Search

In the study of stochastic systems, the committor function describes the probability that a system starting from an initial configuration x will reach a set B before a set A. This paper introduces an efficient and interpretable algorithm for approximating the committor, called the "fast committor machine" (FCM). The FCM uses simulated trajectory data to build a kernel-based model of the committor. The kernel function is constructed to emphasize low-dimensional subspaces that optimally describe the A to B transitions. The coefficients in the kernel model are determined using randomized linear algebra, leading to a runtime that scales linearly with the number of data points. In numerical experiments involving a triple-well potential and alanine dipeptide, the FCM yields higher accuracy and trains more quickly than a neural network with the same number of parameters. The FCM is also more interpretable than the neural net. [ABSTRACT FROM AUTHOR]

To address the challenge of performance portability and facilitate the implementation of electronic structure solvers, we developed the basic matrix library (BML) and Parallel, Rapid O(N), and Graph-based Recursive Electronic Structure Solver (PROGRESS) library. The BML implements linear algebra operations necessary for electronic structure kernels using a unified user interface for various matrix formats (dense and sparse) and architectures (CPUs and GPUs). Focusing on density functional theory and tight-binding models, PROGRESS implements several solvers for computing the single-particle density matrix and relies on BML. In this paper, we describe the general strategies used for these implementations on various computer architectures, using OpenMP target functionalities on GPUs, in conjunction with third-party libraries to handle performance critical numerical kernels. We demonstrate the portability of this approach and its performance in benchmark problems. [ABSTRACT FROM AUTHOR]

Lattice dynamics (LD) enables the calculation of mode-resolved transmittance of phonons passing through an interface, which is essential for understanding and controlling the thermal boundary conductance (TBC). However, the original LD method may yield unphysical transmittance over 100% due to the absence of the constraint of energy conservation. Here, we present a robust LD algorithm that utilizes linear algebra transformations and projection gradient descent iterations to ensure energy conservation. Our approach demonstrates consistency with the original LD method on the atomically smooth Si/Ge interface and exhibits robustness on rough Si/Ge interfaces. The evanescent modes and localized effects at the interface are revealed. In addition, bottom-up analysis of the phonon transmittance shows that the anisotropy in the azimuth angle can be ignored, while the dependency on the frequency and polar angle can be decoupled. The decoupled expression reproduces the TBC precisely. This work provides comprehensive insights into the mode-resolved phonon transmittance across interfaces and paves the way for further research into the mechanism of TBC and its relation to atomic structures. [ABSTRACT FROM AUTHOR]

Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this work, we propose a novel method for solving linear systems. Our approach leverages binary optimization, making it particularly well-suited for problems with large condition numbers. We transform the linear system into a binary optimization problem, drawing inspiration from the geometry of the original problem and resembling the conjugate gradient method. This approach employs conjugate directions that significantly accelerate the algorithm's convergence rate. Furthermore, we demonstrate that by leveraging partial knowledge of the problem's intrinsic geometry, we can decompose the original problem into smaller, independent sub-problems. These sub-problems can be efficiently tackled using either quantum or classical solvers. Although determining the problem's geometry introduces some additional computational cost, this investment is outweighed by the substantial performance gains compared to existing methods. [ABSTRACT FROM AUTHOR]

We study the Schrödinger-Poisson type system: { − Δ u + λ u + (μ 11 ϕ u − μ 12 ϕ v) u = 1 2 π ∫ 0 2 π | u + e i θ v | p − 1 (u + e i θ v) d θ in R 3 , − Δ v + λ v + (μ 22 ϕ v − μ 12 ϕ u) v = 1 2 π ∫ 0 2 π | v + e i θ u | p − 1 (v + e i θ u) d θ in R 3 , where 1 < p < 3 with parameters λ , μ i j > 0. Novel approaches are employed to prove the existence of a positive solution for 1 < p < 3 including, particularly, the finding of a ground state solution for 2 ≤ p < 3 using established linear algebra techniques and demonstrating the existence of two distinct positive solutions for 1 < p < 2. The analysis here, by employing alternative techniques, yields additional and improved results to those obtained in the study of Jin and Seok (2023) [14]. [ABSTRACT FROM AUTHOR]

In the cognitive process of establishing a diagnosis, the performance of a diagnostician can be characterized in terms of sensitivity and specificity. The aims of the present study are to analyze in quantitative terms how cognitive bias affects the performance of a diagnostician, and how a diagnostician's biased decision making is further influenced by personal cost-benefit considerations. The test matrices of two sequential diagnostic tests are manipulated according to the rules of linear algebra, using multiplication of the second with the first test matrix to calculate their joint test characteristics. The decision tree and receiver operating characteristic (ROC) of a biased and unbiased diagnostician are used to calculate which combination of test characteristics maximizes the expected utility value. Biased diagnosticians cannot establish a diagnosis beyond their own limited or distorted level of understanding. An unbiased and a biased diagnostician alike adjust their choice of test characteristics according to their different cost-benefit estimation of the various test outcomes. From the perspective of an unbiased diagnostician, the choices made by a biased diagnostician appear to invert reality. However, the same appearance of inverted reality is perceived by the biased diagnostician, judging the choices made by the unbiased diagnostician. As a general principle, human testers cannot test beyond their own level of understanding. They only see what they know. As they base their judgment on preconceived notions about the utilities associated with different test outcomes, human testers also tend to only know what they want to know. [ABSTRACT FROM AUTHOR]

Over the course of interactions with various application teams, the need for batched sparse linear algebra functions has emerged in order to make more efficient use of the GPUs for many small and sparse linear algebra problems. In this paper, we present our recent work on a batched sparse direct solver for GPUs. The sparse LU factorization is computed by the levels of the elimination tree, leveraging the batched dense operations at each level and a new batched Scatter GPU kernel. The sparse triangular solve is computed by the level sets of the directed acyclic graph (DAG) of the triangular matrix. Batched operations overcome the large overhead associated with launching many small kernels. For medium sized matrix batches with not-so-small bandwidth, using an NVIDIA A100 GPU, our new batched sparse direct solver is orders of magnitude faster than a batched banded solver and uses less than one-tenth of the memory. [ABSTRACT FROM AUTHOR]

Elastic collisions of gliders on linear air tracks are often used to explore conservation of energy and momentum. If one is interested in the glider behavior over a long time span, the analysis involves repeated collisions and is complicated by reflections from the track end stops. Here we analyze elastic collisions on a novel circular air track; since such a track lacks end stops, the mathematical analysis of repeated collisions is amenable to our students. Our analysis uncovers a variety of interesting behaviors which depend on the ratio of the glider masses. We examine periodic sequences where the gliders return to their initial conditions and progressions where (when plotted in polar coordinates) the collision positions take on the locus of a spiral. One set of initial conditions produces an "angle trap" where one glider remains within a certain angular range. We also explore making one glider's mass hypothetically negative, which results in a novel "chasing" motion. Our results were obtained using a 3D interactive simulation (created using C++ within the Unreal engine), which we make available as supplementary material. Editor's Note: Students in introductory courses often analyze one-dimensional elastic collisions between carts on wheels or gliders on an air track. If the track possesses end stops, and the system is left undisturbed after the initial contact, multiple collisions will occur. The analysis of repeated collisions can be somewhat complicated due to the end-stop reflections. In this article, the authors consider a clever simplification involving a circular track without end stops. This renders the mathematical analysis more tractable for students, and the system presents a variety of interesting behaviors to explore. Students and instructors will enjoy this entertaining application of linear algebra and can freely make use of online computer simulations provided as supplementary material. [ABSTRACT FROM AUTHOR]

A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra g is splittable if and only if g = a + n , where a is an abelian subalgebra of g composed of semisimple elements and n is the ideal of all nilpotent matrices of g. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some explicit and recursive formulas for the n$$ n $$th q$$ q $$‐derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between commutative algebras and the properties of q$$ q $$‐differential operator. One of the formulas involves the determinant of the functions and their q$$ q $$‐derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, we used formulas to obtain some interesting q$$ q $$‐identities. [ABSTRACT FROM AUTHOR]

In this paper, we solve a classical counting problem for non-degenerate forms of symplectic and hermitian type defined on a vector space: given a subspace π , we find the number of non-singular subspaces that are trivially intersecting with π and span a non-singular subspace with π. Lower bounds for the quantity of such pairs where π is non-singular were first studied in "Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)", which was later improved in "Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)" and generalised in "Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)". In this paper, we derive explicit formulae, which allow us to give the exact proportion and improve the known lower bounds. [ABSTRACT FROM AUTHOR]

In this tutorial, we summarize four important matrix decompositions commonly used in quantum optics, namely the Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson decompositions. The first two of these decompositions are specialized versions of the singular-value decomposition when applied to symmetric or symplectic matrices. The third factors any symplectic matrix in a unique way in terms of matrices that belong to different subgroups of the symplectic group. The last one instead gives the symplectic diagonalization of real, positive definite matrices of even size. While proofs of the existence of these decompositions exist in the literature, we review explicit constructions to implement these decompositions using standard linear algebra packages and functionalities such as singular-value, polar, Schur, and QR decompositions, and matrix square roots and inverses. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we will consider parallel summable operators on Hilbert space. Operators A and B are said to be parallel summable if and . We will prove that the parallel sum can be represented as A: B = A(A + B)† B without additional assumption of the closedness of the range of the operator A + B. Furthermore, we will derive the equalities C (A : B) = C A : C B and (A : B) : C = A : (B : C ) under weaker conditions than the ones represented in [X. Tian, S. Wang, C. Deng, On parallel sum of operators, Linear Algebra Appl. 603 (2020) 57–83] and [W. Luo, C. Song, Q. Xu, The parallel sum for adjointable operators on Hilbert C* -modules, Acta Math. Sin. 62 (2019) 541–552]. Finally, we will extend some recent result for Hermitian positive semi-definite matrices to bounded linear operators. [ABSTRACT FROM AUTHOR]

Copyright of Formación Universitaria is the property of Centro de Informacion Tecnologica (CIT) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

In this project, we investigated how designing their own tutorial questions could help students negotiate the often-challenging transition to university mathematics. One student from the first linear algebra course at the University of Aberdeen volunteered to create practice questions on topics of linear algebra they selected. Through analysis of the participant's interview, we showed, in accordance with the literature, that the activity benefited them in terms of study skills (motivation, focus, independence), and in terms of mathematics learning (deep learning and mathematical knowledge construction). We also found that the activity had contributed to improving the volunteer's resilience to learning mathematics, and their sense of legitimate participation in the mathematics community. This has not been discussed in the literature and is a significant finding considering the known difficulty for mathematics students, in particular women, to feel they 'belong'. This research further suggests that if the activity took place in groups, it could also help develop a peer-support community which students could rely on to negotiate the transition to university mathematics. [ABSTRACT FROM AUTHOR]

We revisit tensor algebras of subproduct systems with Hilbert space fibers, resolving some open questions in the case of infinite-dimensional fibers. We characterize when a tensor algebra can be identified as the algebra of uniformly continuous noncommutative functions on a noncommutative homogeneous variety or, equivalently, when it is residually finite-dimensional: this happens precisely when the closed homogeneous ideal associated with the subproduct system satisfies a Nullstellensatz with respect to the algebra of uniformly continuous noncommutative functions on the noncommutative closed unit ball. We show that – in contrast to the finite-dimensional case – in the case of infinite-dimensional fibers, this Nullstellensatz may fail. Finally, we also resolve the isomorphism problem for tensor algebras of subproduct systems: two such tensor algebras are (isometrically) isomorphic if and only if their subproduct systems are isomorphic in an appropriate sense. [ABSTRACT FROM AUTHOR]

Mesh processing algorithms are often communicated via concise mathematical notation (e.g., summation over mesh neighborhoods). However, conversion of notation into working code remains a time-consuming and error-prone process, which requires arcane knowledge of low-level data structures and libraries—impeding rapid exploration of high-level algorithms. We address this problem by introducing a domain-specific language (DSL) for mesh processing called I MESH, which resembles notation commonly used in visual and geometric computing and automates the process of converting notation into code. The centerpiece of our language is a flexible notation for specifying and manipulating neighborhoods of a cell complex, internally represented via standard operations on sparse boundary matrices. This layered design enables natural expression of algorithms while minimizing demands on a code generation backend. In particular, by integrating I MESH with the linear algebra features of the I LA DSL and adding support for automatic differentiation, we can rapidly implement a rich variety of algorithms on point clouds, surface meshes, and volume meshes. [ABSTRACT FROM AUTHOR]

The Gram–Schmidt process is a standard topic in beginning linear algebra courses. The computations, while straightforward, often require working with square roots, which can be difficult for students. We demonstrate how rational orthogonal matrices can be used to design questions examining the Gram–Schmidt method which avoid the appearance of square roots. Additionally, we show how these matrices are useful for the creation of geometry questions, which also maintain rational arithmetic. Finally, we discuss ways in which rational orthogonal matrices may be generated as well as give details of databases of generated matrices. [ABSTRACT FROM AUTHOR]

Affine subspaces are translates of linear subspaces, and hyperplanes are well-known instances of affine subspaces. In basic linear algebra, one encounters the explicit formula for projecting onto a hyperplane. An interesting—and relevant for applications—question is whether or not there is a formula for projecting onto the intersection of two hyperplanes. The answer turns out to be yes, as demonstrated recently by Behling, Bello-Cruz, and Santos, by López, by Needell and Ward, and by Ouyang. Most of these authors also provided formulas for projecting onto the intersection of an affine subspace and a hyperplane. In this note, we present an alternative approach which has the advantage of being more explicit and more elementary. Our results also provide useful information in the case when the two sets don't intersect. Luckily, the material is fully accessible to readers with a basic background in linear algebra and analysis. Finally, we demonstrate the computational efficiency of our formula when applied to an image reconstruction problem arising in Computed Tomography, and we also present a new formula for the projection onto the set of generalized bistochastic matrices with a moment constraint. [ABSTRACT FROM AUTHOR]

Topic models are mathematical and statistical models used to analyze textual data. The objective of topic models is to gain information about the latent semantic space of a set of related textual data. The semantic space of a set of textual data contains the relationship between documents and words and how they are used. Topic models are becoming more common in educational measurement research as a method for analyzing students' responses to constructed-response items. Two popular topic models are latent semantic analysis (LSA) and latent Dirichlet allocation (LDA). LSA uses linear algebra techniques, whereas LDA uses an assumed statistical model and generative process. In educational measurement, LSA is often used in algorithmic scoring of essays due to its high reliability and agreement with human raters. LDA is often used as a supplemental analysis to gain additional information about students, such as their thinking and reasoning. This article reviews and compares the LSA and LDA topic models. This article also introduces a methodology for comparing the semantic spaces obtained by the two models and uses a simulation study to investigate their similarities. [ABSTRACT FROM AUTHOR]

The given document is a research paper titled "Quasiregular Radicals of Nonassociative Algebras" written by A. Yu. Golubkov. The paper discusses various mathematical concepts related to Lie algebras, Jordan linear pairs, and linear triple systems. It explores the construction of quasiregular radicals and their applications in the Amitsur-Procesi theorem. The document also discusses the properties and relationships of different radicals in these algebraic structures. The author acknowledges funding support and provides their contact information for further inquiries. [Extracted from the article]

Recently, W. Berndt and S. Sra (Hlawka-Popoviciu inequalities on positive definite tensors. Linear Algebra Appl. 2015;486:317–327) proved a Popoviciu type inequality for any arbitrary finite number of positive definite matrices. In this paper, we proved a quantitative Popoviciu type inequality for four positive semi-definite matrices, which is stronger than Berndt-Sra's corresponding result and also a generalization of Hong-Qi's (Refinements of two determinantal inequalities for positive semidefinite matrices. Math Inequal Appl. 2022;25(3):673–678) result. As applications, we partially recovered our early result in F. Wang (A quantitative Popoviciu type inequality, submitted.] and obtained a quantitative Hartfiel inequality. [ABSTRACT FROM AUTHOR]

Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expense to compute. In this paper, we derive analytic solutions to a class of nonlinear, homogeneous ODEs with linear and quadratic terms on the right-hand side. We formulate a power series expansion of each state variable, whose function depends on the eigenvalues of the linearized system, and solve for the coefficients using some linear algebra and combinatorics. Various experiments exhibit quickly decaying coefficients, such that a good approximation to the true solution consists of just a few terms. [ABSTRACT FROM AUTHOR]

Modons, or dipolar vortices, are common and long-lived features of the upper ocean, consisting of a pair of counter-rotating monopolar vortices moving through self-advection. Such structures remain stable over long times and may be important for fluid transport over large distances. Here, we present a semi-analytical method for finding fully nonlinear modon solutions in a multi-layer quasi-geostrophic model with arbitrarily many layers. Our approach is to reduce the problem to a multi-parameter linear eigenvalue problem which can be solved using numerical techniques from linear algebra. The method is shown to replicate previous results for one- and two-layer models and is applied to a three-layer model to find a solution describing a mid-depth propagating, topographic vortex. [ABSTRACT FROM AUTHOR]

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]

The limited capabilities of current quantum hardware significantly constrain the scale of experimental demonstrations of most quantum algorithmic primitives. This makes it challenging to perform benchmarking of the current hardware using useful quantum algorithms, i.e., application-oriented benchmarking. In particular, the Harrow-Hassidim-Lloyd (HHL) algorithm is a critical quantum linear algebra primitive, but the majority of the components of HHL are far out of the reach of noisy intermediate-scale quantum devices, which has led to the proposal of hybrid classical-quantum variants. The goal of this work is to further bridge the gap between proposed near-term friendly implementations of HHL and the kinds of quantum circuits that can be executed on noisy hardware. Our proposal adds to the existing literature of hybrid quantum algorithms for linear algebra that are more compatible with the current scale of quantum devices. Specifically, we propose two modifications to the Hybrid HHL algorithm proposed by Lee et al., leading to our algorithm Hybrid HHL + + : (1) propose a novel algorithm for determining a scaling factor for the linear system matrix that maximizes the utility of the amount of ancillary qubits allocated to the phase estimation component of HHL, and (2) introduce a heuristic for compressing the HHL circuit. We demonstrate the efficacy of our work by running our modified Hybrid HHL on Quantinuum System Model H-series trapped-ion quantum computers to solve different problem instances of small-scale portfolio optimization problems, leading to the largest experimental demonstrations of HHL for an application to date. [ABSTRACT FROM AUTHOR]

We obtain a characterization of bijective maps preserving the trace of products on the set of tensor products of rank‐1 idempotents, which may be considered as a generalization of Molnár's Wigner‐type theorem in multipartite systems. The proof is based on the studies in linear preserver problems on tensor products. Moreover, a corresponding result on the finite‐dimensional case is presented. [ABSTRACT FROM AUTHOR]

The limited capabilities of current quantum hardware significantly constrain the scale of experimental demonstrations of most quantum algorithmic primitives. This makes it challenging to perform benchmarking of the current hardware using useful quantum algorithms, i.e., application-oriented benchmarking. In particular, the Harrow-Hassidim-Lloyd (HHL) algorithm is a critical quantum linear algebra primitive, but the majority of the components of HHL are far out of the reach of noisy intermediate-scale quantum devices, which has led to the proposal of hybrid classical-quantum variants. The goal of this work is to further bridge the gap between proposed near-term friendly implementations of HHL and the kinds of quantum circuits that can be executed on noisy hardware. Our proposal adds to the existing literature of hybrid quantum algorithms for linear algebra that are more compatible with the current scale of quantum devices. Specifically, we propose two modifications to the Hybrid HHL algorithm proposed by Lee et al., leading to our algorithm Hybrid HHL + + : (1) propose a novel algorithm for determining a scaling factor for the linear system matrix that maximizes the utility of the amount of ancillary qubits allocated to the phase estimation component of HHL, and (2) introduce a heuristic for compressing the HHL circuit. We demonstrate the efficacy of our work by running our modified Hybrid HHL on Quantinuum System Model H-series trapped-ion quantum computers to solve different problem instances of small-scale portfolio optimization problems, leading to the largest experimental demonstrations of HHL for an application to date. [ABSTRACT FROM AUTHOR]

Data-parallel computations, such as linear algebra routines and stencil computations, constitute one of the most relevant classes in parallel computing, e.g., due to their importance for deep learning. Efficiently de-composing such computations for the memory and core hierarchies of modern architectures and re-composing the computed intermediate results back to the final result—we say (de/re)-composition for short—is key to achieve high performance for these computations on, e.g., GPU and CPU. Current high-level approaches to generating data-parallel code are often restricted to a particular subclass of data-parallel computations and architectures (e.g., only linear algebra routines on only GPU or only stencil computations), and/or the approaches rely on a user-guided optimization process for a well-performing (de/re)-composition of computations, which is complex and error prone for the user. We formally introduce a systematic (de/re)-composition approach, based on the algebraic formalism of Multi-Dimensional Homomorphisms (MDHs). Our approach is designed as general enough to be applicable to a wide range of data-parallel computations and for various kinds of target parallel architectures. To efficiently target the deep and complex memory and core hierarchies of contemporary architectures, we exploit our introduced (de/re)-composition approach for a correct-by-construction, parametrized cache blocking, and parallelization strategy. We show that our approach is powerful enough to express, in the same formalism, the (de/re)-composition strategies of different classes of state-of-the-art approaches (scheduling-based, polyhedral, etc.), and we demonstrate that the parameters of our strategies enable systematically generating code that can be fully automatically optimized (auto-tuned) for the particular target architecture and characteristics of the input and output data (e.g., their sizes and memory layouts). Particularly, our experiments confirm that via auto-tuning, we achieve higher performance than state-of-the-art approaches, including hand-optimized solutions provided by vendors (such as NVIDIA cuBLAS/cuDNN and Intel oneMKL/oneDNN), on real-world datasets and for a variety of data-parallel computations, including linear algebra routines, stencil and quantum chemistry computations, data mining algorithms, and computations that recently gained high attention due to their relevance for deep learning. [ABSTRACT FROM AUTHOR]

MAGMA (Matrix Algebra for GPU and Multicore Architectures) is a pivotal open-source library in the landscape of GPU-enabled dense and sparse linear algebra computations. With a repertoire of approximately 750 numerical routines across four precisions, MAGMA is deeply ingrained in the DOE software stack, playing a crucial role in high-performance computing. Notable projects such as ExaConstit, HiOP, MARBL, and STRUMPACK, among others, directly harness the capabilities of MAGMA. In addition, the MAGMA development team has been acknowledged multiple times for contributing to the vendors' numerical software stacks. Looking back over the time of the Exascale Computing Project (ECP), we highlight how MAGMA has adapted to recent changes in modern HPC systems, especially the growing gap between CPU and GPU compute capabilities, as well as the introduction of low precision arithmetic in modern GPUs. We also describe MAGMA's direct impact on several ECP projects. Maintaining portable performance across NVIDIA and AMD GPUs, and with current efforts toward supporting Intel GPUs, MAGMA ensures its adaptability and relevance in the ever-evolving landscape of GPU architectures. [ABSTRACT FROM AUTHOR]

Many students who enroll in a first course in linear algebra major in STEM disciplines other than mathematics. Teachers who serve such students may find it difficult to provide authentic problems from these broader areas that ignite students' interest in linear algebra. In this paper, we highlight an interdisciplinary learning activity that engages students in using linear systems of equations to model the behavior of practical electric circuits. This exercise fits nicely into standard introductory linear algebra curricula and is designed to excite students majoring in engineering, physics, or applied mathematics. We also include references to a collection of open-access resources to support instructors who want to use this material in project-based, flipped-learning, inquiry-oriented, or independent-study environments. [ABSTRACT FROM AUTHOR]

The objective of this paper is to establish comparisons between the Linear Algebra (LA) approach practiced in Engineering courses offered by Chilean and Brazilian Higher Education Institutions and, for that, teachers from both countries were interviewed. From a methodological point of view, the research consists of two case studies. For the analysis of the interviews, we used the precepts of Content Analysis. The data were qualitatively analyzed considering the MTSK theoretical model. In this analysis, we sought to identify what specialized knowledge these teachers have regarding AL and about its teaching and learning. The results have shown that, in relation to the Mathematical Knowledge (MK) domain, which refers to content knowledge, the Knowledge of Topics subdomain (KoT), linked to the way that teachers know the topics they teach, is the most present among interviewees. In the Pedagogical Content Knowledge (PCK) domain, covering specific pedagogical knowledge linked to Mathematics, the Knowledge of Mathematics Teaching (KMT), regarding strategies, teaching theories, resources and materials, and Knowledge of Mathematics Learning Standards (KMLS) subdomains, concerning the curricular specifications and the moments in which the student must or can learn certain content and with what level of depth prevail. [ABSTRACT FROM AUTHOR]

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]

We investigated university students' engagement with digital textbooks by analysing their real-time viewing and use in a linear algebra course taught at four different universities in the United States. Viewing data over a semester was complemented with responses to bi-weekly open-ended surveys and other course artefacts, such as course syllabi. With our approach to ask students about their viewing data in surveys, we were able to relate student majors to textbook engagement and to document more and different uses of textbook elements than what has been reported so far. We used descriptive statistics and frequency graphs to analyse the viewing data and a grounded analytical approach to survey data. By using the didactical tetrahedron, we found that textbook engagement dropped when the algebraic language and content of the textbook was not institutionally legitimized and when the textbook did not directly satisfy interests associated with earning a degree. Textbook engagement was higher than what could have been anticipated by syllabi for students majoring in disciplines other than mathematics. We propose areas for further research. [ABSTRACT FROM AUTHOR]

In order to explain the Lorentz transformation, it is advantageous to use its eigencoordinates rather than Cartesian coordinates. In this manner, the matrix can be diagonalized. Moreover, the diagonal entries have a direct physical meaning—permitting their determination without the need of algebraic equations. This yields a proof that is easier to remember and reproduce. Editor's Note: Physics instructors know there are many ways to derive the Lorentz equations, and we all have our favorites. But take a look at this one, and you may find a new favorite. Appropriate for special relativity courses, as long as the students have some familiarity with linear algebra. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Pell-Lucas and the symmetric Pell-Lucas matrices. The study delves into the linear algebra aspects of these matrices, analyzing their mathematical properties and relationships. We construct decompositions for both the Pell-Lucas matrix and its inverse matrix. We present the Cholesky factorization of the symmetric Pell-Lucas matrices. Furthermore, we derive some valuable identities and bounds for the eigenvalues of these symmetric matrices through the application of majorization notation. [ABSTRACT FROM AUTHOR]

Quantum computers promise to surpass the most powerful classical supercomputers in tackling critical practical problems, such as designing pharmaceuticals and fertilizers, optimizing supply chains and traffic, and enhancing machine learning. Since quantum computers operate fundamentally differently from classical ones, their emergence will give rise to a new evolutionary branch of statistical and data analytics methodologies. This review aims to provide an introduction to quantum computing accessible to statisticians and data scientists, equipping them with a comprehensive framework, the basic language, and building blocks of quantum algorithms, as well as an overview of existing quantum applications in statistics and data analysis. Our objective is to empower statisticians and data scientists to follow quantum computing literature relevant to their fields, collaborate with quantum algorithm designers, and ultimately drive the development of the next generation of statistical and data analytics tools. [ABSTRACT FROM AUTHOR]

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]

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]

Sound, with all its manifestations, presents itself as a form of energy with properties and capabilities that remain partially shrouded in mystery. The art of composing music, the artful assemblage and fusion of sounds, still lingers in a realm of obscurity, fraught with complexities. The techniques wielded by composers often stem from an intrinsic gift that defies replication or sharing. However, akin to other art forms, music can be deciphered through the lenses of mathematics and geometry. Through the utilization of linear algebra, a paramount mathematical discipline, a practical and efficient framework can be established for the comprehension and manipulation of musical compositions. Within this discourse, we delve into the essence of music and its fundamental principles, subsequently exploring the application of mathematical concepts; linear algebra in particular; in the analysis of music. The findings put forth in this manuscript hold the potential to enlighten researchers and enthusiasts within the realm of music and mathematical sciences, enhance comprehension and knowledge, refine and enhance analytical methodologies and musical procedures across various facets of the music industry, and foster a deeper bond between the realms of mathematics and artistic expression. [ABSTRACT FROM AUTHOR]

Innovations are proposed in the courses of mathematical analysis, linear algebra and discrete mathematics taught in the first year of study. Innovations are aimed at improving methods of teaching basic mathematical disciplines in higher education.

Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition operator are useful for visualization, and they can provide an efficient basis for computing statistics, such as the likelihood and average time of events (predictions). Here, we develop inexact iterative linear algebra methods for computing these eigenfunctions (spectral estimation) and making predictions from a dataset of short trajectories sampled at finite intervals. We demonstrate the methods on a low-dimensional model that facilitates visualization and a high-dimensional model of a biomolecular system. Implications for the prediction problem in reinforcement learning are discussed. [ABSTRACT FROM AUTHOR]

For a locally compact group G and p ∈ (1,8), let Bp(G) is the multiplier algebra of the Figà-Talamanca-Herz algebra Ap(G). For p = 2 and G amenable, the algebra B(G) := B2(G) is the usual Fourier-Stieltjes algebra. In this paper, we show that Ap(G) is a Bochner-Schoenberg-Eberlin (BSE) algebra and every clopen subset of G is a synthetic set for Ap(G). Furthermore, we characterize idempotent elements of the Banach algebra Bp(G). This result generalizes the Cohen-Host idempotent theorems for the case of Figà-Talamanca-Herz algebras. Characterization of idempotent elements of Bp(G) is of paramount importance to study homomorphisms in Figà-Talamanca-Herz algebras. [ABSTRACT FROM AUTHOR]

A tuple (Z 1 , ... , Z p) of matrices of size r × r is said to be a commuting extension of a tuple (A 1 , ... , A p) of matrices of size n × n if the Z i pairwise commute and each A i sits in the upper left corner of a block decomposition of Z i (here, r and n are two arbitrary integers with n < r). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called "quantum Zeno dynamics." Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results: (i) Theorems on the uniqueness of commuting extensions for three matrices or more. (ii) Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r = 4 n / 3 , and are apparently the first provably efficient algorithms for this problem applicable beyond r = n + 1. (iii) A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices. [ABSTRACT FROM AUTHOR]

This article details the design and implementation of a didactic intervention strategy aimed at enhancing the pedagogical skills of mathematics education students in the teaching of linear algebra. Given the challenges students often face in grasping abstract mathematical concepts, the intervention leverages the Anthropological Theory of Didactics and Collaborative Learning as its theoretical framework. The phases of the intervention's execution in the classroom, along with the key elements for successful implementation, are thoroughly outlined. The results indicate significant improvements in academic performance in the units’ covering determinants, matrices, and systems of linear equations, while challenges persist in the unit on vector spaces. In terms of task genres, students showed proficiency in defining and calculating, although there was a lower level of mastery in demonstrating. The study concludes that, despite a general unfamiliarity with autonomous learning, the future educators were able to become active and effective contributors to their own mathematical knowledge construction through collaborative group work.

This article extends topics about linear algebra and operator theoretic linear transformations on complex vector spaces to those on bicomplex spaces. For example, Definition 3 for the first time defines algebraically idempotent vector spaces, which generalizes the standard definition of a vector space and which includes bicomplex vector spaces as a special case, along with its dimension and its basis in terms of a corresponding vectorial idempotent representation. The article also shows how an n×nn\times n bicomplex matrix’s idempotent representation leads to a bicomplex Jordan form and a description of its bicomplex invariant subspace lattice diagram. Similarly, in a new way, the article rigorously defines “bicomplex Banach and Hilbert” spaces, and then it expands, for the first time, the theory of compact operators on complex Banach spaces to those on bicomplex Banach spaces. In these ways, the article indicates that the idempotent representation extends complex linear algebra and operator theory in a surprisingly generalized and straightforward way to vector space results with bicomplex and multicomplex scalars.