service mathematics teachers regarding absolute value and the associated thinking strategies. The researchers constructed a comprehension test consisting of thirty-two items, all based on the concept of absolute value. A total of 60 Mathematics majors enrolled in Jordanian universities took the test. After purposeful selection, interviews were conducted with fifteen of the respondents in order to obtain in-depth information and confirm their written test responses. Upon thorough analysis of the data, it was concluded that most subjects encountered difficult in solving the absolute value equations and that there were clear discrepancies in the definitions of absolute value held by the participants. Moreover, it was found that the common errors committed by the students fell into one of six categories: (i) removing the two absolute value symbols; (ii) focusing deeply on rules; (iii) converting the two absolute value symbols into parentheses; (iv) ignoring the number within the two absolute value bars; (v) lack of algebraic treatments; and (vi) inability to graph the absolute value. These findings indicate that there are fundamental pedagogical errors made during the teaching and learning process for certain maths topics.

A numerical representation of hyperelliptic KdV solutions

The periodic and quasi-periodic solutions of the integrable system have been studied for four decades based on the Riemann theta functions. However, there is a fundamental difficulty in representing the solutions numerically because the Riemann theta function requires several transcendental parameters. This paper presents a novel method for the numerical representation of such solutions from the algebraic treatment of the periodic and quasi-periodic solutions of the Baker–Weierstrass hyperelliptic ℘ functions. We demonstrate the numerical representation of the hyperelliptic ℘ functions of genus two.

Møller Maps for Dirac Fields in External Backgrounds

In this paper we study the foundations of the algebraic treatment of classical and quantum field theories for Dirac fermions under external backgrounds following the initial contributions already present in various places in the literature. The treatment is restricted to contractible spacetimes of globally hyperbolic nature in dimensions d≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 4$$\\end{document} and to external fields modelled with trivial principal bundles. In particular, we construct the classical Møller maps intertwining the configuration spaces of charged and uncharged fermions, and we show some of its properties in the case of a U(1) gauge charge. In the last part, as a first step towards a quantization of the theory, we explore the combination of the classical Møller maps with Hadamard bidistributions and prove that they are involutive isomorphisms (algebraically and topologically) between suitable (formal) algebras of functionals (observables) over the configuration spaces of charged and uncharged Dirac fields.

The Rough with the Smooth of the Light Cone String

The polynomials in the generators of a unitary representation of the Poincaré group constitute an algebra which maps the dense subspace D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal D$$\\end{document} of smooth, rapidly decreasing wavefunctions to itself. This mathematical result is highly welcome to physicists, who previously just assumed their algebraic treatment of unbounded operators be justified. The smoothness, however, has the side effect that a rough operator R, which does not map a dense subspace of D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal D$$\\end{document} to itself, has to be shown to allow for some other dense domain which is mapped to itself both by R and all generators. Otherwise their algebraic product, their concatenation, is not defined. Canonical quantization of the light cone string postulates operators -iX1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\ extrm{i}X^1$$\\end{document} and P-=(P0-Pz)/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^-=(P^0 - P^z)/2$$\\end{document} and as their commutator the multiplicative operator R=P1/(P0+Pz)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=P^1/(P^0 + P^z)$$\\end{document}. This is not smooth but rough on the negative z-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z-$$\\end{document}axis of massless momentum. Using only the commutation relations of Pm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^m$$\\end{document} with the generators -iMiz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\ extrm{i}M_{iz}$$\\end{document} of rotations in the Pi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^i$$\\end{document}-Pz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^{z}$$\\end{document}-plane we show that on massless states the operator R is inconsistent with a unitary representation of SO(D-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(D-1)$$\\end{document}. This makes the algebraic determination of the critical dimension, D=26\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=26$$\\end{document}, of the bosonic string meaningless: if the massless states of the light cone string admit R then they do not admit a unitary representation of the subgroup SO(D-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(D-1)$$\\end{document} of the Poincaré group. With analogous arguments we show: Massless multiplets are inconsistent with a translation group of the spatial momentum which is generated by a self-adjoint spatial position operator X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{X}$$\\end{document}.

Constructing the “home-side” of a scientific legacy: Mary Everest Boole, pedagogy, and domesticity

The Victorian writer Mary Everest Boole (1832–1916) developed an idiosyncratic pedagogical treatment of arithmetic, algebra, and logic. Her pedagogy favored active, child-directed learning, and is now generally admired as ahead of its time, though it must be deciphered through fairly eccentric delivery. A recurring theme in Mrs. Boole’s prolific writing is the misunderstood legacy of her late husband, the renowned mathematician and logician George Boole (1815–1864). As existing literature has shown, she worked to promote a morally and religiously charged understanding of his work. More fundamentally, she presented an all-encompassing pedagogical perspective on Mr. Boole’s life and work. Across her voluminous publications, Mrs. Boole filtered everything—mathematics, logic, religion, morality, and homelife—through the lens of pedagogy. She used this expansive conception of teaching to span the gulf between professional and domestic work, thereby claiming a privileged domestic perspective on her husband’s intellectual output and enlisting his legacy as a resource for her own writing. The Booles’ entangled careers show how particular ways of practicing domesticity could shape and be shaped by mathematical identities and ideas.

Bruno Schulz's 1936 book "Methodology of medical genetic research particularly with regard to psychiatry".

In 1936, Bruno Schulz published the first detailed, book-length review of the methodology of psychiatric genetic research, based on his experiences at the German Research Institute of Psychiatry. Emphasis is placed on proper selection of relatives and the ascertainment corrections required for Mendelian transmission models. Twin studies are considered as is the impact of reduced fertility on patterns of risk. For the field work, Schulz emphasizes the importance of trust-building, confidentiality, collateral informants, and the use of medical and other administrative records, all ideally stored in personal files. Several methods of age-correction are reviewed. Schulz provides detailed algebraic treatments of these and other problems, including tests for etiologic homogeneity, with worked examples. He emphasizes two fundamental concerns in psychiatric genetics research: (i) its inter-dependency with the optimal diagnostic boundaries, which are rarely known and (ii) the genetic homogeneity of clinical samples. Given these problems, he is pessimistic about finding Mendelian transmission patterns. He assesses the predominant 19th-century method of psychiatric genetic investigation-"hereditary burden"-to be crude and biased by family size. Although written at a time of consolidation of Nazi power in Germany, this book nowhere endorses their racial/eugenic policies and can be seen as subtly questioning them.

Structural masonry under flexural and compressive loads

Pathological problems arise in structural masonries walls due to fails in details and design erros along lifetime. The study becames important when is focused loads and strength of walls. This paper aims to evaluate pathologies which arise in walls due to a bad behavior of the walls in structural masonry. Flexural strength, shear forces, compressive loads on walls are analysed in a Symbolic Algebraic Computational treatment. Analytical methods are used in compliance with International and National Codes. Procedures are suggested in order to avoid several anomalies that arise due to design failures.

An algebraic treatment of the Pastro polynomials on the real line

The properties of the Pastro polynomials on the real line are studied with the help of a triplet of q q -difference operators. The q q -difference equation and recurrence relation these polynomials obey are shown to arise as generalized eigenvalue problems involving the triplet of operators, with the Pastro polynomials as solutions. Moreover, a discrete biorthogonality relation on the real line for the Pastro polynomials is obtained and is then understood using adjoint operators. The algebra realized by the triplet of q q -difference operators is investigated.

A flow of quantum genetic Lotka-Volterra algebras on [formula omitted

In the present paper, we define a quantum analog of Lotka-Volterra algebras using a coalgebra scheme. This new framework provides a fresh perspective for the treatment of generic algebras. Additionally, a flow of quantum analogs of Lotka-Volterra genetic algebras is investigated. It's worth mentioning that such types of algebras are first introduced in this work. We observe that a flow of algebras is a particular type of continuous-time dynamical system, with states that are algebras and a structural constant matrix that depends on time and satisfies certain analogs of the Kolmogorov-Chapman equations. Using quantum quadratic operators, it is constructed a flow of quantum Lotka-Volterra algebras for the given multiplication. Furthermore, time-dependent behavior properties of these flow algebras are examined. The algebraic properties of the introduced flows are also studied, such as finding idempotents and examining an algebra generated by a pair of idempotents. It is shown that the later one is associative, while the flow is not associative. Additionally, derivations of the flow of the algebras are described.

Student Semiotic Representation Skills in Solving Mathematics Problems

Representational transformation skills significantly influence students' success in problem-solving. Students who struggle with representational transformation skills are often less adept at utilizing mathematical ideas and relationships, and vice versa. Therefore, this study used a qualitative, descriptive-interpretive approach to examine students' semiotic representation skills when solving mathematical problems. The research was conducted in a Year 9 classroom in a public school in Bandung, Indonesia, with 30 participants divided into high, middle, and low-ability groups based on their level of mathematical ability. Data was collected using both test and interview techniques. The results indicated that students in the high and middle ability groups had adequate skills in algebraic treatment and conversion from algebra to geometry and verbal expression skills for constructing algebraic expressions and converting verbal statements into mathematical equations. In contrast, the low-ability group demonstrated a lack of semiotic representation skills in problem-solving. These findings highlight the importance of transformation and conversion skills in mathematical problem-solving activities and can be valuable information for teachers and observers of mathematics education.

Algebraic approach to the frozen formalism problem of time

The long-standing problem of time in canonical quantum gravity is the source of several conceptual and technical issues. Here, recent mathematical results are used to provide a consistent algebraic formulation of dynamical symplectic reduction that avoids difficult requirements such as the computation of a complete set of Dirac observables or the construction of a physical Hilbert space. In addition, the new algebraic treatment makes it possible to implement a consistent realization of the gauge structure off the constraint surface. As a consequence, previously unrecognized consistency conditions are imposed on deparametrization---the method traditionally used to unfreeze evolution in completely constrained systems. A detailed discussion of how the new formulation extends previous semiclassical results shows that an internal time degree of freedom need not be semiclassical in order to define a consistent quantum evolution.

Reconstructing a rotor from initial and final frames using characteristic multivectors: With applications in orthogonal transformations

If an initial frame of vectors is related to a final frame of vectors by, in geometric algebra (GA) terms, a rotor, or in linear algebra terms, an orthogonal transformation, we often want to find this rotor given the initial and final sets of vectors. One very common example is finding a rotor or orthogonal matrix representing rotation, given knowledge of initial and transformed points.In this paper, we discuss methods in the literature for recovering such rotors and then outline a GA method, which generalises to cases of any signature and any dimension, and which is not restricted to orthonormal sets of vectors. The proof of this technique is both concise and elegant and uses the concept of characteristic multivectors as discussed in the book by Hestenes and Sobczyk, which contains a treatment of linear algebra using geometric algebra. Expressing orthogonal transformations as rotors, enables us to create fractional transformations and we discuss this for some classic transforms. In real applications, our initial and/or final sets of vectors will be noisy. We show how to use the characteristic multivector method to find a ‘best fit’ rotor between these sets and compare our results with other methods.

Quantum monadic algebras

We introduce quantum monadic and quantum cylindric algebras. These are adaptations to the quantum setting of the monadic algebras of Halmos, and cylindric algebras of Henkin, Monk and Tarski, that are used in algebraic treatments of classical and intuitionistic predicate logic. Primary examples in the quantum setting come from von Neumann algebras and subfactors. Here we develop the basic properties of these quantum monadic and cylindric algebras and relate them to quantum predicate logic.

Semantic analysis of normalisation by evaluation for typed lambda calculus

AbstractThis paper studies normalisation by evaluation for typed lambda calculus from a categorical and algebraic viewpoint. The first part of the paper analyses the lambda definability result of Jung and Tiuryn via Kripke logical relations and shows how it can be adapted to unify definability and normalisation, yielding an extensional normalisation result. In the second part of the paper, the analysis is refined further by considering intensional Kripke relations (in the form of Artin–Wraith glueing) and shown to provide a function for normalising terms, casting normalisation by evaluation in the context of categorical glueing. The technical development includes an algebraic treatment of the syntax and semantics of the typed lambda calculus that allows the definition of the normalisation function to be given within a simply typed metatheory. A normalisation-by-evaluation program in a dependently typed functional programming language is synthesised.

Entropy Treatment of Evolution Algebras.

In this paper, by introducing an entropy of Markov evolution algebras, we treat the isomorphism of S-evolution algebras. A family of Markov evolution algebras is defined through the Hadamard product of structural matrices of non-negative real S-evolution algebras, and their isomorphism is studied by means of their entropy. Furthermore, the isomorphism of S-evolution algebras is treated using the concept of relative entropy.

Encountering Spurious Elements in Electrical Impedance Spectroscopy Data Fitting

The aim is to present situations in which collected impedance spectrum analysis data contain information that reflects not only the studied sample, but pertains also to the fundamental instrument properties or to the method in which the data is fitted or evaluated. Fitting of such data can lead to spurious, phantom elements in equivalent circuits and to frustration in data fitting. While illustrative experiments were done on specific instruments, the conclusions are generic and apply to any instrument with current supplying and potential probing inputs. The described spurious response stems from the existing unavoidable impedance of the reference inputs and from additional resistance, such as reference electrodes, connected to the inputs. Additionally is discussed a purely mathematical situation, when response artifacts are generated from correct impedance spectrum data, but incorrectly applied algebraic treatment.

Algebraic Properties of Paraorthomodular Posets

Abstract Paraorthomodular posets are bounded partially ordered sets with an antitone involution induced by quantum structures arising from the logico-algebraic approach to quantum mechanics. The aim of the present work is starting a systematic inquiry into paraorthomodular posets theory both from algebraic and order-theoretic perspectives. On the one hand, we show that paraorthomodular posets are amenable of an algebraic treatment by means of a smooth representation in terms of bounded directoids with antitone involution. On the other, we investigate their order-theoretical features in terms of forbidden configurations. Moreover, sufficient and necessary conditions characterizing bounded posets with an antitone involution whose Dedekind–MacNeille completion is paraorthomodular are provided.

Algebraic treatment of the Bateman Hamiltonian

We apply the algebraic method to the Bateman Hamiltonian and obtain its natural frequencies and ladder operators from the adjoint or regular matrix representation of that operator. Present analysis shows that the eigenfunctions compatible with the complex eigenvalues obtained earlier by other authors are not square integrable. In addition to this, the ladder operators annihilate an infinite number of such eigenfunctions.

A Note on Simulating Predecessor-Successor Relationships in Critical Path Models

For any n entities, we exhaust all possible ordered relationships, from rank (or the highest number of connections in a linear chain, comparable to matrix rank) 0 to (n - 1). As an example, we use spreadsheets with the “RAND” function to simulate the case of n = 8 with the order-length = 3, as from a total of 10000possibilities by the number of combinations of selecting 2 (a pair of predecessor-successor) out of 5 (= card{A, B, C, D} + 1) matching- destinations followed by an exponentiation of 4 (= 8 – card{A, B, C, D}). Since the essence of this paper is about ordered structures of networks, our findings here may serve multi-disciplinary interests, in particular, that of the critical path method (CPM) in operations with management. In this connection, we have also included, toward the end of this exposition, a linear algebraic treatment that renders a deterministic mathematical programming for optimal predecessor-successor network structures.

A Generalized Solution to the Point-to-Target Problem Using the Minimum Curvature Method

Summary An explicit solution to the general 3D point-to-target problem based on the minimum curvature method has been sought for more than four decades. The general case involves the trajectory's start and target points connected by two circular arcs joined by a straight line with the position and direction defined at both ends. It is known that the solutions are multivalued, and efficient iterative schemes to find the principal root have been established. This construction is an essential component of all major trajectory construction packages. However, convergence issues have been reported in cases where the intermediate tangent section is either small or vanishes and rigorous mathematical conditions under which solutions are both possible and are guaranteed to converge have not been published. An implicit expression has now been determined that enables all the roots to be identified and permits either exact or polynomial-type solution methods to be used. Most historical attempts at solving the problem have been purely algebraic, but a geometric interpretation of related problems has been attempted, showing that a single circular arc and a tangent section can be encapsulated in the surface of a horn torus. These ideas have now been extended, revealing that the solution to the general 3D point-to-target problem can be represented as a 10th-order self-intersecting geometric surface, characterized by the trajectory's start and end points, the radii of the two arcs, and the length of the tangent section. An outline of the solution's derivation is provided in the paper together with complete details of the general expression and its various degenerate forms so that readers can implement the algorithms for practical application. Most of the degenerate conditions reduce the order of the governing equation. Full details of the critical and degenerate conditions are also provided, and together these indicate the most convenient solution method for each case. In the presence of a tangent section, the principal root is still most easily obtained using an iterative scheme, but the mathematical constraints are now known. It is also shown that all other cases degenerate to quadratic forms that can be solved using conventional methods. It is shown how the general expression for the general point-to-target problem can be modified to give the known solutions to the 3D landing problem and how the example in the published works on this subject is much simplified by the geometric, rather than algebraic treatment.