Optimal Lehmer Mean Bounds for the Toader Mean

We find the greatest value p and least value q such that the double inequality L _p (a, b)?<?T(a, b)?<?L _q (a, b) holds for all a, b?>?0 with a?≠ b, and give a new upper bound for the complete elliptic integral of the second kind. Here ${T(a,b)=\frac{2}{\pi}\int\nolimits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}d\theta}$ and L _p (a, b)?=?(a ^p+1?+?b ^p+1)/(a ^p ?+?b ^p ) denote the Toader and p-th Lehmer means of two positive numbers a and b, respectively.     

Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes’ equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by Maz’ya (DFG-Kolloquium des DFG-Forschungsschwerpunktes Randelementmethoden, 1991) and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes’ equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström’s method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

Mean value estimates in lattice point theory

LetQ(u) be a positive definite quadratic form inr2 variables with a real symmetric coefficient matrix of determinantD. Given a real vectorb with 0b _j <1, forx>0 letA(x) be the number of lattice points in the ellipsoidQ(u+b)x, letV(x) be the volume of this ellipsoid andP(x)=A(x)–V(x). Let . By introduction of a parameter we shall show how the treatment of estimates onP(x) and onM(x) can be unified.     

A problem of D. H. Lehmer and its mean value

Let p be an odd prime and let a be an integer coprime to p. Denote by N (a, p) the number of pairs of integers b, c with bc ≡ a (mod p), 1 ≤ b, c < p and with b, c having different parity. The main purpose of this paper is to deduce an asymptotic formula for (N (2^sad, p) – 1/2 (p – 1)), and obtain some interesting results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Functional a posteriori error estimates for the reaction-convection-diffusion problem

In this paper, a general form of functional type a posteriori error estimates for linear reaction-convection-diffusion problems is presented. It is derived by purely functional arguments without attracting specific properties of the approximation method. The estimate provides a guaranteed upper bound of the difference between the exact solution and any conforming approximation from the energy functional class. It is also proved that the derived error majorants give computable quantities, which are equivalent to the error evaluated in the energy and combined primal-dual norms. Bibliography: 14 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 127–146.     

A posteriori error estimates for the generalized Stokes problem

The paper concerns a posteriori estimates of functional type for the difference between exact and approximate solutions to a generalized Stokes problem. The estimates are derived by transformations of the basic integral identity defining a generalized solution to the problem using the method suggested by the first author. The estimates obtained can be classified into two types. Estimates of the first type are valid only for solenoidal functions, while estimates of the second type are applicable for any functions that belong to the energy space of the respective problem and satisfy the boundary conditions. In the second case, the estimates include an additional penalty term with a multiplier defined by the constant in the Ladyzhenskaya-Babuška-Brezzi condition. It is proved that a posteriori estimates for the velocity field yield computable estimates of the difference between exact and approximate pressure functions in the L₂-norm. It is shown that the estimates provide sharp upper and lower bounds of the error and their practical computation requires to solve only finite-dimensional problems. Bibliography: 34 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 89–101.     

Generalization of a problem of Lehmer

Given a prime number p, Lehmer raised the problem of investigating the number of integers for which a and are of opposite parity, where is such that . We replace the pair by a point lying on a more general irreducible curve defined mod p and instead of the parity conditions on the coordinates more general congruence conditions are considered. An asymptotic result is then obtained for the number of such points. Received: 12 July 2000 / Revised version: 7 November 2000     

On a posteriori error estimates for the stationary Navier-Stokes problem

We obtain a computable upper bound for the difference between a solution to the stationary Navier-Stokes problem and any solenoidal vector-valued function satisfying the boundary condition and possessing necessary differentiability properties. For sufficiently small velocities this estimate implies an estimate of the deviation from exact solution in the energy norm and the uniqueness of a weak solution. Bibliography: 3 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 89–92.     

A problem of D.H. Lehmer and its mean square value formula

Let q be an odd positive integer and let a be an integer coprime to q. For each integer b coprime to q with 1?b<q, there is a unique integer c coprime to q with 1?c<q such that . Let N(a,q) denote the number of solutions of the congruence equation with 1?b,c<q such that b,c are of opposite parity. The main purpose of this paper is to use the properties of Dedekind sums, the properties of Cochrane sums and the mean value theorem of Dirichlet L-functions to study the asymptotic property of the mean square value , and give a sharp asymptotic formula.     

A posteriori error estimates for elliptic problems with Dirac delta source terms

The aim of this paper is to introduce residual type a posteriori error estimators for a Poisson problem with a Dirac delta source term, in L ^p norm and W^1,p seminorm. The estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence.     

On a generalisation of a Lehmer problem

We consider a generalisation of the classical Lehmer problem about the distribution of modular inverses in arithmetic progression, introduced by E. Alkan, F. Stan and A. Zaharescu. Using bounds for multiplicative character sums instead of bounds for Kloosterman sums traditionally applied to this kind of problem, we improve their results in several directions.     

On a kind of generalized Lehmer problem

For 1 ? c ? p ? 1, let E ₁,E ₂, …,E _m be fixed numbers of the set {0, 1}, and let a ₁, a ₂, …, a _m (1 ? a _i ? p, i = 1, 2, …,m) be of opposite parity with E ₁,E ₂, …,E _m respectively such that a ₁ a ₂…a _m ≡ c (mod p). Let $$N(c,m,p) = {1 \over {{2^{m - 1}}}}\mathop {\sum\limits_{{a_1} = 1}^{p - 1} {\sum\limits_{{a_2} = 1}^{p - 1} \ldots } }\limits_{{a_1}{a_2} \ldots \equiv c{\rm{ (}}\bmod {\rm{ }}p)} \sum\limits_{{a_m} = 1}^{p - 1} {(1 - {{( - 1)}^{{a_1} + {E_1}}})(1 - {{( - 1)}^{{a_2} + {E_2}}}) \ldots } (1 - {( - 1)^{{a_m} + {E_m}}}).$$ We are interested in the mean value of the sums $$\sum\limits_{c = 1}^{p - 1} {{E^2}} (c,m,p),$$ where E(c, m, p) = N(c,m, p)?((p ? 1) ^m?1)/(2 ^m?1) for the odd prime p and any integers m ? 2. When m = 2, c = 1, it is the Lehmer problem. In this paper, we generalize the Lehmer problem and use analytic method to give an interesting asymptotic formula of the generalized Lehmer problem.     

Recovery type a posteriori error estimates for the conduction convection problem

Numerical Algorithms - In this paper, we construct a recovery type a posteriori error estimator based on the recovered gradient method to the stationary conduction convection equations. Besides,...     

A backward error for the inverse singular value problem

A backward error for inverse singular value problems with respect to an approximate solution is defined, and an explicit expression for the backward error is derived by extending the approach described in [J.G. Sun, Backward errors for the inverse eigenvalue problem, Numer. Math. 82 (1999) 339-349]. The expression may be useful for testing the stability of practical algorithms.     

A note on the Lehmer problem over short intervals

Let p be an odd prime, c be an integer with (c, p) = 1, and let N be a positive integer with N ≤ p − 1. Denote by r(N, c; p) the number of integers a satisfying 1 ≤ a ≤ N and 2 ∤ a + ā, where ā is an integer with 1 ≤ ā ≤ p − 1, aā ≡ c (mod p). It is well known that r(N, c; p) = 1/2N + O(p ^1/2log² p). The main purpose of this paper is to give an asymptotic formula for Σ _c=1 ^p−1(r(N, c; p) − 1/2N)².     

A posteriori error estimates for discontinuous Galerkin method to the elasticity problem

This work concerns with the discontinuous Galerkin (DG) method for the time‐dependent linear elasticity problem. We derive the a posteriori error bounds for semidiscrete and fully discrete problems, by making use of the stationary elasticity reconstruction technique which allows to estimate the error for time‐dependent problem through the error estimation of the associated stationary elasticity problem. For fully discrete scheme, we make use of the backward‐Euler scheme and an appropriate space‐time reconstruction. The technique here can be applicable for a variety of DG methods as well.     

Interpolation error estimates for mean value coordinates over convex polygons

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in Gillette et al. (Adv Comput Math 37(3), 417–439, 2012), we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach π.     

New a posteriori error estimates for singular boundary value problems

In this paper, we discuss the asymptotic properties and efficiency of several a posteriori estimates for the global error of collocation methods. Proofs of the asymptotic correctness are given for regular problems and for problems with a singularity of the first kind. We were also strongly interested in finding out which of our error estimates can be applied for the efficient solution of boundary value problems in ordinary differential equations with an essential singularity. Particularly, we compare estimates based on the defect correction principle with a strategy based on mesh halving. AMS subject classification 65L05Supported in part by the Austrian Research Fund (FWF) grant P-15072-MAT and SFB Aurora.