A variational principle for block operator matrices and its application to the angular part of the Dirac operator in curved spacetime

The operator associated to the angular part of the Dirac equation in the Kerr-Newman background metric is a block operator matrix with bounded diagonal and unbounded off-diagonal entries. The aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above. In the last section, these analytic bounds are compared with numerical values from the literature.     

Gap functions and global error bounds for set-valued variational inequalities

The set-valued variational inequality problem is very useful in economics theory and nonsmooth optimization. In this paper, we introduce some gap functions for set-valued variational inequality problems under suitable assumptions. By using these gap functions we derive global error bounds for the solution of the set-valued variational inequality problems. Our results not only generalize the previously known results for classical variational inequalities from single-valued case to set-valued, but also present a way to construct gap functions and derive global error bounds for set-valued variational inequality problems.     

Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices

In this paper, we obtain lower and upper bounds for the entries of the inverses of diagonally dominant tridiagonal matrices. First of all we derive the bounds for off-diagonal elements of the inverse as a function of the diagonal ones, then we improve the two-sided bounds for the diagonal entries obtaining sharper lower and upper bounds for all the elements of the inverse.     

Sato-Tate Conjectures and Chebyshev Polynomials

Results of Shahidi on the analytic continuation of certain L-functions have been used by Serre to obtain partial information on the distribution of Sato-Tate angles of the Ramanujan function. Employing the Christoffel numbers associated to the orthogonal polynomials used by Serre we derive upper bounds on the lower density of primes with small angles. We also make effective his lower bounds on the upper density of such primes. Assuming that more analytic information is available, we sharpen similar estimates of Ram Murty, and, using the Christoffel numbers attached to the second order Chebyshev polynomials, derive general bounds on the discrepancy between the empirical angles distribution and the Sato-Tate distribution.     

Recursive lower and dual upper bounds for Bermudan-style options

Although Bermudan options are routinely priced by simulation and least-squares methods using lower and dual upper bounds, the latter are hardly optimized. In this paper, we optimize recursive upper bounds, which are more tractable than the original/nonrecursive ones, and derive two new results: (1) An upper bound based on (a martingale that depends on) stopping times is independent of the next-stage exercise decision and hence cannot be optimized. Instead, we optimize the recursive lower bound, and use its optimal recursive policy to evaluate the upper bound as well. (2) Less time-intensive upper bounds that are based on a continuation-value function only need this function in the continuation region, where this continuation value is less nonlinear and easier to fit (than in the entire support). In the numerical exercise, both upper bounds improve over state-of-the-art methods (including standard least-squares and pathwise optimization). Specifically, the very small gap between the lower and the upper bounds derived in (1) implies the recursive policy and the associated martingale are near optimal, so that these two specific lower/upper bounds are hard to improve, yet the upper bound is tighter than the lower bound.     

Bounds and Numerical Results for Homogenized Degenerated p-Poisson Equations

In this paper we derive upper and lower bounds on the homogenized energy density functional corresponding to degenerated p-Poisson equations. Moreover, we give some non-trivial examples where the bounds are tight and thus can be used as good approximations of the homogenized properties. We even present some cases where the bounds coincide and also compare them with some numerical results.     

Sufcient Conditions for Error Bounds and Linear Regularity in Banach Spaces

In this paper,we study error bounds for lower semicontinuous functions defned on Banach space and linear regularity for fnitely many closed subset in Banach spaces.By using Clarke's subdiferentials and Ekeland variational principle,we establish several sufcient conditions ensuring error bounds and linear regularity in Banach spaces.     

Sufficient conditions for error bounds and linear regularity in Banach spaces

In this paper, we study error bounds for lower semicontinuous functions defined on Banach space and linear regularity for finitely many closed subset in Banach spaces. By using Clarke＇s subd- ifferentials and Ekeland variational principle, we establish several sufficient conditions ensuring error bounds and linear regularity in Banach spaces.     

On characterizing the set of possible effective tensors of composites: The variational method and the translation method

A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization-problem simplifies the derivation of the Hashin-Shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method, due to Tartar, Murat, Lurie, and Cherkaev, uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant non-local projection operator, T₁. An important class of bounds, namely trace bounds on the effective tensors of two-component media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the two-components are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. The variational principles of Gibiansky and Cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by Golden and Papanicolaou. An isomorphism is shown to underlie their variational principles. This isomorphism is used to obtain Dirichlet-type variational principles and bounds for the effective tensors of general non-selfadjoint problems. It is anticipated that these variational principles, which stem from the work of Gibiansky and Cherkaev, will have applications in many fields of science.     

有限变形的极限分析定理

本文建立了有限变形的极限分析的变分原理(定理1),证明了它与有限变形的极限分析的全套方程和条件等价.本文又证明了:根据此变分原理求得的有限变形的极限载荷乘子,介于有限变形的极限分析的上限定理和下限定理所分别给出的上限解和下限解之间.     

The stationary solution of a one-dimensional bipolar quantum hydrodynamic model

In this paper, we consider the existence and uniqueness of stationary solution to the bipolar quantum hydrodynamic model in one dimensional space with general non-constant doping profile. The existence of the stationary solution is proved by Leray-Schauder fixed-point theorem and a crucial truncation technique is used to derive the positive upper and lower bounds of the stationary solution. The uniqueness of the stationary solution is shown by a delicate energy estimate.     

Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: the translation method

We deal with the problem of estimating the volume of inclusions using a small number of boundary measurements in electrical impedance tomography. We derive upper and lower bounds on the volume fractions of inclusions, or more generally two phase mixtures, using two boundary measurements in two dimensions. These bounds are optimal in the sense that they are attained by certain configurations with some boundary data. We derive the bounds using the translation method which uses classical variational principles with a null Lagrangian. We then obtain necessary conditions for the bounds to be attained and prove that these bounds are attained by inclusions inside which the field is uniform. When special boundary conditions are imposed the bounds reduce to those obtained by Milton and these in turn are shown here to reduce to those of Capdeboscq–Vogelius in the limit when the volume fraction tends to zero. The bounds of this article, and those of Milton, work for inclusions of arbitrary volume fractions. We then perform some numerical experiments to demonstrate how good these bounds are.     

Zero-flee regions for a rational function with applications

In this paper, we study a rational function which plays an important role in several problems of interest (eigenvalue problems, linear control theory, ... ). Our main interest is to determine zero-free regions. We also derive upper and lower bounds for this function. Communicated by T.L. Freeman     

A New Variant of Ekeland's Variational Principle for Set-Valued Maps

In this article, following the idea used by Göpfert et al . [A. Göpfert, Chr. Tammer and C. Zalinescu (2000). On the vectorial Ekeland's variational principle and minimal points in product spaces. Nonlinear Analysis, Theory, Methods & Applications , 39 , 909-922] to derive an Ekeland's variational principle for vector-valued functions, we derive a new variant of Ekeland's variational principle for set-valued maps. Finally, we apply this variational principle to obtain an approximate necessary optimality condition for a class of set-valued optimization problems.     

Zero-flee regions for a rational function with applications

In this paper, we study a rational function which plays an important role in several problems of interest (eigenvalue problems, linear control theory, ... ). Our main interest is to determine zero-free regions. We also derive upper and lower bounds for this function.     

线性模型中加权混合估计相对于最小二乘估计的两种相对效率

本文提出了线性模型中加权混合估计相对于最小二乘估计的两种相对效率,并给出了这些相对效率的上下界.     

Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian

Summary. In this paper, we derive quasi-norm a priori and a posteriori error estimates for the Crouzeix-Raviart type finite element approximation of the p-Laplacian. Sharper a priori upper error bounds are obtained. For instance, for sufficiently regular solutions we prove optimal a priori error bounds on the discretization error in an energy norm when . We also show that the new a posteriori error estimates provide improved upper and lower bounds on the discretization error. For sufficiently regular solutions, the a posteriori error estimates are further shown to be equivalent on the discretization error in a quasi-norm. Received January 25, 1999 / Revised version received June 5, 2000 Published online March 20, 2001     

Strong planning and forecast horizons for a model with simultaneous price and production decisions

In this paper, we have solved a general inventory model with simultaneous price and production decisions. Both linear and non-linear (strictly convex) production cost cases are treated. Upper and lower bounds are imposed on state as well as control variables. The problem is solved by using the Lagrangian form of the maximum principle. Strong planning and strong forecast horizons are obtained. These arise when the state variable reaches its upper or lower bound. The existence of these horizons permits the decomposition of the whole problem into a set of smaller problems, which can be solved separately, and their solutions put together to form a complete solution to the problem. Finally, we derive a forward branch and bound algorithm to solve the problem. The algorithm is illustrated with a simple example.     

Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed on an abstract level, we present a general technology for constructing computable sharp upper bounds for the global error for various particular classes of elliptic problems. Here the global error is understood as a suitable energy type difference between the true and computed solutions. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions, and are sharp in the sense that they can be, in principle, made as close to the true error as resources of the used computer allow. The latter can be achieved by suitably tuning the auxiliary parameter functions, involved in the proposed upper error bounds, in the course of the calculations.     

The boundedness for solutions of a certain two-dimensional fractional differential system with delay

In this paper, we study the components-wise upper bounds for solutions of two-dimensional fractional differential system with delay. Prior to the main results, we derive some results on two-dimensional nonlinear integral inequalities, then we investigate upper bounds of solutions basing on the obtained inequalities, finally, an example is given to illustrate the theoretical results.