Matrix Formulation Research Articles

The Pauli-Poisson equation and its semiclassical limit

. The Pauli-Poisson equation is a semi-relativistic model for charged spin-1∕2-par-ticles in a strong external magnetic field and a self-consistent electric potential computed from the Poisson equation in three space dimensions. It is a system of two magnetic Schrödinger equations for the two components of the Pauli 2-spinor, representing the two spin states of a fermion, coupled by the additional Stern-Gerlach term representing the interaction of magnetic field and spin. We study the global well-posedness in the energy space and the semiclassical limit of the Pauli-Poisson to the magnetic Vlasov-Poisson equation with Lorentz force and the semiclassical limit of the linear Pauli equation to the magnetic Vlasov equation with Lorentz force. We use Wigner transforms and a density matrix formulation for mixed states, extending the work of P. L. Lions & T. Paul as well as P. Markowich & N.J. Mauser on the semiclassical limit of the non-relativistic Schrödinger-Poisson equation.

The 1925 revolution of matrix mechanics and how to celebrate it in modern quantum mechanics classes

In 1925, Heisenberg, Born, and Jordan developed matrix mechanics as a strategy to solve quantum-mechanical problems. While finite-sized matrix formulations are commonly taught in quantum instruction, following the logic and detailed steps of the original matrix mechanics has become a lost art. In preparation for the 100th anniversary of the discovery of quantum mechanics, we present a modernized discussion of how matrix mechanics is formulated, how it is used to solve quantum-mechanical problems, and how it can be employed as the starting point for a postulate-based formulation of quantum-mechanics instruction. We focus on the harmonic oscillator to describe how quantum mechanics advanced from the Bohr–Sommerfeld quantization condition, to matrix mechanics, to the current abstract ladder-operator approach. We also describe a number of different activities that can be included in the quantum mechanics classroom to celebrate this centennial.

Use of high-dose matrix formulation of vitamin D for the prevention and treatment of vitamin D deficiency, including in comorbid patients

On May 22, 2024, an Interdisciplinary Expert Council, "A new chapter in the treatment of vitamin D deficiency and insufficiency: 50,000 IU in one matrix tablet – discovering the potential together", was held in Moscow, dedicated to the advantages and features of the innovative high-dose vitamin D recently launched on the Russian market (Devilam 5000 and 50,000 IU in one matrix tablet), with the participation of leading experts of various medical specialties. During the Council, new opportunities for the use of high-dose vitamin D in comorbid patients in need of rapid and effective correction of preexisting vitamin D deficiency or insufficiency were discussed.

Surface Effects Study: A Continuum Approach From Fundamental Modes to Higher Modes and Topological Polarization in Orthotropic Piezoelectric Materials

Abstract The primary goal of the current work is to investigate how wave propagation influences the performance of surface acoustics wave (SAW) macro- and nano-sensors. Therefore, shear horizontal (SH) waves use the surface piezoelectricity theory to explore SH waves in an orthotropic piezoelectric quasicrystal (PQC) layer overlying an elastic framework (Model I), a piezoelectric substrate, and an orthotropic PQC substrate (model II). This study employs a variable-separable technique. The theoretical forms are constructed and used to present the wavenumber of surface waves in any direction of the piezoelectric medium, based on the differential equations and matrix formulation. In addition, we take into account the surface elasticity theory in order to obtain the phase velocity equation. Two configurations are examined: an orthotropic piezoelectric material layer over an elastic framework and a piezoelectric material half-space with a nanosubstrate. Analytical expressions for frequency equations are derived for both symmetric and antisymmetric waves. This study investigates the effects of surface elastic constants, surface density, anisotropic piezoelectric constant, and symmetric and antisymmetric modes on phase velocity. This study is confined to only linear wave propagation. Additionally, the analysis is based on idealized material properties, surface properties, and characteristic length of the material.

Matrix formulation of the wheel-rail contact element in train-track dynamic analysis

Matrix formulation of the wheel-rail contact element in train-track dynamic analysis

From Fossil to Bio-Based AESO-TiO2 Microcomposite for Engineering Applications.

Environmental issues and the reduction of fossil fuel resources will lead to the partial or total substitution of petroleum-based materials with natural, raw, renewable ones. One expanding domain is the obtaining of engineering materials from vegetable oils for sustainable, eco-friendly polymers for different applications. Herein, the authors propose a simplified and green synthesis pathway for a thermally curable, acrylated and epoxidized soybean oil matrix formulation containing only epoxidized soybean oil, acrylic acid, a reactive diluent (5%) and just 0.15 mL of catalyst. The small amount of reactive diluent significantly reduced the initial system viscosity while eliminating the need for adding solvent, hardener, activator, etc. Both the thermally cured composite with a 2% TiO2 microparticle filler and its pristine matrix were comparably characterized in terms of structural, thermal, morphological, dielectric and wettability by Fourier transform infrared spectroscopy, differential scanning calorimetry, thermogravimetry, scanning electron microscopy, broadband dielectric spectrometry and contact angle measurements. The 2% filler in the composite generated superior thermal stability via lower mass loss (48.89% vs. 57.14%) and higher degradation temperatures (395 °C vs. 387 °C), increased the glass transition temperature from -20 °C to -10 °C, rendered the microcomposite hydrophobic by increasing the contact angle from 88° to 96° and enhanced dielectric properties compared to the pristine matrix. All investigations recommend the microcomposite for protective coatings, capacitors, sensors and electronic circuits. This study brings new contributions to green chemistry and sustainable materials.

Extended formulation of macro-element based modelling – Application to single-lap bonded joints

An extended formulation of the macro-element (ME) based models, representing for both adherends and adhesive along the entire overlap in only one four-node element, is presented. Compared to earlier modelling, continuum ME (CME) and discrete ME (DME) based models, the adherend parts are also modelled as plane continuum media, for which high order displacement fields are freely supposed. Both extended CME (ECME) and extended DME (EDME) based modelling can have their displacement field orders of each continuum layer be set individually, and can be enriched using springs for interfaces or boundary conditions modelling, allowing the stresses to vanish at the free edges. The methodology and formulation of the stiffness matrix for both the adherend without adhesive and the bonded overlap is presented. The assessment is performed for a single-lap joint geometry by comparison results from a plane strain finite element (FE) model with extended ME-based models. Good agreements are shown for both thin and thick adhesive case studies.

Optimum selection of nature-inspired texture pattern for cutting tool surface using an integrated multi-criteria decision-making approach: a comparative analysis

Purpose This paper aims to focus on the selection of an appropriate nature-inspired texture pattern for cutting tool tribological surface. The selection process uses the recognized skin textures of different snakes scrolling on highly rough and projected surface conditions to analyze suitability of texture based on the texture geometry and machining conditions. The work also aims to propose a texture pattern selection process to incorporate on cutting tool tribological surface. Design/methodology/approach The selection of alternative nature-inspired texture patterns based on the texture pattern geometry and machining properties leads to a multi-criteria decision-making problem. Thirteen criteria are considered for selecting an appropriate texture pattern among 14 alternatives, i.e. nature-inspired texture patterns. In the present work, an integrated analytical hierarchy process (AHP)-TOPSIS, AHP-multi-objective optimization on the basis of ratio analysis (MOORA) and AHP-Vlse Kriterijumska Optimizacija Kompromisno Resenje (VIKOR) approaches have been proposed for the selection of an appropriate nature-inspired texture pattern. AHP is used for the formulation of decision-making matrix and criteria weight calculations and ranking of alternatives is done by three methods. Spearman’s correlation compared and found positive relations between rank assigned by methods. Experimental validation is done in Lathe for selected texture effects. Findings The texture parameters C-1 (Width of texture) and C-2 (Depth of texture) are found significant, while T-2 (Blended Krait) and T-6 (Banded Racer-1) texture is found optimal to generate on cutting tool surface. Research limitations/implications Only some nature-inspired texture patterns have been recognized before the selection; an infinite number of textures are available in nature. The size of the texture pattern is difficult to identify by the selection process because each texture pattern may have different effects on tribological surfaces. Practical implications The proposed selection methodology of nature-inspired texture patterns will help identify optimal texture geometry for specific tribological applications. The nature-inspired texture patterned tool has a significant impact on the cutting force and temperature due to its tribological effect on the cutting tool surface; it decreases the power required for machining. The machining characteristics like roughness are found to decrease by using nature-inspired texture patterned tools. Social implications Various nature-inspire texture studies to generate specific effects on the tribological surfaces may be started study for the surface of aircraft, ships, bearings, etc. Small and big fabrication industries may benefit by decreasing the cost of machining using nature-inspired texture-patterned tools. Research society will pay attention to nature’s inspiration. Originality/value Novel snake-skin-inspired texture patterns are recognized and hybrid MCDM methods are proposed to select optimal texture pattern. Proposed method used single time normalization to effectively rank the alternatives. The insights gained from this research can be extrapolated to address similar challenges in selecting nature-inspired textures for various applications. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-05-2024-0163/

Green RP-HPLC method for the estimation of carfilzomib in bulk, protein nanocarriers and human plasma: Application of chemometrics and Monte-Carlo simulations

Carfilzomib is a tetrapeptide epoxyketone that has shown potential clinical outcomes in the treatment of multiple myeloma. However, inaccuracies in quantifying such peptide drug products have arisen due to poor stability, low solubility, time-consuming techniques, complex physicochemical properties, and use of non-green solvents with less recyclability. This provides a substantial urge to develop an ecological and sensitive analytical method for quantifying peptide drugs from matrix formulation and biological samples in early as well as lateral stages of product development in pharma industries. As a result, the study aimed to develop a robust ecological method for estimation of carfilzomib via Green RP-HPLC using analytical quality by design (AQbD) paradigms with specific application in protein nanoparticles and biological matrix. Initially, an appropriate wavelength for quantification of carfilzomib was chosen using principal component analysis (PCA) as a chemometric tool.Risk assessment followed by factor screening studies using 8-factor Placket-Burman Design aided in earmarking critical method parameters (CMPs) affecting critical analytical attributes (CAAs). Further, Central Composite Design (CCD) was employed for design space optimisation to demarcate optimum chromatographic conditions, which were corroborated for robustness using Monte-Carlo simulations. The method was validated as per ICH Q2 (R2), followed by quantifying the greenness of the method using Green Assessment tools. The method optimisation resulted in the optimal chromatographic conditions using Green RP-HPLC. The chromatographic system was equipped with a Phenomenex Aeris Peptide-XC C18 column (150 × 4.6 mm × 5 µm), and the mobile phase was composed of isopropanol:methanol:0.1 M PBS (pH 5.5 adjusted using 0.1 % formic acid) (35:45:20v/v), with a 1 ml/min flow rate at a 210 nm ʎmax. The optimised chromatographic conditions resulted in a short retention time (RT) of 4.95 mins, 0.87 tailing factor (TF), 4,875,122 peak area (PA), and 8995 theoretical plate count (TPC). The method demonstrated linearity in a wide range of concentrations (0.1–20 µg/ml) with a correlational coefficient of 0.997 and < 2 % RSD. The method unearthed a high precision rate with more than 95 % of drug recovery in protein nanoparticles and human plasma, thereby confirming the accuracy and sensitivity of the developed method. Chemometrics and Monte-Carlo simulations ratified the robustness and sensitivity of the developed analytical method of Carfilzomib with established greenness and a high degree of practical utility in protein-based nano formulations and human plasma matrix for life cycle product development.

A Matrix-Based Approach to Unified Synthesis of Planar Four-Bar Mechanisms for Motion Generation With Position, Velocity, and Acceleration Constraints

Abstract This paper introduces a novel matrix-based approach for the simultaneous type and dimensional synthesis of planar four-bar linkage mechanisms, accommodating various practical constraints, including position, velocity, acceleration, and joint placements. Traditional design processes segregate type synthesis, the determination of joint and link configurations, from dimensional synthesis, which involves specifying link sizes and pivot locations. This segregation often leads to complexities in addressing the complete design challenge. The novel methodology proposed in this paper departs from the conventional sequential design approach by concurrently evaluating type and dimensional parameters using a data-driven matrix formulation. The crux of the paper’s methodology involves formulating a singular design equation through a transformation matrix, parameterized by the Cartesian parameters of the mechanism’s dyads. This formulation linearly expresses a broad range of constraints, facilitating the identification of viable solutions through singular value decomposition and null space analysis. This integrated approach not only simplifies the synthesis process but also provides direct insights into the mechanism’s parameters, encompassing both type and dimensions, thereby obviating the need for further interpretative steps common to the use of quaternions and kinematic mapping. In essence, the paper presents two main contributions: the development of a unified design equation capable of encompassing a wide array of constraints within the mechanism synthesis process, and the introduction of an algorithm that effectively identifies all potential planar four-bar linkage mechanisms by accurately satisfying up to five constraints. This approach promises to enhance the design and optimization of mechanical systems by offering a more holistic and efficient pathway to mechanism synthesis.

Investigation on the Hydrothermal Condition in Synthesis of Active Matrix from Metakaolin: Physicochemical Properties and Intrinsic Cracking Activities

The current trends in research and development of FCC catalyst is focused on the formulation of active matrices that serve as pre-crackers, with the objective of reducing the diffusional resistance of the longer chain hydrocarbon molecule in the feed. In this study, an aluminosilicate active matrix was synthesised from metakaolin using hydrothermal method. The experimental variables that were varied were hydrothermal temperature, in the range of 80 to 110 °C, and hydrothermal time, in the range of 12 to 72 hours, to investigate the best conditions for synthesising the active matrix. Subsequently, the active matrix was subjected to a series of analyses, including X-ray fluorescence, X-ray diffraction, N2 physisorption, NH3-temperature programmed desorption, Fourier transform infrared spectroscopy, scanning electron microscopy and thermogravimetry, with the objective of determining its composition, crystal characteristics, surface characteristics, acidity, functional groups, material structure, and thermal characteristics. Additionally, the active matrix was tested for its intrinsic cracking activity using the micro activity test (MAT). The results indicate that the best temperature for hydrothermal synthesis of the active matrix is 80 °C. The active matrix synthesised with a heating time of 24 hours demonstrated the highest light cycle oil yield, reaching 38.9 wt%. Meanwhile, the active matrix synthesised at 48 hours exhibited the most favourable characteristics, with a specific surface area of 144.23 m2/g and a pore volume of 0.9933 cm3/g, as well as the highest cracking conversion of 70.0 wt%. Copyright © 2024 by Authors, Published by BCREC Publishing Group. This is an open access article under the CC BY-SA License (https://creativecommons.org/licenses/by-sa/4.0).

An injectable hydrogel of porcine cholecyst extracellular matrix for accelerated wound healing.

Hydrogel formulations of xenogeneic extracellular matrices have been widely used for topical wound care because of their exceptional tunability over other formulations like lyophilized sheets, powders, non-injectable gels, pastes, and ointments. This is important in the treatment of wounds with irregular shapes and depth. This study identified an injectable hydrogel formulation of porcine cholecyst extracellular matrix (60%) in medical-grade carboxymethyl cellulose (40%) as vehicle and evaluated its biomaterial properties. Further, an in-depth evaluation of in vivo wound healing efficacy was conducted in a rat full-thickness skin excision wound healing model, which revealed that the hydrogel formulation accelerated wound healing process compared to wounds treated with a commercial formulation and untreated wounds. The hydrogel appeared to have promoted a desirable pro-regenerative tissue reaction predominated by Th2 helper lymphocytes and M2 macrophages as well as an effective collagen remodeling indicative of diminished scarring. In conclusion, the porcine cholecyst extracellular matrix injectable hydrogel formulation appeared to be a promising candidate formulation as an advanced wound care biomaterial for faster healing of skin wounds with minimal scarring.

Acoustic Black Hole silencers for broadband dissipation in ducted geometries

The design of novel absorbers avoiding the use of porous or fibrous materials for ventilation and air-conditioning systems constitutes a challenging issue. Recently, the development of metamaterials has provided new noise control perspectives for this problem. Traditional Acoustic Black Holes (ABHs) have been derived with close-ended configurations to achieve broadband absorption. These solutions are suitable as anechoic termination but they cannot be used as silencer as they require a mean flow going through the axis of the acoustic treatment. In this study different configurations of opened ABHs will be considered. Nominal ABHs build on straight cavities of increasing depth along the axial direction. To extend the bandwidth of acoustic performance towards the low frequencies, two different types of ABH silencers will be studied. Cavities with coiled geometries will be considered as a first instance, while the entrance of these cavities will be covered by micro-perforated consequently. The transfer matrix formulation will be adapted to these configurations and used to provide the ABH acoustical performance in terms of reflection and transmission of the silencer to find out the best configuration. These results will be validated against a Finite Element method and experiments will be carried out for the nominal configuration.

Evaluation of carrageenans extracted by an eco-friendly technology as source for gelled matrices with potential food application

Red macroalgae are considered an immense source of hydrocolloids (agar and carrageenan) that are gaining momentum in the food industry as an alternative to animal-based ones, like gelatin. This work evaluates carrageenans extracted from four different red macroalgae (Chondrus crispus, Mastocarpus stellatus, Sarcopeltis skottsbergii and Gigartina pistillata) by an eco-friendly process (hydrothermal extraction), for their possible employment as food additives considering purity requirements stated by the European Regulation. In general, carrageenans presented a suitable composition, although some sample presented lower sulfate content than 15 % and higher As content than 3 mg/kg, being only carrageenans from Chondrus crispus and Sarcopeltis skottsbergii appropriate for gelled matrices formulation. Different concentrations of hydrocolloids (1–5 %) and salts (0.1–1 M NaCl, CaCl2 and KCl) were evaluated to reach a desired consistency. Rheological behavior of said gels revealed a gel-like behavior, with G' > G” and practically frequency independency of the parameters. Overall, gels formulated with KCl achieved higher G' with maximum values of 100–1000 Pa, whereas the commercial gelled dessert (used as control) only achieved values of around 10 Pa. After 3 months of cold storage, all gels exhibited a strengthening of the gelled matrix, without water syneresis. The colorimetric parameters were also evaluated, showing higher inclination for red and yellow tones with modest lightness values (around 60 %). In this work, hydrothermally extracted carrageenans from Chondrus crispus and Sarcopeltis skottsbergii were assessed, laying the groundwork for further studies in this area.

On a Matrix Formulation of the Sequence of Bi-Periodic Fibonacci Numbers

In this study, we investigate some new properties of the sequence of bi-periodic Fibonacci numbers with arbitrary initial conditions, through an approach that combines the matrix aspect and the fundamental Fibonacci system. Indeed, by considering the properties of the eigenvalues of their related 2×2 matrix, we provide a new approach to studying the analytic representations of these numbers. Moreover, the similarity of the associated 2×2 matrix with a companion matrix, allows us to formulate the bi-periodic Fibonacci numbers in terms of a homogeneous linear recursive sequence of the Fibonacci type. Therefore, the combinatorial aspect and other analytic representations formulas of the Binet type for the bi-periodic Fibonacci numbers are achieved. The case of bi-periodic Lucas numbers is outlined, and special cases are exposed. Finally, some illustrative examples are given.

Directionally Asymmetric in Orbital Angular Momentum Generation Using Single‐Layer Dielectric Janus Metasurfaces

AbstractJanus metasurfaces, offering versatile light manipulation contingent on incident direction, are explored across microwave to mid‐infrared spectra. Nonetheless, prior designs often resort to spatial multiplexing or vertical stacking, entailing complex fabrication. Metallic Janus metasurfaces, owing to material properties, suffer notable Ohmic losses in the visible spectrum. In this study, Single‐layer TiO2‐based Janus metasurfaces with arbitrary polarization control is experimentally demonstrated, exhibiting directionally asymmetric functionalities in spin and orbital angular momentum (OAM). A novel Jones matrix formulation tailored for Janus metasurfaces is presented, enabling efficient generation of two asymmetric, high‐purity OAM states of vortex beams at wavelength of 532 nm, contingent on incidence direction. This innovation facilitates compact and versatile phase manipulation, encompassing applications such as lasers and optical combiners, thereby expanding its utility across diverse domains.

A Fast Matrix Solver for Linear Boundary-Value Problem in the PEM Fuel Cell Impedance Model

A core physics-based model for PEM fuel cell impedance reduces to the numerical solution of a boundary value problem for two coupled, linear, complex–valued ordinary differential Eqs. with variable coefficients. We report a matrix formulation of a previously developed recurrent method for the fast numerical solution of this problem. This formulation does not require iterations and it utilizes only elementary functions and 4 × 4 matrix products, resulting in one to two orders of magnitude faster code compared to the standard Python boundary–value problem solver. This significantly improves the computational speed of the physics-based impedance model, making it competitive with the popular equivalent circuit modeling approach. The numerical calculation of cell impedance is demonstrated based on the model that includes proton transport in the catalyst layer and oxygen transport in the catalyst and gas-diffusion layers. Least–squares fitting of a real experimental spectrum shows that the matrix solver returns the cell transport parameters 50 times faster than the classic boundary–value problem solver.

Reduced Density Matrix Formulation of Quantum Linear Response.

The prediction of spectral properties via linear response (LR) theory is an important tool in quantum chemistry for understanding photoinduced processes in molecular systems. With the advances of quantum computing, we recently adapted this method for near-term quantum hardware using a truncated active space approximation with orbital rotation, named quantum linear response (qLR). In an effort to reduce the classic cost of this hybrid approach, we here derive and implement a reduced density matrix (RDM) driven approach of qLR. This allows for the calculation of spectral properties of moderately sized molecules with much larger basis sets than so far possible. We report qLR results for benzene and R-methyloxirane with a cc-pVTZ basis set and study the effect of shot noise on the valence and oxygen K-edge absorption spectra of H2O in the cc-pVTZ basis.

Computational Methods for Stationary and Nonstationary Fokker–Planck–Kolmogorov Equations with Random Coefficients

In this paper, methodological approaches for distribution functions of nonlinear stochastic differential equations (SDEs) with uncertain parameters are elaborated. The stationary and nonstationary Fokker–Planck–Kolmogorov (FPK) equations associated with the considered random equations are investigated. A semi-analytical approach based on the extended exponential closure handling the parameters uncertainty effects is presented. A new numerical method coupling the polynomial chaos with the differential quadrature method (DQM) is elaborated for the stationary case. A compact matrix formulation is established allowing considering a large number of generalized polynomial chaos. For nonstationary random FPK equations, the polynomial chaos is coupled with the quadrature discretization. A symmetric discretization of the main Fokker–Planck operator using a tensor product of the Hilbert state space and the Hilbert probabilistic space has resulted. The obtained eigenvalues and random eigenfunctions are used for a spectral decomposition. The time solution is obtained by the spectral expansion as well as by the random Euler stochastic method. The accuracy, effectiveness and advantages of the developed procedure in analyzing the stationary and nonstationary probabilistic solutions of nonlinear stochastic dynamical systems with uncertain parameters excited by a Gaussian white noise are demonstrated.

Ice cream cone fortified with spent coffee ground: Chemical composition, quality and sensory characteristics, and in vitro starch digestibility

The objective of the study was to evaluate the effect of the incorporation of spent coffee grounds in ice cream cones on the quality, sensory characteristics, and in vitro starch digestibility. The incorporation of spent coffee grounds in ice cream cones increased the content of dietary fiber and phenolic compounds. However, the quality and texture characteristics decreased with the addition of spent coffee grounds. The in vitro starch digestibility decreased significantly, resulting in a significant increase in resistant starch content. Fitting starch digestibility using the LOS-plot model revealed the presence of two sequential first-order digestion rates. Sensory analysis revealed that the panelists well accepted ice cream cones fortified with spent coffee grounds. The results suggest that spent coffee grounds are a potential ingredient for the formulation of food matrices with reduced starch digestibility, which contributes to the prevention of chronic degenerative diseases such as type II diabetes.