Expansion Of Function Research Articles

Composition stability of single fcc phase in Cr–Fe–Mn–Ni alloys: First-principles prediction and experimental validation

The relative phase stability of fcc and bcc Cr–Fe–Mn–Ni alloys has been investigated using a combination of density functional theory, cluster expansion (CE), and Monte Carlo (MC) simulations. The MC simulations for fcc and bcc Cr–Fe–Mn–Ni alloys performed using the CE models enabled computing the Gibbs free energies of formation of 1767 fcc and bcc alloys in the whole range of compositions of each element and analyzed as a function of temperature. In order to determine regions of stability of single fcc/bcc structures and atomic fractions of these phases in two-phase regions, we constructed the thermodynamic databases (TDBs) derived from MC simulations with their application in the OpenCalphad calculations. The results obtained using this approach are in qualitative agreement with the available experimental data from the literature and the experiments performed within this work for the samples of [CrFeMn]100−xNix (x = 20, 25 and 35 at.%) alloys synthesized using arc-melting and annealed at 1273 K for 48 h. The analysis of fcc fractions of the near-equiatomic alloy compositions has enabled the identification of the alloys that are predicted to be single fcc phase for a wide range of temperatures.

Bioinformatics analysis of PLA2G7 as an immune-related biomarker in COPD by promoting expansion and suppressive functions of MDSCs

BackgroundImmune mechanism is involved in the pathogenesis of chronic obstructive pulmonary disease (COPD). However, the exact immune pathogenesis still remains unclear. This study aimed to identify the immune-related biomarkers in COPD through bioinformatics analysis and its potential molecular mechanism. MethodsGSE76925 was downloaded from Gene Expression Omnibus (GEO) database. Differentially expressed genes (DEGs) were screened, and enrichment analysis was performed. Single sample gene enrichment analysis (ssGSEA) was conducted to score the infiltration levels of immune cells. Weighted gene co-expression network analysis (WGCNA) was applied to identify trait-related modules and to further determine the key module-related DEGs. Moreover, the correlations between the key genes and clinical parameters and infiltration levels of immune cells were analyzed. Furthermore, expression of the selected one key gene, PLA2G7, the frequency of MDSCs, and the expression of MDSCs-related immunosuppressive mediators were determined among healthy, smokers and COPD patients. Finally, effects of PLA2G7 abnormal expression on the frequency of MDSCs and the expression of MDSCs-related immunosuppressive mediators were examined. ResultsA total of 352 DEGs were observed. These DEGs were mainly related to RNA metabolism and positive regulation of organelle organization. In addition, the black module was the most correlated with COPD. Six key genes (ADAMDEC1, CCL19, CHIT1, MMP9, PLA2G7, and TM4SF19) were identified between the black module and DEGs. Serum Lp-PLA2 and mRNA levels of PLA2G7, MDSCs, and MDSCs-related immunosuppressive mediators were found to be upregulated in COPD patients compared to the controls. The expression of PLA2G7 represented positive impact on the frequency of MDSCs and the expression of MDSCs-related immunosuppressive mediators. ConclusionPLA2G7 may serve as a potential immune-related biomarker contributing to the progression of COPD by promoting expansion and suppressive functions of MDSCs.

New benchmarks in the modelling of X-ray atomic form factors.

Analytical representations of X-ray atomic form factor data have been determined. The original data, f0(s;Z), are reproduced to a high degree of accuracy. The mean absolute errors calculated for all s = sin θ/λ and Z values in question are primarily determined by the precision of the published data. The inverse Mott-Bethe formula is the underlying basis with the electron scattering factor expressed by an expansion in Gaussian basis functions. The number of Gaussians depends upon the element and the data and is in the range 6-20. The refinement procedure, conducted to obtain the parameters of the models, is carried out for seven different form factor tables published in the span Cromer & Mann [(1968), Acta Cryst. A24, 321-324] to Olukayode et al. [(2023), Acta Cryst. A79, 59-79]. The s ranges are finite, the most common span being [0.0, 6.0] Å-1. Only one function for each element is needed to model the full range. This presentation to a large extent makes use of a detailed graphical account of the results.

Expansion and Reduction of Functionality in Secondary Grammaticalization: The Case of the Negative Laa in Rural Jordanian Arabic

Expansion and Reduction of Functionality in Secondary Grammaticalization: The Case of the Negative Laa in Rural Jordanian Arabic

Exact Solutions v.s. Perturbative Calculations of Finite Φ3-Φ4 Hybrid-Matrix-Model

There is a matrix model corresponding to a scalar field theory on noncommutative spaces called Grosse-Wulkenhaar model (Φ4 matrix model), which is renormalizable by adding a harmonic oscillator potential to scalar Φ4 theory on Moyal spaces. There are more unknowns in Φ4 matrix model than in Φ3 matrix model, for example, in terms of integrability. We then construct a one-matrix model (Φ3-Φ4 Hybrid-Matrix-Model) with multiple potentials, which is a combination of a 3-point interaction and a 4-point interaction, where the 3-point interaction of Φ3 is multiplied by some positive definite diagonal matrix M. This model is solvable due to the effect of this M. In particular, the connected ∑i=1BNi-point function G|aN11⋯aN11|⋯|a1B⋯aNBB| of Φ3-Φ4 Hybrid-Matrix-Model is studied in detail. This ∑i=1BNi-point function can be interpreted geometrically and corresponds to the sum over all Feynman diagrams (ribbon graphs) drawn on Riemann surfaces with B boundaries (punctures). Each |a1i⋯aNii| represents Ni external lines coming from the i-th boundary (puncture) in each Feynman diagram. First, we construct Feynman rules for Φ3-Φ4 Hybrid-Matrix-Model and calculate perturbative expansions of some multipoint functions in ordinary methods. Second, we calculate the path integral of the partition function Z[J] and use the result to compute exact solutions for 1-point function G|a| with 1-boundary, 2-point function G|ab| with 1-boundary, 2-point function G|a|b| with 2-boundaries, and n-point function G|a1|a2|⋯|an| with n-boundaries. They include contributions from Feynman diagrams corresponding to nonplanar Feynman diagrams or higher genus surfaces.

Tree tensor network state approach for solving hierarchical equations of motion.

The hierarchical equations of motion (HEOM) method is a numerically exact open quantum system dynamics approach. The method is rooted in an exponential expansion of the bath correlation function, which in essence strategically reshapes a continuous environment into a set of effective bath modes that allow for more efficient cutoff at finite temperatures. Based on this understanding, one can map the HEOM method into a Schrödinger-like equation, with a non-Hermitian super-Hamiltonian for an extended wave function being the tensor product of the central system wave function and the Fock state of these effective bath modes. In this work, we explore the possibility of representing the extended wave function as a tree tensor network state (TTNS) and the super-Hamiltonian as a tree tensor network operator of the same structure as the TTNS, as well as the application of a time propagation algorithm using the time-dependent variational principle. Our benchmark calculations based on the spin-boson model with a slow-relaxing bath show that the proposed HEOM+TTNS approach yields consistent results with those of the conventional HEOM method, while the computation is considerably sped up. In addition, the simulation with a genuine TTNS is four times faster than a one-dimensional matrix product state decomposition scheme.

Semiclassical Asymptotic Expansions for Functions of the Bochner–Schrödinger Operator

Semiclassical Asymptotic Expansions for Functions of the Bochner–Schrödinger Operator

On Survey of the Some Wave Solutions of the Non-Linear Schrödinger Equation (NLSE) in Infinite Water Depth

In this work, we use two different analytic schemes which are the Sine-Gordon expansion technique and the modified exp -expansion function technique to construct novel exact solutions of the non-linear Schrödinger equation, describing gravity waves in infinite deep water, in the sense of conformable derivative. After getting various travelling wave solutions, we plot 3D, 2D and contour surfaces to present behaviours obtained exact solutions.

Stability analysis and novel optical pulses to Kundu–Mukherjee–Naskar model in birefringent fibers

The concern of this study is to investigate the optical solitons pluses. The [Formula: see text]-dimensional Kundu–Mukherjee–Naskar (KMN) mathematical model in birefringent fibers is taken under consideration for this sake because this model has great importance in optics and delineate the propagation of soliton dynamics in optical fiber communication system and the rogue waves in the oceans and the bending of the light beam. Two newly developed approaches first extended rational sinh-Gordon method which is formulated from the well-known sinh-Gordon equation and second [Formula: see text]-expansion function method are manipulated for the construction of optical solitons in distinct formats. Bright, dark, and singular along their combined forms, periodic and plane wave solutions are extracted with the aid of proposed methods. Moreover, the modulation instability of investigated model is also carried out via linear stability theory. To endorse the physical compatibility of the results, the 2D, 3D, contour, and density plots have been delineated using appropriate parametric values. According to our literature research, these two methods that we are working on have not been applied to the KMN equation in birefringent fibers before, and we believe that the new solutions we have obtained will be useful to researchers working in modeling in this field. The evaluated results suggested that the techniques employed in this research to recover inclusive and standard solutions are approachable, efficient, and speedier in computing and can be considered a handy tool in solving more complex phenomena with the assistance of symbolic computation.

Bayesian interpolation with deep linear networks

Characterizing how neural network depth, width, and dataset size jointly impact model quality is a central problem in deep learning theory. We give here a complete solution in the special case of linear networks with output dimension one trained using zero noise Bayesian inference with Gaussian weight priors and mean squared error as a negative log-likelihood. For any training dataset, network depth, and hidden layer widths, we find non-asymptotic expressions for the predictive posterior and Bayesian model evidence in terms of Meijer-G functions, a class of meromorphic special functions of a single complex variable. Through novel asymptotic expansions of these Meijer-G functions, a rich new picture of the joint role of depth, width, and dataset size emerges. We show that linear networks make provably optimal predictions at infinite depth: the posterior of infinitely deep linear networks with data-agnostic priors is the same as that of shallow networks with evidence-maximizing data-dependent priors. This yields a principled reason to prefer deeper networks when priors are forced to be data-agnostic. Moreover, we show that with data-agnostic priors, Bayesian model evidence in wide linear networks is maximized at infinite depth, elucidating the salutary role of increased depth for model selection. Underpinning our results is a novel emergent notion of effective depth, given by the number of hidden layers times the number of data points divided by the network width; this determines the structure of the posterior in the large-data limit.

Bias in the Representative Volume Element method: Periodize the Ensemble Instead of Its Realizations

We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior ahom\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_{\ extrm{hom}}$$\\end{document} of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨·⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\cdot \\rangle $$\\end{document} and the corresponding linear elliptic operator -∇·a∇\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\ abla \\cdot a\ abla $$\\end{document}. In line with the theory of homogenization, the method proceeds by computing d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document} correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨·⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\cdot \\rangle $$\\end{document}. We make this point by investigating the bias (or systematic error), i.e., the difference between ahom\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_{\ extrm{hom}}$$\\end{document} and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O(L-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(L^{-1})$$\\end{document}. In case of a suitable periodization of ⟨·⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\cdot \\rangle $$\\end{document}, we rigorously show that it is O(L-d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(L^{-d})$$\\end{document}. In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨·⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\cdot \\rangle $$\\end{document} of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function.

Equivalence between radial quadrature and finite series for spherical wave expansion of Bessel beams.

The radial quadrature method was recently proposed for formulating the beam shape coefficients (BSCs) for shaped beams. A new deduction of BSCs using the R-quadrature method is presented in this paper, using the integral of the spherical Bessel functions in the interval ranging from zero to infinity. Based on the scalar description of the Bessel beam, the equivalence between the R-quadrature and the finite series (FS) method is confirmed. The spherical wave expansion of the scalar function allows us to simplify the formulation of the BSCs in the R-quadrature and the FS and to speed up the numerical BSC calculation. As a by-product, FS expansions of the associated Legendre functions are established, which we do not find in the literature.

Optical solitons with DNA dynamics arising in oscillator-chain of Peyrard–Bishop model

In this paper, we introduce the Noether approach for studying the Peyrard–Bishop (PB) DNA dynamic model equation. The Jacobi elliptic function expansion (JEFE) method is used to analyze this model. The obtained soliton solutions are hyperbolic, trigonometric, and rational functions. The dynamics of the obtained solutions conform to the properties of shock waves, periodic waves, solitary waves, double periodic waves, and rational solutions. A comparison between the results obtained by the JEFE method and the results existing in the literature shows that our results are more general, and novel, and have never been reported earlier. The wave profiles are investigated by Wolfram Mathematica simulations for the physical wave behavior.

Strain fields of Ms >6.0 earthquakes in Menyuan, Qinghai, China

In predicting earthquakes, it is a major challenge to capture the time factor and spatial isoline anomalies, and understand their physical processes, of the seismic strain field before a strong earthquake. In this study, the seismic strain field was used as representative of seismic activity. The natural orthogonal function expansion method was used to calculate the seismic strain field before the Menyuan Ms 6.4 earthquakes in 1986 and 2016, and the Ms 6.9 earthquake in 2022. Time factor and spatial isoline anomaly of the strain field before each earthquake was extracted. We also compared the evolution of the strain field with numerical simulation results under the tectonic stress system at the source. The results showed that the time factor before the earthquakes had high or low value anomalies, exceeding the mean square error of the stable background. The anomalies were concentrated in the first four typical fields of the strain field, which has multiple components. The abnormal contribution rate of the first typical field is the largest (accounting for 42%–49% of the total field). The long- and medium-term anomalies appear 3-4, and 1-2 years before the earthquake, respectively. There were no short or immediate-term anomalies within 3 months of the earthquake. In addition, during the evolution of the strain field, the abnormal area of the spatial isoline changed with the change in time. Usually, the intersection area of the two isoseismic lines of strain accumulation and strain release becomes a potential location for strong earthquakes. Finally, we found that the high strain field values of the 1986 and 2016 Ms 6.4 earthquakes were equivalent to the numerical simulation results, while the high strain field values of the 2022 Menyuan Ms 6.9 earthquakes were slightly different, but within the accepted error range. These results indicate that the two methods are consistent. We have shown that the natural orgthagonal method can be used to obtain the spatiotemporal anomaly information of strain field preceding strong earthquakes.

Dynamic fracture behavior for inclusion-initiated cracks in graded non-homogeneous magnetic-electric-elastic material

In this paper, dynamic anti-plane problem in graded non-homogeneous magnetic-electric-elastic material with a conductive cylindrical rigid inclusion is studied. Cracks are generated from the inclusion edge under the action of SH guided waves. By applying a pair of reversed shear stresses to the crack surface, the mechanical model of the crack is achieved, and Green’s function method is used in this process. The scattering fields of displacement, electric potential and magnetic potential are constructed by the wave function expansion method. Numerical results clarified the factors that affect the dynamic stress intensity factor of the crack tips, including material parameters, inclusion parameters, wave number and incident angle. It is expected that the investigation of magnetic-electric-elastic material with inclusion-initiated crack defects can provide some references for the project designing, manufacture and application of non-homogeneous multi-fields coupling materials.

High-Frequency Time Series Data Regression Based on the Functional Bayesian Model Averaging

Regression analysis of high-frequency time series data has been an important problem in the field of statistical learning. In this paper, we propose a novel approach based on multiple basis function expansions and Bayesian model averaging when the predictor variable is a high-frequency time series and the response variable is continuous scalar data. On the one hand, the proposed method avoids the curse of dimensionality and extracts the functional information of the original sequence by transforming the discrete data into continuous functions in a function space consisting of some fixed basis functions. On the other hand, for the choice of basis functions, this paper proposes to adaptively determine the optimal solution using Bayesian model averaging, effectively balancing the variance and bias of the predictive model. The real data analysis shows that the proposed method has smaller mean squared error and absolute error, and is robust compared to other methods. Finally, the proposed method can be further extended to time series data applications such as weather forecasting and stock price prediction.

Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving q-Trigonometric Functions with Applications in Computer Modeling

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results.

Holographic thermal correlators from supersymmetric instantons

We present an exact formula for the thermal scalar two-point function in four-dimensional holographic conformal field theories. The problem of finding it reduces to the analysis of the wave equation on the AdS-Schwarzschild background. The two-point function is computed from the connection coefficients of the Heun equation, which can be expressed in terms of the Nekrasov-Shatashvili partition function of an SU(2)SU(2) supersymmetric gauge theory with four fundamental hypermultiplets. The result is amenable to numerical evaluation upon truncating the number of instantons in the convergent expansion of the partition function. We also examine it analytically in various limits. At large spin the instanton expansion of the thermal two-point function directly maps to the light-cone bootstrap analysis of the heavy-light four-point function. Using this connection, we compute the OPE data of heavy-light double-twist operators. We compare our prediction to the perturbative results available in the literature and find perfect agreement.

Controlling Optomechanical Libration with the Degree of Polarization.

Control of the potential energy and free evolution lie at the heart of levitodynamics as key requirements for sensing, wave function expansion, and mechanical squeezing protocols. Here, we experimentally demonstrate versatile control over the optical potential governing the libration motion of a levitated anisotropic nanoparticle. This control is achieved by introducing the degree of polarization as a new tool for rotational levitodynamics. We demonstrate thermally driven free rotation of a levitated anisotropic scatterer around its short axis and we use the rotational degrees of freedom to probe the local spin of a strongly focused laser beam.

On the Pinsky Phenomenon for B-Elliptic Operators

Necessary conditions for the summability of spectral expansions in eigenfunctions of an elliptic operator with a Bessel operator in one of the variables in an arbitrary-dimensional domain adjacent to the singularity hypersurface are obtained. It is proved that if the spectral expansion of an arbitrary function at some point of this hypersurface is summable by Riesz means, then its average value over the half-ball centered at the specified point has generalized smoothness.