Inhomogeneous Term Research Articles

On the modular structure of the genus-one Type II superstring low energy expansion

The analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions up to order D**10 R*4 are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values.

Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II

This is a continuation of Ikoma and Ishii (Ann Inst H Poincare Anal Non Lineaire 29:783–812, 2012) and we study the eigenvalue problem for fully nonlinear elliptic operators, positively homogeneous of degree one, on finite intervals or balls. In the multi-dimensional case, we consider only radial eigenpairs. Our eigenvalue problem has a general first-order boundary condition which includes, as a special case, the Dirichlet, Neumann and Robin boundary conditions. Given a nonnegative integer n, we prove the existence and uniqueness, modulo multiplication of the eigenfunction by a positive constant, of an eigenpair whose eigenfunction, as a radial function in the multi-dimensional case, has exactly n zeroes. When an eigenfunction has n zeroes, we call the corresponding eigenvalue of nth order. Furthermore, we establish results concerning comparison of two eigenvalues, characterizations of nth order eigenvalues via differential inequalities, the maximum principle for the boundary value problem in connection with the principal eigenvalue, and existence of a solution having n zeroes, as a radial function in the multi-dimensional case, of the boundary value problem with an inhomogeneous term.

Cauchy problems of pseudo-parabolic equations with inhomogeneous terms

This paper deals with Cauchy problems of pseudo-parabolic equations with inhomogeneous terms. The aim of the paper is to study the influence of the inhomogeneous term on the asymptotic behavior of solutions. We at first determine the critical Fujita exponent and then give the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity. Furthermore, the precise estimate of life span for the blow-up solution is obtained. Our results show that the asymptotic behavior of solutions is seriously affected by the inhomogeneous term.

Global gradient estimates for nonlinear obstacle problems with nonstandard growth

Abstract We investigate an optimal W 1 , p ⁢ ( ⋅ ) ⁢ q ${W^{1,p(\,\cdot\,)q}}$ -regularity theory for a nonlinear elliptic obstacle problem with nonstandard growth p ⁢ ( ⋅ ) ${p(\,\cdot\,)}$ . With a sufficient small log-Hölder constant on p ⁢ ( ⋅ ) ${p(\,\cdot\,)}$ , under a suitable smallness condition in BMO on the nonlinearity and under a sufficient flatness condition on the boundary of the domain, we establish a global Calderón–Zygmund estimate for such an irregular obstacle problem by proving that the gradient of the weak solution is as integrable as both the gradient of the obstacle and the inhomogeneous term in the variable exponent space L p ⁢ ( ⋅ ) ⁢ q ${L^{p(\,\cdot\,)q}}$ for every q ∈ ( 1 , ∞ ) ${q\in(1,\infty)}$ .

Regularity and analyticity of solutions in a direction for elliptic equations

In this paper, we study the regularity and analyticity of solutions to linear elliptic equations with measurable or continuous coefficients. We prove that if the coefficients and inhomogeneous term are Holder-continuous in a direction, then the second-order derivative in this direction of the solution is Holder-continuous, with a different Holder exponent. We also prove that if the coefficients and the inhomogeneous term are analytic in a direction, then the solution is analytic in that direction.

Boundary [formula omitted] estimates for Monge–Ampère type equations

In this paper, we obtain global second derivative estimates for solutions of the Dirichlet problem of certain Monge–Ampère type equations under some structural conditions, while the inhomogeneous term is only assumed to be Hölder continuous and bounded away from zero and infinity. These estimates correspond to those for the standard Monge–Ampère equation obtained by Trudinger and Wang (2008) [28] and by Savin (2013) [24], and have natural applications in optimal transportation and prescribed Jacobian equations.

Antiperiodic XXZ Chains with Arbitrary Spins: Complete Eigenstate Construction by Functional Equations in Separation of Variables

Generic inhomogeneous integrable XXZ chains with arbitrary spins are studied by means of the quantum separation of variables (SOV) method. Within this framework, a complete description of the spectrum (eigenvalues and eigenstates) of the antiperiodic transfer matrix is derived in terms of discrete systems of equations involving the inhomogeneity parameters of the model. We show here that one can reformulate this discrete SOV characterization of the spectrum in terms of functional T − Q equations of Baxter’s type, hence proving the completeness of the solutions to the associated systems of Bethe-type equations. More precisely, we consider here two such reformulations. The first one is given in terms of Q-solutions, in the form of trigonometric polynomials of a given degree \({\mathsf{N}_s}\), of a one-parameter family of T − Q functional equations with an extra inhomogeneous term. The second one is given in terms of Q-solutions, again in the form of trigonometric polynomials of degree \({\mathsf{N}_s}\) but with double period, of Baxter’s usual (i.e., without extra term) T − Q functional equation. In both cases, we prove the precise equivalence of the discrete SOV characterization of the transfer matrix spectrum with the characterization following from the consideration of the particular class of Q-solutions of the functional T − Q equation: to each transfer matrix eigenvalue corresponds exactly one such Q-solution and vice versa, and this Q-solution can be used to construct the corresponding eigenstate.

Equilibrium similarity solution of the turbulent transport equation along the centreline of a round jet

A novel similarity-based form is derived of the transport equation for the second-order velocity structure function of$\langle ({\it\delta}q)^{2}\rangle$along the centreline of a round turbulent jet using an equilibrium similarity analysis. This self-similar equation has the advantage of requiring less extensive measurements to calculate the inhomogeneous (decay and production) terms of the transport equation. It is suggested that the normalised third-order structure function can be uniquely determined when the normalised second-order structure function, the power-law exponent of$\langle q^{2}\rangle$and the decay rate constants of$\langle u^{2}\rangle$and$\langle v^{2}\rangle$are available. In addition, the current analysis demonstrates that the assumption of similarity, combined with an inverse relation between the mean velocity$U$and the streamwise distance$x-x_{0}$from the virtual origin (i.e. $U\propto (x-x_{0})^{-1}$), is sufficient to predict a power-law decay for the turbulence kinetic energy ($\langle q^{2}\rangle \propto (x-x_{0})^{m}$), rather than requiring a power-law decay ($m=-2$) as an additionalad hocassumption. On the basis of the current analysis, it is suggested that the mean kinetic energy dissipation rate,$\langle {\it\epsilon}\rangle$, varies as$(x-x_{0})^{m-2}$. These theoretical results are tested against new experimental data obtained along the centreline of a round turbulent jet as well as previously published data on round jets for$11\,000\leqslant \mathit{Re}_{D}\leqslant 184\,000$over the range$10\leqslant x/D\leqslant 90$. For the present experiments,$\langle q^{2}\rangle$exhibits power-law behaviour with$m=-1.83$. The validity of this solution is confirmed using other experimental data where$\langle q^{2}\rangle$follows a power law with$-1.89\leqslant m\leqslant -1.78$. The present similarity form of the transport equation for$\langle ({\it\delta}q)^{2}\rangle$is also shown to be closely satisfied by the experimental data.

Blowup for the nonlinear Schrödinger equation with an inhomogeneous damping term in the L2-critical case

We consider the nonlinear Schrödinger equation with L2-critical exponent and an inhomogeneous damping term. By using the tools developed by Merle and Raphael, we prove the existence of blowup phenomena in the energy space H1(ℝ).

Son-Yamamoto relation and holographic renormalization group flows

Motivated by the Son-Yamamoto (SY) relation which connects the three point and two-point correlators we consider the holographic RG flows in the bottom-up approach to holographic QCD via the Hamilton-Jacobi (HJ) equation with respect to the radial coordinate . It is shown that the SY relation is diagonal with respect to the RG flow in the 5d YM-CS model while the RG equation acquires the inhomogeneous term in the model with the additional scalar field which encodes the chiral condensate.

Solutions to a class of parabolic inhomogeneous normalized p-laplace equations

In this article, we prove that viscosity solutions of the parabolic inhomogeneous equations n+pput−ΔpNu=f(x,t)can be characterized using asymptotic mean value properties for all p ≥ 1, including p = 1 and p = ∞. We also obtain a comparison principle for the non-negative or non-positive inhomogeneous term f for the corresponding initial-boundary value problem and this in turn implies the uniqueness of solutions to such a problem.

The existence of global attractors for a class of reaction–diffusion equations with distribution derivatives terms in [formula omitted

We consider the existence of global attractors for a class of semilinear reaction–diffusion equations{∂u∂t−Δu+u−V(x)g(u)+f(u)=Dihi+hin Rn×R+,u(x,0)=u0(x)in Rn, in the whole space, with weighted terms and some distribution derivatives in inhomogeneous terms. Using the measure of non-compactness method, we prove the existence of global attractors in proper spaces.

Electrical and photovoltaic properties of Gaussian distributed inhomogeneous barrier based on tris(8-hydroxyquinoline) indium/p-si interface

Hybrid organic/inorganic heterojunction solar cells are fabricated for the first time by depositing tris(8-hydroxyquinoline) indium (Inq3) films onto p-type silicon (p-Si) single crystals via a thermal evaporation technique. The structural characterization of the films was achieved by X-ray diffraction and scanning electron microscopy. The electrical and photovoltaic properties of the Au/Inq3/p-Si/Al heterojunction were investigated in terms of the current–voltage (I–V) measurements. The dark I–V characteristics of the device are measured at different temperatures ranging from 300 to 400K in which the operating conduction mechanisms and the basic diode parameters such as ideality factor, barrier height and series resistance are determined. At forward bias voltages V≤0.4V, the I–V characteristics have been analyzed on the basis of the thermionic emission theory. With increasing temperature, the ideality factor is found to decrease, while the barrier height is found to increase. This behavior has been explained in terms of the barrier inhomogeneities at the Inq3/p-Si interface. At forward bias voltages 0.4<V≤2.5V, the conduction mechanism is found to be due to the space charge limited current characterized by exponential distribution of traps. Under illumination, the heterojunction showed photovoltaic behavior from which the photovoltaic parameters are evaluated.

The shock wave solution to the Riemann problem for the Burgers equation with the linear forcing term

In this note, the Riemann problem for the inviscid Burgers equation with a linear forcing term is considered. The shock wave solution is obtained by combining the generalized Rankine–Hugoniot jump condition together with the method of characteristics, which reflects the impact of the inhomogeneous forcing term on the shock front. In addition, some interesting phenomena are also observed during the construction process of Riemann solution.

Analytical Solution of General Bagley-Torvik Equation

Bagley-Torvik equation appears in viscoelasticity problems where fractional derivatives seem to play an important role concerning empirical data. There are several works treating this equation by using numerical methods and analytic formulations. However, the analytical solutions presented in the literature consider particular cases of boundary and initial conditions, with inhomogeneous term often expressed in polynomial form. Here, by using Laplace transform methodology, the general inhomogeneous case is solved without restrictions in boundary and initial conditions. The generalized Mittag-Leffler functions with three parameters are used and the solutions presented are expressed in terms of Wiman’s functions and their derivatives.

The Galerkin finite element method for a multi-term time-fractional diffusion equation

We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one- and two-dimensional problems confirm the theoretical convergence rates.

A singular boundary value problem for a degenerate elliptic PDE

We consider a singular boundary value problem associated with the ∞-Laplacian and having nonseparable inhomogeneous term. We establish the existence of ground state solutions to such a problem in bounded domains as well as in the whole Euclidean space. The investigation is based on subsolution and supersolution methods and existence of principal eigenfunctions to the eigenvalue problem of Dirichlet type corresponding to the ∞-Laplacian.

Modified nonlinearities distribution Homotopy Perturbation method as a tool to find power series solutions to ordinary differential equations

In this article, modified non-linearities distribution homotopy perturbation method (MNDHPM) is used in order to find power series solutions to ordinary differential equations with initial conditions, both linear and nonlinear. We will see that the method is particularly relevant in some cases of equations with non-polynomial coefficients and inhomogeneous non-polynomial terms

Dosimetric effects of residual uncertainties in carbon ion treatment of head chordoma

PurposeTo investigate dose distribution variations due to setup errors and range uncertainties in image-guided carbon ion radiotherapy of head chordoma. Materials and methodsTen treatment plans were retrospectively tested with TRiP98 against ±1.0mm and ±1.0° setup errors, as observed in clinical routine, and 2.6% range uncertainty when 2mm CTV-to-PTV margins were applied. Single-fraction simulations were compared with the total treatment dose in terms of DVH bands, conformity and inhomogeneity. The contribution of image processing artifacts on reported results was also discussed, as a function of the imaging dataset resolution. ResultsResults showed that safety margins grant the conformal target coverage in presence of setup errors with D95CTV variations below 10% in 7 patients out of 10. Instead, the inclusion of range uncertainty yielded to appreciable dose degradation, reporting larger effects for CTV and dose conformity, whereas reduced impact is found on the organ-at-risk. The fractionation scheme positively affects dose conformity and inhomogeneity; conversely its influence on DVH bands is strongly related to the patient anatomy. ConclusionBesides safety margins, setup and range uncertainties lead to non-negligible combined contribution. Systematical treatment plan robustness assessment against expected uncertainties is thus encouraged, selecting beam settings and fractionation schemes where homogeneity is preserved.

Effect of geometric parameters on strain, strain inhomogeneity and peak pressure in equal channel angular pressing – A study based on 3D finite element analysis

Equal channel angular pressing (ECAP) is currently being widely investigated because of its potential to produce ultra-fine grained microstructures in metals and alloys. Considerable research has been reported on finite element analysis (FEA) of this process, assuming 2D plane strain condition. The 2D models do not give details of strain distribution in the work-piece. Reports of all the researches are on effect of one-parameter-at-a-time. Combined effect of all geometric parameters is not reported. This paper aims in fulfilling the gap. In the present work 3D FEA of ECAP process was carried out for different combinations of channel angle, inner and outer corner radii. Results in terms of peak pressure, strain and strain inhomogeneity were obtained and analyzed by analysis of mean (ANOM). Main effects and interaction effect of all geometric parameters were quantified by analysis of variance (ANOVA). From the analysis it was found that the peak pressure is largely influenced by channel angle. To obtain desired strain the most important factors are channel angle and outer corner radius. Outer corner radius has the largest influence followed by channel angle on the strain inhomogeneity. There exists an optimum outer corner for which strain inhomogeneity is minimum, which depends on the channel angle. Inner corner alone has no influence on the strain inhomogeneity but its interaction with channel angle has some influence.