Homogeneous Problem Research Articles

The Unruh Effect and the Equivalence Principle

The Unruh effect attracts the attention of researchers because it opens up new possibilities in connection with the ongoing attempts to test Einstein's theory of gravity. A special place among these attempts is occupied by the study of the equivalence principle, since in a number of works on the study of the Unruh effect, doubts have been expressed about its validity. The article shows that there is no reason for this. For this purpose, an infinite homogeneous plane is considered as the source of a stationary gravitational field in the article. The article considers solutions of Einstein's gravitational equations for the infinite homogeneous plane problem. It is shown that along with static solutions corresponding to a stationary gravitational field, there are its non-stationary solutions. Explicit analytical expressions for nonstationary solutions are obtained. It is shown that the non-stationary solution in asymptotic turns into a stationary one. The presence of such a solution explains the birth of particles and removes doubts about the possibility of the existence of the Unruh effect in a static homogeneous gravitational field due to the birth of particles at the stage of establishing a stationary gravitational field of the plane. In addition, considerations are given about the source of the singular gravitational field for the problem of hypercomputation.

Modeling of the elastoplastic deformation process of single crystal superalloys

The aim of the research is the development and verification of a micromechanically motivated model of elastoplastic deformation of two-phase single-crystal nickel-based alloys, predicting behavior under high-temperature thermomechanical actionswith taking into account the presence of γ and γ' phases. The model is relevant for computations of the stress-strain state of cooled single crystal blades of gas turbine units. The constitutive equations for each of the phases took into account the anisotropy of elastic and plastic properties, the presence of octahedral slip systems, features of the cubic system, and various hardening mechanisms, including kinematic, isotropic and latent ones. The identification of the elastic and plastic constants of the material for the γ and γ 'phases was carried out on the basis of the known stress-strain curves for each phase. The determination of the effective properties and deformation diagrams of a two-phase single-crystal alloy, taking into account the presence of γ-γ'phases, was carried out both on the basis of finite element homogenization for the representative volume element, and using the simplest rheological (structural) models of the material, considering serial and parallel connection of phases. The dependences of the elastoplastic properties of two-phase single-crystal nickel-based alloys on the volume fraction of the γ'phase are determined by computational experiments and analytical estimates. In order to determine the optimal strategy for solving the class of problems under consideration, multivariant computational experiments were carried out for various types of boundary conditions of the homogenization problem, the number of periodicity cells, forms of inclusion of the γ'phase, volume fractions of the γ' phase, types of hardening, variants of rheological models and appropriate recommendations were given. The simulation results using the proposed two-level microstructural model of the material demonstrate a good agreement with the experimental data for the single-crystal superalloy CMSX-4.

Singular boundary behaviour and large solutions for fractional elliptic equations.

We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equationsposed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace-Poisson equationwith zero boundarydata. In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that, for equationsdriven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Greenfunction.

Gradient estimates for non-uniformly elliptic problems with BMO nonlinearity

We provide a new approach to obtain Calderón-Zygmund type estimates for non-uniformly elliptic equations with discontinuous nonlinearities of double phase growth. This approach, which is based on a small higher integrability result for the gradient of weak solutions to the associated homogeneous problems together with extrapolation from Muckenhoupt weights, enables us to find a proper comparison estimate of approximation by imposing merely a small BMO assumption on the nonlinearity with respect to the x-variable. As a consequence, we are able to prove an optimal regularity theory for a larger class of double phase problems with discontinuous nonlinearities in the literature.

Homogenization of Evolutionary Incompressible Navier–Stokes System in Perforated Domains

In this paper, we consider the homogenization problems for evolutionary incompressible Navier-Stokes system in three dimensional domains perforated with a large number of small holes which are periodically located. We first establish certain uniform estimates for the weak solutions. To overcome the extra difficulties coming from the time derivative, we use the idea of Temam and consider the equations by integrating in time variable. After suitably extending the weak solutions to the whole domain, we employ the generalized cell problem to study the limit process.

A Multigroup Homogeneous Flux Reconstruction Method Based on the ANOVA-HDMR Decomposition

In this paper, we formulate a new flux reconstruction method based on the High Dimensional Model Representation (HDMR) of the nodal flux shape. The method requires the conventional nodal parameters obtained as a result of the transverse-integration nodal solution procedure as well as corner values calculated by a combination of finite-volume and finite-difference approximations. The proposed flux reconstruction procedure is applicable to rectangular nodes and more than two energy groups. The flux reconstruction method is applied to a number of homogeneous and heterogeneous test problems in order to evaluate its accuracy.

Convergence Properties of a Linear Diffusion-Acceleration Method for k-Eigenvalue Transport Problems

Most methods that use low-order operators to accelerate the iterative solution of transport eigenvalue problems employ nonlinear functionals of the transport solution (such as Eddington tensors) in their low-order equations, which are themselves standard eigenvalue problems. Here, we discuss linear diffusion synthetic acceleration (DSA) for -eigenvalue problems, which belongs to a family of methods that has received less attention than its nonlinear counterparts. We review the history of these linear methods as far as we know it and describe theoretical questions that to our knowledge have remained unanswered. With these methods, a low-order problem is solved after each transport step for an updated eigenvalue and an additive correction to the eigenfunction. These low-order problems are not standard eigenvalue problems, for they contain residuals as fixed sources. The low-order problems admit infinitely many solutions (updated and additive correction to the eigenfunction), and the solution that is obtained depends on the initial guess and iterative method chosen for the low-order problems. Experience has shown that when the low-order problems are solved with a powerlike iteration method and certain initial guesses, they yield solutions that cause rapid convergence to the correct high-order solution. We study the convergence properties of this algorithm applied to two model problems: an infinite homogeneous medium and a one-cell problem. For the infinite homogeneous problem, we present a Fourier analysis of the linear DSA method, which demonstrates that when the low-order problems are solved using a powerlike iteration scheme, the linear DSA scheme provides immediate convergence of the -eigenvalue and rapid convergence of the eigenfunction (much like DSA applied to scattering iterations in fixed-source problems). For the one-cell problems, we find that the linear scheme for -eigenvalue problems performs approximately as well as DSA for fixed-source problems. The latter analysis reveals a quantitative bound on the consistency between low- and high-order operators that is necessary and sufficient for convergence of those problems. With some theoretical foundations for the linear methods now established, we turn to numerical testing. We find, as others have before us using different low-order operators, that the method works well in practice. We provide numerical results from reactor problems in which our linear DSA is approximately as effective as the more widely used nonlinear methods. Our theoretical and numerical results add to the body of evidence that the linear methodology offers a simple path to rapid convergence of -eigenvalue problems, especially for codes that already employ linear low-order operators to converge scattering iterations.

Organic dyestuff modifier enhancing energy storage performance of PEI-based nanocomposites

In recent years, with the increasing demand for power supply in harsh environments, high temperature-resistant dielectrics have attracted more and more attention. However, polymer-based composites with high glass-transition-temperature polymers still suffer fillers’ homogeneous dispersion problems. In this work, polyetherimide (PEI)-based nanocomposites filled with modified BT nanoparticles were designed and prepared. The surface modifier is an organic dye, celestial blue, which contains ortho-hydroxyl functional groups capable of chelating with metal oxides and a conjugated molecular structure that is well compatible with the PEI polymer matrix. The nanocomposites mBT/PEI exhibit improved dielectric and energy storage performance. At 2 vol%, the maximum energy density reaches 4.2 J cm−3, which was 31 % higher than that of the nanocomposite BT/PEI without surface modification.

Duality of optimization problems with gauge functions

ABSTRACT Recently, Yamanaka and Yamashita proposed the so-called positively homogeneous optimization problem, which includes many important problems, such as the absolute-value and the gauge optimization problems. They presented a closed form of the dual formulation for the problem, and showed weak duality and the equivalence to the Lagrangian dual under some conditions. In this work, we focus on a special positively homogeneous optimization problem, whose objective function and constraints consist of some gauge and linear functions. We prove not only weak duality but also strong duality. We also study necessary and sufficient optimality conditions associated to the problem. Moreover, we give sufficient conditions under which we can recover a primal solution from a Karush-Kuhn-Tucker point of the dual formulation. Finally, we discuss how to extend the above results to general optimization problems by considering the so-called perspective functions.

Effective elastic properties and wave surfaces of rock materials containing multiple cavities and cracks (effective field approach)

The paper is devoted to simulation of the effective elastic properties of rock materials containing two substantially different types of defects simultaneously: random sets of ellipsoidal pores (cavities) and elliptical cracks. For solution of the homogenization problem and calculation of the effective elastic properties of such materials, the self-consistent effective field method is used. For composites with one type of heterogeneities, the method coincides with the Mori–Tanaka method. The method allows deriving analytical expressions for the effective elastic stiffness tensors of the materials containing pores and cracks of various scales. These tensors have correct symmetry with respect to tensor indices and provide physically reasonable values of the effective elastic constants in wide regions of porosity and crack density. The materials with pores and cracks of substantially different sizes and of close sizes are considered. The method predicts substantially different effective elastic constants of the materials with these microstructures. Comparison of the components of the effective elastic stiffness tensors for the materials with various values of porosity and crack density are presented. Wave surfaces of acoustical waves in the materials containing pores and cracks are constructed. It is shown that for materials with spherical pores and circular cracks, the shapes of these surfaces are close to ellipsoidal. The results of the paper can be used for determination of fracture level in damaged rock materials by acoustical methods.

Corrector results for space-time homogenization of nonlinear diffusion

The present paper concerns a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of corrector results (i.e., strong convergences of solutions with corrector terms) for gradients, diffusion fluxes and time-derivatives without assumptions for smoothness of coefficients. Proofs of the main results are based on the space-time version of the unfolding method, which is deeply concerned with the strong two-scale convergence theory.

Boundary homogenization with large reaction terms on a strainer-type wall

We consider a homogenization problem for the Laplace operator posed in a bounded domain of the upper half-space, a part of its boundary being in contact with the plane {x_3=0}. On this part, the boundary conditions alternate from Neumann to nonlinear-Robin, being of Dirichlet type outside. The nonlinear-Robin boundary conditions are imposed on small regions periodically placed along the plane and contain a Robin parameter that can be very large. We provide all the possible homogenized problems, depending on the relations between the three parameters: period varepsilon , size of the small regions r_varepsilon and Robin parameter beta (varepsilon ). In particular, we address the convergence, as varepsilon tends to zero, of the solutions for the critical size of the small regions r_varepsilon =O(varepsilon ^{ 2}). For certain beta (varepsilon ), a nonlinear capacity term arises in the strange term which depends on the macroscopic variable and allows us to extend the usual capacity definition to semilinear boundary conditions.

Deep-Learning-Based Complex Scene Text Detection Algorithm for Architectural Images

With the advent of smart cities, the text information in an image can be accurately located and recognized, and then applied to the fields of instant translation, image retrieval, card surface information recognition, and license plate recognition. Thus, people’s lives and work will become more convenient and comfortable. Owing to the varied orientations, angles, and shapes of text, identifying textual features from images is challenging. Therefore, we propose an improved EAST detector algorithm for detecting and recognizing slanted text in images. The proposed algorithm uses reinforcement learning to train a recurrent neural network controller. The optimal fully convolutional neural network structure is selected, and multi-scale features of text are extracted. After importing this information into the output module, the Generalized Intersection over Union algorithm is used to enhance the regression effect of the text bounding box. Next, the loss function is adjusted to ensure a balance between positive and negative sample classes before outputting the improved text detection results. Experimental results indicate that the proposed algorithm can address the problem of category homogenization and improve the low recall rate in target detection. When compared with other image detection algorithms, the proposed algorithm can better identify slanted text in natural scene images. Finally, its ability to recognize text in complex environments is also excellent.

Comprehensive investigation of the Ronen method in slab geometry

A comprehensive investigation of the Ronen method is performed in homogeneous and heterogeneous slab problems from the Sood benchmark, considering isotropic and linearly-anisotropic problems. Three finite differences implementations are exercised and compared. The results are compared to reference solutions using one and two energy groups. The validation is performed for the criticality eigenvalue and the fundamental neutron flux distribution. The results demonstrate the significantly improved accuracy achievable by the Ronen method using a broad set of problems. For standard convergence tolerances, the maximal deviation in criticality eigenvalue is less than ten pcm, and the maximal deviation in the spatial distribution of the flux is less than 2%, always located near sharp interfaces or vacuum boundaries.

Effects of Chinese Firms’ Innovation on New Energy Vehicles Purchases

As energy shortages and environmental pollution intensify, innovation in new energy vehicles is considered a major priority. They present an important opportunity to change the direction of China’s automobile industry while reducing greenhouse gas emissions. Following various launches of new energy vehicles, the problem of product homogenization is emerging, and automobile companies are contemplating how to increase their market share. This study investigates a few questions that have been left unanswered in previous research, which are mainly focused on the value and price of new energy vehicles. We focus on the effect of new energy vehicle companies’ innovative behaviors on consumers’ purchasing intention in the Chinese market. Innovative behaviors are defined here as the rationalization or modernization choices firms make with respect to products, marketing, service, technological, and cultural factors. The study verified the structural equation model constructed using survey data. Our study of 479 surveyed customers shows that consumers’ perceived value is positively (+) correlated with a higher degree of innovation by a firm. In general, the relationship between innovation behavior and consumers’ perceived risk shows negative results, however, the relationship in this research showed contradictory results. The only negative (−) effects on product purchase were observed in marketing and technological innovation; these factors increase perceived risk by increasing the tendency of consumers to choose to purchase a new energy vehicle despite feeling uneasy about the innovation despite uncertainties about the innovation of new energy vehicles. Our results present the relationship between innovative behaviors of new energy vehicle firms, consumers’ perceived value, consumers’ perceived risk, consumers’ innovative affinities and characteristics, and finally, purchase intention to explore the influence of innovation factors on consumers’ purchase intention of new energy vehicles. These findings should assist new vehicle firms in understanding Chinese consumers’ purchase intentions.

The Boltzmann fair division for distributive justice

Fair division is a significant, long-standing problem and is closely related to social and economic justice. The conventional division methods such as cut-and-choose are hardly applicable to real-world problems because of their complexity and unrealistic assumptions about human behaviors. Here we propose a fair division method from a completely different perspective, using the Boltzmann division. The mathematical model of the Boltzmann division was developed for both homogeneous and heterogeneous cake-cutting problems, and the realistic human factors (contributions, needs, and preferences) of the multiple participating players could be successfully integrated. The Boltzmann division was then optimized by maximizing the players’ total utility. We show that the Boltzmann fair division is a division method favorable to the socially disadvantaged or underprivileged, and it is drastically simple yet highly versatile and can be easily fine-tuned to directly apply to a variety of social, economic, and political division problems.

Cosine and sine families on time scales and abstract second order dynamic equations on time scales

Abstract cosine and sine functions defined on a Banach space are useful tools in the study of wide classes of abstract evolution equations. In this paper, we introduce a definition of cosine and sine functions on time scales, which unify the continuous, discrete and the cases “in between.” Our definition includes several types of time scales such as real numbers set, integers numbers set, quantum scales, among others. We investigate the relationship between the cosine function on time scales and its infinitesimal generator, proving several properties concerning it. Also, we investigate the sine functions on time scales, presenting their main properties. Finally, we apply our theory to study the homogeneous and inhomogeneous abstract Cauchy problem on time scales in Banach spaces.

A blockchain anonymity solution to prevent location homogeneity attacks

A blockchain anonymity solution to prevent location homogeneity attacks

On a Predator-Prey Model Involving Age and Spatial Structure

In this paper, we study the mathematical analysis of a nonlinear age-dependent predator–prey system with diffusion in a bounded domain with a non-standard functional response. Using the fixed point theorem, we first show a global existence result for the problem with spatial variable. Other results of existence concerning the spatial homogeneous problem and the stationary system are discussed. At last, numerical simulations are performed by using finite difference method to validate the results.

Classification of blow-up and global existence of solutions to an initial Neumann problem

The aim of this paper is to apply the modified potential well method and some new differential inequalities to study the asymptotic behavior of solutions to the initial homogeneous Neumann problem of a nonlinear diffusion equation driven by the p(x)-Laplace operator. Complete classification of global existence and blow-up in finite time of solutions is given when the initial data satisfies different conditions. We first obtain a threshold result for the solution to exist globally or to blow up in finite time when the initial energy is subcritical or critical. We also establish a decay rate of the L2 norm for global solutions. Furthermore, we provide some sufficient conditions for the existence of global and blow-up solutions for supercritical initial energy. We finally give two-sided estimates of asymptotic behavior when the diffusion term dominates the source. This is a continuation of our previous work [17].