Coincidence-site lattice (CSL) theory provides a fundamental framework for classifying commensurate crystal orientations and plays a central role in describing grain boundaries. In particular, Ranganathan's formula determines all CSL orientation relationships. However, it is not possible to avoid duplication and also to determine the equivalence of the duplicates, and number theoretical analysis can be required to address these issues. In this study, we provide a simple necessary condition to find all CSL orientation relationships in the cubic system by only using linear algebra. The necessary condition is given by calculating whether the (hkl) plane is commensurate to the (001) plane. We also found that the number of orientation relationships without symmetric tilt grain boundaries (GBs) increases as the Σ value increases by comparing the number of solutions of the equation for constructing symmetric tilt GBs with that of the necessary condition.

Abstract We present a hybrid a–priori/a–posteriori goal–oriented error estimator for a combination of dynamic iteration based solution of linear ordinary differential equations discretized by finite elements. Our novel error estimator combines estimates from classical dynamic iteration methods, usually used to enable splitting–based distributed simulation, and from the dual weighted residual method to be able to evaluate and balance both, the dynamic iteration error and the discretization error in desired quantities of interest. The obtained error estimators are used to conduct refinements of the computational mesh and as a stopping criterion for the dynamic iteration. In particular, we allow for an adaptive and flexible discretization of the time domain, where variables can be discretized differently to match both goal and solution requirements, e.g. in view of multiple time scales. We endow the scheme with efficient solvers from numerical linear algebra to ensure its applicability to complex problems. Numerical experiments compare the adaptive approach to a uniform refinement.

MulAtoLEG (Multi-Atom Liouville Equation Generator) is a source-available Mathematica package for generating Liouville superoperators and Liouville equations, specialized for multilevel atomic systems comprising an arbitrary number of atoms. This scheme is based on an extension to multilevel atomic systems, originally developed by Lehmberg [R. H. Lehmberg, Phys. Rev. A 2, 883 (1970)] as an adjoint master equation for ensembles of two-level emitters and later reformulated by Genes [M. Reitz, C. Sommer and C. Genes, PRX Quantum 3, 010201 (2022)] as a master equation. The package facilitates the generation of equations for complex transition configurations in alkali atoms. Although primarily designed for atomic systems, it can also generate the master and adjoint master equations for general Hamiltonians and Lindbladians. In addition, it includes functionalities to construct the differential equations in the dressed-state basis, where, in many cases, the non-unitary evolution operator can be determined explicitly. To maximize computational efficiency, the package leverages Mathematica’s vectorization and sparse linear algebra capabilities. Since MulAtoLEG produces exact equations without approximations, the feasible system size is naturally limited by the available computational resources.

Computed Tomography (CT) is fundamentally an inverse problem combining linear operators, regularization and discrete inference. Artificial intelligence has improved reconstruction, denoising and segmentation. Quantum Computing (QC) is typically discussed in terms of computational speed. For CT, the more relevant question is structural compatibility. Several CT subproblems admit formulations that are aligned with quantum-native primitives for structured linear algebra and quadratic optimization. This short paper identifies three directions: reconstruction, denoising and segmentation. It briefly formalizes each component and details a QUBO-based formulation for the segmentation stage implemented in a prototype demonstrator, QUBOSegment. We also report ongoing development of a production-oriented system, NextGenSegment (NGS), built on our Modular Adaptive Processing Infrastructure (MAPI).This work is intentionally scoped as a formulation- and workflow-oriented proof of concept rather than a benchmarking study claiming quantum performance advantage. The aim is to expose the CT community to technically grounded QC formulations and to encourage systematic benchmarking in realistic synchrotron and laboratory settings.

We introduce CuPyMag , an open-source, Python-based framework for large-scale micromagnetic simulations with magnetostriction. CuPyMag solves micromagnetics with finite elements in a GPU-resident workflow in which key operations, such as right-hand-side assembly, spatial derivatives, and volume averages, are tensorized using CuPy’s BLAS-accelerated backend. Benchmark tests show that the GPU solvers in CuPyMag achieve a speedup of up to two orders of magnitude compared to the CPU codes. Its runtime grows linearly/sublinearly with problem size, demonstrating high efficiency. Additionally, CuPyMag uses the Gauss-Seidel projection method for time integration, which not only allows stable time steps (up to 11 ps) but also solves each governing equation with only 1–3 conjugate-gradient iterations without preconditioning. CuPyMag accounts for magnetoelastic coupling and far-field effects arising from the boundary of the magnetic body, both of which play an important role in magnetization reversal in the presence of local defects. CuPyMag solves these computationally-intensive multiphysics simulations with a high-resolution mesh (up to 3M nodes) in under three hours on an NVIDIA H200 GPU. This acceleration enables micromagnetic simulations with non-trivial defect geometries and resolves nanoscale magnetic structures. It expands the scope of micromagnetic simulations towards realistic, large-scale problems that can guide experiments. More broadly, CuPyMag is developed using widely adopted Python libraries, which provide cross-platform compatibility, ease of installation, and accessibility for adaptations to diverse applications. Program Title: CuPyMag CPC Library link to program files: https://doi.org/10.17632/pmkz4vzz7w.1 Developer’s repository link: https://github.com/hongyiguan/CuPyMag OSF repository link: https://osf.io/6nmsv/?view_only=dbd8fa339f8740a6bd009c03499ea23f Licensing provisions: Apache-2.0 Programming language: Python 3 Nature of problem: The program addresses large-scale micromagnetic simulations that are computationally demanding due to the solution of magnetoelastic coupling (mechanical equilibrium with magnetostriction at each time step), far-field demagnetization effects, and arbitrarily shaped inclusion-like defects that require unstructured finite-element meshes. Solution method: CuPyMag implements a GPU-resident, tensorized workflow built on CuPy’s BLAS-accelerated backend. This design exploits highly optimized linear algebra kernels, enabling efficient parallel execution and maintaining high GPU utilization. It uses an ellipsoid theorem to account for far-field demagnetization effects, and the Gauss-Seidel projection method for stable and efficient time integration.

Fast Homomorphic Linear Algebra with BLAS

Multi-point function secret sharing (FSS) is a building block for pseudo-random correlation generators used in novel silent correlation generation methods for various secure multi-party computation applications. However, the main construction used so far is the naive approach to combining several point functions. In this paper, we propose an efficient and natural generalisation of the point function FSS scheme of Boyle et al. 2016 using a tree structure, a pseudorandom generator and systems of linear equations. We propose a new notion of distributed random multi-point function. Our construction splits the distributed multi-point function scheme into a random multi-point function scheme and an algorithm to transform a random output into the desired output value. The resulting scheme, which we call SLAMP-FSS, improves upon the state of the art in terms of calls to a pseudorandom generator (PRG).

We present a general, constructive procedure to find the basis for tensors of arbitrary order subject to linear constraints by transforming the problem to that of finding the nullspace of a linear operator. The proposed method derives from well-established representation-theoretic foundations and utilizes standard numerical linear algebra techniques that are highly optimized and well-behaved. Our primary applications are in mechanics where modulus tensors and so-called structure tensors can be used to characterize anisotropy of functional dependencies on other inputs such as strain. Like modulus tensors, structure tensors are defined by their invariance to transformations by symmetry group generators but have more general applicability. The fully automated method is an alternative to classical, more intuition-reliant methods such as the Pipkin–Rivlin polynomial integrity basis construction. We demonstrate the utility of the procedure by: (a) enumerating elastic modulus tensors for common symmetries and (b) finding the lowest-order structure tensors that can represent all common point groups/crystal classes. Furthermore, we employ these results in two calibration problems using neural network models following classical function representation theory: (a) learning the symmetry class and orientation of a hyperelastic material given stress–strain data and (b) representing strain-dependent anisotropy of the stress response of a soft matrix-stiff fiber composite in a sequence of uniaxial loadings. These two examples demonstrate the utility of the method in model selection and calibration by: (a) determining structural tensors of a selected order across multiple symmetry groups and (b) determining a basis for a given group that allows the characterization of all subgroups. Using a common order in both cases allows sparse regression to operate on a common function representation to select the best-fit symmetry group for the data.

We present HiRep v2, an open-source software suite for high-performance lattice field theory simulations with dynamical Wilson fermions in higher representations of SU ( N g ) gauge groups. This new version fully supports graphics processing unit (GPU) acceleration, optimizing both gauge configuration generation and measurements for NVIDIA and AMD GPUs. HiRep v2 integrates improved gauge and fermionic lattice actions, advanced inverters, and Monte Carlo algorithms, including the (Rational) Hybrid Monte Carlo ((R)HMC) with Hasenbusch acceleration. It exhibits excellent scalability across multiple GPUs and nodes with minimal efficiency loss, making it a robust tool for large-scale simulations in physics beyond the Standard Model. Program Title: HiRep CPC Library link to program files: (to be added by Technical Editor) Developer’s repository link: https://github.com/claudiopica/hirep Licensing provisions(please choose one): GPLv2 Programming language: C, CUDA C, C++ Supplementary material: Journal reference of previous version: * Does the new version supersede the previous version?: * Reasons for the new version:* Summary of revisions: * Nature of problem(approx. 50-250 words): Lattice Field Theory has proven indispensable for the quantitative understanding of strongly coupled quantum field theories, specifically in providing non-perturbative input to phenomenological models describing the dynamics of Quantum Chromodynamics (QCD) for precision tests of the Standard Model. Simulation software libraries for lattice calculations in QCD are readily available and optimized to run on heterogeneous CPU-GPU architectures with good scaling properties on modern supercomputers. In direct searches for physics beyond the Standard Model, software is needed that can simulate gauge groups other than SU (3) and allow for fermions in higher representations, catering, among other things, to classes of composite Higgs and technicolor theories [1], and predictions in the large- N g limit [2]. There exists no other open-source library that implements the option for higher representations of Wilson fermions with general numbers of colors, that has as many capabilities in terms of actions and measurement code as HiRep . Solution method(approx. 50-250 words): A central element of HiRep is the implementation of a Dirac operator and optimized linear algebra routines that generalize to higher representations and general gauge groups. Since the application of the Dirac operator is one of the main bottlenecks of the numerical simulation, optimizations of the Dirac operator are a central part of any high-performance software implementations. In this work, we present a series of significant developments and enhancements to the HiRep suite. These include but are not limited to, the porting of the code to GPUs, see also [3, 4] for previous progress reports, improvements in computational efficiency, and the introduction of new features to further support advanced lattice simulations. In particular, we show that independent of the theory chosen, our implementation of the Dirac operator reaches excellent performance on GPUs and that the software scales well on state-of-the-art supercomputers to a large number of compute nodes, and HiRep is suitable for simulations of light fermionic masses on large lattices. Another recent performance improvement was achieved in [5], optimizing OpenMP support. Additional comments including restrictions and unusual features (approx. 50-250 words): None. * Items marked with an asterisk are only required for new versions of programs previously published in the CPC Program Library.

A review of self-scheduled strategies for numerical linear algebra on GPU

From LSA to LLM: Evolution and limitations of topic modelling methods for biodiversity conservation

The bakery industry faces challenges in optimizing production due to limited raw materials and multiple product types. This study aims to analyze a system of linear equations representing the relationship between product quantities and raw material usage using the Gaussian elimination method. A case study involving bread, sweet bread, and cake production was conducted based on the availability of flour, sugar, and eggs. The system was formulated and solved using Gaussian elimination to obtain its general solution. The results show that the system has infinitely many solutions, indicating multiple feasible production combinations that fully utilize available resources. One practical solution identified is the production of 20 units of bread, 30 units of sweet bread, and 20 units of cake. This combination ensures optimal use of raw materials without waste. The findings demonstrate that Gaussian elimination is an effective method for supporting production planning and decision-making. Overall, linear algebra provides a reliable approach for optimizing resource allocation in the bakery industry.

Abstract For iteratively computing the smallest eigenpair of a huge-scale symmetric matrix, we construct a randomized admissible block coordinate descent (BCD) method by first partitioning the matrix into a number of blocks with respect to its columns, then computing its next iterate through updating the current iterate along with a randomly selected block sub-vector of the affine coordinate direction, and finally obtaining the step-length through minimizing the Rayleigh quotient of the next iterate. This iteration method is indeed a blockwise variant of the admissibly randomized coordinate descent (CD) method proposed and analyzed recently by Bai & Chen (2025, Admissibly randomized coordinate descent methods for computing extreme eigenpairs of symmetric matrices. Numer. Linear Algebra Appl., 32, e70016:1–15), and it can also be considered as a randomized variant of the block CD method. For this class of iteration methods, we rigorously analyze its local and semilocal convergence properties, and solidly demonstrate its computational advantages over the admissibly randomized CD method, as well as the BCD method by numerical experiments.

Quantification of the creatine kinase (CK) forward reaction rate (kf) in the human heart using phosphorus magnetic resonance spectroscopy is clinically important; however, it is limited by long acquisition times, operator subjectivity in analysis, and potential skeletal muscle contamination. This study evaluates if combining compartment-based reconstruction techniques with acquisition-weighted (AW) Triple Repetition Time Saturation Transfer (TRiST) acquisitions could overcome these challenges. Healthy volunteers were scanned with a fully weighted (FW) TRiST protocol twice, and once with an AW TRiST protocol on a 3T MRI. The resulting spectra were reconstructed with conventional Fourier Transform (FT), as well as compartment-based reconstruction techniques: Spectroscopy with Linear Algebra Modeling (SLAM), Spectral Localization by IMaging (SLIM), and an unweighted mean of the FT spectra (ROI-FT). kf values were calculated and compared across reconstruction methods and acquisition types. The cardiac kf values from FW TRiST were 0.21 ± 0.07 s-1 (FT), 0.26 ± 0.08 s-1 (SLAM), 0.26 ± 0.07 s-1 (SLIM), and 0.30 ± 0.10s-1 (ROI-FT). Corresponding values from AW TRiST were 0.27 ± 0.07 s-1, 0.25 ± 0.05 s-1, 0.25 ± 0.04 s-1, and 0.24 ± 0.08 s-1, respectively. No significant differences were observed between FW and AW results. A significant decrease in cardiac PCr/ATP ratios was observed for SLAM and SLIM reconstructed data, suggesting decreased signal contamination from skeletal muscle. Compartment-based reconstruction techniques minimize the operator subjectivity present in the current FT method of analyzing TRiST experiments, in addition to reducing skeletal muscle contamination. When combined with an AW acquisition, scan times were reduced by 47% without compromising kf accuracy. This method provides a more robust and efficient evaluation of in vivo cardiac metabolism.

Modons are a type of dipolar vortex consisting of two counter-rotating regions of fluid, moving together through self-advection. Here, we present a semi-analytical method for finding modon solutions to the three-dimensional quasi-geostrophic equations. This method works by deriving a system of integral equations governing three-dimensional modon solutions and reducing this system to a linear algebra problem which may be solved using standard methods. We show that these solutions reduce to known analytical solutions in some simple setups, and present new fully-three-dimensional modon solutions consisting of elevated buoyancy signatures on the top and bottom surface.

We investigate the flat-foldability of 2 × $n$ map folding patterns through the lens of order theory and linear algebra. We model the problem as determining the existence of a linear extension of a poset compatible with topological constraints, viewing the 2 × $n$ map folding as an order extension of the 1 × 2$n$ strip folding. Building on the fact that the configuration space of a strip forms a lattice, we treat the valid map foldings as a specific subset defined by geometric obstructions. To rigorously detect these obstructions (self-intersections), we introduce an algebraic invariant over the field $\mathbb{F}_2$. We represent the overlapping order as a transitive closure matrix over the Boolean semiring and formulate the intersection-avoidance condition as the vanishing of a bilinear form over $\mathbb{F}_2$. This framework yields a purely combinatorial and linear-algebraic characterization of flat-foldability, allowing for the efficient enumeration of all valid 2 × $n$ patterns by filtering the strip lattice.

Unmanned Aerial Vehicles (UAVs) are complex nonlinear systems characterized by high dimensionality. They are prone to aerodynamic effects, structural dynamics, actuation constraints, and networked interactions, requiring advanced mathematical models and precise control. Their governing equations involve nonlinear rigid-body dynamics coupled with fluid and elasticity models, while modern architectures introduce redundancy that creates constrained mappings between generalized forces and actuator inputs. Coordinated UAV teams add another layer of mathematical structure through graph-based interaction models that determine consensus, formation keeping, and distributed stability. These characteristics give rise to several interconnected challenges. High-fidelity aerodynamic and aeroelastic solvers provide accurate results; however, these are computationally intensive, motivating the development of reduced-order models and data-driven approximations that preserve dominant physical behavior. Methods for quantifying uncertainty support robustness assessments by characterizing the effects of parametric variation and model form error. At the actuation level, control allocation problems rely on constrained linear algebra, convex optimization, and dynamic formulations to ensure feasible and stable realization of command forces and moments. In multi-agent systems, the spectral properties of adjacency and Laplacian matrices govern convergence and cooperative behavior. This article reviews the state of the art in these areas, highlights the mathematical foundations that relate them, and provides a coherent perspective on the methods that enable reliable modeling and control of modern UAV systems.

ABSTRACT The (signed) projective cubes, as a special class of graphs closely related to the hypercubes, are on the crossroad of geometry, algebra, discrete mathematics and linear algebra. Defined as Cayley graphs on binary groups, they represent basic linear dependencies. Capturing the four‐color theorem as a homomorphism target they show how mapping of discrete objects, namely graphs, may relate to special mappings of plane to projective spaces of higher dimensions. In this work, viewed as a signed graph, first we present a number of equivalent definitions each of which leads to a different development. In particular, the new notion of common product of signed graphs is introduced which captures both Cartesian and tensor products of graphs. With a nonstandard use of homomorphism between signed projective cubes, we show that they all have circular chromatic number 4. Observing that the 4‐color theorem is about mapping planar graphs into , we study some conjectures in extension of 4CT and show the importance of extended double cover operation in formulating such conjectures. As a particular corollary we build a highly symmetric triangle‐free graph on 12 vertices admitting a homomorphism from every planar graph of odd girth at least 7. Considering their connection to algebraic geometry, we observe that , widely known as the Clebsh graph, also known as Greenwood‐Gleason graph, is the intersection graph of the 16 straight lines of an algebraic surface known as Segre surface, which is a Del Pezzo surface of degree 4. Noting that the Clebsch surface is one of the most symmetric presentations of a cubic surface all of which contains 27 lines. Hence, from hereafter, we believe, a proper name for should be Segre graph.

The growth problem in Gaussian elimination (GE) remains a foundational question in numerical analysis and numerical linear algebra. Wilkinson resolved the growth problem in GE with partial pivoting (GEPP) in his initial analysis from the 1960s, while he was only able to establish an upper bound for the GE with complete pivoting (GECP) growth problem. The GECP growth problem has seen a spike in recent interest, culminating in improved lower and upper bounds established by Bisain, Edelman, and Urschel in 2023, but still remains far from being fully resolved. Due to the complex dynamics governing the location of GECP pivots, analysis of GECP growth for particular input matrices often estimates the actual growth rather than computes the growth exactly. We present a class of dense random butterfly matrices for which we can compute the exact GECP growth. We extend previous results that established exact growth computations for butterfly matrices when using GEPP and GE with rook pivoting (GERP) to now also include GECP for structured subclasses of inputs. Moreover, we present a new method to construct random Hadamard matrices using butterfly matrices.

Exploiting intra-core heterogeneity for high-performance dense linear algebra on Huawei Ascend 910 NPUs