Unitary Matrix Research Articles

Novel Implicit Integration Algorithms With Identical Second‐Order Accuracy and Flexible Dissipation Control for First‐Order Transient Problems

ABSTRACTFor the solutions of first‐order transient problems with optimal efficiency, conventional single‐step implicit methodologies, unaided by additional post‐processing techniques, encounter limitations in concurrently attaining identical second‐order accuracy and controllable numerical dissipation. Addressing this challenge, the present study contributes not only a comprehensive analytical framework for formulating implicit integration algorithms but also leverages the auxiliary variable and the composite sub‐step technique to propose two distinct types of implicit integration algorithms. Each type is characterized by self‐initiation, unconditional stability, identical second‐order accuracy, controllable numerical dissipation, and zero‐order overshoots. Recognizing that single‐step implicit methods can be conceptualized as composite single‐sub‐step ones, both algorithm types achieve identical effective stiffness matrices, thereby reducing the computational effort for solving linear problems. The utilization of auxiliary variables endows the proposed single‐step method with numerical attributes akin to established counterparts, while achieving identical second‐order accuracy. Conversely, the adoption of the composite sub‐step technique in the proposed two‐sub‐step methods surpasses the published algorithms by significantly reducing the error constants. This superiority persists when enforcing the same sub‐step sizes and sub‐step dissipation levels. Numerical simulations affirm that the proposed methods consistently outperform existing alternatives without incurring additional computational expenses.

Infinite quantum signal processing

Quantum signal processing (QSP) represents a real scalar polynomial of degree d using a product of unitary matrices of size 2×2, parameterized by (d+1) real numbers called the phase factors. This innovative representation of polynomials has a wide range of applications in quantum computation. When the polynomial of interest is obtained by truncating an infinite polynomial series, a natural question is whether the phase factors have a well defined limit as the degree d→∞. While the phase factors are generally not unique, we find that there exists a consistent choice of parameterization so that the limit is well defined in the ℓ1 space. This generalization of QSP, called the infinite quantum signal processing, can be used to represent a large class of non-polynomial functions. Our analysis reveals a surprising connection between the regularity of the target function and the decay properties of the phase factors. Our analysis also inspires a very simple and efficient algorithm to approximately compute the phase factors in the ℓ1 space. The algorithm uses only double precision arithmetic operations, and provably converges when the ℓ1 norm of the Chebyshev coefficients of the target function is upper bounded by a constant that is independent of d. This is also the first numerically stable algorithm for finding phase factors with provable performance guarantees in the limit d→∞.

Efficient on-chip training of large-scale optical neural network through block adjoint training algorithm

MZI-based block optical neural networks (BONNs), which utilize block matrix multiplication to achieve large-scale network models, have garnered significant attention but still lack efficient training algorithms. In this article, by calculating the original field and adjoint field for the block matrices in BONNs and directly updating the phase values of all phase shifters within the optical mesh, we propose an on-chip block adjoint training (BAT) algorithm for large-scale BONNs. To demonstrate the effectiveness of our proposed algorithm, the trained BONNs are applied in image classification tasks for MNIST and SVHN datasets. The calculated results demonstrate that the performance of the BAT algorithm (95.915% for the MNIST dataset and 82.64% for the SVHN dataset) is competitive with the traditional gradient algorithm based on artificial neural networks (96.238% and 84.182%), but the BONNs can infer 1.5 times and 1.3 times faster than artificial neural networks, respectively. By studying the influence of the block size and the inputted position of the padded zero signals, we demonstrate that the BAT algorithm based on the BONNs with 12 block sizes can achieve higher performance by adding the padded zero signals to the same side beside the normal inputted signals. Additionally, we demonstrate that substituting the complete weight matrices with unitary matrices to construct BONNs is an efficient way to reduce both the system area and the required trainable parameters. Finally, we demonstrate the relatively good robustness of the BAT algorithm and the imprecision alleviation method by using on-chip retraining. Notably, our proposed BAT algorithm shows excellent potential for more complex tasks and network models.

Infinite Dimensional Maximal Torus Revisited

Let Tm be the maximal torus of a set of m×m unitary diagonal matrices. Let T be a collection of all maps that rigidly rotate every circle of latitude of the sphere with a fixed angle. T is also a maximal torus, and we shall prove in this paper that T is the topological limit inf of Tm.

Multiple color information cryptosystem using Hessenberg-decomposition-modulated chaotic and face biometric phase encoding

A novel, to the author’s knowledge, multiple color information cryptosystem based on Hessenberg decomposition (HD)-modulated chaotic and face biometric phase encoding is introduced. The face biometric phase mask (FPM) and chaotic phase mask (CPM) are modulated by HD to obtain an upper Hessenberg matrix, an upper triangular matrix, and two unitary matrices for the first time. Each original color image is decomposed into R, G, and B channels. Each channel is individually modulated by an upper triangular matrix and two unitary matrices. The upper Hessenberg matrix and modulated R, G, and B channels are inverse discrete wavelet transformed to produce a fused image. In the same way, fused images for multiple color images are generated and combined into a single complex image, which is bonded with a first chaotic face biometric phase mask (CFPM) and fractional Fourier transformed. The resultant image is amplitude- and phase-truncated to generate the first common decryption key and preliminary encrypted image. The encrypted image is bonded with a second CFPM and fractional Fourier transformed. The obtained image is amplitude- and phase-truncated to generate the second common decryption key and final encrypted image. The proposed method utilizes the fused image as a covered image to conceal the modulated R, G, and B channels of each original color image. Furthermore, six decryption keys (three matrix decryption keys, one individual decryption key, two common decryption keys), and six encryption keys (two CFPMs and four orders of fractional Fourier transform) provide resistance against various types of potential attacks. A hybrid optoelectronic system can be utilized to implement the proposed cryptosystem. Numerical simulation results validate the feasibility and efficiency of the proposed scheme.

Retraction Note: Improving the reconstruction of dental occlusion using a reconstructed-based identical matrix point technique

Retraction Note: Improving the reconstruction of dental occlusion using a reconstructed-based identical matrix point technique

Beyond the Blackwell order in dichotomies

I establish a translation invariance property of the Blackwell order for dichotomies and show that the norm of the distance from the identity matrix may be interpreted as a measure of informativeness. The better experiment is closer to the fully revealing experiment; this measure extends the Blackwell order, is complete, and prior-independent.

NFGCL: A Negative-sampling-free Graph Contrastive Learning Framework for Recommendation

Recently, Graph Contrastive Learning (GCL) has seen rapid growth in recommender systems. GCL-based Collaborative Filtering (CF) methods typically integrate the primary recommendation task with auxiliary CL tasks, which benefit from two key properties: alignment and uniformity. However, these methods often rely on graph augmentation and negative sampling, which are time-consuming and risk learning incorrect signals. To remedy these issues, we propose a novel Negative-sampling-Free Graph Contrastive Learning (NFGCL) framework to achieve high-quality representations without negative sampling or graph augmentation. Specifically, we use in-batch positive user-item pairs to construct a preference matrix where each element represents cosine similarity between a user and an item. We introduce a novel contrastive objective to make the preference matrix close to an identity matrix, drawing positive pairs closer and distancing negative pairs to achieve better alignment while improving uniformity. Additionally, we propose a simple yet effective representation-level augmentation method that integrates the normalized first-layer Graph Convolutional Networks (GCN) output into the final embeddings, further enhancing alignment. We also employ the uniformity loss to regulate the uniformity of user/item representation distribution. Extensive experiments on three public datasets demonstrate that our method outperforms baseline methods and effectively balances alignment and uniformity.

Partially unitary learning.

The problem of an optimal mapping between Hilbert spaces IN of |ψ〉 and OUT of |ϕ〉 based on a set of wavefunction measurements (within a phase) ψ_{l}→ϕ_{l},l=1,⋯,M, is formulated as an optimization problem maximizing the total fidelity ∑_{l=1}^{M}ω^{(l)}|〈ϕ_{l}|U|ψ_{l}〉|^{2} subject to probability preservation constraints on U (partial unitarity). The constructed operator U can be considered as an IN to OUT quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension dim(OUT)×dim(IN) transforming operators as A^{OUT}=UA^{IN}U^{†}. An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.

Commutators of skew-involutions

Let $\mathrm{SL}_n(\mathbb{F})$ be the group of all $n\times n$ matrices over a field $\mathbb{F}$ with determinant $1$. Denote by $I$ ($I_n$) the ($n\times n$) identity matrix. A matrix $A$ is called skew-involution if $A^2=-I$. It is proved that every matrix in $\mathrm{SL}_{2n}(\mathbb{F})$ is a product of at most three commutators of skew-involutions if $\mathbb{F}\ne\mathbb{Z}_3$ and $\mathrm{SL}_{2n}(\mathbb{F})\ne\mathrm{SL}_{2}(\mathbb{Z}_2)$, and at most four commutators of skew-involutions if $\mathbb{F}=\mathbb{Z}_3$ and $n>1$. Every complex symplectic matrix is a product of two commutators of complex symplectic skew-involutions, and every real symplectic matrix is a product of not more than four commutators of real symplectic skew-involutions.

Local certification of unitary operations

In this work, we analyze the local certification of unitary quantum channels, which is a natural extension of quantum hypothesis testing. A particular case of a quantum channel operating on two systems corresponding to product states at the input, is considered. The goal is to minimize the probability of the type II error, given a specified maximum probability of the type I error, considering assistance through entanglement with auxiliary systems. Our result indicates connection of the local certification problem with a product numerical range of unitary matrices. We show that the optimal local strategy does not need usage of auxiliary systems and requires only single round of one-way classical communication. Moreover, we compare local and global certification strategies and show that typically local strategies are optimal, yet in some extremal cases, where global strategies make no errors, local ones may fail miserably. Finally, some application for local certification of von Neumann measurements are discussed as well.

Comprehensive classification of the algebra generated by two idempotent matrices

For two idempotent matrix P,Q∈Cn×n, let alg(In,P,Q) denote the smallest subalgebra of Cn×n that contains P,Q and the identity matrix In. This paper provides a complete classification of alg(In,P,Q) without imposing any restrictions on P and Q. As a result of this classification, the issue of group invertibility within alg(In,P,Q) is fully resolved.

Exploring Bounded Component Analysis Using an ℓ∞ Norm Criterion

In this paper we propose a new criterion for the Blind Source Separation (BSS) of antisparse bounded sources, based on the sum of the ℓ∞-norm of the sources. Based on the observation that the mixing process of bounded sources with any mixing matrix with unitary Frobenius norm will increase the ℓ∞-norm of the sources, unless it is the identity matrix, the minimization of the sum of the ℓ∞-norm of the sources can be used for the estimation of a separation matrix. To that, a Principle Component Analysis technique followed by a Givens Rotations based optimization method can be used for the separation of independent bounded sources. Also, the Givens Rotations based optimization method can be used for the separation of correlated bounded sources mixed by a rotation matrix. We theoretically analyze the proposed criterion and assess its performance through numerical simulations involving three distinct types of bounded signals. Our theoretical and experimental findings underscore the efficacy of the ℓ∞ norm as a suitable contrast function for antisparse bounded sources, showcasing its superior performance relative to a state-of-the-art algorithm.

The Relationship Between Institutional Pedagogy and the Brand Identity Formation of Higher Education Institutions

This study sought to facilitate insight into the potential role of pedagogy in the brand identity formation of higher education institutions (HEIs) through a study of selected HEIs in The Gambia. Specifically, the study sought to address the following research question: What role do pedagogical practices play in building the brand identity of selected universities in The Gambia? The intra-paradigm qualitative mixed method of data collection underpinned the research design. This research design facilitated a preliminary analysis of the contents of institutional documents and social media postings. This process was followed by telephonic and virtually mediated in-depth interviews through which this researcher explored the interactionist interpretations, recollections, experiences, and opinions of 54 participants (students and staff) on the themes of institutional brand management practices, institutional pedagogical practices, institutional brand identity, and the links between pedagogical practices and institutional brand identity. The study used the Corporate Brand Identity Matrix as a supporting analysis framework. The findings show a relationship between pedagogical practices and institutional brand identity formation. The evidence suggests that the selected HEIs use hardly differentiated production-style portfolios of academic courses to pursue largely unengaged students, prospective students, and other stakeholders. Further findings indicate that teaching and learning practice is dominated by academic staff's discretionary use of transmissive pedagogy. This insight emerged against the background of additional evidence, which shows a link between pedagogy policy and practices of HEIs and stakeholder impressions. A synthesis of these findings culminated in the emergence of the pedagogy-based higher education brand identity matrix (P-HEBIM), which this study proposes as a novel framework for the branding of HEIs. The study sets out a practitioner guide on how higher education managers can pursue the institutional brand management priorities of branding strategy development, competitor intelligence, and brand communication using the P-HEBIM as a framework.

Structural Analysis of Virus Regulatory N6-Methyladenosine (m6A) Machinery of the Black Flying Fox (Pteropus alecto) and the Egyptian Fruit Bat (Rousettus aegyptiacus) Shows Evolutionary Conservation Amongst Mammals.

N6-methyladenosine (m6A) is an abundant RNA epitranscriptomic modification in eukaryotes. The m6A machinery includes cellular writer, eraser and reader proteins that regulate m6A. Pteropus alecto (P. alecto) (the Australian black flying fox) and Rousettus aegyptiacus (R. aegyptiacus) (the Egyptian fruit bat) are bats associated with several viral zoonoses yet neglected in the field of m6A epigenetics studies. This study utilises various bioinformatics and in silico tools to genetically identify, characterise and annotate the m6A machinery in P. alecto and R. aegyptiacus. A range of bioinformatic tools were deployed to comprehensively characterise all known m6A-associated proteins of P. alecto and R. aegyptiacus. Results: Phylogenetically, the m6A fat mass and obesity-associated protein (FTO) eraser placed the order Chiroptera (an order including all bat species) in a separate clade. Additionally, it showed the lowest identity matrices in P. alecto and R. aegyptiacus when compared to other mammals (74.1% and 72.8%) and Homo sapiens (84.0% and 76.1%), respectively. When compared to humans, genetic loci-based analysis of P. alecto and R. aegyptiacus showed syntenic conservation in multiple flanking genes of 8 out the 10 m6A-associated genes. Furthermore, amino acid alignment and protein tertiary structure of the two bats' m6A machinery demonstrated conservation in the writers but not in erasers and readers, compared to humans. These studies provide foundational annotation and genetic characterisation of m6A machinery in two important species of bats which can be exploited to study bat-virus interactions at the interface of epitranscriptomics.

Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions

For positive integers $d$ and $p$ such that $d \ge p$, let $\mathbb{R}^{d \times p}$ denote the set of $d \times p$ real matrices, $I_p$ be the identity matrix of order $p$, and $V_{d,p} = \{x \in \mathbb{R}^{d \times p} \mid x'x = I_p\}$ be the Stiefel manifold in $\mathbb{R}^{d \times p}$. Complete asymptotic expansions as $d \to \infty$ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on $V_{d,p}$. The accuracy of each truncated expansion is strictly increasing in $d$; also, for sufficiently large $d$, the accuracy is strictly increasing in $m$, the number of terms in the truncated expansion. Lower bounds are obtained for the truncated expansions when the matrix parameters of the matrix Bingham distribution are positive definite and when the matrix parameter of the matrix Langevin distribution is of full rank. These results are applied to obtain the rates of convergence of the asymptotic expansions as both $d \to \infty$ and $p \to \infty$. Values of $d$ and $p$ arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as $d$ or $m$ increases. These results extend recently-obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions.

Structural Changes in Copper Slags During Slow Cooling

The objects of the study were converter slags from the Balkhash copper plant in their initial state and after heat treatment. Using mineralogical and X-ray phase analysis, scanning electron microscopy (SEM), and electron probe microanalysis (EPMA), it was found that the initial converter slag and its thermally treated samples have identical matrices with almost complete coincidence in mineral and phase compositions. The distinguishing feature is the quantitative ratio of mineral components in the slag mass. Almost all of the iron is oxidized and present in the form of fayalite, magnetite, and magnetite, with other elements (silicon, copper, zinc, and aluminum) incorporated into its lattice. The structure of all slag samples indicates an association of sulfur exclusively with copper. Copper in the slags was identified in both metallic and sulfide forms. Slow cooling of the converter slag after its remelting contributes to the reduction in the sulfide–metal suspension in the volume of the melt and its coarsening. During slow cooling, structural changes occur not only in the main oxide part of the slag but also in the polymetallic globules.

One-step ahead short-term hourly global solar radiation forecasting with a dynamical system based on classification of days

The data-driven method is a common approach to solar irradiation prediction. However, its weakness is that it is highly dependent on the characteristics of the prior data, which leads to a lag delay in forecasting. Therefore, this paper proposes a prediction model that considers data-driven, data-cycle variations, and statistical information for solving such problems. Firstly, the periodicity of the data is removed using the Fourier transform. Secondly, monthly daily characterization matrices are constructed to predict from a global perspective without including information from previous time series. Finally, the coupled autoregressive and dynamical system (CARDS) model is improved using the identity matrix. Comparing the model proposed in this paper with seven other popular prediction models, the error can be reduced by 30.2 % on average at four locations in the northern and southern hemispheres. Moreover, the proposed model does not increase the computational burden of existing models.

Elimination and Factorization

Summary If a matrix A has rank r, then its row echelon form (from elimination) contains the identity matrix in its first r independent columns. How do we intepret the matrix F that appears in the remaining columns of that echelon form? The matrix F multiplies those first r independent columns of A to give its n–r dependent columns. Then F reveals bases for the row space and the nullspace of the original matrix A. And F is the key to the column-row factorization A = CR .

Oxidation Potential of 2,6-Dimethyl-1,4-dihydropyridine Derivatives Estimated by Structure Descriptors

Linear relationships, expressing the electrochemical properties of molecules as functions of structure, give insight into the associated electrochemical process and are a tool for prediction. Many biological activities rely on water-based dissociation, making electrochemical properties a bridge between structure and activity. Motivated by a previous study, a replica is made here on a different dataset in order to validate/invalidate the previously reported results. There are several methods for obtaining structure-based descriptors. Some of the methods have been devised to account for molecular topology, some to account for molecular geometry, and others to account for both. Two methods are involved here to derive structure-based descriptors and further obtain structure–property relationships (FMPI and ChPE). In order to express structure descriptors, both FMPI and ChPE express first the topology of the molecule, using the heavy atoms identity matrix and the heavy atoms adjacency matrix, both square symmetric matrices in the belief that symmetry is one major factor of molecular stability. A set of 2,6-dimethyl-1,4-dihydropyridine derivatives with oxidation peak potentials and coulometrically determined number of electrons experimental data is subjected to the search for structure–activity relationships. Even if the 2,6-dimethyl-1,4-dihydropyridine is a symmetric compound (of Cs point group), their derivatives are generally not symmetric (9 out of 24 are asymmetric). The dataset is subjected to descriptive and inferential statistics in order to filter out the most relevant structure–activity relationship. The geometry is built using three levels of theory (one from molecular mechanics and two others from density functionals, of which one accounts for the interaction with water as solvent). One challenge of picking one out of two reported measured values is dealt with by calculating the likelihood associated with the two choices. Relevant structure–activity models are extracted and discussed. The use of in vivo (in water, SM8 model) models in geometry optimization (from MMFF94 and B3LYP and to M06 + Water SM8) results in a precision gain, but this is, in most of the cases, not statistically significant, and this can be considered a negative result.