Basic Algebra Research Articles

Graphs of Wajsberg Algebras via Complement Annihilating

In this paper, W-graph, called the notion of graphs on Wajsberg algebras, is introduced such that the vertices of the graph are the elements of Wajsberg algebra and the edges are the association of two vertices. In addition to this, commutative W-graphs are also symmetric graphs. Moreover, a graph of equivalence classes of Wajsberg algebra is constructed. Meanwhile, new definitions as complement annihilator and ∆-connection operator on Wajsberg algebras are presented. Lemmas and theorems on these notions are proved, and some associated results depending on the graph’s algebraic properties are presented, and supporting examples are given. Furthermore, the algorithms for determining and constructing all these new notions in each section are generated.

Non-complementary strand commutation as a fundamental alternative for information processing by DNA and gene regulation.

The discovery of the DNA double helix has revolutionized our understanding of data processing in living systems, with the complementarity of the two DNA strands providing a reliable mechanism for the storage of hereditary information. Here I reveal the 'strand commutation' phenomenon-a fundamentally different mechanism of information storage and processing by DNA/RNA based on the reversible low-affinity interactions of essentially non-complementary nucleic acids. I demonstrate this mechanism by constructing a memory circuit, a 5-min square-root circuit for 4-bit inputs comprising only nine processing ssDNAs, simulating a 572-input AND gate (surpassing the bitness of current electronic computers), and elementary algebra systems with continuously changing variables. Most importantly, I show potential pathways of gene regulation with strands of maximum non-complementarity to the gene sequence that may be key to the reduction of off-target therapeutic effects. This Article uncovers the information-processing power of the low-affinity interactions that may underlie major processes in an organism-from short-term memory to cancer, ageing and evolution.

Professional Development of Mathematics Teachers’ Pedagogical Content Knowledge in Teaching Basic Algebra

This article presents the findings of the mathematics teachers’ professional development (PD) in teaching algebra from a continuously implemented two months PD program. The program was conducted as a case study for a group of 20 secondary level mathematics teachers who are following a two-year in-service teacher training course. This program targeted to improve the secondary level mathematics teachers’ pedagogical content knowledge (PCK) in teaching algebra. The content of the PD program included the nature of algebraic concepts, psychological aspects related to PCK, algebraic thinking and the algebraic learning styles. The result of the paired t-test indicated 0.345, a positive correlation between the post-test marks and pre-test marks. It interprets a favourable progress of the professional development (PD) program. From the interview findings, teacher trainees asserted the importance of algebraic thinking and the psychological foundation for making mathematics education effective and quality. The results asserted that the mathematics teachers’ PCK in algebra could be developed toa considerable extent from this PD program. The findings further revealed that mathematics teachers could develop their algebraic thinking from the program and it isa must. Since mathematics teachers’ PCK directly influence on students’ algebra learning and mathematics achievement, the findings of the study deepens the understanding that the mathematics teachers PCK must be developed to increase the students’ mathematics learning and understanding of the concepts. Therefore, I recommend that teacher education courses and continuous education programs must be developed in the teacher education curriculum with reforms for enhancing the quality of mathematics education in Sri Lanka.

Extension of the generalized n-strong Drazin inverse

The aim of this paper is to present an extension of the generalized n-strong Drazin inverse for Banach algebra elements using a g-Drazin invertible element rather than a quasinilpotent element in the definition of the generalized n-strong Drazin inverse. Thus, we introduce a new class of generalized inverses which is a wider class than the classes of the generalized n-strong Drazin inverse and the extended generalized strong Drazin inverses. We prove a number of characterizations for this new inverse and some of them are based on idempotents and tripotents. Several generalizations of Cline?s formula are investigated for the extension of the generalized n-strong Drazin inverse.

Mathematical Maturity of In-Service Junior High School Mathematics Teachers in Imus City, Cavite, Philippines

This study determined the in-service junior high school mathematics teachers’ level of mathematical maturity in terms of expertise in knowledge of the subject matter. Validated questionnaires, interviews, and statistical tools were used to address the objectives. The results revealed that the respondents were found to have outstanding performance in Trigonometry, very satisfactory performance in Basic Math and Algebra, satisfactory performance in Geometry, and did not meet the expectation in Statistics. In addition, the teacher respondents have high confidence level in the use of general computer processes and telecommunications and average level of confidence in the use of other technologies in teaching and learning. In addition, a high confidence level in the use of general computer processes and telecommunications and an average level of confidence in the use of other technologies in teaching and learning were shown by the respondents. Given these findings, it is recommended that in-service mathematics teachers should have intensive trainings and seminars on Mathematics to improve their knowledge in the subject, and a special training on the use of Information and Communication Technology (ICT) to maximize its use in teaching and learning.

Complete hierarchy of linear systems for certifying quantum entanglement of subspaces

We introduce a hierarchy of linear systems for showing that a given subspace of pure quantum states is entangled (i.e., contains no product states). This hierarchy outperforms known methods already at the first level, and it is complete in the sense that every entangled subspace is shown to be so at some finite level of the hierarchy. It straightforwardly generalizes to the case of higher Schmidt rank, as well as the multipartite cases of completely and genuinely entangled subspaces. These hierarchies work extremely well in practice even in very large quantum systems, as they can be implemented via elementary linear algebra techniques rather than the semidefinite programming techniques that are required by previously known hierarchies.

DEVELOPMENT OF ELEMENTARY LINEAR ALGEBRA COURSE E-MODULES BASED ON FLIP BOOK MAKER WITH INTEGRATED ISLAMIC VALUES

The aims of this study were (1) to find out the process of developing an e-module for elementary linear algebra courses based on a flip book maker that integrates Islamic values. (2) to determine the characteristics of the e-module. (3) to determine the level of validity, practicality, and effectiveness of e-modules. The type of research used is Research and Development concerning the Plomp development model which consists of 5 stages. The product developed is an e-module course on elementary linear algebra based on a flip book maker. The test subjects in this study were second-semester students at UIN Alauddin Makassar, in the academic year 2021/2022. The instruments used were expert validation sheets, lecturer response questionnaires, observation sheets on the ability of lecturers to manage to learn, learning implementation observation sheets, student response questionnaires, student activity sheets, and learning achievement tests. Based on the trial results, it was found that (1) the product development process referred to as the Plomp development model was not fully implemented until the implementation stage due to time constraints, (2) the characteristics of the e-module are digitally designed with the flip book maker application which can be accessed using Android or Android. laptop, load learning videos, load examples of contextual questions integrated with Islamic values, and presentation of material using a scientific approach, (3) e-modules are categorized as valid with an average validity of 4.25, declared practical because all aspects are fully implemented, and declared effective because the level of completeness reached 96.3%.

Homogeneous approximation for minimal realizations of series of iterated integrals

In the paper, realizable series of iterated integrals with scalar coefficients are considered and an algebraic approach to the homogeneous approximation problem for nonlinear control systems with output is developed. In the first section we recall the concept of the homogeneous approximation of a nonlinear control system which is linear w.r.t.\ the control and the concept of the series of iterated integrals. In the second section the statement of the realizability problem is given, a criterion for realizability and a method for constructing a minimal realization of the series are recalled. Also we recall some ideas of the algebraic approach to the description of the homogeneous approximation: the free graded associative algebra, which is isomorphic to the algebra of iterated integrals, the free Lie algebra, the Poincar\'{e}-Birkhoff-Witt basis, the dual basis and its construction by use of the shuffle product, the definition of the core Lie subalgebra, which defines the homogeneous approximation of a control system. In the third section we show how to find the core Lie subalgebra of the systems that is a realization of the one-dimensional series of iterated integrals without finding the system itself. The result obtained is illustrated by the example, in which we demonstrate two methods for finding the core Lie subalgebra of the realizing system. In the last section it is shown that for any graded Lie subalgebra of finite codimension there exists a one-dimensional homogeneous series such that this Lie subalgebra is the core Lie subalgebra for its minimal realization. The proof is constructive: we give a method of finding such a series; we use the dual basis to the Poincar\'{e}-Birkhoff-Witt basis of the free associative algebra, which is built by the core Lie subalgebra, and the shuffle product in this algebra. As a consequence, we get a classification of all possible homogeneous approximations of systems that are realizations of one-dimensional series of iterated integrals.

Bases for upper cluster algebras and tropical points

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of their bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland’s representation-theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all known cluster algebras arising from representation theory or higher Teichmüller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds.

Neural network acceleration methods via selective activation

AbstractThe increase in neural network recognition accuracy is accompanied by a significant increase in the scales of networks and computations. To make deep learning frameworks widely used on mobile platforms, model acceleration has become extremely important in computer vision. In this study, a novel neural network acceleration method based on selective activation is proposed. First, as the algebraic basis for selective activation, mask general matrix multiplication is used to reduce matrix multiplication calculations. Second, to screen and remove activated neurons and reduce the number of calculations, we introduce an Activation Management Unit that includes two different strategies, Selective Activation with Primary Weights (SAPW) and Selective Activation with Primary Inputs (SAPI). SAPW greatly reduces the number of calculations of the fully connected layer and self‐attention and better guarantees detection accuracy. SAPI has the best performance on convolutional architectures, which can significantly reduce the amount of convolutional computation while maintaining the image classification accuracy. We present result of extensive experiments on computational and accuracy tradeoffs and show strong performance for CIFAR‐10 classification and Pascal VOC2012 detection. Compared with the dense method, the proposed selective activation method significantly reduces the number of neural network calculations with equal accuracy.

On p-standard table algebras of order p3, II

The classification is here completed for the p-standard table algebras of order p3 for any rational prime p. These include the adjacency algebras of p-schemes of order p3. The remaining case, where the thin radical of the distinguished basis of the table algebra has order p2, is resolved. The algebras in this case are explicitly determined up to isomorphism as wreath products, partial wreath products, or as members of a more complex family called hexagonal standard table algebras. These are parametrized by certain correspondences introduced in this article called hexagonal functions, that are defined in general from a subset of a group to an arbitrary set.

Algebraic flux correction finite element method with semi-implicit time stepping for solute transport in fractured porous media

This work is concerned with the numerical modeling of the Darcy flow and solute transport in fractured porous media for which the fractures are modeled as interfaces of codimension one. The hybrid-dimensional flow and transport problems are discretizaed by a lumped piece-wise linear finite element method, combined with the algebraic correction of the convective fluxes. The resulting transport discretization can be interpreted as a conservative finite volume scheme that satisfies the discrete maximum principle, while introducing a very limited amount of numerical diffusion. In the context of fractured porous media flow the CFL number may vary by several order of magnitude, which makes explicit time stepping unfeasible. To cope with this difficulty we propose an adaptive semi-implicit time stepping strategy that reduces to the low order linear implicit discretization in the high CFL regions that include, but may not be limited to the fracture network. The performance of the fully explicit and semi-implicit variants of the method are investigated through the numerical experiment.

NONRECURSIVE CANONICAL BASIS COMPUTATIONS FOR LOW RANK KASHIWARA CRYSTALS OF TYPE A

We give an improved algorithm for the action of the divided power of a Chevalley basis element of an affine Lie algebra of type A on canonical basis elements satisfying an easily checked uniformity condition and compare calculation times for our algorithm against the standard algorithm. For symmetric Kashiwara crystals of affine type A and rank e=2, and for the canonical basis elements that we call external, corresponding to weights on the outer skin of the Kashiwara crystal, we construct the canonical basis elements in a nonrecursive manner. In particular, for a symmetric crystal with Λ=aΛ0+aΛ1, we give formulae for the canonical basis elements for all the e-regular multipartitions with defects either k(a−k) or k(a−k)+2a, for 0≤k≤a.

An Energy-Efficient Domain-Specific Reconfigurable Array Processor With Heterogeneous PEs for Wearable Brain–Computer Interface SoCs

Recently, there is increasing demand for energy-efficient signal processing in wearable visual-stimuli-based brain-computer interface (V-BCI) devices. For the better accuracy and the reduced latency of the V-BCI system, the target identification (TI) algorithm that analyzes brain signals is being advanced, and the importance of an energy-efficient accelerating chip that processes various linear algebra operations constituting the TI algorithms is growing. In this paper, we propose a domain-specific reconfigurable array processor (RAP) with a dynamically reconfigurable and scalable array including 5-heterogeneous processing elements (PEs) for the energy-efficient acceleration of basic linear algebra subprograms (BLAS) and matrix decompositions. The system-on-chip (SoC), including the proposed RAP, was fabricated in 130-nm CMOS technology with an area of 16.87-mm2 and measured at 1.0 V 90 MHz. The RAP achieved an information transfer rate (ITR) of 139.9-bits/min and a TI accuracy of 95.4% on a fabricated chip through an optimized TI algorithm and scalable array processing. In addition, the RAP has <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$16.8\times $ </tex-math></inline-formula> higher TI energy efficiency than prior work and achieved an energy efficiency of 2144.2-bits/min/mW for information transfer processing rate with the proposed TI algorithm. The RAP supports a greater variety of linear algebra operations and data sizes with hardware reconfiguration than the prior accelerators.

Distillability Problem of 4×4 Monomial Matrices

It is a fundamental problem in quantum information whether a particular quantum state of a composite system is entangled. It has enormous potential in quantum error correction, quantum cryptography, and quantum teleportation applications. This problem can be transferred in the form of a mathematical conjecture called the distillation conjecture. In the first section of this paper, relevant physical and mathematical information is presented, including basic linear algebra knowledge, the statement, and concrete applications of multiple mathematical knowledge like conjugate, eigenvalue, and singular value. Then, we introduce the distillation conjecture in a mathematical version for a more precise mathematical analysis. In an effort to make more significant headway in proving the conjecture, we selected some theories and findings relating to the Kronecker product, Kronecker sum, eigenvalue, and singular value, then evaluated and grouped them. In addition, we provided multiple proofs of the conjecture under varying conditions and made numerous attempts and hypotheses regarding how to establish the conjecture.

AMCA Memorial Lecture Honoring Lewis T. Nielsen, PhD

The 2021 AMCA Memorial Lecture honors Lewis T. Nielsen, Ph.D., 1920–2014 on the centennial of his birth. Dr. Nielsen was born August 6, 1920, in Salt Lake City, UT. Other than the four years he served in the US Army during World War II, Nielsen lived his entire life in Salt Lake City. He was the eldest of four children, with two sisters and a brother. His father, Knute Lester Nielsen, was injured during World War I, and Nielsen credited his father's disability pension for supporting the family during the Great Depression and allowing “Lew” to pursue his education rather than having to work to support the family.Nielsen's lifelong love for the study of mosquito biology and systematics evolved serendipitously. After failing a junior high school algebra class, he was kept after school writing equations on a blackboard. At about this time a teacher by the name of Archibald took an interest in him and urged him to think seriously about his future and to continue his education. It was at this early age that Nielsen began developing an interest in natural history. While earning a nearly straight-A average in high school, he took botany and zoology, which further fueled his interest in natural history.Following high school graduation at the age of 17, he entered the University of Utah in the fall of 1937, which began a lifelong association with the “U.” While working his way through college, he was assigned a position in the Zoology Department pinning mosquitoes for Don Rees. That led to a summer job at the Salt Lake City Mosquito Abatement and ultimately his life's work with mosquitoes. His three degrees in biology included a B.S. (1941), M.A. (1947) “On the Biology of Aedes dorsalis,” and Ph.D. (1955) “The Taxonomy, Biology and Control of the Rocky Mountain Aedes Mosquito Species.” The intervals between degrees included a tour of duty in the US Army, 1942–1946, and the difficult task of teaching introductory biology classes as an instructor at the university while working on his doctorate. In 1956, Nielsen was promoted to Associate Professor and taught Entomology, Medical Entomology, Insect Morphology, Insect Sociality, Mosquito Systematics, and a general education course he developed called Insects and Man. In 1965, he became the Director of General Biology and was promoted to the rank of Professor in 1966. During his academic career, Nielsen had 29 students that received their Master of Science degrees and 10 students that received their doctorates. He published 82 scientific publications and was the lead author on 41 of those publications. At retirement in 1989, after 43 years teaching and research at the University of Utah, he received the honor of being named Professor Emeritus of Biology.Nielsen played a major role in local mosquito research and control. No one knew more about the biology and taxonomy of Utah mosquitoes. His favorite mosquitoes were the snowmelt mosquitoes of the higher mountainous elevations. He twice served as the President of the Utah Mosquito Abatement Association (UMAA) (1958 and 1990) and was awarded the UMAA Don M. Rees Award in 1988 and the Meritorious Service Award in 2005. For more than 40 years, he organized the association's annual spring training workshop. From the inception of the UMAA in 1948 until his death, he never missed an Annual Meeting of the UMAA.Nielsen had a very close relationship with the American Mosquito Control Association (AMCA). He served as the AMCA President (1977) and received the Medal of Honor Award (1988), was made an Honorary Member (1994), and was awarded the John Belkin Award (2000). He was the Editor for AMCA's Mosquito Systematics publication from 1979 to 1992, Coeditor 1992–1993, and Editor Emeritus 1993–1995. In 1997, Nielsen gave the Memorial Lecture entitled “In Honor of Stanley Carpenter.”Lew's love for the outdoors continued unabated into his nineties including hunting and fishing trips with life-long friends who shared his appreciation for the beauty of nature.

How Well Do High-Achieving Undergraduate Students Understand School Algebra?

The aim of this research is to investigate how well high-achieving students entering tertiary-level education in Ireland understand school algebra. As part of a larger project, a 31-item test was developed to assess first-year undergraduate students’ understanding of basic algebraic concepts. The test was administered online to students studying at least one mathematics module at tertiary level and received 327 responses. In this article, we study how the subset of high-achieving undergraduates in our sample performed on the test. The results demonstrated a very high level of understanding among students, as befits their level of study and prior achievement relative to the difficulty of the test. However, one subsection of the test stood out as being disproportionately difficult for these students. The section focused on valid solutions of equations and inequalities. The items in question are described in detail in this article as is the associated data. Our analysis shows that this topic is an area of concern even for high-achieving undergraduates and so deserves further attention. We conclude with a discussion of the implications of this research and details of the larger project.

Companions of fields of rational and real algebraic numbers

Companions of the field of rational numbers and a real-closed algebraic expansion of the field of rational numbers are studied. The description of existentially closed companions of a real-closed algebraic expansion of a field of rational numbers refers to the field of study of classical algebraic structures. The general theory of companions and existentially closed companions, built on the basis of Fraisse's classes in the works of A.T. Nurtazin, is included in the classical field of existentially closed theories in model theory. The basic concept of a companion: two models of the same signature are called companions if for any finite submodel of one of them, there is an isomorphic finite submodel in the other. This approach, applied to specific classical structures and their theories, provides new tools for the study of these objects. The study of the companion class of rational and algebraic real number fields reveals companion fields containing transcendental and possibly algebraic elements with special properties of polynomials defining these elements.

Existence and Uniqueness of Algebraic Closures

Summary This is the second part of a two-part article formalizing existence and uniqueness of algebraic closures, using the Mizar [2], [1] formalism. Our proof follows Artin’s classical one as presented by Lang in [3]. In the first part we proved that for a given field F there exists a field extension E such that every non-constant polynomial p ∈ F [X] has a root in E. Artin’s proof applies Kronecker’s construction to each polynomial p ∈ F [X]\F simultaneously. To do so we needed the polynomial ring F [X 1, X 2, ...] with infinitely many variables, one for each polynomal p ∈ F [X]\F. The desired field extension E then is F [X 1, X 2, …]\I, where I is a maximal ideal generated by all non-constant polynomials p ∈ F [X]. Note, that to show that I is maximal Zorn’s lemma has to be applied. In this second part this construction is iterated giving an infinite sequence of fields, whose union establishes a field extension A of F, in which every non-constant polynomial p ∈ A[X] has a root. The field of algebraic elements of A then is an algebraic closure of F. To prove uniqueness of algebraic closures, e.g. that two algebraic closures of F are isomorphic over F, the technique of extending monomorphisms is applied: a monomorphism F → A, where A is an algebraic closure of F can be extended to a monomorphism E → A, where E is any algebraic extension of F. In case that E is algebraically closed this monomorphism is an isomorphism. Note that the existence of the extended monomorphism again relies on Zorn’s lemma.

Arbitrarily large Morita Frobenius numbers

We construct blocks of finite groups with arbitrarily large Morita Frobenius numbers, an invariant which determines the size of the minimal field of definition of the associated basic algebra. This answers a question of Benson and Kessar. This also improves upon a result of the second author where arbitrarily large $\mathcal{O}$-Morita Frobenius numbers are constructed.