Inhomogeneous Case Research Articles

Long Term Average Cost Control Problems Without Ergodicity

We consider a stochastic control problem with time-inhomogeneous linear dynamics and a long-term average quadratic cost functional. We provide sufficient conditions for the problem to be well-posed. We describe an explicit optimal control in terms of a bounded and non-negative solution of a Riccati equation on [0, infty ), without an initial and terminal condition. We show that, in contrast to the time-homogeneous case, in the inhomogeneous case the optimally controlled state dynamics are not necessarily ergodic.

Distribution Density of the First Exit Point of a Two-Dimensional Diffusion Process from a Circle Neighborhood of Its Initial Point: The Inhomogeneous Case

Distribution Density of the First Exit Point of a Two-Dimensional Diffusion Process from a Circle Neighborhood of Its Initial Point: The Inhomogeneous Case

Existence, uniqueness and asymptotic behavior of solutions to two-term fractional differential equations

In this paper, we give a systematic analysis for solutions to two-term fractional differential equations. Our main contribution is in Section 5. In this section we have shown the global attractiveness, the Mittag-Leffler stability and the local attractiveness of solutions to some classes of two-term fractional differential equations. To do these, our approach is combining Cauchy’s argument principle in complex analysis (to obtain an explicit necessary and sufficient condition for the asymptotic stability of linear equations), the Hankel type contour (to estimate the decay rate of special functions arising in the elementary solutions) and a linearization method developed recently for fractional-order differential equations (to prove the Mittag-Leffler stability of the trivial solution to nonlinear equations and the local attractivity of the origin for the inhomogeneous nonlinear case). Numerical simulations are finally presented to confirm the theoretical findings.

Profile decomposition in Sobolev spaces and decomposition of integral functionals I: Inhomogeneous case

The present paper is devoted to analyzing the lack of compactness of bounded sequences in inhomogeneous Sobolev spaces, where bounded sequences might fail to be compact due to an isometric group action, that is, translation. It will be proved that every bounded sequence (un) has (possibly infinitely many) profiles, and then the sequence is asymptotically decomposed into a sum of translated profiles and a double-suffix residual term, where the residual term becomes arbitrarily small in appropriate Lebesgue or Sobolev spaces of lower order. To this end, functional analytic frameworks are established in an abstract way by making use of a group action G, in order to characterize profiles by (un) with G. One also finds that a decomposition of the Sobolev norm into profiles is bounded by the supremum of the norm of un. Moreover, the profile decomposition leads to results of decomposition of integral functionals of subcritical order. It is noteworthy that the space where the decomposition of integral functionals holds is the same as that where the residual term is vanishing.

Interpolation inequalities in function spaces of Sobolev-Lorentz type

Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters of the function spaces. Several results on necessary conditions are also provided. Next, utilizing the interpolation inequalities together with some embedding results, we prove Gagliardo-Nirenberg inequalities for fractional derivatives in Lorentz spaces, which do hold even for the limiting case when one of the parameters is equal to 1 or ∞.

NEW RESULTS FOR PARABOLIC EQUATION ON THE SPHERE WITH CAPUTO–FABRIZIO OPERATOR

In this paper, we are interested in studying the initial value problem for parabolic problem associated with the Caputo–Fabrizio derivative. We deal the problem in two cases: linear inhomogeneous case and nonlinearity source term. For the linear case, we derive the convergence result of the mild solution when the fractional order [Formula: see text] under some various assumptions on the initial datum. For the nonlinear problem, we show the existence and uniqueness of the mild solution using Banach fixed point theory. We also prove the convergence result of the mild solution when the fractional order [Formula: see text].

Effects of structural inhomogeneity on equilibration processes in Langevin dynamics.

In recent decades, computer experiments have led to an accurate and fundamental understanding of atomic and molecular mechanisms in fluids, such as different kinds of relaxation processes toward steady physical states. In this paper, we investigate how exactly the configuration of initial states in a molecular-dynamics simulation can affect the rates of decay toward equilibrium for the widely known Langevin canonical ensemble. For this purpose, we derive an original expression relating the system relaxation time τ_{sys} and the radial distribution function g(r) in the near-zero and high-density limit. We found that, for an initial state which is slightly marginally inhomogeneous in the number density of atoms, the system relaxation time τ_{sys} is much longer than that for the homogeneous case and an increasing function of the Langevin coupling constant, γ. We also found, during structural equilibration, g(r) at large distances approaches 1 from above for the inhomogeneous case and from below for the macroscopically homogeneous one.

Inhomogeneous affine Volterra processes

We extend recent results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel K(t,s) and inhomogeneous drift and diffusion coefficients b(s,Xs) and σ(s,Xs). In the case of affine b and σσT we show how the conditional Fourier–Laplace functional can be represented by a solution of an inhomogeneous Riccati–Volterra integral equation. For a kernel of convolution type K(t,s)=K¯(t−s) we establish existence of a solution to the stochastic inhomogeneous Volterra equation. If in addition b and σσT are affine, we prove that the conditional Fourier–Laplace functional is exponential–affine in the past path. Finally, we apply these results to an inhomogeneous extension of the rough Heston model used in mathematical finance.

Quantum vacuum fluctuations and the principle of virtual work in inhomogeneous backgrounds

We discuss several aspects of the stress-energy tensor for a quantum scalar field in an inhomogeneous background, the latter being modeled by a variable mass. Using a perturbative approach, dimensional regularization, and adiabatic subtraction, we present all-order formal expressions for the stress-energy tensor. Importantly, we provide an explicit proof of the principle of virtual work for Casimir forces, taking advantage of the conservation law for the renormalized stress-energy tensor. We discuss also discontinuity-induced divergences. For the particular case of planar inhomogeneities, we corroborate the perturbative results with a WKB-inspired expansion.

An assessment of phase relationships between heat fluxes and ground surface temperatures in a diurnal cycle based on monitoring studies at the Verkhnee Dubrovo meteorological station

Research subject. The relationships between heat fluxes and ground surface temperatures in a diurnal cycle.Methods. Experimental studies, including actinometric observations and monitoring of surface temperatures and surface heat fluxes for a soil and an artificial layer, were performed at the Verkhnee Dubrovo meteorological station in 2020.Results. The surface heat flux varies synchronously both with the total solar radiation near the earth’s surface and insolation at the upper boundary of the atmosphere. Maximal values of these heat fluxes are observed at the solar noon. The temperature response lags behind them by the time interval determined by the soil’s thermal heterogeneity. In this study, we extend our model of the ground surface temperature response to external radiative forcing, which was developed earlier, to the case of a thermal inhomogeneous half-space. An analytical expression for the simplest case of inhomogeneity (the presence of an upper layer with thermal properties different from those of the underlying rocks) is given and investigated. If the upper layer demonstrates a reduced thermal conductivity, the phase shift between the heat flux and the temperature response decreases in comparison with the value for a homogeneous half-space (45°).Conclusion. The soil studies conducted at the “Verkhnee Dubrovo” meteorological station allowed us to construct a thermophysical section and to verify the previously developed model using experimental data. The errors of theoretical estimates, in general, do not exceed those of the phase estimate at a 10-minute sampling rate. The obtained results can be applied when conducting climatic (including palaeoclimatic) and environmental studies, as well as when investigating heat exchange processes on artificial urban surfaces and their role in the formation of urban heat islands.

Simulation of Gaussian random field in a ball

Abstract We address the problem of statistical simulation of a scalar real Gaussian random field inside the unit 3D ball. Two different methods are studied: (i) the method based on the known homogeneous isotropic power spectrum developed by Meschede and Romanowicz [M. Meschede and B. Romanowicz, Non-stationary spherical random media and their effect on long-period mantle waves, Geophys. J. Int. 203 2015, 1605–1625] and (ii) the method based on known radial and angular covariance functions suggested in this work. The first approach allows the extension of the simulation technique to the inhomogeneous or anisotropic case. However, the disadvantage of this approach is the lack of accurate statistical characterization of the results. The accuracy of considered methods is illustrated by numerical tests, including a comparison of the estimated and analytical covariance functions. These methods can be used in many applications in geophysics, geodynamics, or planetary science where the objective is to construct spatial realizations of 3D random fields based on a statistical analysis of observations collected on the sphere or within a spherical region.

Correlations and commuting transfer matrices in integrable unitary circuits

We consider a unitary circuit where the underlying gates are chosen to be \check{R}Ř-matrices satisfying the Yang-Baxter equation and correlation functions can be expressed through a transfer matrix formalism. These transfer matrices are no longer Hermitian and differ from the ones guaranteeing local conservation laws, but remain mutually commuting at different values of the spectral parameter defining the circuit. Exact eigenstates can still be constructed as a Bethe ansatz, but while these transfer matrices are diagonalizable in the inhomogeneous case, the homogeneous limit corresponds to an exceptional point where multiple eigenstates coalesce and Jordan blocks appear. Remarkably, the complete set of (generalized) eigenstates is only obtained when taking into account a combinatorial number of nontrivial vacuum states. In all cases, the Bethe equations reduce to those of the integrable spin-1 chain and exhibit a global SU(2) symmetry, significantly reducing the total number of eigenstates required in the calculation of correlation functions. A similar construction is shown to hold for the calculation of out-of-time-order correlations.

Distribution density of the first exit point of a two-dimensional diffusion process from a circle neighborhood of its initial point: the inhomogeneous case

Рассматривается двумерный диффузионный процесс. Распределение точки первого выхода такого процесса из произвольной области его значений как функция от начальной точки процесса определяется эллиптическим дифференциальным уравнением второго порядка и соответствует решению задачи Дирихле этого уравнения (случай непостоянных коэффициентов). Исследуется плотность распределения точки первого выхода процесса из малой круговой окрестности его начальной точки и ее связь с задачей Дирихле. В терминах этой асимптотики доказаны две теоремы. Первая теорема о достаточных и вторая теорема о необходимых условиях того, что распределение точки первого выхода как функция от начальной точки процесса удовлетворяет частному виду эллиптического дифференциального уравнения второго порядка, которое соответствует стандартному винеровскому процессу со сносом и обрывом. Определены устранимые члены второго порядка разложения по степеням радиуса малой круговой окрестности начальной точки процесса. В терминах устранимых членов эти две теоремы превращены в одну теорему о необходимом и достаточном условии соответствия этому винеровскому процессу.

Dynamical effective field model for interacting ferrofluids: I. Derivations for homogeneous, inhomogeneous, and polydisperse cases

Quite recently I have proposed a nonperturbative dynamical effective field model (DEFM) to quantitatively describe the dynamics of interacting ferrofluids. Its predictions compare very well with the results from Brownian dynamics simulations. In this paper I put the DEFM on firm theoretical ground by deriving it within the framework of dynamical density functional theory, taking into account nonadiabatic effects. The DEFM is generalized to inhomogeneous finite-size samples for which the macroscopic and mesoscopic scale separation is nontrivial due to the presence of long-range dipole–dipole interactions. The demagnetizing field naturally emerges from microscopic considerations and is consistently accounted for. The resulting mesoscopic dynamics only involves macroscopically local quantities such as local magnetization and Maxwell field. Nevertheless, the local demagnetizing field essentially couples to magnetization at distant macroscopic locations. Thus, a two-scale parallel algorithm, involving information transfer between different macroscopic locations, can be applied to fully solve the dynamics in an inhomogeneous sample. I also derive the DEFM for polydisperse ferrofluids, in which different species can be strongly coupled to each other dynamically. I discuss the underlying assumptions in obtaining a thermodynamically consistent polydisperse magnetization relaxation equation, which is of the same generic form as that for monodisperse ferrofluids. The theoretical advances presented in this paper are important for both qualitative understanding and quantitative modeling of the dynamics of ferrofluids and other dipolar systems.

Global Well-Posedness to the 3D Cauchy Problem of Nonhomogeneous Heat Conducting Navier–Stokes Equations with Vacuum and Large Oscillations

We revisit the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic (MHD) equations in $\mathbb{R}^2$. For the initial density allowing vacuum at infinity, we derive the global existence and uniqueness of strong solutions provided that the initial density and the initial magnetic decay not too slowly at infinity. In particular, the initial data can be arbitrarily large. This improves our previous work where the initial density has non-vacuum states at infinity. The result could also be viewed as an extension of the study in L{\"u}-Xu-Zhong for the inhomogeneous case to the full inhomogeneous situation. The method is based on delicate spatial weighted estimates and the structural characteristic of the system under consideration. As a byproduct, we get the global existence of strong solutions to the 2D Cauchy problem for nonhomogeneous heat conducting Navier-Stokes equations with vacuum at infinity.

Normal modes of a waveguide as eigenvectors of a self-adjoint operator pencil

A waveguide with a constant, simply connected section S is considered under the condition that the substance filling the waveguide is characterized by permittivity and permeability that vary smoothly over the section S, but are constant along the waveguide axis. Ideal conductivity conditions are assumed on the walls of the waveguide. On the basis of the previously found representation of the electromagnetic field in such a waveguide using 4 scalar functions, namely, two electric and two magnetic potentials, Maxwells equations are rewritten with respect to the potentials and longitudinal components of the field. It appears possible to exclude potentials from this system and arrive at a pair of integro-differential equations for longitudinal components alone that split into two uncoupled wave equations in the optically homogeneous case. In an optically inhomogeneous case, this approach reduces the problem of finding the normal modes of a waveguide to studying the spectrum of a quadratic self-adjoint operator pencil.

Remark on Algorithm 982: Explicit Solutions of Triangular Systems of First-order Linear Initial-value Ordinary Differential Equations with Constant Coefficients

Algorithm 982: Explicit solutions of triangular systems of first-order linear initial-value ordinary differential equations with constant coefficients provides an explicit solution for an homogeneous system, and a brief description of how to compute a solution for the inhomogeneous case. The method described is not directly useful if the coefficient matrix is singular. This remark explains more completely how to compute the solution for the inhomogeneous case and for the singular coefficient matrix case.

On the dynamics of DNA molecules with an-harmonics potential in the normal and damaged states

DNA damage is due to any change that introduces a deviation from the usual double-helical structure. At this state, the study of the motion of DNA molecules requires novel dynamical models. We propose a discrete model to describe the nonlinear dynamics DNA molecule with inharmonic potential. We consider two physical situations, the ideal or homogeneous state, and the inhomogeneous or damage state. The problem of the homogeneous (normal) case with ANHP was considered in the literature, but the dynamics of DNA molecules were not inspected. Here, our objective is to distinguish between the dynamics of DNA in the normal and damaged states. The novelty of the present work stems from a conjuncture that the DNA state can be depicted from the dynamics of the molecules. The unified and extended unified methods are used to find the exact solutions of the systems in both cases, respectively. Numerically, it is found that, in the ideal case, the motion of DNA molecules is periodic and stable. Considering the inhomogeneous case due to damage, the small or moderate values of the backbone rigidity make the motion of the molecule periodic and stable, but for high values, the motion becomes aperiodic and unstable. But for small or moderate values of the backbone rigidity, the motion becomes periodic and stable. These results are of interest in vulgarizing the physical properties of DNA molecules.

Explicit solutions and asymptotic behaviors of Caputo discrete fractional-order equations with variable coefficients

A new discrete fractional-order Peano-Baker series is established in this paper. Based on this series, the explicit solutions of Caputo linear discrete fractional-order equations (DFOEs) with variable matrix coefficients involving the homogeneous and inhomogeneous cases are obtained. On the basis of the comparison theorem and the explicit solutions of Caputo linear DFOEs with variable coefficients, the asymptotic behaviors of the solutions of the homogeneous and inhomogeneous Caputo linear DFOEs with variable coefficients are presented. Finally, the applicability and effectiveness of the proposed results are illustrated by four examples.

Influence of Using a Contrast-Enhanced CT Image as the Primary Image on CyberKnife Brain Radiosurgery Treatment Plans.

Background and PurposeThis study aimed to quantify the differences between pre- and post-contrast agent (CA) CT for CyberKnife brain SRS plans.Materials and MethodsTwenty-five patients were retrospectively analyzed. They were divided into two categories, inhomogeneous cases (13 patients) and homogeneous cases (12 patients), according to whether the tumor was close to the cavity and inhomogeneous tissues or not. The pre-CA and post-CA plans were designed and calculated using the same monitor unit and paths as those in the ray-tracing algorithm, respectively.ResultsThe CT number difference of tumor between pre- and post-CA was significant (on average, 24.78 ± 18.56 HU, P-value < 0.01). The deviation value of the target was the largest at approximately 37 HU (inhomo-) and 13 HU (homo-) (P < 0.01), and the values of the organs at risk (OARs) were not statistically significant (P-value > 0.05). However, it was not statistically significant for the dose difference between the two groups with the injection of CA (P-value > 0.05). The absolute effective depth difference generally remained at a level of 1 mm, but the dose difference was quitely fluctuated sometimes more than 20%. The absolute effective depth difference of the inhomo-case (0.62 mm) was larger than that of the homo-case (0.37 mm) on median, as well as the variation amplitude (P-value < 0.05). Moreover, the relative dose differences between the two cases were 0.38% (inhomo-) and 0.2% (homo-), respectively (P-value < 0.05). At the criterion of 1 mm/1%, the gamma pass rate of the homo-case (95.89%) was larger than that of the inhomo-case (93.79%). For the OARs, except for the cochlea, the two cases were almost the same (>98.85%). The tumor control probability of the target was over 99.99% before and after injection of a CA, as well as the results for the homo-case and inhomo-case.ConclusionsConsidering the difference of evaluation indexes between pre- and post-CA images, we recommended plain CT to be employed as the primary image for improving the CK treatment accuracy of brain SRS, especially when the target was close to CA-sensitive OARs and cavity.