Basic Algebra Research Articles

Indefinite linear quadratic optimal control for continuous-time rectangular descriptor Markov jump systems: infinite-time case

The indefinite linear quadratic (ILQ) optimal control problem has many important applications in financial, economic systems, etc., which has been widely researched in stochastic systems and descriptor systems, etc. However, for ILQ problem of rectangular descriptor Markov jump systems (DMJSs), because there are impulses simultaneously in descriptor subsystems and at the switching time, it is theoretically difficult to research and there is no result yet. This paper discusses the ILQ optimal control problem for continuous-time linear rectangular DMJSs. Firstly, under some rank conditions and inequality conditions, the ILQ problem for rectangular DMJSs can be equivalently transformed into standard LQ problem for Markov jump systems (MJSs) by using elementary linear algebra method. Then based on the LQ theory of MJSs, the solvable sufficient condition of the ILQ problem for rectangular DMJSs and the non-negative optimal cost value are obtained. The optimal control can be synthesised as state feedback, and the resulting optimal closed-loop system has the stochastically stable solution. In addition, with some rank inequality assumptions, the differential subsystem of the resulting optimal closed-loop system can be ensured to have a unique solution. Finally, two numerical examples are provided to illustrate the effectiveness of the methods proposed in this paper.

A BLIS-like matrix multiplication for machine learning in the RISC-V ISA-based GAP8 processor

We address the efficient realization of matrix multiplication (gemm), with application in the convolution operator for machine learning, for the RISC-V core present in the GreenWaves GAP8 processor. Our approach leverages BLIS (Basic Linear Algebra Instantiation Software) to develop an implementation that (1) re-organizes the gemm algorithm adapting its micro-kernel to exploit the hardware-supported dot product kernel in the GAP8; (2) explicitly orchestrates the data transfers across the hierarchy of scratchpad memories via DMA (direct memory access); and (3) operates with integer arithmetic.

Fast multiplicative algorithms for symmetric nonnegative tensor factorization

Symmetric nonnegative tensor factorization (SNTF) is an important tool for clustering analysis. To date, most of algorithms for SNTF are based on multiplicative update rules, which have many attractive properties, e.g., they are often simple to implement and can enforce nonnegativity without extra projection steps. However, the existing multiplicative algorithms often converge slowly due to the conservative multiplicative learning steps or the use of low-level BLAS (basic linear algebra subprograms) in their implementation. In this paper, three new multiplicative algorithms are proposed for SNTF to overcome the drawback of slow convergence. First, a parallel multiplicative algorithm, which can be implemented with high-level BLAS, is derived by auxiliary optimization. To further accelerate the convergence, two new parallel multiplicative algorithms, which enable larger learning steps for improving efficiency, are developed based on weighted geometric mean and weighted arithmetic mean, respectively. Finally, we apply the proposed algorithms to multiway probabilistic clustering, where a new hyper-stochastic normalization scheme based on Euclidean distance is developed for better data preprocessing. The experiment results on both synthetic and real-world data show that the proposed SNTF algorithms converge faster than the state-of-the-art algorithms.

Viscoelastic and Viscoplastic Glucose Theory (VGT #17): Selecting 12 Cases and Applying Theories of Viscoelasticity and Viscoplasticity from Engineering and Perturbation Theory from Quantum Mechanics to Predict the Behavior of an Output Biomarker as the Strain (ε or Behavior, Symptom) by Using the Strain Rate (dε/dt) Multiplying with a Correlated Input Biomarker of Stress Value (σ or Stressor, Cause) as

The author has collected ~3 million data regarding his health condition and lifestyle details over the past 12 years. He spent the entire year of 2014 to develop a metabolism index (MI) model using topology concept, nonlinear algebra, algebraic geometry, and finite element method. This MI model contains various measured biomarkers and recorded lifestyle details along with their induced new biomedical variables for an additional ~1.5 million data. Body weight, glucose, blood pressure, heart rate, lipids, body temperature, and blood oxygen level, along with important lifestyle details, including diet, exercise, sleep, stress, water intake, and daily life routines are included in his MI database. His developed MI model has a total of 10 categories covering approximately 500 detailed elements that constitute his defined “metabolism model” which are the building blocks or root causes for diabetes and other chronic disease complications, including but not limited to cardiovascular disease (CVD), chronic heart disease (CHD), chronic kidney disease (CKD), retinopathy, neuropathy, foot ulcer, and hypothyroidism. The end result of the MI development work is a combined MI value within any selected time period with 73.5% as its dividing line between a healthy and unhealthy state. The MI serves as the foundation to many of his follow-up medical research work. During the period from 2015 to 2017, he focused his research on type 2 diabetes (T2D), especially glucoses, including fasting plasma glucose (FPG), postprandial plasma glucose (PPG), estimated average glucose (eAG), and hemoglobin A1C (HbA1C). During the following period from 2018 to 2022, he concentrated on researching medical complications resulting from diabetes, chronic diseases, and metabolic disorders which include heart problems, stroke, kidney problems, retinopathy, neuropathy, foot ulcer, diabetic skin fungal infection, hypothyroidism, and diabetic constipation, cancer, and dementia. He also developed a few mathematical risk models to calculate the probability percentages of developing various diabetic complications. Recently, he has applied theories of elasticity, plasticity, viscoelasticity, and viscoplasticity from engineering along with the theories of wave and energy from physics to conduct his research on output biomarkers (symptoms or behaviors) resulting from a suspected or identified input biomarkers (causes or stressors). As a result, he has written 16 different articles in this research area. In this article, he utilizes the perturbation theory of quantum mechanics in modern physics to develop a set of predicted output biomarkers using the identified input biomarkers from his previous research work. Next, he compares his predicted biomarkers against his measured biomarkers using their calculated correlation coefficients and prediction accuracies.

Viscoelastic and Viscoplastic Glucose Theory (VGT #18): Applying Theories of Viscoelasticity and Viscoplasticity from Engineering and Perturbation Theory from Quantum Mechanics from 9 Cases to Predict the Behavior or Symptom of an Output Biomarker as the Strain (ε), by Selecting a Highly Correlated Input Biomarker as the Stress (σ), and using the Stress Rate (dσ/dt) Multiplied with the Stress (σ)

The author has collected ~3 million data regarding his health condition and lifestyle details over the past 12 years. He spent the entire year of 2014 to develop a metabolism index (MI) model using topology concept, nonlinear algebra, algebraic geometry, and finite element method. This model contains various measured biomarkers and recorded lifestyle details along with their induced new biomedical variables for an additional ~1.5 million data. Body weight, glucose, blood pressure, heart rate, lipids, body temperature, and blood oxygen level, along with important lifestyle details, including diet, exercise, sleep, stress, water intake, and daily life routines are included in his MI database. The developed MI model has a total of 10 categories covering approximately 500 detailed elements that constitute his defined “metabolism model” which are the building blocks or root causes for diabetes and other chronic disease complications, including but not limited to cardiovascular disease (CVD), chronic heart disease (CHD), chronic kidney disease (CKD), retinopathy, neuropathy, foot ulcer, and hypothyroidism. The end result of the MI development work is a combined MI value over a selected time period with 73.5% as its dividing line between a healthy and unhealthy state. The MI serves as the foundation to many of his follow-up medical research work. During the period from 2015 to 2017, he focused his research on type 2 diabetes (T2D), especially glucoses, including fasting plasma glucose (FPG), postprandial plasma glucose (PPG), estimated average glucose (eAG), and hemoglobin A1C (HbA1C). During the following period from 2018 to 2022, he concentrated on researching medical complications resulting from diabetes, chronic diseases, and metabolic disorders which include heart problems, stroke, kidney problems, retinopathy, neuropathy, foot ulcer, diabetic skin fungal infection, hypothyroidism, diabetic constipation, cancer, and dementia. He also developed a few mathematical risk models to calculate the probability percentages of developing various diabetic complications. Recently, he applied the theories of elasticity, plasticity, viscoelasticity, and viscoplasticity from engineering along with the theories of wave and energy from physics to conduct his research on output biomarkers (symptoms or behaviors) resulting from a suspected or identified input biomarkers (causes or stressors). As a result, he has written 17 different articles in this research area.

Resource guide for teaching post-quantum cryptography

Public-key cryptography has become a popular way to motivate the teaching of concepts in elementary number theory, abstract algebra, and introduction to proof courses, as well as in cryptography courses. Unfortunately, many experts expect quantum computers to make common forms of public-key cryptography obsolete in the near future. Fortunately, there are several systems being evaluated to replace RSA and the other systems we currently use. While some of the systems are too complicated to be good examples in introductory courses, others are either quite manageable or have simplified versions that are manageable. This article gives a tour of the main types of systems under consideration and the teaching resources available for instructors who want to teach them.

Construction and application of provable positive and exact cubature formulas

Abstract Many applications require multi-dimensional numerical integration, often in the form of a cubature formula (CF). These CFs are desired to be positive and exact for certain finite-dimensional function spaces (and weight functions). Although there are several efficient procedures to construct positive and exact CFs for many standard cases, it remains a challenge to do so in a more general setting. Here, we show how the method of least squares (LSs) can be used to derive provable positive and exact formulas in a general multi-dimensional setting. Thereby, the procedure only makes use of basic linear algebra operations, such as solving an LSs problem. In particular, it is proved that the resulting LSs CFs are ensured to be positive and exact if a sufficiently large number of equidistributed data points is used. We also discuss the application of provable positive and exact LSs CFs to construct nested stable high-order rules and positive interpolatory formulas. Finally, our findings shed new light on some existing methods for multi-variate numerical integration and under which restrictions these are ensured to be successful.

Instructional Design: Teaching Algebraic Equations to Grade 8 Students with Involvement of Mathematical Reasoning in Cambridge IGCSE Curriculum

Mathematics tends to be a subject that is not favored due to a misunderstanding of the whole concept. The author believes that through involvement of mathematical reasoning skills and proper instructional strategies, students understanding can be transformed and students will be able to relearn the concept. The aim of this paper is to help teachers to recreate their lesson plans in a form of an instructional design with the involvement of mathematical reasoning, specifically in learning and relearning algebraic equations (constructivism), that can further diminish the anxiety and frustration students experience due to misconceptions in their algebraic comprehension. Through a combination of theories related to the teaching and learning of algebra in secondary school and frameworks regarding instructional strategies, the betterment of mathematical reasoning is expected to go along the relearning process, and the students will be equipped to move forward in learning mathematics. This instructional design will provide a variety of learning activities ideas, including a sample lesson plan along with other supporting documents for learning activities that teachers can use in the classroom.

An analysis of errors for pre-service teachers in first order ordinary differential equations

Literature has established that some learners encountered difficulties solving first order ordinary differential equations (ODEs). The use of error analysis in teaching ODEs is believed to make essential contribution towards calculus knowledge development. This paper therefore focuses on analyzing pre-service teachers’ (PSTs) errors and misconceptions apropos of first order ODEs. The paper analyzed the nature of errors made in a test which was written by PSTs on the above topic. The test comprised various types of first order differential equations such as ODEs with separable variables, exact ODEs, ODEs that needed integrating factors, linear ODEs, and homogeneous ODEs. The purpose was to investigate the challenges faced by PSTs in various types of ODEs and the nature of misconceptions that they had in each particular type. This is a qualitative study that involved 63 PSTs who wrote a test on ODEs after being taught the topic for two weeks. The authors marked the work in order to ascertain the misconceptions and errors exhibited by the participants in the test. The PSTs’ performance in the test was analyzed using the SOLO taxonomy and the Newman’s theory mistake analysis. The study established that the topic was rather difficult for PSTs due to various reasons that included, among others, knowledge gaps in integration rules, algebraic computations and, in rare cases, differentiation, as well as misapplication of the rules of natural logarithms. This research therefore recommends that mathematics teacher educators ought to rather focus on the concept of integration and basic algebra before introducing the topic on ODEs to teachers on training.

Boolean and Elementary Algebra with a Roll‐To‐Roll Printed Electrochemical Memristor (Adv. Mater. Technol. 5/2022)

Information Storage and Processing A non-volatile, conjugated polymeric electrochemical memristor is developed in article number 2101108 by Stephen H. Foulger and colleagues, that combines both information storage and processing. The three-terminal device was fabricated by means of gravure roll-to-roll printing on a flexible substrate with a water-soluble, self-doped conjugated polymer and could perform both elementary algebra and Boolean logic-based operations through very low power programming pulses.

Finite Generation of Lie Derived Powers of Skew Lie Algebras

Let [Formula: see text] be a finitely generated associative algebra over a field of characteristic different from 2. Herstein asked when the Lie algebra [Formula: see text] is finitely generated. Recently, it was shown that for a finitely generated nil algebra [Formula: see text] all derived powers of [Formula: see text] are finitely generated Lie algebras. Let [Formula: see text] be the Lie algebra of skew-symmetric elements of an associative algebra with involution. We consider all derived powers of the Lie algebra [Formula: see text] and prove that for any finitely generated associative nil algebra with an involution, all derived powers of [Formula: see text] are finitely generated Lie algebras.

The Application of Synchronous and Asynchronous Learning using E-learning on Elementary Linear Algebra

Learning activities were changed from face-to-face to full-online due to the COVID-19 pandemic at the end of 2019. Online learning through WhatsApp Group (WAG) was used by a mathematics teacher in higher education program in the early 2020. However, the learning result showed that 75% of preservice teacher were unable to accomplish their elementary linear algebra material. The combination of online learning through asynchronous method and synchronous learning was an alternative solution to solve the problem and enable preservice teachers having virtual face-to-face interactions with their lecturers. Therefore, this study was conducted to determine the improvement in preservice teachers' learning outcomes and responses to synchronous and asynchronous learning. The research design used was one group posttest-only with 23 subjects were selected by clustered random sampling. Data were analyzed by descriptive statistical analysis. The test results showed that learning outcomes after synchronous and asynchronous learning was effective. While the questionnaire results expressed most preservice teachers gave a positive response to both learning methods. Thus, the combination of the two methods could improve the effectiveness of online learning during the Covid-19 pandemic. It is recommended that higher education institutions provide innovative e-learning platforms for preservice teachers in developing further online learning methods.

Teleparallelism in the algebraic approach to extended geometry

Extended geometry is based on an underlying tensor hierarchy algebra. We extend the previously considered L∞ structure of the local symmetries (the diffeomorphisms and their reducibility) to incorporate physical fields, field strengths and Bianchi identities, and identify these as elements of the tensor hierarchy algebra. The field strengths arise as generalised torsion, so the naturally occurring complex in the L∞ algebra is…←torsionBI’s←torsion←vielbein←diffeomorphism parameters←…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\dots \\leftarrow \\mathrm{torsion}\\ \\mathrm{BI}'\\mathrm{s}\\leftarrow \\mathrm{torsion}\\leftarrow \\mathrm{vielbein}\\leftarrow \\mathrm{diffeomorphism}\\ \\mathrm{parameters}\\leftarrow \\dots $$\\end{document}In order to obtain equations of motion, which are not in this complex, (pseudo-)actions, quadratic in torsion, are given for a large class of models. This requires considering the dual complex. We show how local invariance under the compact subgroup locally defined by a generalised metric arises as a “dual gauge symmetry” associated with a certain torsion Bianchi identity, generalising Lorentz invariance in the teleparallel formulation of gravity. The analysis is performed for a large class of finite-dimensional structure groups, with E5 as a detailed example. The continuation to infinite-dimensional cases is discussed.

Rees algebra and special fiber ring of binomial edge ideals of closed graphs

In this paper, I compute the regularity of the Rees algebra of binomial edge ideals of closed graphs. I obtain a lower bound for the regularity of the Rees algebra of binomial edge ideals. I also present some algebraic properties of the Rees algebra and special fiber ring of binomial edge ideals of closed graphs via algebraic properties of their initial algebra and Sagbi basis theory. I obtain an upper bound for the regularity of the special fiber ring of binomial edge ideals of closed graphs.

Combinatorics of F-Polynomials

Abstract We use the stabilization functors to study the combinatorial aspects of the $F$-polynomial of a representation of any finite-dimensional basic algebra. We characterize the vertices of their Newton polytopes. We give an explicit formula for the $F$-polynomial restricting to any face of its Newton polytope. For acyclic quivers, we give a complete description of all facets of the Newton polytope when the representation is general. We also prove that the support of the $F$-polynomial is saturated for any rigid representation. We provide many examples and counterexamples and pose several conjectures.

Breaches of the didactic contract as a driving force behind learning and non-learning: a story of flaws and wants

AbstractIn agreement with the main tenets of the anthropological theory of the didactic (ATD), this study uncovers dependencies between what students can learn, the established curriculum and the current state of mathematicians’ mathematics (‘scholarly mathematics’). One main result is that the mathematics taught, too often taken for granted by curriculum developers and teachers, is in fact problematic not only to students but also to teachers and curriculum developers and is sometimes a challenge even to current scholarly mathematics. The mathematics taught during a given historical period within a given institution contains flaws that, when they cease to go unnoticed, generate crises, in the form of breaches of the prevailing didactic contract. The resolution of these crises allows the institution and its actors—in particular students and teachers—to learn new contents and often also leads to the more or less damaging unlearning of old contents. This key phenomenon is illustrated, at the triple level of the classroom, the curriculum and scholarly mathematics, with regard to elementary algebra and mathematical analysis, most importantly in the case of maxima and minima problems.

The PBW Basis of Uq,q¯(\U0001d58c\U0001d591̈n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$U_{q,\\overline {q}}(\\ddot {\\mathfrak {g}\\mathfrak {l}}_{n})$\\end{document}

We consider the PBW basis of the quantum toroidal algebra of gln\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathfrak {g}\\mathfrak {l}}_{n}$\\end{document}, which was developed in Neguț (Adv. Math. 372, 2020), and prove commutation relations between its generators akin to the ones studied in Burban and Schiffmann (Duke Math. J. 161(7):1171–1231, 2012) for n = 1. This gives rise to a new presentation of the quantum toroidal algebra of type A.

Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges

The main contribution of this review is to show some relevant relationships between three geometric structures on a connected Lie group G, generated by the same dynamics. Namely, Linear Control Systems, Almost Riemannian Structures, and Degenerate Dynamical Systems. These notions are generated by two ordinary differential equations on G: linear and invariant vector fields. A linear vector field on G is determined by its flow, a 1-parameter group of Aut(G), the Lie group of G-automorphisms. An invariant vector field is just an element of the Lie algebra g of G. The Jouan Equivalence Theorem and the Pontryagin Maximum Principal are instrumental in this setup, allowing the extension of results from Lie groups to arbitrary manifolds for the same kind of structures which satisfy the Lie algebra finitude condition. For each structure, we present the first given examples; these examples generate the systems in the plane. Next, we introduce a general definition for these geometric structures on Euclidean spaces and G. We describe recent results of the theory. As an additional contribution, we conclude by formulating a list of open problems and challenges on these geometric structures. Since the involved dynamic comes from algebraic structures on Lie groups, symmetries are present throughout the paper.

Optimal Method of Integer Factorization

The object of the research is performance of integer factorization algorithms and possibility of mathematical methods using in these algorithms. The subject of the research is cryptographic properties GNFS method. Methods of research are methods of the theory of elliptic curves, finite fields, abstract algebra and advanced the theory of factorization algorithms. As a result of this work, the dependence of the properties of a minimal time of factorization and choice of algebraic factor bases over the ring Zn, where n=pq was established. Moreover, we have implemented the general number field sieve (GNFS), which is the most efficient classical algorithm known for factoring integers.

An Exploration of Synthetic Division for Non-Linear Polynomial Divisors

As a goal of developing alternative algorithm on division of polynomials whose dividend is 𝑎1𝑥 𝑛 + 𝑎2𝑥 𝑛−1 + 𝑎3𝑥 𝑛−2 + ⋯ + 𝑎𝑛𝑥 + 𝑎𝑛+1 and the divisor is 𝑏1𝑥 𝑚 + 𝑏2𝑥 𝑚−1 + 𝑏3𝑥 𝑚−2 … + 𝑏𝑚𝑥 + 𝑏𝑚+1, where 𝑛 > 𝑚, 𝑎1 ≠ 0, 𝑏1 ≠ 0, and 𝑎𝑖 ′𝑠and 𝑏𝑖 ′𝑠including 𝑎𝑛+1 and 𝑏𝑚+1 are constant, the researcher explored the synthetic division in compact form.The researcher believes that the algorithm in this study is a good alternative in dividing polynomials of higher degrees. In the said compact form, division of only some distinct pairs of non-linear polynomials were presented and served as reference problems for the researcher during the initial exploration. The Division Algorithm for Polynomials theorem was applied in exploring the problems with quadratic, cubic and quartic divisors. This resulted to the development of formulas for the coefficients 𝑡1of the quotient 𝑄(𝑥) and coefficients 𝑟1 of the remainder 𝑅(𝑥) which were considered important parts of the algorithm. Aside from the condition for the inapplicability for non-linear divisors, additional conditions were provided to the problems where the usual synthetic division is inappropriate. At the end, the algorithm on division of polynomials of higher degrees by non-linear divisors was developed using basic research. Illustrative examples of dividing polynomials using the developed algorithms with monic (𝑏1 = 1) and non-monic (𝑏1 ≠ 1) divisors were provided. Results were verified through other existing methods: long division and synthetic division in compact form. Keywords: synthetic division, high school algebra, short division