273 results
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2. On Some Distance Spectral Characteristics of Trees.
- Author
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Hayat, Sakander, Khan, Asad, and Alenazi, Mohammed J. F.
- Subjects
DATA transmission systems ,LINEAR algebra ,GRAPH theory ,SPECTRAL theory ,GRAPH connectivity - Abstract
Graham and Pollack in 1971 presented applications of eigenvalues of the distance matrix in addressing problems in data communication systems. Spectral graph theory employs tools from linear algebra to retrieve the properties of a graph from the spectrum of graph-theoretic matrices. The study of graphs with "few eigenvalues" is a contemporary problem in spectral graph theory. This paper studies graphs with few distinct distance eigenvalues. After mentioning the classification of graphs with one and two distinct distance eigenvalues, we mainly focus on graphs with three distinct distance eigenvalues. Characterizing graphs with three distinct distance eigenvalues is "highly" non-trivial. In this paper, we classify all trees whose distance matrix has precisely three distinct eigenvalues. Our proof is different from earlier existing proof of the result as our proof is extendable to other similar families such as unicyclic and bicyclic graphs. The main tools which we employ include interlacing and equitable partitions. We also list all the connected graphs on ν ≤ 6 vertices and compute their distance spectra. Importantly, all these graphs on ν ≤ 6 vertices are determined from their distance spectra. We deliver a distance cospectral pair of order 7, thus making it a distance cospectral pair of the smallest order. This paper is concluded with some future directions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Tools and algorithms for the construction and analysis of systems: a special issue for TACAS 2019.
- Author
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Vojnar, Tomáš and Zhang, Lijun
- Subjects
ALGORITHMS ,SOFTWARE development tools ,TECHNOLOGY transfer ,LINEAR algebra ,CONFERENCES & conventions - Abstract
Automated techniques and tools for the construction and analysis of systems are inevitable to manage the complexity of the current systems. Such techniques and tools are the subject of interest of the International Conference on Tools and Algorithms for the Construction and Analysis of Systems—TACAS. This special issue of Software Tools for Technology Transfer presents extended versions of five selected papers from the 25th edition of TACAS that took place in 2019. All of the papers included into this special issue aim at various aspects of automated design and formal verification and hence contribute to development of more reliable computer systems. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
4. COMPLETE CONSISTENCY AND ASYMPTOTIC NORMALITY FOR THE WEIGHTED ESTIMATOR IN A NONPARAMETRIC REGRESSION MODEL UNDER DEPENDENT ERRORS.
- Author
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SAMURA, SALLIEU KABAY, SHIJIE WANG, LING CHEN, XUEJUN WANG, and FEI ZHANG
- Subjects
POLYNOMIALS ,NORMAL operators ,LINEAR algebra ,MATHEMATICAL formulas ,MATHEMATICAL inequalities - Abstract
In this paper, we investigate the effect of dependent errors in the fixed design nonparametric regression models. Under some mild conditions, we obtain the complete consistency and asymptotic normality for the weighted estimator in the fixed design nonparametric regression models. In addition, a simulation study is undertaken to investigate finite sample behavior of the estimator. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. WIGNER-YANASE-DYSON FUNCTION AND LOGARITHMIC MEAN.
- Author
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SHIGERU FURUICHI
- Subjects
MATHEMATICAL inequalities ,NORMAL operators ,LINEAR algebra ,POLYNOMIALS ,MATHEMATICAL formulas - Abstract
The ordering betweenWigner-Yanase-Dyson function and logarithmic mean is known. Also bounds for logarithmic mean are known. In this paper, we give two reverse inequalities for Wigner-Yanase-Dyson function and logarithmic mean. We also compare the obtained results with the known bounds of the logarithmic mean. Finally, we give operator inequalities based on the obtained results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
6. An algebraic approach to circulant column parity mixers
- Author
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Subroto, Robert Christian
- Published
- 2024
- Full Text
- View/download PDF
7. A generalized adaptive Monte Carlo algorithm based on a two-step iterative method for linear systems and its application to option pricing.
- Author
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Aalaei, Mahboubeh
- Subjects
MONTE Carlo method ,ITERATIVE methods (Mathematics) ,LINEAR systems ,OPTIONS (Finance) ,LINEAR algebra - Abstract
In this paper, we present a generalized adaptive Monte Carlo algorithm using the Diagonal and Off-Diagonal Splitting (DOS) iteration method to solve a system of linear algebraic equations (SLAE). The DOS method is a generalized iterative method with some known iterative methods such as Jacobi, Gauss-Seidel, and Successive Overrelaxation methods as its special cases. Monte Carlo algorithms usually use the Jacobi method to solve SLAE. In this paper, the DOS method is used instead of the Jacobi method which transforms the Monte Carlo algorithm into the generalized Monte Carlo algorithm. we establish theoretical results to justify the convergence of the algorithm. Finally, numerical experiments are discussed to illustrate the accuracy and efficiency of the theoretical results. Furthermore, the generalized algorithm is implemented to price options using the finite difference method. We compare the generalized algorithm with standard numerical and stochastic algorithms to show its efficiency. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
8. Hands-On Fundamentals of 1D Convolutional Neural Networks—A Tutorial for Beginner Users.
- Author
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Cacciari, Ilaria and Ranfagni, Anedio
- Subjects
CONVOLUTIONAL neural networks ,GENERATIVE adversarial networks ,RECURRENT neural networks ,DEEP learning ,TRANSFORMER models - Abstract
In recent years, deep learning (DL) has garnered significant attention for its successful applications across various domains in solving complex problems. This interest has spurred the development of numerous neural network architectures, including Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), Generative Adversarial Networks (GANs), and the more recently introduced Transformers. The choice of architecture depends on the data characteristics and the specific task at hand. In the 1D domain, one-dimensional CNNs (1D CNNs) are widely used, particularly for tasks involving the classification and recognition of 1D signals. While there are many applications of 1D CNNs in the literature, the technical details of their training are often not thoroughly explained, posing challenges for those developing new libraries in languages other than those supported by available open-source solutions. This paper offers a comprehensive, step-by-step tutorial on deriving feedforward and backpropagation equations for 1D CNNs, applicable to both regression and classification tasks. By linking neural networks with linear algebra, statistics, and optimization, this tutorial aims to clarify concepts related to 1D CNNs, making it a valuable resource for those interested in developing new libraries beyond existing ones. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
9. Closure Operations on Intuitionistic Linear Algebras.
- Author
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Tenkeu Jeufack, Y.L., Alomo Temgoua, E.R., and Heubo-Kwegna, O.A.
- Subjects
LINEAR algebra ,LATTICE theory ,HEYTING algebras ,QUOTIENT rings - Abstract
In this paper, we introduce the notions of radical filters and extended filters of Intuitionistic Linear algebras (IL-algebras for short) and give some of their properties. The notion of closure operation on an IL-algebra is also introduced as well as the study of some of their main properties. The radical of filters and extended filters are examples of closure operations among several others provided. The class of stable closure operations on an IL-algebra L is used to study the unifying properties of some subclasses of the lattice of filters of L. In particular, we obtain that for a stable closure operation c on an IL-algebra, the collection of c-closed elements of its lattice of filters forms a complete Heyting algebra. Hyperarchimedean IL-algebras are also characterized using closure operations. It is shown that the image of a semi-prime closure operation on an IL-algebra is isomorphic to a quotient IL-algebra. Some properties of the quotients induced by closure operations on an IL-algebra are explored. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
10. Complete Set of Bounds for the Technical Moduli in 3D Anisotropic Elasticity.
- Author
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Vannucci, Paolo
- Subjects
LINEAR algebra ,PROBLEM solving ,ELASTICITY ,ENGINEERING - Abstract
The paper addresses the problem of finding the necessary and sufficient conditions to be satisfied by the engineering moduli of an anisotropic material for the elastic energy to be positive for each state of strain or stress. The problem is solved first in the most general case of a triclinic material and then each possible case of elastic syngony is treated as a special case. The method of analysis is based upon a rather forgotten theorem of linear algebra and, in the most general case, the calculations, too much involved, are carried out using a formal computation code. New, specific bounds, concerning some of the technical constants, are also found. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
11. Abridged spectral matrix inversion: parametric fitting of X-ray fluorescence spectra following integrative data reduction
- Author
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Simon J. George, Ben Huntsman, Ingrid J. Pickering, Graham N. George, Cheyenne D. Kiani, Olena Ponomarenko, Monica Y. Weng, and Andrew M. Crawford
- Subjects
Nuclear and High Energy Physics ,Astrophysics::High Energy Astrophysical Phenomena ,X-ray fluorescence ,01 natural sciences ,Fluorescence ,Spectral line ,law.invention ,010309 optics ,Matrix (mathematics) ,law ,0103 physical sciences ,010306 general physics ,Instrumentation ,Parametric statistics ,Physics ,Radiation ,X-Rays ,Research Papers ,Synchrotron ,Computational physics ,Radiography ,Linear algebra ,Algorithms ,Synchrotrons ,Energy (signal processing) ,Data reduction - Abstract
Recent improvements in both X-ray detectors and readout speeds have led to a substantial increase in the volume of X-ray fluorescence data being produced at synchrotron facilities. This in turn results in increased challenges associated with processing and fitting such data, both temporally and computationally. Herein an abridging approach is described that both reduces and partially integrates X-ray fluorescence (XRF) data sets to obtain a fivefold total improvement in processing time with negligible decrease in quality of fitting. The approach is demonstrated using linear least-squares matrix inversion on XRF data with strongly overlapping fluorescent peaks. This approach is applicable to any type of linear algebra based fitting algorithm to fit spectra containing overlapping signals wherein the spectra also contain unimportant (non-characteristic) regions which add little (or no) weight to fitted values, e.g. energy regions in XRF spectra that contain little or no peak information.
- Published
- 2021
12. Conforming finite element function spaces in four dimensions, part II: The pentatope and tetrahedral prism.
- Author
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Williams, David M. and Nigam, Nilima
- Subjects
- *
FUNCTION spaces , *FINITE element method , *PRISMS , *LINEAR algebra , *DEGREES of freedom - Abstract
In this paper, we present explicit expressions for conforming finite element function spaces, basis functions, and degrees of freedom on the pentatope (the 4-simplex) and tetrahedral prism elements. More generally, our objective is to construct high-order finite element function spaces that maintain conformity with infinite-dimensional spaces of a carefully chosen de Rham complex in four dimensions. This paper is a natural extension of the companion paper entitled "Conforming finite element function spaces in four dimensions, part I: Foundational principles and the tesseract" by Nigam and Williams (2024). In contrast to Part I, in this paper we focus on two of the most popular elements which do not possess a full tensor-product structure in all four coordinate directions. We note that these elements appear frequently in existing space-time finite element methods. In order to build our finite element spaces, we utilize powerful techniques from the recently developed 'Finite Element Exterior Calculus'. Subsequently, we translate our results into the well-known language of linear algebra (vectors and matrices) in order to facilitate implementation by scientists and engineers. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
13. Solution of the Elliptic Interface Problem by a Hybrid Mixed Finite Element Method.
- Author
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Wang, Yuhan, Wang, Peiyao, Zhang, Rongpei, and Liu, Jia
- Subjects
FINITE element method ,LAGRANGE multiplier ,LAGRANGE equations ,TRIANGULATION ,ELLIPTIC equations ,LINEAR algebra - Abstract
This paper addresses the elliptic interface problem involving jump conditions across the interface. We propose a hybrid mixed finite element method on the triangulation where the interfaces are aligned with the mesh. The second-order elliptic equation is initially decomposed into two equations by introducing a gradient term. Subsequently, weak formulations are applied to these equations. Scheme continuity is enforced using the Lagrange multiplier technique. Finally, we derive an explicit formula for the entries of the matrix equation representing Lagrange multiplier unknowns resulting from hybridization. The method yields approximations of all variables, including the solution and gradient, with optimal order. Furthermore, the matrix representing the final linear algebra systems is not only symmetric but also positive definite. Numerical examples convincingly demonstrate the effectiveness of the hybrid mixed finite element method in addressing elliptic interface problems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
14. CHARACTERIZATIONS OF SLICE BESOV--TYPE AND SLICE TRIEBEL--LIZORKIN--TYPE SPACES AND APPLICATIONS.
- Author
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YUAN LU and JIANG ZHOU
- Subjects
NORMAL operators ,POLYNOMIALS ,LINEAR algebra ,MATHEMATICAL formulas ,MATHEMATICAL inequalities - Abstract
Let ... and t, r, p ∈ (0, ∞) . In this paper, we introduce the slice Besov-type space ... and the slice Triebel-Lizorkin-type space ..., and establish their φ-transform characterizations in the sense of Frazier and Jawerth. The embedding properties, characterizations via the Peetre maximal function, the Lusin area function, smooth atomic and molecular decompositions of these spaces are also obtained. As applications, we obtain the boundedness on these spaces of Fourier multipliers with symbols satisfying some generalized Hörmander condition. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
15. IMPROVEMENTS OF A--NUMERICAL RADIUS FOR SEMI-HILBERTIAN SPACE OPERATORS.
- Author
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HONGWEI QIAO, GUOJUN HAI, and ALATANCANG CHEN
- Subjects
NORMAL operators ,LINEAR algebra ,MATHEMATICAL inequalities ,MATHEMATICAL formulas ,INTEGRALS - Abstract
Let A be a bounded positive operator on a complex Hilbert space ... . The semi-product ..., induces a semi-norm ... on H. Let ω
A (T) amd ... denote the A-numerical radius and the A-operator semi-norm of an operator T in semi-Hilbertian space ..., respectively. In this paper, some new bounds for the A-numerical radius of operators in semi-inner product space induced by A are derived. In particular, for ... and α ≥ 0, we prove that ... and ... . It is worth noting that our results improve the existing A-numerical radius inequalities. Further, we also give a refinement inequality of A-operator semi-norm triangle inequality. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
16. FRACTIONAL INTEGRAL OPERATORS ON GRAND MORREY SPACES AND GRAND HARDY-MORREY SPACES.
- Author
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KWOK-PUN HO
- Subjects
INTEGRALS ,EUCLIDEAN geometry ,MATHEMATICAL formulas ,MATHEMATICAL inequalities ,LINEAR algebra ,NORMAL operators - Abstract
This paper establishes the mapping properties of the fractional integral operators on the grand Morrey spaces and the grand Hardy-Morrey spaces defined on the Euclidean spaces. We obtain our results by refining the Rubio de Francia extrapolation method as the existing extrapolation method cannot be directly applied to the grand Morrey spaces. This method also yields the mapping properties of nonlinear operators. In particular, we establish the Sobolev embedding, the Poincaré inequality and the mapping properties of the fractional geometric maximal functions on the grand Morrey spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
17. SHARPENING OF INEQUALITIES CONCERNING POLYNOMIALS.
- Author
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DEVI, MAISNAM TRIVENI, CHANAM, BARCHAND, and SINGH, THANGJAM BIRKRAMJIT
- Subjects
MATHEMATICAL formulas ,MATHEMATICAL inequalities ,LINEAR algebra ,NORMAL operators ,INTEGRALS - Abstract
Let ... be a polynomial of degree n having all its zeros in ..., then Aziz [Proc. Am. Math. Soc., 89, (1983) 259--266] proved ... . In this paper, we prove a polar derivative extension which sharpens the above inequality. As a consequence, we also derive a result on Bernstein type inequality for the class of polynomials having all its zeros in ... . [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
18. SPLITTING INEQUALITIES FOR DIFFERENCES OF EXPONENTIALS.
- Author
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MARCHENKO, VITALII
- Subjects
MATHEMATICAL inequalities ,MATHEMATICAL formulas ,LINEAR algebra ,NORMAL operators ,PROBABILITY theory - Abstract
The paper is focused on two-sided splitting inequalities for differences of complex exponentials ... for large ... is real unbounded sequence clustering with appropriate speed. Moreover, it is shown that if ... is a Riesz basis of a Hilbert space H, then for any k ≥ 1 the system ... is complete, minimal but not uniformly minimal in H. Also some properties of systems of functions of real argument t, ... where ..., are discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
19. LOWER BOUNDS FOR THE SMALLEST SINGULAR VALUE VIA PERMUTATION MATRICES.
- Author
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CHAOQIAN LI, XUELIN ZHOU, and HEHUI WANG
- Subjects
PERMUTATION groups ,NORMAL operators ,MATHEMATICAL inequalities ,LINEAR algebra ,POLYNOMIALS ,MATHEMATICAL formulas - Abstract
We in this paper improve the well-known C. R. Johnson's lower bound for the smallest singular value via permutation matrices. A direct algorithm is also given to compute the new lower bound. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. REVERSE OSTROWSKI'S TYPE WEIGHTED INEQUALITIES FOR CONVEX FUNCTIONS ON LINEAR SPACES WITH APPLICATIONS.
- Author
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DRAGOMIR, SILVESTRU SEVER, JLELI, MOHAMED, and SAMET, BESSEM
- Subjects
CONVEX functions ,VECTOR spaces ,NORMAL operators ,MATHEMATICAL inequalities ,LINEAR algebra ,POLYNOMIALS ,MATHEMATICAL formulas - Abstract
In this paper we provide several upper and lower bounds for the Ostrowski difference ... where f : C → R is a convex function, C is a convex subset of a vector space X and w is integrable and nonnegative a.e. on [0,1] . A perturbed version under some natural assumptions on the weight function w is also considered. These results are then employed to obtain several weighted integral inequalities for norms and semi-inner products. The particular case of inner product spaces is analyzed and refinements of the weighted integral midpoint inequality for norms are provided. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
21. SOME REFINEMENTS OF YOUNG TYPE INEQUALITIES.
- Author
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CHANGSEN YANG and GEGE ZHANG
- Subjects
MATHEMATICAL inequalities ,NORMAL operators ,LINEAR algebra ,POLYNOMIALS ,HILBERT space - Abstract
In this paper, we give some new improvements and reverse improvements of Young type inequalities. The conclusion proved by Yang and Wang [J. Math. Inequal., 17 (2023), 205-217] involved the monotonicity of ..., where ... and ... . This article demonstrates the monotonicity of ..., where ... and ... . And this implies a main conclusion as follows: ..., where ... and ... . Furthermore, we can get some related results about operator, Hilbert-Schmidt norm, trace norm by these scalars results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. APPROXIMATION BY PERTURBED BASKAKOV--TYPE OPERATORS.
- Author
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ACU, ANA MARIA, ÇETIN, NURSEL, and TACHEV, GANCHO
- Subjects
NORMAL operators ,MATHEMATICAL inequalities ,LINEAR algebra ,POLYNOMIALS ,MATHEMATICAL formulas - Abstract
In this paper, we introduce a new Baskakov-type operator. Firstly, we obtain the rate of convergence by using modulus of continuity and then Voronovskaja type asymptotic formula for these operators. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. FRAME INEQUALITIES IN HILBERT SPACES: TWO--SIDED INEQUALITIES WITH NEW STRUCTURES.
- Author
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ZHONG-QI XIANG, CHUN-XIA LIN, and XIANG-CHUN XIAO
- Subjects
HILBERT space ,MATHEMATICAL inequalities ,POLYNOMIALS ,LINEAR algebra ,DISCRETE mathematics - Abstract
This paper is devoted to establishing frame inequalities in Hilbert spaces. By using operator theory methods, several two-sided inequalities for frames are presented, which, comparing to previous inequalities on frames and generalized frames, admit new structures. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. STEVIĆ--SHARMA TYPE OPERATORS FROM H∞ INTO THE BLOCH SPACE.
- Author
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QINGHUA HU and XIAOJING ZHOU
- Subjects
NORMAL operators ,MATHEMATICAL inequalities ,LINEAR algebra ,POLYNOMIALS ,DISCRETE mathematics - Abstract
In this paper, we give some characterizations for the boundedness and compactness of some Stević-Sharma type operators called the polynomial differentiation composition operators from H
∞ into the Bloch space on the unit disk. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
25. SOME WEIGHTED DYNAMIC INEQUALITIES OF HARDY TYPE WITH KERNELS ON TIME SCALES NABLA CALCULUS.
- Author
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AWWAD, ESSAM and SAIED, A. I.
- Subjects
CALCULUS ,POLYNOMIALS ,MATHEMATICAL inequalities ,DISCRETE mathematics ,LINEAR algebra - Abstract
In this paper, we present some properties of the time scale nabla calculus and how to apply it for proving the dynamic inequalities. Also, we prove some weighted dynamic inequalities of Hardy type with kernels on time scales nabla calculus and also, we study the characterizations of the weights for these inequalities in different spaces and for the exponent p > 1. The Holder inequality, Jensen inequality, and Minkowski inequality are used to prove our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. GENERALIZATIONS OF HARDY--TYPE INEQUALITIES BY THE HERMITE INTERPOLATING POLYNOMIAL.
- Author
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HIMMELREICH, KRISTINA KRULIĆ, PEČARIĆ, JOSIP, POKAZ, DORA, and PRALJAK, MARJAN
- Subjects
POLYNOMIALS ,MATHEMATICAL inequalities ,DISCRETE mathematics ,LINEAR algebra ,STATISTICAL correlation - Abstract
In this paper we obtain generalizations of Hardy-type inequalities for convex functions of the higher order by applying Hermite interpolating polynomials. The results for particular cases: Lagrange, (m,n-m) and two-point Taylor interpolating polynomials are also considered. Finally, we derive the Grüss and Ostrowski type inequalities related to these generalizations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. JENSEN--MARSHALL--KY FAN--TYPE INEQUALITIES AND THEIR APPLICATIONS IN BUSINESS PROFIT MANAGEMENT MODEL.
- Author
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RU LIU, JIAJIN WEN, TIANYONG HAN, and JUN YUAN
- Subjects
LINEAR algebra ,DISCRETE mathematics ,MATHEMATICAL inequalities ,COST functions ,STATISTICAL correlation - Abstract
Abstract. This paper will introduce the theory of ϕ-Jensen coefficient. By means of the functional analysis, linear algebra, discrete mathematics and inequality theories with proper hypotheses, the Jensen-type inequality, Marshall-type inequality and the Ky Fan-type inequality are obtained as follows: ... respectively, as well as we also displayed the applications of our main results in business profit management model, and some conditions such that ... hold are obtained, where p is the profit function and e is the cost function. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. Hopf algebra structures on generalized quaternion algebras.
- Author
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Chen, Quanguo and Deng, Yong
- Subjects
LINEAR algebra ,HOPF algebras ,QUATERNIONS ,MATHEMATICAL models ,MATHEMATICAL formulas - Abstract
In this paper, we use elementary linear algebra methods to explore possible Hopf algebra structures within the generalized quaternion algebra. The sufficient and necessary conditions that make the generalized quaternion algebra a Hopf algebra are given. It is proven that not all of the generalized quaternion algebras have Hopf algebraic structures. When the generalized quaternion algebras have Hopf algebraic structures, we describe all the Hopf algebra structures. Finally, we shall prove that all the Hopf algebra structures on the generalized quaternion algebras are isomorphic to Sweedler Hopf algebra, which is consistent with the classification of 4-dimensional Hopf algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. An efficient polynomial-based verifiable computation scheme on multi-source outsourced data.
- Author
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Zhang, Yiran, Geng, Huizheng, Su, Li, He, Shen, and Lu, Li
- Subjects
INFORMATION technology security ,CLOUD computing ,DATA security failures ,LINEAR algebra ,TRUST ,NUMERICAL analysis - Abstract
With the development of cloud computing, users are more inclined to outsource complex computing tasks to cloud servers with strong computing capacity, and the cloud returns the final calculation results. However, the cloud is not completely trustworthy, which may leak the data of user and even return incorrect calculations on purpose. Therefore, it is important to verify the results of computing tasks without revealing the privacy of the users. Among all the computing tasks, the polynomial calculation is widely used in information security, linear algebra, signal processing and other fields. Most existing polynomial-based verifiable computation schemes require that the input of the polynomial function must come from a single data source, which means that the data must be signed by a single user. However, the input of the polynomial may come from multiple users in the practical application. In order to solve this problem, the researchers have proposed some schemes for multi-source outsourced data, but these schemes have the common problem of low efficiency. To improve the efficiency, this paper proposes an efficient polynomial-based verifiable computation scheme on multi-source outsourced data. We optimize the polynomials using Horner's method to increase the speed of verification, in which the addition gate and the multiplication gate can be interleaved to represent the polynomial function. In order to adapt to this structure, we design the corresponding homomorphic verification tag, so that the input of the polynomial can come from multiple data sources. We prove the correctness and rationality of the scheme, and carry out numerical analysis and evaluation research to verify the efficiency of the scheme. The experimental indicate that data contributors can sign 1000 new data in merely 2 s, while the verification of a delegated polynomial function with a power of 100 requires only 18 ms. These results confirm that the proposed scheme is better than the existing scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
30. -Dimensional Generalizations of a Thébault Conjecture.
- Author
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Tran, Q. H. and Herrera, B.
- Subjects
- *
LOGICAL prediction , *LINEAR algebra , *GENERALIZATION - Abstract
This paper presents some generalizations of a Thébault conjecture, provides an analog of the Thébault conjecture for the -simplex, and also solves a conjecture in a 2022 paper by the authors by using linear algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. On Matrices with Only One Non-SDD Row.
- Author
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Doroslovački, Ksenija and Cvetković, Dragana
- Subjects
MATRIX inversion ,LINEAR algebra ,MATRICES (Mathematics) ,MATRIX norms ,DYNAMICAL systems ,UNIVALENT functions ,LOCALIZATION (Mathematics) - Abstract
The class of H-matrices, also known as Generalized Diagonally Dominant (GDD) matrices, plays an important role in many areas of applied linear algebra, as well as in a wide range of applications, such as in dynamical analysis of complex networks that arise in ecology, epidemiology, infectology, neurology, engineering, economy, opinion dynamics, and many other fields. To conclude that the particular dynamical system is (locally) stable, it is sufficient to prove that the corresponding (Jacobian) matrix is an H-matrix with negative diagonal entries. In practice, however, it is very difficult to determine whether a matrix is a non-singular H-matrix or not, so it is valuable to investigate subclasses of H-matrices which are defined by relatively simple and practical criteria. Many subclasses of H-matrices have recently been discussed in detail demonstrating the many benefits they can provide, though one particular subclass has not been fully exploited until now. The aim of this paper is to attract attention to this class and discuss its relation with other more investigated classes, while showing its main advantage, based on its simplicity and elegance. This new approach, which we are presenting in this paper, will be compared with the existing ones, in three possible areas of applications, spectrum localization; maximum norm estimation of the inverse matrix in the point, as well as the block case; and error estimation for LCP problems. The main conclusion is that the importance of our approach grows with the matrix dimension. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
32. A survey on machine learning in array databases.
- Author
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Villarroya, Sebastián and Baumann, Peter
- Subjects
MACHINE learning ,DATABASES ,DATABASE design ,LINEAR algebra ,SQL - Abstract
This paper provides an in-depth survey on the integration of machine learning and array databases. First,machine learning support in modern database management systems is introduced. From straightforward implementations of linear algebra operations in SQL to machine learning capabilities of specialized database managers designed to process specific types of data, a number of different approaches are overviewed. Then, the paper covers the database features already implemented in current machine learning systems. Features such as rewriting, compression, and caching allow users to implement more efficient machine learning applications. The underlying linear algebra computations in some of the most used machine learning algorithms are studied in order to determine which linear algebra operations should be efficiently implemented by array databases. An exhaustive overview of array data and relevant array database managers is also provided. Those database features that have been proven of special importance for efficient execution of machine learning algorithms are analyzed in detail for each relevant array database management system. Finally, current state of array databases capabilities for machine learning implementation is shown through two example implementations in Rasdaman and SciDB. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
33. An Implementation of Image Secret Sharing Scheme Based on Matrix Operations.
- Author
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Ren, Zihan, Li, Peng, and Wang, Xin
- Subjects
MATRIX multiplications ,MATRIX inversion ,SHARING ,CRYPTOGRAPHY ,LINEAR algebra ,MATRICES (Mathematics) - Abstract
The image secret sharing scheme shares a secret image as multiple shadows. The secret image can be recovered from shadow images that meet a threshold number. However, traditional image secret sharing schemes generally reuse the Lagrange's interpolation in the recovery stage to obtain the polynomial in the sharing stage. Since the coefficients of the polynomial are the pixel values of the secret image, it is able to recover the secret image. This paper presents an implementation of the image secret sharing scheme based on matrix operations. Different from the traditional image secret sharing scheme, this paper does not use the method of Lagrange's interpolation in the recovery stage, but first identifies the participants as elements to generate a matrix and calculates its inverse matrix. By repeating the matrix multiplication, the polynomial coefficients of the sharing stage are quickly derived, and then the secret image is recovered. By theoretical analysis and the experimental results, the implementation of secret image sharing based on matrix operation is higher than Lagrange's interpolation in terms of efficiency. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
34. Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its Extension.
- Author
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Palanikumar, M., Al-Shanqiti, Omaima, Jana, Chiranjibe, and Pal, Madhumangal
- Subjects
NONCOMMUTATIVE rings ,PRIME ideals ,PROGRAMMING languages ,LINEAR algebra - Abstract
In computer programming languages, partial additive semantics are used. Since partial functions under disjoint-domain sums and functional composition do not constitute a field, linear algebra cannot be applied. A partial ring can be viewed as an algebraic structure that can process natural partial orderings, infinite partial additions, and binary multiplications. In this paper, we introduce the notions of a one-prime partial bi-ideal, a two-prime partial bi-ideal, and a three-prime partial bi-ideal, as well as their extensions to partial rings, in addition to some characteristics of various prime partial bi-ideals. In this paper, we demonstrate that two-prime partial bi-ideal is a generalization of a one-prime partial bi-ideal, and three-prime partial bi-ideal is a generalization of a two-prime partial bi-ideal and a one-prime partial bi-ideal. A discussion of the m p b 1 , (m p b 2 , m p b 3) systems is presented. In general, the m p b 2 system is a generalization of the m p b 1 system, while the m p b 3 system is a generalization of both m p b 2 and m p b 1 systems. If Φ is a prime bi-ideal of ℧, then Φ is a one-prime partial bi-ideal (two-prime partial bi-ideal, three-prime partial bi-ideal) if and only if ℧ \ Φ is a m p b 1 system ( m p b 2 system, m p b 3 system) of ℧. If Θ is a prime bi-ideal in the complete partial ring ℧ and Δ is an m p b 3 system of ℧ with Θ ∩ Δ = ϕ , then there exists a three-prime partial bi-ideal Φ of ℧, such that Θ ⊆ Φ with Φ ∩ Δ = ϕ . These are necessary and sufficient conditions for partial bi-ideal Θ to be a three-prime partial bi-ideal of ℧. It is shown that partial bi-ideal Θ is a three-prime partial bi-ideal of ℧ if and only if H Θ is a prime partial ideal of ℧. If Θ is a one-prime partial bi-ideal (two-prime partial bi-ideal) in ℧, then H Θ is a prime partial ideal of ℧. It is guaranteed that a three-prime partial bi-ideal Φ with a prime bi-ideal Θ does not meet the m p b 3 system. In order to strengthen our results, examples are provided. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
35. Mod2VQLS: A Variational Quantum Algorithm for Solving Systems of Linear Equations Modulo 2.
- Author
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Aboumrad, Willie and Widdows, Dominic
- Subjects
LINEAR systems ,COST functions ,QUANTUM computers ,MATRIX multiplications ,ALGORITHMS ,LINEAR algebra - Abstract
This paper presents a system for solving binary-valued linear equations using quantum computers. The system is called Mod2VQLS, which stands for Modulo 2 Variational Quantum Linear Solver. As far as we know, this is the first such proposal. The design is a classical–quantum hybrid. The quantum components are a new circuit design for implementing matrix multiplication modulo 2, and a variational circuit to be optimized. The classical components are the optimizer, which measures the cost function and updates the quantum parameters for each iteration, and the controller that runs the quantum job and classical optimizer iterations. We propose two alternative ansatze or templates for the variational circuit and present results showing that the rotation ansatz designed specifically for this problem provides the most direct path to a valid solution. Numerical experiments in low dimensions indicate that Mod2VQLS, using the custom rotations ansatz, is on par with the block Wiedemann algorithm, which is the best-known to date solution for this problem. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. A hybrid model for data visualization using linear algebra methods and machine learning algorithm.
- Author
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Ali, Mohsin, Choudhary, Jitendra, and Kasbe, Tanmay
- Subjects
LINEAR algebra ,EIGENVECTORS ,DATA visualization ,RECEIVER operating characteristic curves ,PRINCIPAL components analysis ,MACHINE learning ,DATA modeling ,DIMENSION reduction (Statistics) - Abstract
The t-distributed stochastic neighbor embedding (t-SNE) is a powerful technique for visualizing high-dimensional datasets. By reducing the dimensionality of the data, t-SNE transforms it into a format that can be more easily understood and analyzed. The existing approach is to visualize high-dimensional data but not deeply visualize. This paper proposes a model that enhances visualization and improves the accuracy. The proposed model combines the non-linear embedding technique t-SNE, the linear dimensionality reduction method principal component analysis (PCA), and the QR decomposition algorithm for discovering eigenvalues and eigenvectors. In Addition, we quantitatively compare the proposed model QRPCA-t-SNE with PCA-t-SNE using the following criteria: data visualization with different perplexity and different principal components, confusion matrix, model score, mean square error (MSE), training, testing accuracy, receiver operating characteristic curve (ROC) score, and AUC score. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
37. Numerical Linear Algebra for the Two-Dimensional Bertozzi–Esedoglu–Gillette–Cahn–Hilliard Equation in Image Inpainting.
- Author
-
Awad, Yahia, Fakih, Hussein, and Alkhezi, Yousuf
- Subjects
NUMERICAL solutions for linear algebra ,INPAINTING ,LINEAR algebra ,EQUATIONS ,FINITE difference method - Abstract
In this paper, we present a numerical linear algebra analytical study of some schemes for the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation. Both 1D and 2D finite difference discretizations in space are proposed with semi-implicit and implicit discretizations on time. We prove that our proposed numerical solutions converge to continuous solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
38. Row Stochastic Matrices and Linear Preservers of Matrix Majorization T: Rm → Rn.
- Author
-
Mohammadhasani, Ahmad, Dehghanian, Mehdi, and Sayyari, Yamin
- Subjects
MATRICES (Mathematics) ,NONNEGATIVE matrices ,STOCHASTIC analysis ,APPLIED mathematics ,LINEAR algebra - Abstract
A nonnegative square and real matrix R is a row stochastic matrix if the sum of the entries of each row is equal to one. Let x, y 2 Rn. The vector x is said to be matrix majorized by y and denoted by x y if x = yR for some row stochastic matrix R. In the present paper, we characterize the linear preservers of matrix majorization T: Rm ! Rn. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
39. Parallel Optimization of BLAS on a New-Generation Sunway Supercomputer.
- Author
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Ren, Yinqiao and Xu, Yi
- Subjects
SUPERCOMPUTERS ,MATRIX multiplications ,LINEAR algebra - Abstract
The new-generation Sunway supercomputer has ultra-high computing capacity. But due to the unique heterogeneous architecture of the supercomputer, the open-source versions of basic linear algebra subprograms (BLAS) are insufficient for performance or compatibility. In addition, due to the update of the architecture, BLAS based on the previous Sunway could not fully exploit the performance of the successor. To address the challenges, we propose an optimized BLAS on the new-generation Sunway supercomputer in this paper. Specially, for achieving efficient computation, a parallel optimization method based on the new-generation Sunway for the Level-1 BLAS computing between vectors and the Level-2 BLAS computing between vectors and matrices is first proposed. Then, an adaptive scheduling algorithm for various data sizes is proposed, which is used to balance the tasks of core groups. Finally, to achieve highly efficient general matrix multiplication (GEMM) kernels, a parallel optimization method based on the new-generation Sunway for the Level-3 BLAS computing between matrices is proposed, which includes source-level optimization as well as assembly-level optimization. Experimental results show that the memory bandwidth utilization of the optimized Level-1/2 BLAS exceeds 95%, and the computational efficiency of the optimized GEMM kernel exceeds 94%. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
40. Numerical Approximations of Diblock Copolymer Model Using a Modified Leapfrog Time-Marching Scheme.
- Author
-
Chen, Lizhen, Ma, Ying, Ren, Bo, and Zhang, Guohui
- Subjects
LINEAR algebra ,LINEAR systems - Abstract
An efficient modified leapfrog time-marching scheme for the diblock copolymer model is investigated in this paper. The proposed scheme offers three main advantages. Firstly, it is linear in time, requiring only a linear algebra system to be solved at each time-marching step. This leads to a significant reduction in computational cost compared to other methods. Secondly, the scheme ensures unconditional energy stability, allowing for a large time step to be used without compromising solution stability. Thirdly, the existence and uniqueness of the numerical solution at each time step is rigorously proven, ensuring the reliability and accuracy of the method. A numerical example is also included to demonstrate and validate the proposed algorithm, showing its accuracy and efficiency in practical applications. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
41. Global Convergence of Hessenberg Shifted QR I: Exact Arithmetic
- Author
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Banks, Jess, Garza-Vargas, Jorge, and Srivastava, Nikhil
- Published
- 2024
- Full Text
- View/download PDF
42. OPTIMALITY CONDITIONS OF QUASI (α, ε)-SOLUTIONS AND APPROXIMATE MIXED TYPE DUALITY FOR DC COMPOSITE OPTIMIZATION PROBLEMS.
- Author
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DONGHUI FANG, JIAOLANG WANG, XIANYUN WANG, and CHING-FENG WEN
- Subjects
HAUSDORFF spaces ,VECTOR spaces ,LINEAR algebra ,VECTOR analysis ,MATHEMATICS - Abstract
This paper is devoted to the approximate optimality condition and mixed type duality for DC composite optimization problems in locally convex Hausdorff topological vector spaces. By using the properties of the Frechet subdifferential, a new constraint qualification is introduced. Under this ´ constraint qualification, some approximate optimality conditions of the quasi (α, ε)-optimal solution for DC compose optimization problem and associated mixed type duality theorems are established, which extend and improve the corresponding results in the previous papers. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
43. Selected Payback Statistical Contributions to Matrix/Linear Algebra: Some Counterflowing Conceptualizations.
- Author
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Griffith, Daniel A.
- Subjects
EIGENFUNCTIONS ,LINEAR algebra ,NONLINEAR analysis ,REGRESSION analysis ,VECTORS (Calculus) - Abstract
Matrix/linear algebra continues bestowing benefits on theoretical and applied statistics, a practice it began decades ago (re Fisher used the word matrix in a 1941 publication), through a myriad of contributions, from recognition of a suite of matrix properties relevant to statistical concepts, to matrix specifications of linear and nonlinear techniques. Consequently, focused parts of matrix algebra are topics of several statistics books and journal articles. Contributions mostly have been unidirectional, from matrix/linear algebra to statistics. Nevertheless, statistics offers great potential for making this interface a bidirectional exchange point, the theme of this review paper. Not surprisingly, regression, the workhorse of statistics, provides one tool for such historically based recompence. Another prominent one is the mathematical matrix theory eigenfunction abstraction. A third is special matrix operations, such as Kronecker sums and products. A fourth is multivariable calculus linkages, especially arcane matrix/vector operators as well as the Jacobian term associated with variable transformations. A fifth, and the final idea this paper treats, is random matrices/vectors within the context of simulation, particularly for correlated data. These are the five prospectively reviewed discipline of statistics subjects capable of informing, inspiring, or otherwise furnishing insight to the far more general world of linear algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
44. Factor Graphs for Navigation Applications: A Tutorial.
- Author
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Taylor, Clark and Gross, Jason
- Subjects
- *
BAYESIAN analysis , *KALMAN filtering , *BAYES' estimation , *LINEAR algebra , *NAVIGATION - Abstract
This tutorial presents the factor graph, a recently introduced estimation framework that is a generalization of the Kalman filter. An approach for constructing a factor graph, with its associated optimization problem and efficient sparse linear algebra formulation, is described. A comparison with Kalman filters is presented, together with examples of the generality of factor graphs. A brief survey of previous applications of factor graphs to navigation problems is also presented. Source code for the extended Kalman filter comparison and for generating the graphs in this paper is available at https://github.com/cntaylor/ factorGraph2DsatelliteExample. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. Finding All the Strong and Weak Defining Hyperplanes of PPS Without Solving any LPs.
- Author
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Oskoueian, S. and Rezai, H. Zhiani
- Subjects
HYPERPLANES ,RETURNS to scale ,LINEAR programming ,LINEAR algebra ,DATA envelopment analysis - Abstract
The production possibility set (PPS) is defined as the set of all inputs and outputs of a system in which inputs can produce outputs. The frontier the production possibility set can be partitioned to strong defining hyperplanes and weak defining hyperplanes. These hyperplanes are useful in sensitivity and stability analysis, identifying the status of returns to scale of a DMU, incorporating performance into the efficient frontier analysis, and so on. In this paper, by using the basic concepts of Linear Algebra, we propose an algorithm for finding all strong and weak defining hyperplanes of PPS without solving any linear programming problems. The proposed method is applicable to both, PPS under constant and variable returns-to-scale assumptions. Two numerical examples are presented to explain the usage and effectiveness of the proposed algorithm. Our method can be easily implemented using existing packages for mathematical algorithm, such as python. [ABSTRACT FROM AUTHOR]
- Published
- 2022
46. Survey of Linear Algebra Solvers for Exascale Computing.
- Author
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HE Lianhua, XU Shun, and JIN Zhong
- Subjects
LINEAR algebra ,NUMERICAL solutions for linear algebra ,PARALLEL programming ,HETEROGENEOUS computing ,SOFTWARE compatibility ,SCIENTIFIC computing - Abstract
The application of scientific engineering computing based on exascale computing not only offers opportunities but also creates challenges for the development of numerical linear algebra algorithms. Firstly, the characteristics of exascale computing are analyzed, including: parallel programming for large-scale heterogeneous parallel architecture has become the mainstream approach; reducing the extremely high energy costs associated with running large- scale applications is a major concern; multi-precision heterogeneous computing hardware has triggered further research of mixed precision computing. Secondly, the optimization work of mainstream dense and sparse linear algebra solvers for high-performance computing architectures is reviewed, and the characteristics and advantages of each solver are compared. Then, the main technology progress of linear algebra solvers is summarized, mainly including: isolating heterogeneous computing modules and designing a new unified programming framework to achieve performance portability of software algorithms; improving the performance level of numerical computing and data storage using mixed precision methods while ensuring the overall requirements of scientific engineering computing applications; combined with hardware multi-level cache and network communication characteristics, advanced parallel computing algorithms are developed to avoid or reduce inefficient large- scale data communication. Finally, this paper provides an outlook on the future research trends in this direction. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
47. Embedding of Unimodular Row Vectors.
- Author
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Wu, Tao, Liu, Jinwang, and Guan, Jiancheng
- Subjects
LINEAR algebra ,LINEAR equations ,LINEAR systems ,POLYNOMIALS - Abstract
In this paper, we mainly study the embedding problem of unimodular row vectors, focusing on avoiding the identification of polynomial zeros. We investigate the existence of the minimal syzygy module of the ZLP polynomial matrix and demonstrate that the minimal syzygy module has structural properties that are similar to the fundamental solution system of homogeneous linear equations found in linear algebra. Finally, we provide several embedding methods for unimodular vectors in certain cases. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
48. Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows.
- Author
-
Ancourt, Kevin, Peter, Jacques, and Atinault, Olivier
- Subjects
COMPRESSIBLE flow ,EULER equations ,INVISCID flow ,PARTIAL differential equations ,EQUATIONS ,LINEAR algebra - Abstract
The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
49. Two-grid methods of finite element approximation for parabolic integro-differential optimal control problems.
- Author
-
Xu, Changling and Li, Huilai
- Subjects
OPTIMAL control theory ,FINITE element method ,PARABOLIC differential equations ,APPROXIMATION theory ,LINEAR algebra - Abstract
In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
50. Calculating the Segmented Helix Formed by Repetitions of Identical Subunits thereby Generating a Zoo of Platonic Helices †.
- Author
-
Read, Robert L.
- Subjects
THEORY of screws ,HELICAL structure ,ZOOS ,HELICES (Algebraic topology) ,SOLID geometry ,LINEAR algebra - Abstract
Eric Lord has observed: "In nature, helical structures arise when identical structural subunits combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant." This paper proves Lord's observation. Constant-time algorithms are given for the segmented helix generated from the intrinsic properties of a stacked object and its conjoining rule. Standard results from screw theory and previous work are combined with corollaries of Lord's observation to allow calculations of segmented helices from either transformation matrices or four known consecutive points. The construction of these from the intrinsic properties of the rule for conjoining repeated subunits of arbitrary shape is provided, allowing the complete parameters describing the unique segmented helix generated by arbitrary stackings to be easily calculated. Free/Libre open-source interactive software and a website which performs this computation for arbitrary prisms along with interactive 3D visualization is provided. We prove that any subunit can produce a toroid-like helix or a maximally-extended helix, forming a continuous spectrum based on joint-face normal twist. This software, website and paper, taken together, compute, render, and catalog an exhaustive "zoo" of 28 uniquely-shaped platonic helices, such as the Boerdijk–Coxeter tetrahelix and various species of helices formed from dodecahedra. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
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