Basic Algebra Research Articles

Approximately invertible elements in non-unital normed algebras

We introduce the concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibility with concepts of topological divisors of zero and density of (modular) ideals. We exemplify approximate invertibility in the group algebra, Wiener algebras, and operator ideals. For Wiener algebras with approximate identities (in particular, for the Fourier image of the convolution algebra), the approximate invertibility of an algebra element is equivalent to the property that it does not vanish. We also study approximate invertibility and its deeper connection with the Gelfand and representation theory in non-unital abelian Banach algebras as well as abelian and non-abelian C*-algebras.

Structure of blocks with normal defect and abelian inertial quotient

Abstract Let k be an algebraically closed field of prime characteristic p. Let $kGe$ be a block of a group algebra of a finite group G, with normal defect group P and abelian $p'$ inertial quotient L. Then we show that $kGe$ is a matrix algebra over a quantised version of the group algebra of a semidirect product of P with a certain subgroup of L. To do this, we first examine the associated graded algebra, using a Jennings–Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the nonprincipal block in the case of a semidirect product of an extraspecial p-group P of exponent p and order $p^3$ with a quaternion group of order eight with the centre acting trivially. In the case of $p=3$ , we give explicit generators and relations for the basic algebra as a quantised version of $kP$ . As a second example, we give explicit generators and relations in the case of a group of shape $2^{1+4}:3^{1+2}$ in characteristic two.

The Bell Inequality, Inviolable by Data Used Consistently with Its Derivation, Is Satisfied by Quantum Correlations Whose Probabilities Satisfy the Wigner Inequality

It is not generally known that the inequality that Bell derived using three random variables must be identically satisfied by any three corresponding data sets of ±1’s that are writable on paper. This surprising fact is not immediately obvious from Bell’s inequality derivation based on causal random variables, but follows immediately if the same mathematical operations are applied to finite data sets. For laboratory data, the inequality is identically satisfied as a fact of pure algebra, and its satisfaction is independent of whether the processes generating the data are local, non-local, deterministic, random, or nonsensical. It follows that if predicted correlations violate the inequality, they represent no three cross-correlated data sets that can exist, or can be generated from valid probability models. Reported data that violate the inequality consist of probabilistically independent data-pairs and are thus inconsistent with inequality derivation. In the case of random variables as Bell assumed, the correlations in the inequality may be expressed in terms of the probabilities that give rise to them. A new inequality is then produced: The Wigner inequality, that must be satisfied by quantum mechanical probabilities in the case of Bell experiments. If that were not the case, predicted quantum probabilities and correlations would be inconsistent with basic algebra.

A Computer Program For The System Of Weak Fuzzy Complex Numbers And Their Arithmetic Operations Using Python

Weak fuzzy complex number system is considered as a generalization of complex numbers by using an algebraic element with fuzzy property. It is believed that this new numerical system will play many important roles in various computer science disciplines, proceeding from this importance this paper aims to introduce for the first time the algebraic operations defined on weak fuzzy complex numbers using Python and Jupyter Notebook. Hence, by building a new class and its functions in Python for this new set, we can now use weak fuzzy complex numbers and the main arithmetic operations on them.

Algebra of shares, complete bipartite graphs and $\mathfrak{sl}_2$ weight system

A function of chord diagrams is called a weight system if it satisfies the so-called four-term relations. Vassiliev's theory describes finite-order knot invariants in terms of weight systems. In particular, there is a weight system corresponding to the coloured Jones polynomial. This weight system is described in terms of the Lie algebra $\mathfrak{sl}_2$. According to the Chmutov-Lando theorem, the value of this weight system depends only on the intersection graph of the chord diagram. Therefore, it is possible to discuss the values of this weight system at intersection graphs. We obtain formulae for the generating functions of the values of the $\mathfrak{sl}_2$ weight system at complete bipartite graphs. Using these formulae we prove that Lando's conjecture about the degree of the polynomial that is the value of this weight system at the projection of a graph onto the subspace of primitive elements in the Hopf algebra of graphs is true for complete bipartite graphs and for a certain wider class of graphs. We introduce the algebra of shares and the $\mathfrak{sl}_2$ weight system on shares. These are the main tools for our proof. Bibliography: 14 titles.

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.