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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]

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]

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]

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 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]

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]

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]

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]

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.

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]

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.

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]

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]

Neural-network architectures have been increasingly used to represent quantum many-body wave functions. These networks require a large number of variational parameters and are challenging to optimize using traditional methods, as gradient descent. Stochastic reconfiguration (SR) has been effective with a limited number of parameters, but becomes impractical beyond a few thousand parameters. Here, we leverage a simple linear algebra identity to show that SR can be employed even in the deep learning scenario. We demonstrate the effectiveness of our method by optimizing a Deep Transformer architecture with 3 × 105 parameters, achieving state-of-the-art ground-state energy in the J1–J2 Heisenberg model at J2/J1 = 0.5 on the 10 × 10 square lattice, a challenging benchmark in highly-frustrated magnetism. This work marks a significant step forward in the scalability and efficiency of SR for neural-network quantum states, making them a promising method to investigate unknown quantum phases of matter, where other methods struggle. Stochastic reconfiguration (SR) is a method to enhance the optimization of variational functions in quantum many-body systems but becomes impractical beyond a few thousand parameters. Here, the authors combine an alternative SR formulation with a Deep Transformer architecture with 105 parameters, achieving the best variational energy for the J1–J2 Heisenberg model on the 10 × 10 square lattice [ABSTRACT FROM AUTHOR]

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a high-order/ hp –version discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the mesh size h and polynomial degree p. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport. While the convergence of poly-energetic source iteration is independent of h and p , we observe that the number of iterations required to attain convergence when employing GMRES only depends mildly on h and p. Moreover, this latter approach is highly effective in the low energy regime. [ABSTRACT FROM AUTHOR]

The Darcy's equation consists of the mass conservation equation and the Darcy's law that involves the hydraulic potential (or called pressure) and the fluid velocity, which governs the flow of an incompressible fluid through a porous medium. In this paper, we investigate the Legendre Galerkin spectral collocation least squares method for approximating the problem of Darcy flow in homogeneous medium and non-homogeneous medium, respectively. The proposed scheme can be solved the approximate solutions of the hydraulic potential and the average velocity of the fluid simultaneously. A symmetric positive definite coefficient matrix of the corresponding linear algebra equation is obtained by applying our scheme. Numerical examples are presented to validate the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper considers technical systems described by nonlinear dynamic models. The fault tolerance property of such systems is ensured by introducing feedback with full or partial fault decoupling. The solution is based on separating a subsystem insensitive or minimally sensitive to faults and its subsequent analysis. For this purpose, a logical-dynamic approach is used, which operates only linear algebra methods. An illustrative practical example is provided. [ABSTRACT FROM AUTHOR]

Neighborhood Lights Out is a game played on graphs. Begin with a graph and a vertex labeling of the graph from the set {0, 1, 2, . . ., ℓ - 1} for ℓ ∈ N. The game is played by toggling vertices: when a vertex is toggled, that vertex and each of its neighbors has its label increased by 1 (modulo ℓ). The game is won when every vertex has label 0. For any n ≥ 2 it is clear that one cannot win the game on Kn unless the initial labeling assigns all vertices the same label. Given that Kn has the maximum number of edges of any simple graph on n vertices it is natural to ask how many edges can be in a graph so that the Neighborhood Lights Out game is winnable regardless of the initial labeling. We find the maximum number of edges a winnable n-vertex graph can have when at least one of n and ℓ is odd. When n and ℓ are both even we find the maximum size in two additional cases. The proofs of our results require us to introduce a new version of the Lights Out game that can be played given any square matrix. [ABSTRACT FROM AUTHOR]

In the era of heterogeneous computing, a new paradigm called accelerator level parallelism (ALP) has emerged. In ALP, accelerators are used concurrently to provide unprecedented levels of performance and energy efficiency. To reach that there are many problems to be solved, one of the most challenging being co-execution. In this paper, we present a new scheduling framework called POAS, a general method for providing co-execution to applications. Our proposal consists of four steps: predict, optimize, adapt and schedule. With POAS, an unseen application can be executed concurrently in ALP with little effort. We evaluate POAS on a heterogeneous environment consisting of CPUs, GPUs (CUDA cores), and XPUs (Tensor cores) on two different fields, namely linear algebra (matrix multiplication benchmark) and deep learning (convolution benchmark). Our experiments prove that POAS provides excellent performance and completes the tasks within a time very close to the optimal time for the hardware and applications used, with a negligible execution time overhead. Moreover, the POAS predictor performed exceptionally well, achieving very low RMSE values for both use cases. Therefore, POAS can be a valuable tool for fully exploiting ALP and improving overall performance over offloading in heterogeneous settings. [ABSTRACT FROM AUTHOR]

General matrix multiplication (gemm) is a fundamental kernel in scientific computing and current frameworks for deep learning. Modern realisations of gemm are mostly written in C, on top of a small, highly tuned micro-kernel that is usually encoded in assembly. The high performance realisation of gemm in linear algebra libraries in general include a single micro-kernel per architecture, usually implemented by an expert. In this paper, we explore a couple of paths to automatically generate gemm micro-kernels, either using C++ templates with vector intrinsics or high-level Python scripts that directly produce assembly code. Both solutions can integrate high performance software techniques, such as loop unrolling and software pipelining, accommodate any data type, and easily generate micro-kernels of any requested dimension. The performance of this solution is tested on three ARM-based cores and compared with state-of-the-art libraries for these processors: BLIS, OpenBLAS and ArmPL. The experimental results show that the auto-generation approach is highly competitive, mainly due to the possibility of adapting the micro-kernel to the problem dimensions. [ABSTRACT FROM AUTHOR]

Sparse matrix-vector multiplication (SpMV) plays a critical role in a wide range of linear algebra computations, particularly in scientific and engineering disciplines. However, the irregular memory access patterns, extensive memory usage, high bandwidth requirements, and underutilization of parallelism hinder the computational efficiency of SpMV on GPUs. In this paper, we propose a novel approach called block-wise dynamic mixed-precision (BDMP) to address these challenges. Our methodology involves partitioning the original matrix into uniformly sized blocks, with each block's size determined by considering architectural characteristics and accuracy requirements. Additionally, we dynamically assign precision to each block using a precision selection method that takes into account the value distribution of the original sparse matrix. We develop two distinct SpMV computation algorithms for BDMP: BDMP-PBP (Precision-based partitioning) and BDMP-TCKI (Tailored compression and kernel implementation). BDMP-PBP partitions the matrix into two independent matrices for separate computations based on block precision, offering flexibility for integration with other optimization techniques. Meanwhile, BDMP-TCKI focuses on achieving significant thread-level parallelism and memory utilization by tailoring an appropriate compressed storage format and kernel implementation for each block. We compare BDMP with NVIDIA's cuSPARSE library and three state-of-the-art SpMV methods, including SELLP, MergeBase, and BalanceCSR, using matrices from the University of Florida's SuiteSparse dataset collection. BDMP-PBP and BDMP-TCKI show average speedups up to 2.64 × and 2.91 × on Turing RTX 2080Ti, and up to 2.99 × and 3.22 × on Ampere A100. The results demonstrate that BDMP enables the optimization of computation speed without compromising the precision necessary for reliable results. [ABSTRACT FROM AUTHOR]

Since bipolar quantum linear algebra (BQLA), under two operations–-addition and multiplication—demonstrates the properties of semirings, and since ideals play an important role in abstract algebra, our results are compelling for the ideals of a semiring. In this article, we investigate the characteristics of ideals, principal ideals, prime ideals, maximal ideals, and the smallest ideal containing any nonempty subset. By applying elementary real analysis, particularly the infimum, our main result is stated as follows: for any closed set I in BQLA, I is a nontrivial proper ideal if and only if there exists c ∈ (0 , 1 ] such that I = (− x , y) ∈ R 2 | c x ≤ y ≤ x c and x , y ≥ 0 . This shows that its shape has to be symmetric with the graph y = − x . [ABSTRACT FROM AUTHOR]

Flexible systems are linear systems of inclusions in which the elements of the coefficient matrix are external numbers in the sense of nonstandard analysis. External numbers represent real numbers with small, individual error terms. Using Gaussian elimination, a flexible system can be put into a row-echelon form with increasing error terms on the right-hand side. Then parameters are assigned to the error terms and the resulting system is solved by common methods of linear algebra. The solution set may have indeterminacy not only in terms of linear spaces, but also of modules. We determine maximal robustness for flexible systems. [ABSTRACT FROM AUTHOR]

Recently there has been growing interest in using photonics to perform the linear algebra operations of neuromorphic and quantum computing applications, aiming at harnessing silicon photonics' (SiPho) high-speed and energy-efficiency credentials. Accurately mapping, however, a matrix into optics remains challenging, since state-of-the-art optical architectures are sensitive to fabrication imperfections. This leads to reduced fidelity that degrades as the insertion losses of the optical matrix nodes or the matrix dimensions increase. In this work, we present the experimental deployment of a 4 × 4 coherent crossbar (Xbar) as a silicon chip and validate experimentally its theoretically predicted fidelity restoration credentials. We demonstrate the experimental implementation of 10,000 arbitrary linear transformations achieving a record-high fidelity of 99.997% ± 0.002, limited mainly by the measurement equipment. Our work represents an integrated optical circuit providing almost unity and loss-independent fidelity in the realization of arbitrary matrices, highlighting light's credentials in resolving complex computations. The Authors demonstrate a fidelity-restorable universal integrated linear optical circuit that relies on a novel 4 × 4 silicon photonic crossbar architecture. Its experimental characterization yields a fidelity of 99.93 ± 0.06%, calculated over 10,000 matrices. [ABSTRACT FROM AUTHOR]

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]

The objects of study of the humanities and social sciences are intrinsically complex. Because it is philosophically attractive, and because it helps in practice to manage such complexity, one of the most influential central ideas throughout the history and present of these disciplines is the notion that the large number of empirical manifestations that characterize their objects of study are actually expressions of a few factors that influence all other variables. The corresponding statistical methodology to implement these ideas has different names and differs in detail in different disciplines, but one name that can be recognized in many of them is “factor analysis”. The first objective of this work is to present a classical method of linear algebra, known as “Singular Value Decomposition” (SVD), in an intuitive and at the same time rigorous way to the community of human and social sciences. SVD systematizes and generalizes the factorization of any data matrix. In addition, the method is of enormous importance in the era of big data and machine learning, which are increasingly influencing research in all areas of study. The second objective is to invite questioning of certain hypotheses in traditional factor analysis. The SVD reveals that factors are inherent in any matrix-structured data set; what is crucial is how singular values decay. Data will determine this decay, with potentially profoundly transformative theoretical repercussions. [ABSTRACT FROM AUTHOR]

We use the least-squares method for inconsistent systems to find a new allocation rule for rationing and surplus problems. In particular, we study allocation problems considering different priorities to satisfy the agents’ demands, which influence how the distribution is carried out. The new distribution rule is proposed by choosing different inner products defined in linear algebra and providing explicit formulas for assigning resources to the agents. Moreover, we illustrate how it can recover different allocation rules by adequately defining the priorities. As an application of the rationing problem with priority agents, we consider real data for allocating police officers in the states of Mexico. In this example, the agents represent the states of Mexico, and their priorities were established based on the criminal incidence. Also, we compute and compare the resource distribution given by classic allocation rules and the percentage of loss obtained with each one. [ABSTRACT FROM AUTHOR]

Chemical reaction balancing is a fundamental aspect of chemistry, ensuring the conservation of mass and atoms in reactions. This article introduces a specialized Python functions designed for automating the balancing of chemical reactions. Leveraging the versatility and simplicity of Python, the module employs advanced algorithms to provide an efficient and user-friendly solution for scientists, educators, and industry professionals. This article delves into the design, implementation, features, applications, and future developments of the Python functions for automated chemical reaction balancing. The functions thus developed were tested on some typical chemical reactions and the results are the same as that in the literature. [ABSTRACT FROM AUTHOR]

Semilinear maps are a generalization of linear maps between vector spaces where we allow the scalar action to be twisted by a ring homomorphism such as complex conjugation. In particular, this generalization unifies the concepts of linear and conjugate-linear maps. We implement this generalization in Lean’s mathlib library, along with a number of important results in functional analysis which previously were impossible to formalize properly. Specifically, we prove the Fréchet–Riesz representation theorem and the spectral theorem for compact self-adjoint operators generically over real and complex Hilbert spaces, additionally developing the Fourier theory needed to state and prove Parseval’s identity. We also show that semilinear maps have applications beyond functional analysis by formalizing the one-dimensional case of a theorem of Dieudonné and Manin that classifies the isocrystals over an algebraically closed field with positive characteristic. [ABSTRACT FROM AUTHOR]

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]

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]

Let A be a bounded positive operator on a complex Hilbert space ... . The semi-product ..., induces a semi-norm ... on H. Let ωA (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]

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]

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]

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]

This article considers the class of convex functions and derives further Jensen-Mercer'stype inequalities. The obtained results improve and generalize some known inequalities. A reverse of Jesnen-Mercer's inequality for scalars and operators is also given. As an application, we provide a new and non-trivial inequality related to the Wigner-Yanase-Dyson function and the logarithmic mean. [ABSTRACT FROM AUTHOR]

In this article, we investigate a discrete-time risk model. The risk model includes the quota--(α, β) reinsurance contract effect on the surplus process. The premium process and claim process are assumed to be m-dependent sequences of identically distributed non-negative random variables. Using Martingale and inductive methods, We obtained upper bounds for the ultimate ruin probability of an insurance company. Finally, we present a numerical example to show the efficiency of the methods. [ABSTRACT FROM AUTHOR]

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]

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]

This article improves the triangle inequality for complex numbers, using the Hermite- Hadamard inequality for convex functions. Then, applications of the obtained refinement are presented to include some operator inequalities. The operator applications involve numerical radius inequalities and operator mean inequalities. [ABSTRACT FROM AUTHOR]