Precise Formula Research Articles

Effect of homogeneous generalized thermoelasticity on semiconductor layer under magnetic field on Green and Naghdi model without energy dissipation

This paper aims to explore the impact of viscosity and time on the spread of thermoelastic waves within a uniform and isotropic three-dimensional medium subject to a thermal load on its surface. This study utilizes the temperature-rate-dependent thermoelasticity based on the GN model, specifically applying the GN II model of generalized thermoelasticity, which does not account for energy dissipation. The normal mode analysis technique is employed to address the non-dimensional coupled field equations, yielding precise formulas for displacement, stress, temperature distribution, and strain. This issue is further illustrated by graphically depicting the field variables for a material similar to copper alongside the corresponding results. Comparative analyses of numerical data, with and without considering viscosity effects, suggest that the wave propagation speed will be limited.

Employee Attrition: Analysis of Data Driven Models

Companies constantly strive to retain their professional employees to minimize the expenses associated with recruiting and training new staff members. Accurately anticipating whether a particular employee is likely to leave or remain with the company can empower the organization to take proactive measures. Unlike physical systems, human resource challenges cannot be encapsulated by precise scientific or analytical formulas. Consequently, machine learning techniques emerge as the most effective tools for addressing this objective. In this paper, we present a comprehensive approach for predicting employee attrition using machine learning, ensemble techniques, and deep learning, applied to the IBM Watson dataset. We employed a diverse set of classifiers, including Logistic regression classifier, K-nearest neighbour (KNN), Decision Tree, Naïve Bayes, Gradient boosting, AdaBoost, Random Forest, Stacking, XG Boost, “FNN (Feedforward Neural Network)”, and “CNN (Convolutional Neural Network)” on the dataset. Our most successful model, which harnesses a deep learning technique known as FNN, achieved superior predictive performance with highest Accuracy, recall and F1-score of 97.5%, 83.93% and 91.26%.

Optimality conditions and constraint qualifications for cardinality constrained optimization problems

The cardinality constrained optimization problem (CCOP) is an optimization problem where the maximum number of nonzero components of any feasible point is bounded. In this paper, by rewriting the cardinality constraint as a constraint requiring that any feasible point must lie in the union of certain subspaces, we consider CCOP as a mathematical program with disjunctive subspaces constraints (MPDSC). Since a subspace is a special case of a convex polyhedral set, MPDSC is a special case of the mathematical program with disjunctive constraints (MPDC). Using the special structure of subspaces, we are able to obtain more precise formulas for the tangent and (directional) normal cones for the disjunctive set of subspaces. We then obtain first and second order optimality conditions by using the corresponding results from MPDC. Thanks to the special structure of the subspace, we are able to obtain some results for MPDSC that do not hold in general for MPDC. In particular we show that the relaxed constant positive linear dependence (RCPLD) is a sufficient condition for the metric subregularity/error bound property for MPDSC which is not true for MPDC in general. Finally we show that under all constraint qualifications presented in this paper, certain exact penalization holds for CCOP.

Profinite genus of free products with finite amalgamation

A finitely generated residually finite group G is an OEˆ-group if any action of its profinite completion Gˆ on a profinite tree with finite edge stabilizers admits a global fixed point. In this paper, we study the profinite genus of free products G1⁎HG2 of OEˆ-groups G1,G2 with finite amalgamation H. Given such G1,G2,H we give precise formulas for the number of isomorphism classes of G1⁎HG2 and of its profinite completion. We compute the genus of G1⁎HG2 and list various situations when the formula for the genus simplifies.

Convexity of energy functions of harmonic maps homotopic to covering maps of surfaces

We study the strict convexity of the energy function of harmonic maps at their critical points from a Riemann surface to a Riemann surface, or to the product of negatively curved surfaces. When the target is a Riemann surface and when the map is of nonzero degree, we obtain a precise formula for the second derivative of the energy function along a Weil–Petersson geodesic, which implies that the energy function is strictly convex at its critical points. When the target is the product of two surfaces where each projection of the harmonic map is homotopic to a covering map, we also prove the strict convexity of the associated energy function. As an application we prove that the energy function has a unique critical point in these cases.

TO THE CALCULATION OF BANK PROTECTION MEASURES FOR CHANNELS SLOPES EXPOSED TO ALONGSHORE WAVES

Calculation methods for strengthening and covering slopes with paving stones or tiles have been developed, and precise formulas for calculating the diameters of paving stones have been adopted. The three most characteristic locations of the triangular base of the measuring pyramid on the slope are: a) When the stones forming the nest on the slope are positioned in such a way that the measuring stone placed on the tip of the pyramid is most easily overturned in the direction of the tidal current; b) When the measuring stone can be tilted in the direction of the shoreline, towards large widths; c) When the possible direction of overturning makes a large angle of 30° with the shoreline, that is when one of the sides of the equilateral triangle of the base of the pyramid is parallel to the shoreline.

Precise large deviations for a multidimensional risk model with regression dependence structure

AbstractIn this paper, we consider a nonstandard multidimensional risk model, in which the claim sizes $\{\vec{X}_k, k\ge 1\}$ form an independent and identically distributed random vector sequence with dependent components. By assuming that there exists the regression dependence structure between inter-arrival time and the claim-size vectors, we extend the regression dependence to a more practical multidimensional risk model. For the univariate marginal distributions of claim vectors with consistently varying tails, we obtain the precise large deviation formulas for the multidimensional risk model with the regression size-dependent structure.

Reinventing Formulas for Construction Project Delay Index Due to Management and Production

The objective of this study is to construct a precise formula for the management and production delay indicators that are integrated into Ms. Project's dashboard. Two simulation techniques, such as the manual formula computation and calculation integrated into the Ms. Project dashboard, were employed. Due to management and the production team's tardiness, the data were obtained through trial-and-error methods. Excel was used to analyze the data, and Ms. Project was used to enter the calculations. The research showed that the Ms. Project dashboard formula gave more detailed information about the construction project, including: (1) contract value; (2) actual progress value during monitoring; (3) value of plan progress during monitoring; (4) progress deviation; (5) cause of delay; and (6) management delay index and production delay index. The novelty of this study is that project delays have traditionally been held against the production party (contractor), whereas implementation delays have never been taken against the management party (consultant). However, using this method makes it evident who is responsible for a project delay, whether it comes from management (a consultant) or the manufacturing side (a contractor). Doi: 10.28991/HIJ-2023-04-04-06 Full Text: PDF

Exponential moments for disk counting statistics at the hard edge of random normal matrices

We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let n be the number of points. We focus on two regimes: (a) the “hard edge regime” where all disk boundaries are at a distance of order \frac{1}{n} from the hard wall, and (b) the “semi-hard edge regime” where all disk boundaries are at a distance of order \frac{1}{\sqrt{n}} from the hard wall. As n \to + \infty , we prove that the moment generating function enjoys asymptotics of the form \exp (C_{1}n + C_{2}\ln n + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(n^{-\frac{3}{5}})) for the hard edge, \exp (C_{1}n + C_{2}\sqrt{n} + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(\frac{(\ln n)^{4}}{n})) for the semi-hard edge. In both cases, we determine the constants C_{1},\dots,C_{4} explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk counting function, and establish several central limit theorems. Surprisingly, and in contrast to the “bulk”, “soft edge”, and “semi-hard edge” regimes, the second and higher order cumulants of the disk counting function in the “hard edge” regime are proportional to n and not to \sqrt{n} .

A precise asymptotic description of half‐linear differential equations

AbstractWe study asymptotic behavior of solutions of nonoscillatory second‐order half‐linear differential equations. We give (in some sense optimal) conditions that guarantee generalized regular variation of all solutions, where no sign condition on the potential is assumed. For all of these solutions, we establish precise asymptotic formulas, where positive as well as negative potential is considered. We examine, as consequences, also equations with regularly varying coefficients, or with the coefficients viewed as perturbations of exponentials, or the equations under certain critical (double roots) settings. We make also asymptotic analysis of Poincaré–Perron solutions. Many of our results are new even in the linear case.

Precise learning curves and higher-order scaling limits for dot-product kernel regression *

As modern machine learning models continue to advance the computational frontier, it has become increasingly important to develop precise estimates for expected performance improvements under different model and data scaling regimes. Currently, theoretical understanding of the learning curves (LCs) that characterize how the prediction error depends on the number of samples is restricted to either large-sample asymptotics () or, for certain simple data distributions, to the high-dimensional asymptotics in which the number of samples scales linearly with the dimension (). There is a wide gulf between these two regimes, including all higher-order scaling relations , which are the subject of the present paper. We focus on the problem of kernel ridge regression for dot-product kernels and present precise formulas for the mean of the test error, bias and variance, for data drawn uniformly from the sphere with isotropic random labels in the rth-order asymptotic scaling regime with held constant. We observe a peak in the LC whenever for any integer r, leading to multiple sample-wise descent and non-trivial behavior at multiple scales. We include a colab (available at: https://tinyurl.com/2nzym7ym) notebook that reproduces the essential results of the paper.

Convergence to closed-form distribution for the backward [formula omitted] at some random times and the phase transition at [formula omitted

We study a one-dimensional SDE that we obtain by performing a random time change of the backward Loewner dynamics in H. The stationary measure for this SDE has a closed-form expression. We show the convergence towards its stationary measure for this SDE, in the sense of random ergodic averages. The precise formula of the density of the stationary law gives a phase transition at the value κ=8 from integrability to non-integrability, that happens at the same value of κ as the change in behavior of the SLEκ trace from non-space filling to space-filling curve. Using convergence in total variation for the law of this diffusion towards stationarity, we identify families of random times on which the law of the arguments of points under the backward SLEκ flow converge to a closed form expression measure. For κ=4, this gives precise characterization for the random times on which the law of the arguments of points under the backward SLEκ flow converge to the uniform law.

Two-stage site selection of hydrogen refueling stations coupled with gas stations considering cooperative effects based on the CRITIC-ITFAHP-MABAC method: A case study in Beijing

Powering hydrogen-fueled vehicles (HFVs) through hydrogen electrolysis using excess sustainable energy is an effective way to move towards sustainable development. Hydrogen refueling stations (HRSs) coupled with gas stations (GSs) can effectively integrate and utilize the resources in existing GSs, improve operational efficiency, and promote the cooperative income of the two. Aiming at the lack of cooperative effects between gas and hydrogen considered in the current site selection process for HRSs, a general criterion establishment process with refined measurement system are proposed to determine a more comprehensive criterion system, including the cooperative criteria and the precise calculation formula of each criterion. Based on these, a new two-stage site selection framework for HRSs is established. The first stage offers eligible alternatives through veto criteria. Then in the second stage, the Criteria Importance Through Intercriteria Correlation (CRITIC) and Improved Triangular Fuzzy Analytic Hierarchy Process (ITFAHP) methods are used to determine the objective weights and subjective weights of the criteria, and the results are combined according to the minimum deviation estimation (MDE). The rankings of alternatives are finally determined through the Multi-Attributive Border Approximation Area Comparison (MABAC) method. Three scenarios are proposed for further analysis in the case study of Beijing, China. The results show that the two-stage site selection framework with new criterion system makes the evaluation results more differentiated. When the importance coefficient of subjective weights reflecting the importance of the criterion in practice changes from 0 to 1, the score increase of the best alternative A5 (Zhoujiaxiangbei) is 49.14% higher than that of the second-ranked alternative, indicating that this alternative is more in line with people's perception of a good selection. The analysis results also further verify the rationality and robustness of the decision-making framework.

On almost p-standard system of parameters of idealization and applications

Let (R,m) be a Noetherian local ring and M be a finitely generated R-module. In this paper, we construct almost p-standard systems of parameters (a strict subclass of d-sequences) of the idealization R⋉M. As applications, we give a class of Cohen-Macaulay Rees algebras for idealizations, Cohen-Macaulay Rees modules for unmixed modules, and give precise formulas computing all Hilbert coefficients of the idealization with respect to an almost p-standard system of parameters.

Revisiting semileptonic B meson decays at next-to-next-to-leading order

We compute next-to-next-to-leading order corrections to the semileptonic decay rate of B mesons for arbitrary values of the final-state quark mass. For the contribution with one massive quark in the final state, we extend the literature result and obtain analytic expressions in terms of iterated integrals. For the complete contribution, which also includes contributions with three massive quarks in the final state, we present a semi-analytic method, which leads to a precise approximation formula for the decay rate. Our results agree with the expansions available for b → cℓν¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ u} $$\\end{document}ℓ and b → uℓν¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ u} $$\\end{document}ℓ in the literature. The main emphasize of this paper is on the technical aspects of the calculation which are useful for a wider range of applications.

Explicit Formulas of Moving Average Control Chart for Zero Modified Geometric Integer Valued Autoregressive Process

This research presents precise formulas to calculate the average time to signal (ATS) of the moving average control chart (MA chart) for detecting changes in the autocorrelation of count data when the process has zero inflation and zero deflation. Thus, a zero-modified geometric integer value autoregressive order 1 (ZMGINAR (1)) process is a suitable geometrical alternative for autocorrelated count data with an enormous (or shortfall) number of zeros. The average time to signal is a traditional control chart performance; the mean of the observations taken before a process signal that it is beyond the control limit. The numerical results demonstrate the effectiveness of the control limit in detecting changes in the effect of inflation and deflation of zeros. The usefulness of a control chart in detecting variations in the model of the process can be illustrated by the actual data sample of count data.

Calculation of Thermodynamic Quantities of 1D Ising Model with Mixed Spin-(s,(2t − 1)/2) by Means of Transfer Matrix

In this paper, we consider the one-dimensional Ising model (shortly, 1D-MSIM) having mixed spin-(s,(2t−1)/2) with the nearest neighbors and the external magnetic field. We establish the partition function of the model using the transfer matrix. We compute certain thermodynamic quantities for the 1D-MSIM. We find some precise formulas to determine the model’s free energy, entropy, magnetization, and susceptibility. By examining the iterative equations associated with the model, we use the cavity approach to investigate the phase transition problem. We numerically determine the model’s periodicity.

Beyond the Maxwell Garnett approximation for interacting plasmonic nanoparticles: An analytical and numerical study

We revisit the issue of building a precise mixing formula for the effective permittivity of interacting assemblies of plasmonic nanoparticles. More precisely, we reconsider the analytical expressions rendered by the Maxwell Garnett and Torquato et al. approximation formulas and compare them to each other and to a numerical approach based on the boundary element method applied to interacting assemblies of metallic (gold or silver) nanoparticles. For efficient numerical simulations of interacting assemblies of relatively large sizes, we set up an algorithm with adaptive surface meshing that depends on the particle’s position within the assembly. Next, we derive expressions for the resonance frequency of the assembly from the analytical formulas, which are valid for gold and silver particle assemblies embedded in matrices with large optical indices. We then compare the analytical results with our numerical findings. We find that the Maxwell Garnett approximation formula underestimates the resonance wavelength and that its validity range in terms of inclusion fraction strongly depends on the nature of the metal and the embedding matrix. In the case of silver particles embedded in high-permittivity matrices, the Maxwell Garnett formula should only be used for low particle concentrations. Torquato’s formula, on the other hand, which accounts for multipolar interactions and the assembly spatial arrangement, renders a better agreement with the numerical simulations.

Phase transition for the generalized two-community stochastic block model

AbstractWe study the problem of detecting the community structure from the generalized stochastic block model with two communities (G2-SBM). Based on analysis of the Stieljtes transform of the empirical spectral distribution, we prove a Baik–Ben Arous–Péché (BBP)-type transition for the largest eigenvalue of the G2-SBM. For specific models, such as a hidden community model and an unbalanced stochastic block model, we provide precise formulas for the two largest eigenvalues, establishing the gap in the BBP-type transition.

Single-crystal structure, cation distribution, and optical/magnetic properties of Ce3+-doped Gd3Al2Ga3O12 crystal

Ce-doped Gd3Al2Ga3O12 (Ce3+:GAGG) scintillator finds practical application in industrial computed tomography and medical imaging, however a precise understanding of its single-crystal structure has been lacking. In this study, we grew a 1-inch-sized single crystal with the nominal composition 0.5%Ce3+:GAGG using the Czochralski method. To determine the crystal structure, we employed single-crystal XRD along with element analyses using ICP and XRF-WDS techniques. Our analysis revealed the precise formula of the crystal as 0.1% Ce3+:Gd3(Ga0.543Al0.457)2(Ga0.721Al0.279)3O12. Notably, Ga3+ cations exhibit a stronger preference at the tetrahedral sites compared to Al3+ cations. The optical absorption and luminescence resulting from Ce3+ 4f–5d transitions are cross-validated in the transmission and photoluminescence spectra. From 1.8 to 300 K, Ce:GAGG exhibits paramagnetic behavior and obeys the Curie-Weiss law. The 8S7/2 ground state of Gd3+ ion contributes to the overall effective magnetic moment on the one hand and plays a significant role in energy transfer from Gd3+ and Ce3+ upon excitation on the other hand. The magneto-optical Faraday rotation measurement of Ce:GAGG was performed to determine Verdet constants at 532 nm and 633 nm. These results may help a better understanding and the practical exploitation of Ce3+:GAGG as a multifunctional material in optic and magnetic fields.