Matrix Form Research Articles

Short-time and long-time dynamics of the zeolite films in the light of alternative diffusion models

Analysis of the dynamics of diffusion of two mixtures of hydrogen and heavier components through a flat zeolite film was carried out. The concepts of short-time and long-time dynamics were introduced. They derive from the observation of temporal trajectories obtained by applying abrupt perturbations of concentrations on one side of the membrane.An alternative description of the diffusion dynamics was proposed compared to the commonly accepted model that employs a matrix of thermodynamic factors Γ. In this way, the difficulty of analytical determination of this matrix for more complex adsorption equilibria was eliminated.The model equations were expressed in a matrix form, which enables its easy extension to any number of components. The proposed research methodology was illustrated using an example of diffusion of two selected mixtures, i.e. {H2, n-C4H10} and {H2, CO2} in silicalite-1. These gaseous mixtures have applications in specific technological processes.The two models were used to perform quantitative benchmarking of short-time and long-time dynamics of flat zeolite membranes. The assumed boundary conditions correspond to a semi-batch Wicke-Kallenbach diffusion cell. It was demonstrated that the phenomena occurring in short-time and long-time diffusion represent two different dynamic features important from a scientific and practical point of view.A quantitative measure for degree of confinement was proposed based on dynamic adsorption-desorption hysteresis.

Irregular array optimization for beamforming with a polar coordinate-based partition coding approach

Abstract An innovative irregular array configuration optimization method for enhancing beamforming is introduced. This study presents partition coding to optimize sensor positioning and quantity of a non-uniform concentric circular array. This novel approach transcends traditional techniques by integrating structural partitioning and performance optimization to quantify the array’s geometry-performance correlation. Sensor candidate positions are mapped in polar coordinates, with each configuration translated into a sensor position matrix form. A significant innovation lies in the adaptation of the partition coding genetic algorithm to enhance the encoding of candidate positions and to refine crossover and mutation operations, underpinned by an elite retention strategy for selecting the optimal array configuration. Both simulation and experimental results substantiate the method’s effectiveness, achieving high-resolution acoustic mapping with commendably low computational complexity.

Improving data interpretability with new differential sample variance gene set tests.

Gene set analysis methods have played a major role in generating biological interpretations from omics data such as gene expression datasets. However, most methods focus on detecting homogenous pattern changes in mean expression and methods detecting pattern changes in variance remain poorly explored. While a few studies attempted to use gene-level variance analysis, such approach remains under-utilized. When comparing two phenotypes, gene sets with distinct changes in subgroups under one phenotype are overlooked by available methods although they reflect meaningful biological differences between two phenotypes. Multivariate sample-level variance analysis methods are needed to detect such pattern changes. We use ranking schemes based on minimum spanning tree to generalize the Cramer-Von Mises and Anderson-Darling univariate statistics into multivariate gene set analysis methods to detect differential sample variance or mean. We characterize these methods in addition to two methods developed earlier using simulation results with different parameters. We apply the developed methods to microarray gene expression dataset of prednisolone-resistant and prednisolone-sensitive children diagnosed with B-lineage acute lymphoblastic leukemia and bulk RNA-sequencing gene expression dataset of benign hyperplastic polyps and potentially malignant sessile serrated adenoma/polyps. One or both of the two compared phenotypes in each of these datasets have distinct molecular subtypes that contribute to heterogeneous differences. Our results show that methods designed to detect differential sample variance are able to detect specific hallmark signaling pathways associated with the two compared phenotypes as documented in available literature. The results in this study demonstrate the usefulness of methods designed to detect differential sample variance in providing biological interpretations when biologically relevant but heterogeneous changes between two phenotypes are prevalent in specific signaling pathways. Software implementation of the developed methods is available with detailed documentation from Bioconductor package GSAR. The available methods are applicable to gene expression datasets in a normalized matrix form and could be used with other omics datasets in a normalized matrix form with available collection of feature sets.

Local-time formula for dissipation in solid ionic electrolytes

When ions move through solids, they interact with the solid's constituent atoms and cause them to vibrate around their equilibrium points. This vibration, in turn, modifies the potential landscape through which the mobile ions travel. Because the present-time potential depends on past interactions, the coupling is inherently nonlocal in time, making its numerical and analytical treatment challenging. For sufficiently slow-moving ions, we linearize the phonon spectrum to show that these nonlocal effects can be ignored, giving rise to a draglike force. Unlike the more familiar drag coefficient in liquids, the drag takes on a matrix form due to the crystalline structure of the framework. We numerically simulate trajectories and dissipation rates using both the time-local and nonlocal formulas to validate our simplification. The time-local formula dramatically reduces the computational cost of calculating the motion of a mobile particle through a crystalline framework and clearly connects the properties of the material to the drag experienced by the particle. Published by the American Physical Society 2024

A Study on the Sum of ‘m+1’ Consecutive Cullen Numbers

Objectives: To discover new formulae for the matrix form of the sum of m+1 Cullen numbers. This is an effort to explain the Recursive Matrix form formula and a few of its uses. Methods: Conclusions derived from theorems and the matching matrix representations of them. Additionally, a few apps are offered. The terminology of Cullen Numbers are utilised to demonstrate the main theorems. Additionally, algebraic simplifications and mathematical computations are used to derive the findings. Findings: A lemma is used to construct the formula for the Sum of m+1 consecutive Cullen Numbers. Here, the recursive and sum forms are obtained in matrix form. It also obtains the matrix representation of the sums of m+1 successive Cullen Numbers. Several intriguing correlations among Special Numbers, Cullen Numbers, and Carol Numbers are provided in the application section. Novelty: New formulas are obtained for the analysis. Research has recently discovered matrix representation with its recursive versions. Furthermore, there are many relationships between Cullen Numbers along with other peculiar numbers. Keywords: Cullen Numbers, Carol Numbers, Woodall Numbers, Sum of Cullen Number

Centrifuge tests study on settlement and damage modes of bridge approaches using deep-seated slab

The issue of bridge end bumps is a critical concern in the failure of bridge and bridge approaches. A series of novel centrifuge tests utilizing a ring model box were conducted to investigate settlement and its induced damages at the bridge approach. A new mitigation method, the deep-seated slab, for bridge end bumps was modeled in the test. This study analyzed the decisive role of pavement stiffness, soil modulus, and load cycles on deformation from the perspective of structure-soil interaction under standard traffic load conditions. The test results show that when deep-seated slabs are used, the deformation of the bridge approach follows an exponential decay pattern, eventually stabilizing after approximately one slab length. Furthermore, the upper and lower bridges exhibit distinct damage modes, i.e., the bridge damage by wheel collision at the upper bridge and the pavement damage by wheel impact at the lower bridge. The damage zone on the pavement is approximately 1.7 times the wheel width and the damage zone on the bridge 2.6 times. Finally, a predictive model for the deformation of bridge approaches was proposed, considering the effect of pavement stiffness, subgrade soil modulus, and load cycles. The relationship between the deformation and the three normalized variables conforms to the quadratic polynomial function in matrix form.

Graph neural network incorporating time-varying frequency domain features with application in spatial wind speed field prediction

Precise forecasting of wind speed is vital to guarantee safe travel on bridges. Nevertheless, current research primarily concentrates on single-point prediction. When it comes to forecasting spatial wind fields, graph neural networks prove to be powerful methods, but many of them have yet to take into account the impact of frequency domain attributes. To address this, a novel model called the graph neural network with time-varying frequency features (GTF) is proposed. Its core layer employs wavelet transforms in matrix form to extract various frequency sub-series, followed by the use of adaptive adjacency matrices to identify their spatial characteristics. Furthermore, aggregating all frequency band information and exploring spatial features further through multi-kernel convolutions. The predictive performance of different models for wind speed forecasting is assessed by utilizing spatial wind field data collected on a large-span bridge. Experimental results demonstrate that spatiotemporal model GTF outperforms the other models in terms of predictive accuracy. The quantity of wind observation points and their level of correlation exert a certain impact on the GTF model, while hyperparameters such as stacking depth and wavelet decomposition parameters have minimal effects. Finally, recommended parameters for the GTF model are provided.

Pati-Salam models with A4 modular symmetry

The flavor structure of quarks and leptons and quark-lepton unification are studied in the framework of Pati-Salam models with A4 modular symmetry. The three generations of the left-handed and right-handed fermions are assigned to be triplet or singlets of A4. The light neutrino masses are generated through the type-I seesaw mechanism. We perform a systematic classification of Pati-Salam models according to the transformations of matter fields under the A4 modular symmetry, and the general form of the fermion mass matrix is given. We present four phenomenologically viable benchmark models which provide excellent descriptions of masses and flavor mixing of quarks and leptons, including neutrinos. In such models we find that the normal ordered neutrino mass spectrum is preferred over the inverted case, with neutrinoless double beta decay predicted to be too small to be observed by the next generation of experiments.

Emergent Optical Resonances in Atomically Phase-Patterned Semiconducting Monolayers of WS2.

Atomic-scale control of light-matter interactions represents the ultimate frontier for many applications in photonics and quantum technology. Two-dimensional semiconductors, including transition-metal dichalcogenides, are a promising platform to achieve such control due to the combination of an atomically thin geometry and convenient photophysical properties. Here, we demonstrate that a variety of durable polymorphic structures can be combined to generate additional optical resonances beyond the standard excitons. We theoretically predict and experimentally show that atomic-sized patches of the 1T phase within the 1H matrix form unique electronic bands that lead to the emergence of robust optical resonances with strong absorption, circularly polarized emission, and long radiative lifetimes. The atomic manipulation of two-dimensional semiconductors opens unexplored scenarios for light harvesting devices and exciton-based photonics.

Confronting the Lippmann–Schwinger equation and the [formula omitted] method for coupled-wave separable potentials

We study a family of separable potentials with and without added contact interactions by solving the associated Lippmann–Schwinger equation with two coupled partial waves. The matching of the resulting amplitude matrix with the effective-range expansion is studied in detail. When a counterterm is included in the potential we also carefully discuss its renormalization. Next, we use the matrix N/D method and study whether the amplitude matrices from the potentials considered admit an N/D representation in matrix form. As a novel result we show that it is typically not possible to find such matrix representation for the coupled partial-wave case. However, a separate N/D representation for each coupled partial wave — a valid option known in the literature — is explicitly implemented and numerically solved in cases where the matrix N/D method is unavailable.

Structural response reconstruction of beam-like bridge based on equivalent loads under moving forces

Response reconstruction has advantages in enriching monitoring data with high quality and it is of great importance to bridge structural health monitoring. However, there still are many current difficulties with response reconstruction due to the complex operation of bridges, e.g., hard to deal with time-varying positions of vehicles and inefficient reconstruction for cases with a long time. According to the theory of equivalent load identification, this paper proposes a model-based method for reconstructing responses induced by moving forces. The proposed method introduces moving time windows to extract the measured responses in an orderly manner and re-organizes them into a matrix form. The measured responses, expressed in matrix form, then serve as the input data to identify both equivalent loads and structural initial conditions, but not to estimate the actual moving forces. Herein, the equivalent loads represent some concentrated loads with time-invariant acting positions and are employed to approximately re-express the main information of moving forces. Matrix regularization with a sparse penalty term is used to ensure the robustness of identified results. Since the structural inputs have been estimated, the structural responses happening at the validation sensors then can be calculated as a forward problem. In this process, the role of identified structural inputs is an intermediate variable connecting the measuring sensors and the validation sensors so that, the transmissibility function (TF) discussed in the proposed method is expressed in an implicit pattern. Compared to the response reconstruction methods developed from moving force identification (MFI), an obvious characteristic is that information on the actual moving forces is no longer necessary. Therefore, it can bring convenience in practical application such as the time-varying positions of moving vehicles are not needed in advance. In addition, the application of matrix regularization has the potential for efficient computation. To verify the feasibility and effectiveness of the proposed method, both numerical simulations and experimental studies are finally carried out. The illustrated results indicate that the proposed method can accurately reconstruct the structural response. In addition, some related issues are also discussed.

Trading particle shape with fluid symmetry: on the mobility matrix in 3-D chiral fluids

Chiral fluids – such as fluids under rotation or a magnetic field as well as synthetic and biological active fluids – flow in a different way than ordinary ones. Due to symmetries broken at the microscopic level, chiral fluids may have asymmetric stress and viscosity tensors, for example giving rise to a hydrostatic torque or non-dissipative (odd) and parity-violating viscosities. In this article, we investigate the motion of rigid bodies in such an anisotropic fluid in the incompressible Stokes regime through the mobility matrix, which encodes the response of a solid body to forces and torques. We demonstrate how the form of the mobility matrix, which is usually determined by particle geometry, can be analogously controlled by the symmetries of the fluid. By computing the mobility matrix for simple shapes in a three-dimensional (3-D) anisotropic chiral fluid, we predict counterintuitive phenomena such as motion at an angle to the direction of applied forces and spinning under the force of gravity.

A Magnetic Flux Leakage Detector for Ferromagnetic Pipeline Welds with a Magnetization Direction Perpendicular to the Direction of Travel.

For the sake of realizing the safety detection of natural gas and petroleum pipeline welds, this paper designs a ferromagnetic pipeline weld magnetic flux leakage detector based on the calculation of the magnetic circuit of the detection probe, with the magnetization direction perpendicular to the traveling direction. The traditional pipeline magnetic flux leakage detection device uses a detection system mode in which the magnetization direction is parallel to the direction of travel. However, due to the structural characteristics of the weld, the traditional detection system mode is not applicable. Since the weld magnetic flux leakage detector needs to travel along the direction of the weld, the detector designed in this paper rotates the magnetizer 90 degrees along the direction of the weld seam so that the magnetization direction is perpendicular to the direction of travel, breaking through the technical barrier that make traditional magnetic flux leakage detection devices unsuitable for weld detection. The detection device includes a magnetizing structure, a data sampling device, and a driving and traveling device. The magnetic flux leakage signal collected by the detector is converted into a digital image in the form of a grayscale matrix. Using mathematical morphology and chain code algorithms in image processing technology, a pipeline weld defect inversion software system is developed, and a preliminary quantitative analysis of pipeline weld defects is achieved. The application of this technology enables the inspection and protection of oil and gas pipeline welds throughout their life cycle, broadens the scope of existing inspection objects, and is of great safety significance for ensuring national public security.

A Connected Convolutional Neutral Network Protocol for Design of Two-Dimensional Materials Based on Modified Graphdiyne.

In materials science, doping plays a crucial role in manipulating the electronic properties of materials. Conventional screening via a trial-and-error strategy is challenging owing to the enormous chemical space. We proposed a connected convolutional neutral network (CCNN) for quick screening of boron nitrogen (B-N) codoped graphdiyne in terms of band gap. A paired-atomic localized matrix (PALM) descriptor was designed to describe the local chemical environment of materials with the matrix form adapted to a neutral network. An attribution analysis was conducted, and a quantitative relationship between structure and band gap is proposed, which reveals more significant influence of B-N doping at sp2 hybridized sites than at sp hybridized sites on broadening of the band gap of GDY. The accuracy and efficiency of the proposed approach implicate its potential in promoting the design of graphdiyne-based optoelectronic devices and catalysts with expected electronic properties, opening a new avenue for rational design of novel materials.

Numerical Analysis of Dynamic Stability of Flutter Panels in Supersonic Flow

This paper introduces a novel numerical method for investigating the dynamic stability of a flutter panel exposed to a supersonic gas flow and a fluctuating axial excitation force. Initially, the system equation of motion was derived using Lagrange’s equation, where the first two modes are coupled Mathieu–Hill equations with damping, constituting a system of linear second-order differential equations with periodically variable coefficients. Subsequently, a new numerical method was proposed to analyze the dynamic stability of coupled Mathieu–Hill equations with damping. This method involves breaking down an arbitrary parametric load into discrete segments to approximate the variable excitation function using a step function. The system responses of each segment are then accumulated in matrix form. The proposed numerical method proves particularly effective for dynamic systems whose parameters cannot be treated as small. In practical application, the method allows the construction of instability regions corresponding to natural frequencies, subharmonics, and combination frequencies. Dynamic stability diagrams were generated based on dynamic pressure ratio, air/panel density ratio, Mach number, panel thickness—length ratio, and excitation frequency. The results demonstrated general agreement with those obtained through Hsu’s perturbation method, however, our numerical results have proven more accurate. The paper concludes by offering suggestions for suppressing panel flutter through appropriate parameter combinations.

Black-Hole singularity and its possible mitigations: Reformulation of Penrose’s singularity theorem using null Raychaudhuri matrix

In this paper, we introduce a notion of null Raychaudhuri matrix motivated by the matrix representation of tensor fields. The evolution of this matrix gives the matrix form of null Raychaudhuri equation. Using this distinct geometric approach, we have reformulated the original Penrose’s singularity theorem on Black-Hole and have commented on the characteristic of the Raychaudhuri matrix at the singularity. The paper also suggests two possible mitigations for the physical singularity of Schwarzschild black hole via the Wheeler–DeWitt formalism and Bohmian formalism.

Efficient design of complex-valued neural networks with application to the classification of transient acoustic signals.

A paper by the current authors Paul and Nelson [JASA Express Lett. 3(9), 094802 (2023)] showed how the singular value decomposition (SVD) of the matrix of real weights in a neural network could be used to prune the network during training. The paper presented here shows that a similar approach can be used to reduce the training time and increase the implementation efficiency of complex-valued neural networks. Such networks have potential advantages compared to their real-valued counterparts, especially when the complex representation of the data is important, which is the often case in acoustic signal processing. In comparing the performance of networks having both real and complex elements, it is demonstrated that there are some advantages to the use of complex networks in the cases considered. The paper includes a derivation of the backpropagation algorithm, in matrix form, for training a complex-valued multilayer perceptron with an arbitrary number of layers. The matrix-based analysis enables the application of the SVD to the complex weight matrices in the network. The SVD-based pruning technique is applied to the problem of the classification of transient acoustic signals. It is shown how training times can be reduced, and implementation efficiency increased, while ensuring that such signals can be classified with remarkable accuracy.

Multi-stage trajectory tracking control design under comprehensive constraints based on Generalized Udwadia–Kalaba theory

A novel control design based on Generalized Udwadia–Kalaba (GUK) theory is proposed to achieve the trajectory tracking under comprehensive constraints in this paper. To describe the complex task requirements, a constraint framework including equality and inequality constraints is constructed. Thus, the complex trajectory tracking problem is converted into a constraint-following problem. Based on the framework above, GUK equation is introduced to consider the equality and inequality constraints individually. Specifically, the equality constraints are transformed into second-order matrix form, while the inequality constraints are addressed via a diffeomorphism. Then, a novel control method including servo control term and high-order control term is proposed. Through these two control terms, the constraint forces for equality and inequality can be solved separately. The effectiveness of this control method is verified by the stability analysis based on Lyapunov minimax apprach and illustrative application of unmanned surface vehicle (USV). Meanwhile, the superiority of this method is demonstrated through a comparative analysis with Udwadia–Kalaba (UK) method, Linear Quadratic Regulator (LQR) method and sliding mode control (SMC) method.

A formulation of quantum fluid mechanics and trajectories

A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schrödinger equation, respectively, where the Schrödinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schrödinger equations that are a generalization of the Hartree–Fock equations are also obtained. These equations contain a quantum Coulomb’s law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.

Numerical solutions and simulations of the fractional COVID‐19 model via Pell–Lucas collocation algorithm

The aim of this study is to present the evolution of COVID‐19 pandemic in Turkey. For this, the SIR (Susceptible, Infected, Removed) model with the fractional order derivative is employed. By applying the collocation method via the Pell–Lucas polynomials (PLPs) to this model, the approximate solutions of model with fractional order derivative are obtained. Hence, the comments are made about the susceptible population, the infected population, and the recovered population. For the method, firstly, PLPs are expressed in matrix form for a selected number of . With the help of this matrix relationship, the matrix forms of each term in the SIR model with the fractional order derivative are constituted. For implementation and visualization, we utilize MATLAB. Moreover, the outcomes for the Runge–Kutta method (RKM) are obtained using MATLAB, and these results are compared with the results obtained with the Pell–Lucas collocation method (PLCM). From all simulations, it is concluded that the presented method is effective and reliable.