Tensorized multi-dimensional multi-view clustering based on nonnegative matrix factorization.

Computing topological invariants in two-dimensional quasicrystals and supermoiré matter is a remarkable open challenge due to the absence of translational symmetry and the colossal number of sites inherent to these systems. Here, we establish a method to compute local topological invariants of exceptionally large systems using tensor networks, enabling the computation of invariants for Hamiltonians with hundreds of millions of sites, several orders of magnitude above the capabilities of conventional methodologies. Our approach leverages a tensor network representation of the density matrix using a Chebyshev tensor network algorithm, enabling large-scale calculations of topological markers in quasicrystalline and moiré systems. We demonstrate our methodology with two-dimensional quasicrystals featuring C 8 and C 10 rotational symmetries and mosaics of Chern phases. Our Letter establishes a powerful method to compute topological phases in exceptionally large-scale topological systems, providing the required tool to rationalize generic super-moiré and quasicrystalline topological matter.

Advanced measurement techniques in quantum Monte Carlo: The permutation matrix representation approach

This paper proposes a hybrid model applicable to electromagnetic calculation of interior permanent magnet synchronous motors (IPMSMs) with arbitrary rotor topologies. The model integrates the subdomain method with the non-parametric magnetic reluctance network (MRN), and applies periodic or anti-periodic boundary conditions to both sides of the model. The MRN enables accurate prediction of rotor saturation characteristics, while the subdomain method is employed to account for the slotting effect. The block matrix representation of the model is constructed via the bidirectional coupling boundary conditions at the air gap. The proposed model can incorporate motion characteristics under boundary constraints, thereby enabling time-stepping solution of the magnetic field distribution. Two motor topologies are analyzed using this model; its computational accuracy is verified by comparison with the finite element method, and experimental validation is conducted based on a V-type IPMSM prototype.

Purpose The purpose of this study is to develop an efficient and accurate numerical technique for solving nonlinear boundary value problems, with particular focus on the Darcy–Brinkman–Forchheimer equation (DBFE) and the quartic strongly nonlinear heat transfer equation (QSNHTE). Design/methodology/approach The approach is developed by constructing a matrix-based method that employs Pell-Lucas polynomials (PLPs) and utilizes the evenly spaced collocation points. Initially, the solution is formulated in a matrix representation, allowing all terms within the problems to be expressed accordingly. The Pell-Lucas collocation method (PLCM) is established by using the matrix formulation with the evenly spaced collocation points. Through this procedure, the original nonlinear equations are transformed into a system of linear algebraic equations, and solving this system yields the coefficient matrix corresponding to the PLP-based solutions. An error analysis is conducted for both problems. Subsequently, numerical implementations are carried out using MATLAB. Additionally, the L∞ and root mean square error norms are computed for various polynomial degrees and parameter values, providing quantitative validation of the method's accuracy. Findings The numerical results demonstrate that the proposed PLCM provides highly accurate and stable solutions for both DBFE and QSNHTE. The computed error norms decrease significantly with increasing polynomial degree, confirming the convergence and reliability of the method. Comparisons with existing numerical approaches reported in the literature show that the proposed technique achieves competitive accuracy, which is further illustrated through tabulated data and graphical representations. Research limitations/implications The proposed PLCM has the limitation that it is currently formulated and tested for one-dimensional nonlinear problems, and its performance has been demonstrated specifically for the DBFE and QSNHTE models; therefore, further studies are needed to evaluate its applicability and efficiency for higher-dimensional and more complex nonlinear systems. Practical implications The proposed PLCM provides a reliable and computationally efficient tool for solving nonlinear boundary value problems arising in porous media flow and nonlinear heat transfer. Its matrix-based structure and straightforward MATLAB implementation make it suitable for practical engineering applications requiring high accuracy and numerical stability. Social implications By improving the accuracy and stability of numerical simulations for porous media flow and nonlinear heat transfer models, the proposed method may contribute indirectly to more efficient energy systems and environmentally sustainable engineering designs. Originality/value The originality of this work lies in the matrix representation of nonlinear differential operators based on PLPs, enabling the numerical treatment of strongly nonlinear boundary value problems arising from the DBFE and QSNHTE.

In this study, the small and large Morgan-Voyce transformations are applied to generalized Horadam polynomials. We obtain the new sequences from these transformations. In addition, we provide the generating functions, recurrence relations, and some formulas for these polynomial sequences. Moreover, we give matrix representations of Morgan-Voyce transformations of generalized Horadam polynomials. All provided results are reduced to the small and large Morgan-Voyce transformations of well-known integer sequences including Fibonacci, Lucas, Pell, Pell-Lucas, Jacosthal, and Jacobsthal-Lucas numbers. Finally, we present some applications of special cases of these matrix representations in cryptography.

Biological matrix data are essential for computational analysis, providing a structured framework to identify patterns and relationships in biological systems. Many other biological data types, including sequences, networks, and images, can be transformed into matrix representations through feature extraction and encoding. However, their high dimensionality complicates analysis, leading to increased computational complexity and the risk of overfitting, known as the curse of dimensionality. To address these challenges, we developed SIBioX, a matrix-based bioinformatics tool powered by swarm intelligence algorithms. It integrates 54 swarm intelligence methods, 5 conventional feature selection techniques, and 17 machine learning models, enabling comprehensive analysis of biological matrix data. With a user-friendly graphical interface, it supports operations such as feature normalization, selection, classification, clustering, statistical analysis, and data visualization. Additionally, it converts nonmatrix biological data, like gene and protein sequences, into matrix formats for further study. Experimental results demonstrate that SIBioX not only attains high accuracy in feature selection but also effectively reduces dimensionality, thereby streamlining bioinformatics workflows and promoting greater efficiency in biomedical research.

Multiplication is a fundamental operation in many applications, and multipliers are widely adopted in various circuits. However, optimizing multipliers is challenging due to the extensive design space. In this article, we propose a multiplier design optimization framework based on reinforcement learning. We utilize matrix and tensor representations for the compressor tree of a multiplier, enabling seamless integration of convolutional neural networks as the agent network. The agent optimizes the multiplier structure using a Pareto-driven reward customized to balance area and delay. Furthermore, we enhance the original framework with parallel reinforcement learning and design space pruning techniques and extend its capability to optimize fused multiply-accumulate designs. Experiments conducted on different bit widths of multipliers demonstrate that multipliers produced by our approach outperform all baseline designs in terms of area, power, and delay. The performance gain is further validated by comparing the area, power, and delay of processing element arrays using multipliers from our approach and baseline approaches.

Introduction This computational modeling study introduces a novel Explainable Artificial Intelligence (XAI) framework for optimizing single-dose psilocybin treatment protocols through personalized intervention modeling using publicly available mental health datasets. All results presented are derived from novel simulated data and predictive modeling only, not from real-time clinical implementations or actual patient treatments. Methods The mathematical optimization model integrates digital twin technologies, multimodal depression detection systems, and Bayesian optimization algorithms to create comprehensive computational patient profiles with temporal resolution processing capabilities at 250 Hz sampling frequency. Validation employed three publicly available datasets: (1) the Psilocybin Precision Functional Mapping dataset from OpenNeuro containing neuroimaging data from 7 participants, (2) the MODMA multimodal mental disorder dataset with 53 participants including electroencephalography and audio signals, and (3) a meta-analytic psilocybin therapy outcomes dataset containing aggregated results from 10 clinical trials. The framework incorporates pharmacokinetic modeling with an absorption rate constant of 0.45 per hour and an elimination rate constant of 0.23 per hour, receptor occupancy dynamics based on a dissociation constant of 6.3 nanomolar, and simulated real-time monitoring protocols processing physiological parameters including heart rate variability, blood pressure measurements, and cortisol levels at a 1 Hz frequency. Results The simulated computational model demonstrates significant improvements in prediction accuracy, reaching 94.7%, and therapeutic transparency, achieving 89.3% explainability scores. Simulated validation demonstrates computational precision of 92.8% in predicting treatment response patterns across diverse patient populations in silico . The proposed computational methodology addresses key challenges in psychedelic-assisted therapy modeling through interpretable artificial intelligence models, achieving 96.2% computational safety index scores and 91.5% algorithmic compliance metrics in simulation environments. Energy-efficient computational architecture achieves 73.4% carbon footprint reduction through optimized algorithm design and sparse matrix representations. Discussion This study presents a theoretical computational framework for modeling therapeutic outcomes through simulation and prediction, establishing a foundation for future clinical validation through prospective randomized controlled trials. The framework supports sustainable digital mental healthcare delivery systems compatible with renewable energy infrastructure. All findings represent computational predictions and simulated scenarios requiring extensive clinical validation before any practical application.

We present a novel Bayesian model and a corresponding robust, probabilistic calibration procedure for the CORSAIR polarimeter that can be applied to other polarimeters. Our calibration procedure combines existing Mueller matrix representations of polarimeters with Bayesian methods, and computes the posterior distribution of the parameters by collecting data from the polarimeter at different states. We show that the algorithm is able to converge and recover a well-constrained posterior of the free parameters with a credible interval that is consistent with the ground truth values. Posterior predictive checks indicate that our generative model with inferred parameters can reproduce the calibration data within the predictive uncertainty, and captures the dominant systematic effects of the calibration procedure. We further show that we can propagate calibration uncertainties in the distributions to downstream reconstructions of Stokes measurements and magnetic-field estimates. We find that the contribution of calibration uncertainty towards the reconstructed results is minimal relative to that of the photon noise uncertainty, indicating that estimates using our Bayesian calibration algorithm can achieve photon noise-limited measurements in the magnetic-field parameters. Finally, we test the Bayesian calibration algorithm on a lab prototype of the CORSAIR polarimeter, and show that it converges and closely recovers theoretical estimates of the free parameters from real-world measurements.

ABSTRACT The algebraic structures of integrable hierarchies play an important role in the study of soliton equations. In this paper, we use splitting theory to give a matrix representation of a constrained CKP hierarchy, which can be considered as a generalization of the ‐KdV hierarchy and the constrained KP hierarchy. An equivalent construction in terms of the pseudo‐differential operator is discussed. Darboux transformations, scaling transformations, and tau functions for this constrained hierarchy are studied.

The anchor-based clustering method is currently a predominant technique for handling large-scale data. However, in multiview data, existing anchor-based methods face a key challenge: balancing individual anchor graph distinctiveness with final consistency. To address this challenge, we propose a large-scale multiview clustering (MVC) method via joint learning of anchor representation and multigraph alignment (ARMGA). Specifically, ARMGA introduces a unified framework that facilitates the concurrent learning of single-view anchor representations and virtual graph-based multigraph alignment. The approach aims to preserve the adaptability of anchor learning across different views, while ensuring the ultimate consistency of the merged anchor graph. Furthermore, ARMGA employs Schatten- $\boldsymbol {p}$ norm on the tensor formed by the adaptive anchor representation, originating from multigraph alignment, to reinforce cross-view consistency. This technique effectively leverages complementary information preserved across views to bolster the overall structure and consensus information. Ultimately, to attenuate the noise impact on the anchor representation matrix, ARMGA capitalizes on the cosine angle information from the low-rank representation as coefficients within the relationship matrix and efficiently reduces computational complexity through deductions. On nine datasets, ARMGA has exhibited a notable improvement in clustering performance indicators by 2%-10% over other algorithms, while also maintaining lower time complexity.

Abstract We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a planar, simply connected domain. The eigenvalues admit a characterization in terms of a layer potential of the Helmholtz equation. Using the exterior conformal mapping associated with the given domain, we reformulate the layer potential as an infinite-dimensional matrix. Based on this matrix representation, we develop a finite section approach for approximating the Laplacian eigenvalues and provide a convergence analysis by applying the Gohberg–Sigal theory for operator-valued functions. Moreover, we derive an asymptotic formula for the Laplacian eigenvalues on deformed domains that results from the changes in the conformal mapping coefficients.

In this paper, a new class of singular fractional integro-differential equations is defined. The Caputo tempered differentiation is considered to define these equations. By employing the fixed point theorem, the existence and uniqueness of a solution for such equations is proved. The orthonormal discrete Hahn polynomials are considered to generate a computational strategy for these tempered fractional equations. To enable this methodology, a matrix representation of the tempered integral operator is constructed for the polynomial basis. Subsequently, the fractional operators in the governing equation are discretized using the polynomial approximation framework. By leveraging the precomputed integral operator matrix and solving the resulting algebraic system, a numerical approximation to the solution is efficiently computed. The convergence of the introduced strategy is proved, theoretically. The precision of the scheme is checked by solving three examples.

ABSTRACT In this work, we introduce the set of Hurwitz split quaternions, , which is an extension of the split quaternions over the integers, . In comparison to , the Hurwitz splits only have integer or half‐integer coefficients, not both. We demonstrate that it has a ring structure and examine certain features of this notion by pointing out the differences from . We also provide the matrix representation of the ring of Hurwitz split quaternions. In addition, we study some of its prime ideals and show that this ring is Noetherian. Then we describe the set of integer‐valued polynomials over to be . Once we prove forms a ring, we investigate some essential properties of this ring.

This study develops an analytical description of the motion of dilute solid particles in the boundary layer of laminar horizontal flows subjected to weak transverse pulsations. The analysis is formulated for dilute spherical solid particles subjected to transverse velocity pulsations in a laminar boundary-layer flow. A coupled matrix representation of the governing equations is formulated, and closed-form solutions are obtained using Laplace transformation. The analytical expressions capture transient evolution, forced oscillations, resonance effects, and long-term behaviour for particles with different density ratios. Numerical evaluation shows that light particles migrate toward faster regions of the boundary layer and accelerate longitudinally, while heavy particles move toward slower layers and decelerate. Transverse pulsations generate oscillatory trajectories whose amplitude increases near resonance. Impulsive perturbations superimposed on the continuous motion lead to discontinuous transitions consistent with the linear matrix system. The results provide a unified physical interpretation of particle redistribution mechanisms in boundary layers and offer a compact analytical tool for dilute multiphase flow modelling.

We study positive semi-definite (PSD) biquadratic forms and their sum-of-squares (SOS) representations. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is an SOS. This extends the known result for fully symmetric biquadratic forms. Furthermore, we describe an efficient computational procedure for constructing SOS decompositions, exploiting the Kronecker-product structure of the associated matrix representation. We introduce simple biquadratic forms. For m≥2, we provide a explicit example to show the lower bound for sos rank of m×2 biquadratic forms is m+1, and show that previously proved results indicating that a 2×2 PSD biquadratic form can be expressed as the sum of three squares and a 3×2 PSD biquadratic form can be expressed as the sum of four squares are tight. We also present an 3×3 SOS biquadratic form, which can be expressed as the sum of six squares, but not the sum of five squares. Moreover, we establish a universal upper bound mn−1 for any m×n SOS biquadratic form, which improves the trivial bound mn.

This paper introduces the Gaussian Lehmer sequence and presents its representation using a lower triangular Pascal matrix. Building on this matrix form, a new coding method is developed with improved capabilities for error detection and correction. The study explores the use of this coding approach in the realm of complex numbers within coding theory. Further investigation shows that the sequence follows distinct recurrence relations for odd and even terms. To unify these behaviors, a new function is proposed that captures both cases, resulting in the definition of a biperiodic sequence. The paper also provides a thorough analysis of the circulant matrix generated by the sequence, including the computation of its determinant and inverse, highlighting the promise of this matrix-based framework in advancing coding theory.

It is known that every (single-qudit) Clifford operator maps the full set of generalized Pauli matrices (GPMs) to itself under unitary conjugation, which is an important quantum operation and plays a crucial role in quantum computation and information. However, in many quantum information processing tasks, it is required that a specific set of GPMs be mapped to another such set under conjugation, instead of the entire set. We formalize this by introducing local Clifford operator, which maps a given $n$-GPM set to another such set under unitary conjugation. We establish necessary and sufficient conditions for such an operator to transform a pair of GPMs, showing that these local Clifford operators admit a classical matrix representation, analogous to the classical (or symplectic) representation of standard (single-qudit) Clifford operators. Furthermore, we demonstrate that any local Clifford operator acting on an $n$-GPM ($n\geq 2$) set can be decomposed into a product of standard Clifford operators and a local Clifford operator acting on a pair of GPMs. This decomposition provides a complete classical characterization of unitary conjugation mappings between $n$-GPM sets. As a key application, we use this framework to address the local unitary equivalence (LU-equivalence) of sets of generalized Bell states (GBSs). We prove that the 31 equivalence classes of $4$-GBS sets in bipartite system $\mathbb{C}^{6}\otimes \mathbb{C}^{6}$ previously identified via Clifford operators are indeed distinct under LU-equivalence, confirming that this classification is complete.

A reinforcement learning (RL) framework is introduced for the efficient synthesis of quantum circuits that generate specified target quantum states from a fixed initial state, addressing a central challenge in both the Noisy Intermediate-Scale Quantum (NISQ) era and future fault-tolerant quantum computing. The approach utilizes tabular Q-learning, based on action sequences, within a discretized quantum state space, to effectively manage the exponential growth of the space dimension. The framework introduces a hybrid reward mechanism, combining a static, domain-informed reward that guides the agent toward the target state with customizable dynamic penalties that discourage inefficient circuit structures such as gate congestion and redundant state revisits. This is a circuit-aware reward, in contrast to the current trend of works on this topic, which are primarily fidelity-based. By leveraging sparse matrix representations and state-space discretization, the method enables practical navigation of high-dimensional environments while minimizing computational overhead. Benchmarking on graph-state preparation tasks for up to seven qubits, we demonstrate that the algorithm consistently discovers minimal-depth circuits with optimized gate counts. Moreover, extending the framework to a universal gate set still yields low depth circuits, highlighting the algorithm’s robustness and adaptability. The results confirm that this RL-driven approach, with our completely circuit-aware method, efficiently explores the complex quantum state space and synthesizes near-optimal quantum circuits, providing a resource-efficient foundation for quantum circuit optimization.