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Developing numerical exact solvers for open quantum systems is a challenging task due to the non-perturbative and non-Markovian nature when coupling to structured environments. The Feynman–Vernon influence functional approach is a powerful analytical tool to study the dynamics of open quantum systems. Numerical treatments of the influence functional including the quasi-adiabatic propagator technique and the tensor-network-based time-evolving matrix product operator method have proven to be efficient in studying open quantum systems with bosonic environments. However, the numerical implementation of the fermionic path integral suffers from the Grassmann algebra involved. In this work, we present a detailed introduction to the Grassmann time-evolving matrix product operator method for fermionic open quantum systems. In particular, we introduce the concepts of Grassmann tensor, signed matrix product operator, and Grassmann matrix product state to handle the Grassmann path integral. Using the single-orbital Anderson impurity model as an example, we review the numerical benchmarks for structured fermionic environments for real-time nonequilibrium dynamics, real-time and imaginary-time equilibration dynamics, and its application as an impurity solver. These benchmarks show that our method is a robust and promising numerical approach to study strong coupling physics and non-Markovian dynamics. It can also serve as an alternative impurity solver to study strongly correlated quantum matter with dynamical mean-field theory. [ABSTRACT FROM AUTHOR]

The matrix product state (MPS) Ansatz offers a promising approach for finding the ground state of molecular Hamiltonians and solving quantum chemistry problems. Building on this concept, the proposed technique of quantum circuit MPS (QCMPS) enables the simulation of chemical systems using a relatively small number of qubits. In this study, we enhance the optimization performance of the QCMPS Ansatz by employing the variational quantum imaginary time evolution (VarQITE) approach. Guided by McLachlan's variational principle, the VarQITE method provides analytical metrics and gradients, resulting in improved convergence efficiency and robustness of the QCMPS. We validate these improvements numerically through simulations of H2, H4, and LiH molecules. In addition, given that VarQITE is applicable to non-Hermitian Hamiltonians, we evaluate its effectiveness in preparing the ground state of transcorrelated Hamiltonians. This approach yields energy estimates comparable to the complete basis set (CBS) limit while using even fewer qubits. In particular, we perform simulations of the beryllium atom and LiH molecule using only three qubits, maintaining high fidelity with the CBS ground state energy of these systems. This qubit reduction is achieved through the combined advantages of both the QCMPS Ansatz and transcorrelation. Our findings demonstrate the potential practicality of this quantum chemistry algorithm on near-term quantum devices. [ABSTRACT FROM AUTHOR]

It is challenging to simulate open quantum systems that are connected to a reservoir through multiple channels. For example, vibrations may induce fluctuations in both energy gaps and electronic couplings, which represent two independent channels of system–bath couplings. Systems of this kind are ubiquitous in the processes of excited state radiationless decay. Combined with density matrix renormalization group (DMRG) and matrix product states (MPS) methods, we develop an interaction-picture chain mapping strategy for vibrational reservoirs to simulate the dynamics of these open systems, resulting in time-dependent spatially local system–bath couplings in the chain-mapped Hamiltonian. This transformation causes the entanglement generated by the system–bath interactions to be restricted within a narrow frequency window of vibrational modes, enabling efficient DMRG/MPS dynamical simulations. We demonstrate the utility of this approach by simulating singlet fission dynamics using a generalized spin-boson Hamiltonian with both diagonal and off-diagonal system–bath couplings. This approach generalizes an earlier interaction-picture chain mapping scheme, allowing for efficient and exact simulation of systems with multi-channel system–bath couplings using matrix product states, which may further our understanding of nonlocal exciton–phonon couplings in exciton transport and the non-Condon effect in energy and electron transfer. [ABSTRACT FROM AUTHOR]

We introduce a Python package based on matrix product states (MPS) to simulate both the time-dependent Schrödinger equation (TDSE) and the hierarchical equations of motion (HEOM). The wave function in the TDSE or the reduced density operator/auxiliary density operators in the HEOM are represented using MPS. A matrix product operator (MPO) is then constructed to represent the Hamiltonian in the TDSE or the generalized Liouvillian in the HEOM. The fourth-order Runge–Kutta method and the time-dependent variational principle are used to propagate the MPS. Several examples, including the nonadiabatic interconversion dynamics of the pyrazine molecule, excitation energy transfer dynamics in molecular aggregates and photosynthetic light-harvesting complexes, the spin-boson model, a laser driven two-state model, the Holstein model, and charge transport in the Anderson impurity model, are presented to demonstrate the capability of the package. [ABSTRACT FROM AUTHOR]

We investigate the application of matrix product state (MPS) representations of the influence functionals (IFs) for the calculation of real-time equilibrium correlation functions in open quantum systems. Focusing specifically on the unbiased spin-boson model, we explore the use of IF-MPSs for complex time propagation, as well as IF-MPSs for constructing correlation functions in the steady state. We examine three different IF approaches: one based on the Kadanoff–Baym contour targeting correlation functions at all times, one based on a complex contour targeting the correlation function at a single time, and a steady state formulation, which avoids imaginary or complex times, while providing access to correlation functions at all times. We show that within the IF language, the steady state formulation provides a powerful approach to evaluate equilibrium correlation functions. [ABSTRACT FROM AUTHOR]

The MPSDynamics.jl package provides an easy-to-use interface for performing open quantum systems simulations at zero and finite temperatures. The package has been developed with the aim of studying non-Markovian open system dynamics using the state-of-the-art numerically exact Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm based on environment chain mapping. The simulations rely on a tensor network representation of the quantum states as matrix product states (MPS) and tree tensor network states. Written in the Julia programming language, MPSDynamics.jl is a versatile open-source package providing a choice of several variants of the Time-Dependent Variational Principle method for time evolution (including novel bond-adaptive one-site algorithms). The package also provides strong support for the measurement of single and multi-site observables, as well as the storing and logging of data, which makes it a useful tool for the study of many-body physics. It currently handles long-range interactions, time-dependent Hamiltonians, multiple environments, bosonic and fermionic environments, and joint system–environment observables. [ABSTRACT FROM AUTHOR]

Evaluation of the charge transport property of organic semiconductors requires exact quantum dynamics simulation of large systems. We present a numerically nearly exact approach to investigate carrier transport dynamics in organic semiconductors by extending the non-Markovian stochastic Schrödinger equation with complex frequency modes to a forward–backward scheme and by solving it using the matrix product state (MPS) approach. By utilizing the forward–backward formalism for noise generation, the bath correlation function can be effectively treated as a temperature-independent imaginary part, enabling a more accurate decomposition with fewer complex frequency modes. Using this approach, we study the carrier transport and mobility in the one-dimensional Peierls model, where the nonlocal electron–phonon interaction is taken into account. The reliability of this approach was validated by comparing carrier diffusion motion with those obtained from the hierarchical equations of motion method across various parameter regimes of the phonon bath. The efficiency was demonstrated by the modest virtual bond dimensions of MPS and the low scaling of the computational time with the system size. [ABSTRACT FROM AUTHOR]

We describe a general-purpose computational toolkit for simulating open quantum systems, which provides numerically exact solutions for composites of zero-dimensional quantum systems that may be strongly coupled to multiple, quite general non-Markovian environments. It is based on process tensor matrix product operators (PT-MPOs), which efficiently encapsulate environment influences. The code features implementations of several PT-MPO algorithms, in particular Automated Compression of Environments for general environments comprised of independent modes as well as schemes for generalized spin boson models. The latter includes a divide-and-conquer scheme for periodic PT-MPOs, which enable million time step simulations for realistic models. PT-MPOs can be precalculated and reused for efficiently probing different time-dependent system Hamiltonians. They can also be stacked together and combined to provide numerically complete solutions of small networks of open quantum systems. The code is written in C++ and is fully controllable by configuration files, for which we have developed a versatile and compact human-readable format. [ABSTRACT FROM AUTHOR]

A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by n, the algorithm uses O(n1.43) field operations, breaking through the 3/2 barrier in the exponent for the first time. The previous fastest algebraic algorithms, due to Brent and Kung in 1978, require O(n1.63) field operations in general, and n3/2+o(1) field operations in the special case of power series over a field of large enough characteristic. If cubic-time matrix multiplication is used, the new algorithm runs in n5/3+o(1) operations, while previous ones run in O(n2) operations. Our approach relies on the computation of a matrix of algebraic relations that is typically of small size. Randomization is used to reduce arbitrary input to this favorable situation. [ABSTRACT FROM AUTHOR]

We propose an approach to represent the second-quantized electronic Hamiltonian in a compact sum-of-products (SOP) form. The approach is based on the canonical polyadic decomposition of the original Hamiltonian projected onto the sub-Fock spaces formed by groups of spin–orbitals. The algorithm for obtaining the canonical polyadic form starts from an exact sum-of-products, which is then optimally compactified using an alternating least squares procedure. We discuss the relation of this specific SOP with related forms, namely the Tucker format and the matrix product operator often used in conjunction with matrix product states. We benchmark the method on the electronic dynamics of an excited water molecule, trans-polyenes, and the charge migration in glycine upon inner-valence ionization. The quantum dynamics are performed with the multilayer multiconfiguration time-dependent Hartree method in second quantization representation. Other methods based on tree-tensor Ansätze may profit from this general approach. [ABSTRACT FROM AUTHOR]

block2 is an open source framework to implement and perform density matrix renormalization group and matrix product state algorithms. Out-of-the-box it supports the eigenstate, time-dependent, response, and finite-temperature algorithms. In addition, it carries special optimizations for ab initio electronic structure Hamiltonians and implements many quantum chemistry extensions to the density matrix renormalization group, such as dynamical correlation theories. The code is designed with an emphasis on flexibility, extensibility, and efficiency and to support integration with external numerical packages. Here, we explain the design principles and currently supported features and present numerical examples in a range of applications. [ABSTRACT FROM AUTHOR]

In the past decade, the quantum chemical version of the density matrix renormalization group method has established itself as the method of choice for strongly correlated molecular systems. However, despite its favorable scaling, in practice, it is not suitable for computations of dynamic correlation. Several approaches to include that in post-DMRG methods exist; in our group, we focused on the tailored coupled cluster (TCC) approach. This method works well in many situations; however, in exactly degenerate cases (with two or more determinants of equal weight), it exhibits a bias toward the reference determinant representing the Fermi vacuum. Although sometimes it is possible to use a compensation scheme to avoid this bias for energy differences, it is certainly a drawback. In order to overcome this bias of the TCC method, we have developed a Hilbert-space multireference version of tailored CC, which can treat several determinants on an equal footing. We have implemented and compared the performance of three Hilbert-space multireference coupled cluster (MRCC) variants—the state universal one and the Brillouin–Wigner and Mukherjee's state specific ones. We have assessed these approaches on the cyclobutadiene and tetramethyleneethane molecules, which are both diradicals with exactly degenerate determinants at a certain geometry. We have also investigated the sensitivity of the results on the orbital rotation of the highest occupied and lowest unoccupied molecular orbital (HOMO–LUMO) pair, as it is well known that Hilbert-space MRCC methods are not invariant to such transformations. [ABSTRACT FROM AUTHOR]

Tensor networks have historically proven to be of great utility in providing compressed representations of wave functions that can be used for the calculation of eigenstates. Recently, it has been shown that a variety of these networks can be leveraged to make real time non-equilibrium simulations of dynamics involving the Feynman–Vernon influence functional more efficient. In this work, a tensor network is developed for non-perturbatively calculating the equilibrium correlation function for open quantum systems using the path integral methodology. These correlation functions are of fundamental importance in calculations of rates of reactions, simulations of response functions and susceptibilities, spectra of systems, etc. The influence of the solvent on the quantum system is incorporated through an influence functional, whose unconventional structure motivates the design of a new optimal matrix product-like operator that can be applied to the so-called path amplitude matrix product state. This complex-time tensor network path integral approach provides an exceptionally efficient representation of the path integral, enabling simulations for larger systems strongly interacting with baths and at lower temperatures out to longer time. The derivation, design, and implementation of this method are discussed along with a wide range of illustrations ranging from rate theory and symmetrized spin correlation functions to simulation of response of the Fenna–Matthews–Olson complex to light. [ABSTRACT FROM AUTHOR]

Matrix diagonalization is almost always involved in computing the density matrix needed in quantum chemistry calculations. In the case of modest matrix sizes (≲4000), performance of traditional dense diagonalization algorithms on modern GPUs is underwhelming compared to the peak performance of these devices. This motivates the exploration of alternative algorithms better suited to these types of architectures. We newly derive, and present in detail, an existing Chebyshev expansion algorithm [Liang et al., J. Chem. Phys. 119, 4117–4125 (2003)] whose number of required matrix multiplications scales with the square root of the number of terms in the expansion. Focusing on dense matrices of modest size, our implementation on GPUs results in large speed ups when compared to diagonalization. Additionally, we improve upon this existing method by capitalizing on the inherent task parallelism and concurrency in the algorithm. This improvement is implemented on GPUs by using CUDA and HIP streams via the MAGMA library and leads to a significant speed up over the serial-only approach for smaller (≲1000) matrix sizes. Finally, we apply our technique to a model system with a high density of states around the Fermi level, which typically presents significant challenges. [ABSTRACT FROM AUTHOR]

It is known that every singular matrix of a finite degree defined over a field can be written as a product of a finite number of idempotent matrices. Insipired by this result we investigate the same problem for $ \mathbb {N}\times \mathbb {N} $ N × N matrices. It turns out that in this case the analogous theorem does not hold. [ABSTRACT FROM AUTHOR]

This study is a follow-up of two our papers (Appl. Math. Lett. 126 (2022) and MCMA 29:4 (2023)), where we developed a vector randomized algorithms with iterative refinement for large system of linear algebraic equations. We focus in this paper on the application of the vector randomized iterative refinement algorithm to boundary integral equations that solve interior Dirichlet and exterior Neumann boundary value problems for 2D Laplace equation. [ABSTRACT FROM AUTHOR]

In [H. H. Nguyen and R. Van Peski, Universality for cokernels of random matrix products, Adv. Math. 438 2024, Paper No. 109451], Nguyen and Van Peski raised the question of whether the surjective flag of ℤ p {{\mathbb{Z}}_{p}} -modules modeled by cok ⁢ ( M 1 ⁢ … ⁢ M k ) ↠ ⋯ ↠ cok ⁢ ( M 1 ) \mathrm{cok}(M_{1}\dots M_{k})\twoheadrightarrow\cdots\twoheadrightarrow% \mathrm{cok}(M_{1}) for independent random matrices M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p}) satisfies the Cohen–Lenstra heuristic. We answer the question affirmatively when M 1 , … , M k {M_{1},\dots,M_{k}} follow the Haar measure, and our proof demonstrates how classical ideas in Cohen–Lenstra heuristic adapt naturally to the flag setting. We also prove an analogue for non-square matrices. [ABSTRACT FROM AUTHOR]

We show that the quantum approximate optimization algorithm (QAOA) for higher-order, random coefficient, heavy-hex compatible spin glass Ising models has strong parameter concentration across problem sizes from 16 up to 127 qubits for p = 1 up to p = 5, which allows for computationally efficient parameter transfer of QAOA angles. Matrix product state (MPS) simulation is used to compute noise-free QAOA performance. Hardware-compatible short-depth QAOA circuits are executed on ensembles of 100 higher-order Ising models on noisy IBM quantum superconducting processors with 16, 27, and 127 qubits using QAOA angles learned from a single 16-qubit instance using the JuliQAOA tool. We show that the best quantum processors find lower energy solutions up to p = 2 or p = 3, and find mean energies that are about a factor of two off from the noise-free distribution. We show that p = 1 QAOA energy landscapes remain very similar as the problem size increases using NISQ hardware gridsearches with up to a 414 qubit processor. [ABSTRACT FROM AUTHOR]

The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for n × n Boolean matrices of the form O (n 2. 5 7 5) has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the classical computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time Õ (n 2 + λ / 2) = Õ (n 2. 4 3 4) , where λ satisfies the equation ω (1 , λ , 1) = 1 + 1. 5 λ and ω (1 , λ , 1) is the exponent of the multiplication of an n × n λ matrix by an n λ × n matrix. Next, we consider a relaxed version of the MW problem (in the classical model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most ℓ in time Õ ((n / ℓ) n ω (1 , log n ℓ , 1)). Then, by reducing the relaxed problem to the so called k -witness problem, we provide an algorithm that reports for each non-zero entry C [ i , j ] of the product matrix C a witness of rank O (⌈ W C (i , j) / k ⌉) , where W C (i , j) is the number of witnesses for C [ i , j ] , with high probability. The algorithm runs in Õ (n ω k 0. 4 6 5 3 + n 2 + o (1) k) time, where ω = ω (1 , 1 , 1). [ABSTRACT FROM AUTHOR]

The issue of opinion sharing and formation has received considerable attention in the academic literature, and a few models have been proposed to study this problem. However, existing models are limited to the interactions among nearest neighbors, with those second, third, and higher-order neighbors only considered indirectly, despite the fact that higher-order interactions occur frequently in real social networks. In this paper, we develop a new model for opinion dynamics by incorporating long-range interactions based on higher-order random walks that can explicitly tune the degree of influence of higher-order neighbor interactions. We prove that the model converges to a fixed opinion vector, which may differ greatly from those models without higher-order interactions. Since direct computation of the equilibrium opinion is computationally expensive, which involves the operations of huge-scale matrix multiplication and inversion, we design a theoretically convergence-guaranteed estimation algorithm that approximates the equilibrium opinion vector nearly linearly in both space and time with respect to the number of edges in the graph. We conduct extensive experiments on various social networks, demonstrating that the new algorithm is both highly efficient and effective. [ABSTRACT FROM AUTHOR]

In-memory processing is becoming a popular method to alleviate the memory bottleneck of the Von Neumann computing model. With the goal of improving both latency and energy cost associated with such in-memory processing, emerging non-volatile memory technologies, such as Spintronic magnetic memory, are of particular interest, as they can provide a near-SRAM read/write performance and eliminate nearly all static energy without experiencing any endurance limitations. Spintronic Racetrack Memory (RM) further addresses density concerns of spin-transfer torque memory (STT-MRAM). Moreover, it has recently been demonstrated that portions of RM nanowires can function as a polymorphic gate, which can be leveraged to implement multi-operand bulk bitwise operations. With more complex control, they can also be leveraged to build arithmetic integer and floating point processing in memory (PIM) primitives. This article proposes SPIMulator, a Spintronic PIM simulator that can simulate the storage and PIM architecture of executing PIM commands in Racetrack memory. SPIMulator functionally models the polymorphic gate properties recently proposed for Racetrack memory, which allows transverse access that determines the number of "1"s in a segment of each Racetrack nanowire. From this simulation, SPIMulator can report real-time performance statistics such as cycle count and energy. Thus, SPIMulator simulates the multi-operand bit-wise logic operations recently proposed and can be easily extended to implement new PIM operations as they are developed. Due to the functional nature of SPIMulator, it can serve as a programming environment that allows development of PIM-based codes for verification of new acceleration algorithms. We demonstrate the value of SPIMulator through the modeling and estimations of performance and energy consumption of a variety of example applications, including the Advanced Encryption Standard (AES) for encryption primarily based on logical and look-up operations; multiplication of matrices, a frequent requirement in scientific, signal processing, and machine learning algorithms; and bitmap indices, a common search table employed for database lookups. [ABSTRACT FROM AUTHOR]

Blockchain is a technology that combines a set of properties to guarantee network security, transparency, and visibility, including a decentralized structure, distributed notes and storage mechanism, consensus algorithm, intelligent contracts, and asymmetric encryption. Supply chain management activities, including supply chain management provenance, business process reengineering, and security enhancement, have enormous potential to be transformed by blockchain. This research uses the integrated interpretive structural modeling–cross‐impact matrix multiplication applied to classification (ISM‐MICMAC) technique to ascertain the hierarchical relationships and to comprehend the severity of interrelationships among various components in tackling our research questions. In MICMAC analysis, the enablers were classified into four categories based on their dependence and driving powers. The combined ISM‐MICMAC methodology employed for this study relies on experts' individual evaluations and subjective assessments. Therefore, even with extreme caution, it is impossible to guarantee that the results are entirely devoid of personal biases. To further validate the linkages discovered in this study, we suggest employing more multiple‐criteria decision‐making (MCDM) methodologies and comparing the results with those of our research. Another method for confirming the results of the current study is to use an empirical research design based on survey methods. [ABSTRACT FROM AUTHOR]

In 2020, Cao et al. proved that any repeated-root constacyclic code is monomially equivalent to a matrix product code of simple-root constacyclic codes. In this paper, we study a family of matrix product codes with wonderful properties, which is a generalization of linear codes obtained from the [ u + v | u - v ] -construction and [ u + v | λ - 1 u - λ - 1 v ] -construction. Then we show that any λ -constacyclic code (not necessary repeated-root λ -constacyclic code) of length N over the finite field F q with gcd (q - 1 ord (λ) , N) ≥ 2 , where ord (λ) is the order of λ in the cyclic group F q ∗ = F q \ { 0 } , is a matrix product code of some constacyclic codes. It is a highly interesting question that the existence of sequences { C 1 , C 2 , C 3 ,... } of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances, i.e., C i is an [ n (C i) , k (C i) , d (C i) ] q -linear code such that lim i → + ∞ n (C i) = + ∞ and lim i → + ∞ d (C i) n (C i) > 0. Based on the [ u + v | λ - 1 u - λ - 1 v ] -construction, we construct several families of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances by using Reed-Muller codes, projective Reed-Muller codes. And we construct some new Euclidean isodual λ -constacyclic codes with square-root-like minimum Hamming distances from Euclidean self-dual cyclic codes and Euclidean self-dual negacyclic codes by monomial equivalences. [ABSTRACT FROM AUTHOR]

Identifying barriers aligned with the development objectives and their interrelationships in Citizen-Centric Smart City (CCSC) is pivotal for devising tailored developmental strategies. Hence, this article aims to conduct a comprehensive exploration of these elements, with a specific focus on citizens’ sense of gain (CSG), a crucial component of citizen-centricity. The study conceptualizes the CCSC development objectives and identifies 20 barriers that align with them by employing a literature review and the Delphi method. Besides, the barriers are analyzed by adopting the Decision Making Trial and Evaluation Laboratory (DEMATEL)-Interpretative Structural Modeling (ISM)-Cross-impact matrix multiplication applied to classification (MICMAC) method. The study reveals that CCSC in China should focus on making CSG more comprehensive and sustainable in five dimensions decomposed from the objectives, “Lack of departments to interface with citizens’ feedback” is identified as the barrier with the highest causality and centrality, and “Lack of guiding framework and policy system for CCSC” is a fundamental barrier to the China’s CCSC development. By identifying the critical barriers and suggesting practical solutions, this study can help policymakers and practitioners worldwide to prioritize their efforts towards creating more sustainable and CCSCs. [ABSTRACT FROM AUTHOR]

This paper proposes an approach for computing the network's equilibrium point related to the fault clearing time in transient stability studies. The computation of this point is not a trivial task, particularly when the algebraic network's equations are expressed in the power balance form. A natural attempt to solve this problem is using Newton's method. However, convergence issues are found because of the lack of a general strategy for initializing nodal voltages at the clearing time. This problem has not been widely discussed in the existing literature and, therefore, is comprehensively analyzed in this paper. Furthermore, the paper proposes the use of a network's model based on current injections and an extended admittance matrix to overcome the problem. This model is efficiently solved via the fixed-point iteration method, which involves factorization of the extended admittance matrix into the product of a lower triangular matrix [L] and an upper triangular matrix [U]. This solution executes a just once and only forward–backward substitution during the iterative solution process. Case studies clearly demonstrate the proposal's effectiveness in computing the equilibrium point in operating conditions where Newton's method fails to converge. [ABSTRACT FROM AUTHOR]

Sparse matrix–matrix multiplication (SpMM) is essential for deep learning models and scientific computing. Recently, Tensor Cores (TCs) on GPUs, originally designed for dense matrix multiplication with mixed precision, have gained prominence. However, utilizing TCs for SpMM is challenging due to irregular memory access patterns and a varying number of non-zero elements in a sparse matrix. To improve data locality, previous studies have proposed reordering sparse matrices before multiplication, but this adds computational overhead. In this paper, we propose Tensor Core-Adapted SpMM (TCA-SpMM), which leverages TCs without requiring matrix reordering and uses the compressed sparse row (CSR) format. To optimize TC usage, the SpMM algorithm's dot product operation is transformed into a blocked matrix–matrix multiplication. Addressing load imbalance and minimizing data movement are critical to optimizing the SpMM kernel. Our TCA-SpMM dynamically allocates thread blocks to process multiple rows simultaneously and efficiently uses shared memory to reduce data movement. Performance results on sparse matrices from the Deep Learning Matrix Collection public dataset demonstrate that TCA-SpMM achieves up to 29.58× speedup over state-of-the-art SpMM implementations optimized with TCs. [ABSTRACT FROM AUTHOR]

The Heisenberg uncertainty relation, together with Robertson's generalisation, serves as a fundamental concept in quantum mechanics, showing that noncommutative pairs of observables cannot be measured precisely. In this study, we explore the Robertson-type uncertainty relations to demonstrate their effectiveness in establishing a series of thermodynamic uncertainty relations and quantum speed limits in open quantum dynamics. The derivation utilises a scaled continuous matrix product state representation that maps the time evolution of the quantum continuous measurement to the time evolution of the system and field. Specifically, we consider the Maccone–Pati uncertainty relation, a refinement of the Robertson uncertainty relation, to derive thermodynamic uncertainty relations and quantum speed limits. These newly derived relations, which use a state orthogonal to the initial state, yield bounds that are tighter than previously known bounds. Moreover, we consider the Robertson–Schrödinger uncertainty, which extends the Robertson uncertainty relation. Our findings not only reinforce the significance of the Robertson-type uncertainty relations, but also expand its applicability in identifying uncertainty relations in open quantum dynamics. [ABSTRACT FROM AUTHOR]

AbstractStarch-based foods have a large consumer base worldwide. It is known that the quality of freeze-dried starch-based products is subject to deterioration during the production process. The present study was unprecedented carried out to examine the effects of freeze-drying on the cross-linking between components of gelatinized starch-based system. The results showed that the retrogradation of starch in the process of dehydration, the weakening of the binding capacity between the substrate and water of the freeze-dried product and the weakening of the interaction strength between the components of the product were the key mechanisms leading to the quality deterioration of the wheat starch-based conditioned food during the freeze-drying. The addition of lipid and protein can inhibit starch retrogradation during freeze-drying, and the inhibition effect of oil is better. However, the intervention of oil and fat will cause dehydrated starch to re-adsorb the melted oil/water mixture during the freeze-drying process, resulting in the deterioration of product quality. The intervention of protein will improve the binding ability of freeze-dried wheat starch-based product matrix to water. In the ternary system, the intervention of protein will weaken the inhibition effect of lipid on retrogradation of starch in the freeze-drying process. In addition, the lower drying temperature will lead to the decrease of the interaction strength between the components of ternary mixed system. Avoiding the melting of frozen materials during drying is the key control point to improve the quality of freeze-dried wheat starch-based products. [ABSTRACT FROM AUTHOR]

We establish the mathematical fundamentals for a unified description of curvature, torsion, and non-metricity 2-forms in the way extending the so-called Möbius representation of the affine group, which is the method to convert the semi-direct product into the ordinary matrix product, to revive the fertility of gauge theories of gravity. First of all, we illustrate the basic concepts for constructing the metric-affine geometry. Then the curvature and torsion 2-forms are described in a unified manner by using the Cartan connection of the Möbius representation of the affine group. In this unified-description, the curvature and torsion are derived by Cartan’s structure equation with respect to a common connection 1-form. After that, extending the Möbius representation, the dilation and shear 2-forms, or equivalently, the non-metricity 2-form, are introduced in the same unified manner. Based on the unified-description established in this paper, introducing a new group parametrization and applying the Inönü–Wigner group contraction to the full theory, the relationships among symmetries, geometric quantities, and geometries are investigated with respect to the three gauge groups: the metric-affine group and its extension, and an extension of the (anti)-de Sitter group in which the non-metricity exists. Finally, possible applications to theories of gravity are briefly discussed. [ABSTRACT FROM AUTHOR]

Although the market share of domestic children's furniture is increasing annually, some potential problems limit its long-term and stable development, and there is still a gap in China compared with foreign countries. This study focused on the demand preferences for growable children's beds and examined the design features that influence these preferences. This study introduces a combination of Hierarchical Analyses (AHP), Quality Function Development (QFD), and the Platts Conceptual Decision Matrix (PUGH) into the innovative design of a research model for children's furniture (AHP-QFD-PUGH). This study screened and classified the decision-making indicators obtained from the research, ranked their importance by quantitative calculation, and finally proposed an optimal design solution. Additionally, to further study the structural characteristics, the function-behavior-structure (FBS) model served as a supplementary analysis tool to effectively circumvent subjective factors in product design. This integrated model accurately explored user needs and product characteristics, providing substantial guidance and new ideas for optimizing the design of growable children's beds and enhancing growth of the children's furniture industry. [ABSTRACT FROM AUTHOR]

Oscillators exhibiting an Andronov–Hopf bifurcation are candidates to mimic the functionality of the cochlea, since the transfer response of these oscillators is compressive and frequency selective. The former implies that small stimuli are amplified and strong stimuli are attenuated, while the latter means that the oscillator only reacts in a (small) frequency band. However, this implies that many oscillators are needed to cover a relevant frequency band. By introducing the notion of tunable characteristic frequencies, i.e., the characteristic frequency can be adjusted by a controllable input, the number of oscillators can be eventually reduced. Subsequently, the tunability enhancement of coupled oscillators is investigated by analyzing the local dynamics of a network of oscillators. For this, necessary conditions for the emergence of Andronov–Hopf bifurcations are determined for networks consisting of two groups, i.e., a group is a network of identical oscillators. By choosing the eigenvalues of the product of the cross-coupling matrix as bifurcation parameters and exploiting the structure of the transfer matrix of this network, the critical points and, thus, the characteristic frequency at this point can be derived. Tunability of the characteristic frequency is then enabled by controlling the asymmetry between the groups of oscillators. [ABSTRACT FROM AUTHOR]

Although not expected to be used due to restrictions raised on their usage, Persisted Organic Pollutants (POP) such as organochlorinated pesticides (OCPs) can be found in several matrices, even nowadays. The lack of biodegradation and, furthermore, their persistence in the environment result in the possible occurrence of these lipophilic toxins in several matrices, from environmental samples and foods to human milk. The current review focuses on the usage of hyphenated techniques for the determination and monitoring of OCPs in several matrices, such as milk—both animal and human milk. The lipid matrix of milk and dairy products favors the possible bioaccumulation of the above pollutants, and the complex matrix of the dairy products is a challenge for method development. Additionally, spectroscopic methods—mainly Nuclear Magnetic Resonance (NMR)-based metabolomics—for biomonitoring of OCPs persistence, bioaccumulation, and effect of possible exposure, along with NMR usage in several methods developed, are also presented and discussed. Finally, we introduce and present the metabolomic approach for OCPs and other POPs in lipid matrices. [ABSTRACT FROM AUTHOR]

Purpose: The ability of a business to outperform its rivals is known as its competitive edge, and it presents special difficulties in the context of the "digital revolution," or the fourth industrial revolution. To obtain a competitive edge in the startup operations 4.0 era, this study aims to examine the organizational, technological and competence-related challenges presented by Industry 4.0. It does this by concentrating on the tools, competencies, methods, approaches, tools and strategies that are crucial. Using the Total Interpretive Structural Modeling (TISM) technique, the goal is to find, analyze and classify enablers for startup operations 4.0. Design/methodology/approach: A closed-ended questionnaire and planned interviews were used in the data collection process. In startup operations 4.0, the cross-impact matrix multiplication applied to classification method is used to rank and categorize competitive advantage factors, whereas the TISM technique is used to analyze how components interact. Findings: The study highlights the critical significance of the "Internet of Things (IoT)," "information technologies," "technological platforms," "employee empowerment," "augmented reality (AR)" and "operational technologies" in its identification of 12 enablers for startup operations 4.0. Research limitations/implications: The main focus of the study is on the variables that affect startup operations 4.0's competitive advantage. Practical implications: Academics and important stakeholders can better understand the factors influencing competitive advantage in startup operations 4.0 with the help of this research. Originality/value: Large businesses have been profoundly impacted by Industry 4.0 principles; however, startup operations 4.0's competitive advantage has not received as much attention. This paper offers a fresh take on the concept of competitive advantage in startup operations 4.0 research. [ABSTRACT FROM AUTHOR]

Quaternion and biquaternion symmetry transformations have been applied to non-Cartesian reference systems of direct and reciprocal crystal lattices. The transformations performed directly in the sets of crystal reference axes simplify the calculations, eliminate the need for orthogonalization, permit the use of crystallographic vectors for defining the directions of rotations and perform the computations directly in the crystal coordinates. The applications of the general quaternion transformations are envisioned for physical, chemical, crystallographic and engineering applications. The general quaternion multiplication rules for any symmetry-unrestricted lattices have been derived for the triclinic crystallographic system and have been applied to the biquaternion representations of all point-group symmetry elements, including the crystallographic hexagonal system. Cayley multiplication matrices for point-groups, based on the biquaternion symbols of proper and improper symmetry elements, have been exemplified. [ABSTRACT FROM AUTHOR]

An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N, introduced to represent a set of related DNA sequences, also known as a pangenome. The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length m in an ED text. The EDSM problem has recently received some attention by the combinatorial pattern matching community, culminating in an O ~ (n m ω - 1) + O (N) -time algorithm [Bernardini et al., SIAM J. Comput. 2022], where ω denotes the matrix multiplication exponent and the O ~ (·) notation suppresses polylog factors. In the k-EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most k errors. k-EDSM can be solved in O (k 2 m G + k N) time, under edit distance, or O (k m G + k N) time, under Hamming distance, where G denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, G is only bounded by N, and so even for k = 1 , the existing algorithms run in Ω (m N) time in the worst case. In this paper we make progress in this direction. We show that 1-EDSM can be solved in O ((n m 2 + N) log m) or O (n m 3 + N) time under edit distance. For the decision version of the problem, we present a faster O (n m 2 log m + N log log m) -time algorithm. We also show that 1-EDSM can be solved in O (n m 2 + N log m) time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from 1-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the k-errata trees for indexing with errors [Cole et al., STOC 2004]. This is an extended version of a paper presented at LATIN 2022. [ABSTRACT FROM AUTHOR]

Since Transformers show a strong capability of building long-range dependencies, the relevant methods are extensively employed for image deraining tasks. However, the intrinsic limitations of Transformers, including costly computational complexity and insufficient ability to capture high-frequency components of the image, hinder the the utilization of Transformers in high-resolution images and lead to the unsatisfactory recovery of local edges and textures. To overcome these limitations, we propose an simple but effective Frequency Domain Enhanced Transformer (FDEFormer) for the image deraining. Firstly, drawing inspiration from the convolution theorem, we devise an efficient approach called frequency domain enhanced multi-head self-attention. The proposed approach replaces traditional matrix multiplication with element-wise product operations in the frequency domain, leading to a substantial reduction in computational complexity. Secondly, the existing spatial domain Transformers-based methods only focus on low-frequency features but pay less attention to high-frequency components, which can adversely affect the quality of the reconstructed images. Therefore, to integrate the content of different frequency levels, we propose a dual domain-complemented feed-forward network. Besides, we further present an attention feature fusion module to facilitate a more effective fusion of features across different layers. Extensive experiments on several datasets demonstrate that our FDEFormer performs favorably against state-of-the-art methods while taking acceptable computational costs. The source code and pre-trained models are available at https://github.com/bobozy1999/FDEFormer. [ABSTRACT FROM AUTHOR]

How can students effectively learn and transfer mathematical procedures to solve new problems? Here, we tested the effects of deliberately committing and correcting errors during mathematical problem-solving practice on transfer of the learned procedures. In two experiments, university students were instructed on mathematical algorithms (synthetic division and matrix multiplication) and solved practice problems during open-book study. Learners were then tested on flexibly adapting the algorithms to solve novel problems that were structurally more complex or embedded in "real-life" scenarios (i.e., mathematical word problems). Deliberately committing and correcting procedural errors during problem-solving practice yielded better transfer than errorless repeated practice (Experiment 1) or studying incorrect worked examples by finding, explaining, and correcting the errors that one's peers had made (Experiment 2). Yet, most learners failed to accurately predict or recognize the advantage of deliberate erring even after the test, instead misjudging this technique as less effective. This suggests that experiencing the benefit of deliberate erring is insufficient to dispel learners' metacognitive illusion that generating errors is not helpful for their learning. Overall, our results point to the critical role of first-hand errors in mathematical learning. Relative to avoiding errors or even studying others' errors and juxtaposing them with the correct solutions, guiding learners to deliberately commit and correct their own errors after instruction improves mathematical problem solving and transfer. Educational Impact and Implications Statement: Transfer of learning lies at the heart of education, but is often difficult to achieve. Here, we show in the domain of mathematics that deliberately committing and correcting procedural errors during problem-solving practice enhances students' transfer of the learned procedures to solve novel, more challenging problems. Deliberate erring was not only more effective than errorless repeated practice, but also finding, explaining, and correcting others' errors in incorrect worked examples. These results expand our repertoire of approaches to harness the power of errors for improving mathematical problem solving and transfer in education. [ABSTRACT FROM AUTHOR]

We introduce the idea of constructing recursion operators for full-fledged nonlocal symmetries and apply it to the reduced quasi-classical self-dual Yang–Mills equation. It turns out that the discovered recursion operators can be interpreted as infinite-dimensional matrices of differential functions which act on the generating vector functions of the nonlocal symmetries simply by matrix multiplication. To the best of our knowledge, there are no other examples of such recursion operators in the literature so far, so our approach is completely innovative. Further, we investigate the algebraic properties of the discovered operators and discuss the R -algebra structure on the set of all recursion operators for full-fledged nonlocal symmetries of the equation in question. Finally, we illustrate the action of the obtained recursion operators on particularly chosen full-fledged symmetries and emphasize their advantages compared to the action of traditionally used recursion operators for shadows. [ABSTRACT FROM AUTHOR]

The article is concerned with the problem of additivity of the tensor rank. That is, for two independent tensors, we study when the rank of their direct sum is equal to the sum of their individual ranks. The statement saying that additivity always holds was previously known as Strassen's conjecture (1969) until Shitov proposed counterexamples (2019). They are not explicit and only known to exist asymptotically for very large tensor spaces. In this article, we present families of pairs of small three-way tensors for which the additivity holds. For instance, over the base field C , it is the case if both tensors are of rank less or equal 7. This proves that a pair of 2 × 2 matrix multiplication tensors has the rank additivity property. We show also that the Alexeev-Forbes-Tsimerman substitution method preserves the structure of a direct sum of tensors. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this study is to develop a new hybrid method that combines interpretative structural modeling (ISM) and matrix cross-impact multiplication applied to classification (MICMAC) to investigate the influencing factors of sustainable infrastructure vulnerability (SIV). Design/methodology/approach: (1) Literature review and case study were used to identify the possible influencing factors; (2) a semi-structured interview was conducted to identify representative factors and the interrelationships among influencing factors; (3) ISM was adopted to identify the hierarchical structure of factors; (4) MICMAC was used to analyze the driving power (DRP) and dependence power (DEP) of each factor and (5) Semi-structured interview was used to propose strategies for overcoming SIV. Findings: Results indicate that (1) 18 representative factors related to SIV were identified; (2) the relationship between these factors was divided into a five-layer hierarchical structure. The 18 representative factors were divided into driving factors, dependent factors, linkage factors and independent factors and (3) 12 strategies were presented to address the negative effects of these factors. Originality/value: The findings illustrate the factors influencing SIV and their hierarchical structures, which can benefit the stakeholders and practitioners of an infrastructure project by encouraging them to take effective countermeasures to deal with related SIVs. [ABSTRACT FROM AUTHOR]

Optical computing harnesses the speed of light to perform vector-matrix operations efficiently. It leverages interference, a cornerstone of quantum computing algorithms, to enable parallel computations. In this work, we interweave quantum computing with classical structured light by formulating the process of photonic matrix multiplication using quantum mechanical principles such as state superposition and subsequently demonstrate a well-known algorithm, namely, Deutsch–Jozsa's algorithm. This is accomplished by elucidating the inherent tensor product structure within the Cartesian transverse degrees of freedom of light, which is the main resource for optical vector-matrix multiplication. To this end, we establish a discrete basis using localized Gaussian modes arranged in a lattice formation and demonstrate the operation of a Hadamard gate. Leveraging the reprogrammable and digital capabilities of spatial light modulators, coupled with Fourier transforms by lenses, our approach proves adaptable to various algorithms. Therefore, our work advances the use of structured light for quantum information processing. [ABSTRACT FROM AUTHOR]

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Finding new bounds for norms and the numerical radius of certain matrix forms is a renowned topic that has attracted numerous researchers. In this paper, we show some new bounds for unitarily invariant norms and the numerical radius of certain matrix forms that involve products of matrices and sums of products.New arithmetic-geometric mean-type inequalities and a refinement of the celebrated Cauchy–Schwarz inequality are shown among the obtained results.Our results are compared with those in the literature through numerical examples and rigorous approaches. [ABSTRACT FROM AUTHOR]

In this paper, the minimal observability of switching Boolean networks (SBNs) is investigated. Firstly, applying the semi‐tensor product (STP) method of matrices, a parallel extension system is constructed, based on which a necessary and sufficient condition to detect the observability of the SBNs is given. Secondly, when an SBN is unobservable, the specific steps to obtain the required measurements to make the system observable are given using the set reachable method; however, the measurements given in this part are not necessarily the fewest. Then, a criterion for determining the minimum number of measurements is further proposed through a constructed indicator matrix. Lastly, the effectiveness of the new results is verified by an example. [ABSTRACT FROM AUTHOR]

By utilizing the properties of positive definite matrices, mathematical expectations, and positive linear functionals in matrix space, the Kantorovich inequality and Wielandt inequality for positive definite matrices and random variables are obtained. Some novel Kantorovich type inequalities pertaining to matrix ordinary products, Hadamard products, and mathematical expectations of random variables are provided. Furthermore, several interesting unified and generalized forms of the Wielandt inequality for positive definite matrices are also studied. These derived inequalities are then exploited to establish an inequality regarding various correlation coefficients and study some applications in the relative efficiency of parameter estimation of linear statistical models. [ABSTRACT FROM AUTHOR]

This paper develops a parameter tuning method for solving the set restabilization problem of perturbed Boolean control networks (BCNs). First, the absorbable attractor, which we previously proposed, is recalled. Based on the relationship between attractors, a necessary and sufficient restabilizability criterion is derived. This criterion is used to check whether a perturbed BCN can be stabilized to the original target set by modifying the least number of parameters to the old controller. Furthermore, a constructive method for fine-tuning the old controller is provided if the criterion condition derived above is satisfied. Compared with the existing relevant results, ours have clear advantages, since they can address the set restabilization problem of BCNs subject to multi-column function perturbations, which has not been solved yet. Finally, two examples are employed to show the effectiveness and advantages of our results. [ABSTRACT FROM AUTHOR]

Time-domain wave scattering in an unbounded two-dimensional acoustic medium by sound-hard scatterers is considered. Two canonical geometries, namely a split-ring resonator (SRR) and an array of cylinders, are used to highlight the theory, which generalizes to arbitrary scatterer geometries. The problem is solved using the generalized eigenfunction expansion method (GEM), which expresses the time-domain solution in terms of the frequency-domain solutions. A discrete GEM is proposed to numerically approximate the time-domain solution. It relies on quadrature approximations of continuous integrals and can be thought of as a generalization of the discrete Fourier transform. The solution then takes a simple form in terms of direct matrix multiplications. In parallel to the GEM, the singularity expansion method (SEM) is also presented and applied to the two aforementioned geometries. It expands the time-domain solution over a discrete set of unforced, complex resonant modes of the scatterer. Although the coefficients of this expansion are divergent integrals, we introduce a method of regularizing them using analytic continuation. The results show that while the SEM is usually inaccurate at t=0 , it converges rapidly to the GEM solution at all spatial points in the computational domain, with the most rapid convergence occurring inside the resonant cavity. [ABSTRACT FROM AUTHOR]

Sylvester matrix equations play important roles in control and system theory. We consider a class of the generalized coupled Sylvester quaternion matrix equations, which includes a lot of special matrix equations such as Stein matrix equation, Lyapunov matrix equation and Sylvester matrix equation on the quaternion skew field. First, the quaternion Krylov subspace and its F-orthogonal basis are constructed via a new quaternion matrix product and the global quaternion Arnoldi process. Then a global quaternion generalized minimal residual (Gl-QGMRES) method is proposed for solving the generalized coupled Sylvester quaternion matrix equations. The convergence of the proposed method is theoretically analysed. Finally, the effectiveness and feasibility of the proposal are verified by some numerical experiments. As an application of Gl-QGMRES method, we establish a kind of colour image encryption and decryption by embedding watermark images. [ABSTRACT FROM AUTHOR]

Multiple kernel k‐means clustering (MKKC) can efficiently incorporate multiple base kernels to generate an optimal kernel. Many existing MKKC methods all need two‐step operation: learning clustering indicator matrix and performing clustering on it. However, the optimal clustering results of two steps are not equivalent to those of original problem. To address this issue, in this paper we propose a novel method named one‐step multiple kernel k‐means clustering based on block diagonal representation (OS‐MKKC‐BD). By imposing a block diagonal constraint on the product of indicator matrix and its transpose, this method can encourage the indicator matrix to be block diagonal. Then the indicator matrix can produce explicit clustering indicator, so as to implement one‐step clustering, which avoids the disadvantage of two‐step operation. Furthermore, a simple kernel weighting strategy is used to obtain an optimal kernel, which boosts the quality of optimal kernel. In addition, a three‐step iterative algorithm is designed to solve the corresponding optimization problem, where the Riemann conjugate gradient iterative method is used to solve the optimization problem of the indicator matrix. Finally, by extensive experiments on eleven real data sets and comparison of clustering results with 10 MKC methods, it is concluded that OS‐MKKC‐BD is effective. [ABSTRACT FROM AUTHOR]