Nonconstant Term Research Articles

The analogue of Büchi's Problem for function fields

Büchi's n Squares Problem asks for an integer M such that any sequence ( x 0 , … , x M − 1 ) , whose second difference of squares is the constant sequence (2) (i.e. x n 2 − 2 x n − 1 2 + x n − 2 2 = 2 for all n), satisfies x n 2 = ( x + n ) 2 for some integer x. Hensley's Problem for r-th powers (where r is an integer ⩾2) is a generalization of Büchi's Problem asking for an integer M such that, given integers ν and a, the quantity ( ν + n ) r − a cannot be an r-th power for M or more values of the integer n, unless a = 0 . The analogues of these problems for rings of functions consider only sequences with at least one non-constant term. Let K be a function field of a curve of genus g. We prove that Hensley's Problem for r-th powers has a positive answer for any r if K has characteristic zero, improving results by Pasten and Vojta. In positive characteristic p we obtain a weaker result, but which is enough to prove that Büchi's Problem has a positive answer if p ⩾ 312 g + 169 (improving results by Pheidas and the second author).

Predicting EQ-5D Utility Scores from the Seattle Angina Questionnaire in Coronary Artery Disease

The Seattle Angina Questionnaire (SAQ), a descriptive quality of life instrument, is often used in coronary artery disease studies. In its current form, however, it cannot be used in economic evaluations. The investigators sought to create a mapping algorithm that would allow translation of SAQ scores into EQ-5D utility scores. Data from the Alberta Provincial Project for Outcome Assessment in Coronary Heart Disease (APPROACH) database were used to examine the relationship between scores in each of the 5 domains of the SAQ (physical limitation, anginal stability, anginal frequency, treatment satisfaction, and disease perception) and the EQ-5D utility score. The cohort was divided into 80% derivation and 20% validation sets. Mapping algorithms were developed using simple linear regression and Tobit models. To account for the skewed distribution of the EQ-5D scores and the presence of heteroscedasticity, Bayesian extensions were applied to each model by specifying a nonconstant variance for the error term. Model performance was assessed by comparing predicted and observed mean EQ-5D scores in the validation set, and the unadjusted R2. The cohort consisted of 1992 patients. The simple linear regression model had the best predictive performance, with an R2 of 0.38. The nonconstant variance term did not improve overall performance for any of the models. The linear regression model accurately estimated the mean EQ-5D score in the validation set (predicted score 0.81 v. observed score 0.81). Mean EQ-5D utility weights can be accurately estimated from the SAQ using a simple linear regression mapping algorithm.

Some new nonlinear Gronwall-Bellman-Type integral inequalities and applications to BVPs

In this paper we establish some new nonlinear Gronwall-Bellman-Type integral inequalities with two variables, which include a non-constant term outside the integrals. These inequalities generalize the results in Chen et al. (J. Inequal. Appl. 2009:258569, 15 p., 2009). We apply our result to a boundary value problem of a partial differential equation for boundedness, uniqueness and continuous dependence.

EIGENELEMENTS OF A GENERAL AGGREGATION-FRAGMENTATION MODEL

We consider a linear integro-differential equation which arises to describe both aggregation-fragmentation processes and cell division. We prove the existence of a solution (λ, [Formula: see text], ϕ) to the related eigenproblem. Such eigenelements are useful to study the long-time asymptotic behavior of solutions as well as the steady states when the equation is coupled with an ODE. Our study concerns a non-constant transport term that can vanish at x = 0, since it seems to be relevant to describe some biological processes like proteins aggregation. Non-lower-bounded transport terms bring difficulties to find a priori estimates. All the work of this paper is to solve this problem using weighted-norms.

Nonlinear Retarded Integral Inequalities with Two Variables and Applications

We consider some new nonlinear retarded integral inequalities with two variables, which extend the results in the work of W.-S. Wang (2007), and the one in the work of Y.-H. kim (2009). These inequalities include not only a nonconstant term outside the integrals but also more than one distinct nonlinear integrals without assumption of monotonicity. Finally, we give some applications to the boundary value problem of a partial differential equation for boundedness and uniqueness.

An Extension to Nonlinear Sum-Difference Inequality and Applications

We establish a general form of sum-difference inequality in two variables, which includes both more than two distinct nonlinear sums without an assumption of monotonicity and a nonconstant term outside the sums. We employ a technique of monotonization and use a property of stronger monotonicity to give an estimate for the unknown function. Our result enables us to solve those discrete inequalities considered in the work of W.-S. Cheung (2006). Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.

A Generalized Sum-Difference Inequality and Applications to Partial Difference Equations

We establish a general form of sum-difference inequality in two variables, which includes both two distinct nonlinear sums without an assumption of monotonicity and a nonconstant term outside the sums. We employ a technique of monotonization and use a property of stronger monotonicity to give an estimate for the unknown function. Our result enables us to solve those discrete inequalities considered by Cheung and Ren 2006 . Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.

A generalized retarded Gronwall-like inequality in two variables and applications to BVP

In this paper, we establish a generalized retarded integral inequality of Gronwall-like type in two variables, which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals without assumption of monotonicity. Using our result we can solve both the integral inequality in [W.S. Cheung, Some new nonlinear inequalities and applications to boundary value problems, Nonlinear Anal. 64 (2006) 2112–2128] and the one in [R.P. Agarwal, S. Deng, W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications, Appl. Math. Comput. 165 (2005) 599–612]. We apply our result to a boundary value problem of a partial differential equation for boundedness, uniqueness and continuous dependence.

Modified exponential schemes for convection–diffusion problems

The original exponential schemes of the finite volume approach proposed by Spalding [Spalding DB. A novel finite-difference formulation for differential expressions involving both first and second derivatives. Int J Numer Methods Eng 1972;4:509–51] as well as by Raithby and Torrance [Raithby GD, Torrance KE. Upstream-weighted differencing schemes and their application to elliptic problems involving fluid flow. Comput Fluids 1974;2:191–206], on which the well known hybrid and power-law schemes were based, had been derived without considering the non-constant source term which can be linearized as a function of a scalar variable ϕ. Following a similar method to that of Spalding, we derived three modified exponential schemes, corresponding to the average and integrated source terms, with the last scheme involving matching the analytical solutions of the neighbouring sub-regions by assuming the continuity of the first derivative of scalar variable ϕ. To validate the higher accuracy of the modified exponential schemes, as compared to classical schemes, numerical predictions obtained by various discretization schemes were compared with exact analytical solutions for linear problems. For non-linear problems, with non-constant source term, the solutions of the numerical discretization equations were compared with accurate solutions obtained with fine grids. To test the suitability of the proposed schemes in practical problems of computational fluid dynamics, all schemes were also examined by varying the mass flow rate and the coefficient of the non-constant source term. Finally, the best performing scheme is recommended for applications to CFD problems.

Differential realization of pseudo-Hermiticity: A quantum mechanical analog of Einstein’s field equation

For a given pseudo-Hermitian Hamiltonian of the standard form: H=p2∕2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator η satisfying H†=ηHη−1 to the solution of a differential equation. If the configuration space is R, this is a Klein-Gordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of η. We apply our general results to calculate η for the PT-symmetric square well, an imaginary scattering potential, and a class of imaginary delta-function potentials. For the first two systems, our method reproduces the known results in a straightforward and extremely efficient manner. For all these systems we obtain the most general η up to second-order terms in the coupling constants.

Second order phase field asymptotics for multi-component systems

We derive a phase field model which approximates a sharp interface model for solidification of a multicomponent alloy to second order in the interfacial thickness $\varepsilon$. Since in numerical computations for phase field models the spatial grid size has to be smaller than $\varepsilon$ the new approach allows for considerably more accurate phase field computations than have been possible so far. In the classical approach of matched asymptotic expansions the equations to lowest order in $\varepsilon$ lead to the sharp interface problem. Considering the equations to the next order, a correction problem is derived. It turns out that, when taking a possibly non-constant correction term to a kinetic coefficient in the phase field model into account, the correction problem becomes trivial and the approximation of the sharp interface problem is of second order in $\varepsilon$. By numerical experiments, the better approximation property is well supported. The computational effort to obtain an error smaller than a given value is investigated revealing an enormous efficiency gain.

Generalization of a retarded Gronwall-like inequality and its applications

This paper generalizes a Lipovan’s result of Gronwall-like inequalities [J. Math. Anal. Appl. 252 (2000) 389] to a new type of retarded inequalities which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals. From our result Bihari’s result and Pinto’s result can be deduced as some special cases. Our result can be applied to give the global existence and estimate solutions. We also apply it to improve an estimation for almost periodicity of an invariant manifold.

Electroosmotic flow in capillary channels filled with nonconstant viscosity electrolytes: exact solution of the Navier-Stokes equation.

The partial differential equation describing unsteady velocity profile of electroosmotic flow (EOF) in a cylindrical capillary filled with a nonconstant viscosity electrolyte was derived. Analytical solution, based on the general Navier-Stokes equation, was found for constant viscosity electrolytes using the separation of variables (Fourier method). For the case of a nonconstant viscosity electrolyte, the steady-state velocity profile was calculated assuming that the viscosity decreases exponentially in the direction from the wall to the capillary center. Since the respective equations with nonconstant viscosity term are not solvable in general, the method of continuous binding conditions was used to solve this problem. In this method, an arbitrary viscosity profile can be modeled. The theoretical conclusions show that the relaxation times at which an EOF approaches the steady state are too short to have an impact on a separation process in any real systems. A viscous layer at the wall affects EOF significantly, if it is thicker than the Debye length of the electric double layer. The presented description of the EOF dynamics is applicable to any microfluidic systems.

Preconditioning of improved and “perfect” fermion actions

We construct a locally-lexicographic SSOR preconditioner to accelerate the parallel iterative solution of linear systems of equations for two improved discretizations of lattice fermions: (i) the Sheikholeslami-Wohlen scheme Where a non-constant block-diagonal term is added to the Wilson fermion matrix and (ii) renormalization group improved actions which incorporate couplings beyond nearest neighbors of the lattice fermion fields. In case (i) we find the block ll-SSOR-scheme to be more effective by a factor ≈ 2 than odd-even preconditioned solvers in terms of convergence rates, at β = 6.0. For type (ii) actions, we show that our preconditioner accelerates the iterative solution of a linear system of hypercube fermions by a factor of 3 to 4.

Monte Carlo Variance of Scrambled Net Quadrature

Hybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. This paper studies the variance of one such hybrid, scrambled nets, by applying a multidimensional multiresolution (wavelet) analysis to the integrand. The integrand is assumed to be measurable and square integrable but not necessarily of bounded variation. In simple Monte Carlo, every nonconstant term of the multiresolution contributes to the variance of the estimated integral. For scrambled nets, certain low-dimensional and coarse terms do not contribute to the variance. For any integrand in L2, the sampling variance tends to zero faster under scrambled net quadrature than under Monte Carlo sampling, as the number of function evaluations n tends to infinity. Some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the Monte Carlo variance, uniformly over all integrands f. Latin hypercube sampling is a special case of scrambled net quadrature.

Higher-Order Exponential Difference Schemes for the Computations of the Steady Convection–Diffusion Equation

Conventional exponential difference schemes may yield accurate and stable solutions for the one-dimensional, source-free convection–diffusion equation. However, its accuracy will be deteriorated in the presence of a nonconstant source term or in multidimensional problems. Attempts are made to increase the accuracy of exponential difference schemes. First, we propose an exponential difference scheme that retains second-order accuracy in the presence of a source term or in multidimensional situations. Mathematical analysis and numerical experiments are performed to validate this scheme. Second, a local particular solution method is introduced to raise the solution accuracy for problems with a source term. This method locally transforms the original problem to a source-free one, to which an accurate solution can be obtained. Performance of this process is verified by numerical calculations of some test problems. Third, two skew exponential difference schemes are proposed to raise the solution accuracy in multidimensional problems: one is designed to be free of numerical diffusion and the other with minimum numerical diffusion to ensure solution monotonicity. Comparisons with existing schemes are performed by conducting numerical experiments on several test problems. Finally, a simple blending procedure of these two schemes is suggested to yield an accurate and stable representation of the convection–diffusion problem in all possible situations, with or without solution discontinuities.

Bent functions and random boolean formulas

Let α be a nonlinear boolean connective with equal number of zero and unit entries in its table. We study an iterative process of combining randomly chosen boolean functions from some starting set G via the connective α. We suppose that all functions in G are defined on the same finite nonempty domain M. We are interested in the situations, when the process converges to the uniform distribution on {0, 1} M . We classify the boolean connectives α according to the asymptotic rate of this convergence. Although the probability of any function in our process converges to ( 1 2 ) ¦M , there are some differences in the terms of lower order of magnitude. If M is the boolean cube of an even dimension n and G is the set of all linear boolean functions of n variables and the connective α belongs to the class of the lowest possible rate of convergence in the above-mentioned classification, we can express the main nonconstant term of the asymptotic expansion of the probability of occurring of a single function f through the Fourier transform of f. Using this, we prove that the bent functions achieve asymptotically the minimal probability of occurring among all boolean functions. At the same time, the linear functions achieve asymptotically the maximal probability.

A Numerical Procedure for Vehicle-Induced Internal Waves

Abstract An axially marching numerical method is developed for the simulation of the internal waves produced by the translation of a submersed vehicle in a density-stratified ocean. The method provides for the direct solution of the primitive variables [v, p, ρ] for the nonlinear and steady-state three-dimensional Euler’s equation with a nonconstant density term in the vehicle-fixed cartesian coordinate system. By utilizing a known potential flow around the vehicle for an estimate of the axial velocity gradient, the present parabolic algorithm allows local upstream disturbances and an axial velocity variation.