Exact penalty and error bounds in DC programming

In the present paper, we are concerned with conditions ensuring the exact penalty for nonconvex programming. Firstly, we consider problems with concave objective and constraints. Secondly, we establish various results on error bounds for systems of DC inequalities and exact penalty, with/without error bounds, in DC programming. They permit to recast several class of difficult nonconvex programs into suitable DC programs to be tackled by the efficient DCA.     

Exact bounds on average granular pressures in a general bin model

An exact theory of averaged static bin pressures that takes into account the general variations of the pressure ratio, the bulk density, the grain-wall friction coefficient, the cross-sectional area, and the perimeter of the bin, with respect to the depth of the stored grain in a bin is presented.Exact solutions are then presented for the general differential inequalities involved. It is shown that the essential features of Janssen's classical results are retained. However, it is shown that smaller bounds than his lower bounds on bin pressures are quite possible contrary to the presently accepted notion of bin pressures. For a particular case an estimate on an improved lower bound on the overburden pressure is obtained and is shown to be more than 10% lower than that predicted by Janssen's formula.     

Exact bounds for the uniform approximation by spline interpolants

Summary. This paper completes a result of Reimer (1984) concerning -th-degree cardinal and -periodic interpolation. The method of proof is not restricted to the case of and being odd and seems to be more elementary. Received February 1, 1993 / Revised version received September 14, 1993     

Exact capacity results for stable processes

Exact results are proved for the capacity of pullbacks of analytic sets by stable processes. Received: 25 May 1988 / Revised version: 15 September 1997     

Exact bounds for the solution branches of nonlinear eigenvalue problems

Summary A method is constructed which yields a strip containing the full solution sets of nonlinear eigenvalue problems of the formu=Tu.The strip can be narrowed iteratively, and the method applies for both stable and unstable branches. Its high degree of accuracy is demonstrated by numerical examples. In particular, a lower bound is given for the critical value at which criticality is lost in the thermal ignition problem for the unit ball.     

Exact bounds on the order of torsion points of curves of genus 1

Let k be an algebraic number field of degree n on ₂; and , respectively, the curves on k; let, and _m, '_m be the bases of groups of all points of order m on and g, respectively. A proof of the following theorem is sketched: let p>3 be prime; if, then (p^t)6n; if k, then (p^t)4n. The resulting bounds are unimprovable.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 57–65, 1986.     

Exact bounds for the sensitivity analysis of structures with uncertain-but-bounded parameters

Based on interval mathematical theory, the interval analysis method for the sensitivity analysis of the structure is advanced in this paper. The interval analysis method deals with the upper and lower bounds on eigenvalues of structures with uncertain-but-bounded (or interval) parameters. The stiffness matrix and the mass matrix of the structure, whose elements have the initial errors, are unknown except for the fact that they belong to given bounded matrix sets. The set of possible matrices can be described by the interval matrix. In terms of structural parameters, the stiffness matrix and the mass matrix take the non-negative decomposition. By means of interval extension, the generalized interval eigenvalue problem of structures with uncertain-but-bounded parameters can be divided into two generalized eigenvalue problems of a pair of real symmetric matrix pair by the real analysis method. Unlike normal sensitivity analysis method, the interval analysis method obtains informations on the response of structures with structural parameters (or design variables) changing and without any partial differential operation. Low computational effort and wide application rang are the characteristic of the proposed method. Two illustrative numerical examples illustrate the efficiency of the interval analysis.     

Exact bounds for norms of differentiable periodic functions in L metric

Exact values are obtained for the upper bounds on the norms,fL, on the classes W ^(r) H(r =0, 1, 2,...) of r-fold differentiable functions f(x), of periodicity2, for which (f^(r);t)(t),where (t) is a given convex modulus of continuity.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 569–576, December, 1967.     

Exact inequalities for the upper bounds of seminorms on a class of periodic functions

We establish exact inequalities for the upper bounds of seminorms on classes of differentiable periodic functions, and as corollaries of them, exact inequalities between the best approximations by trigonometric polynomials of functions and their derivatives in the metrics of C and L.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 647–654, May, 1973.     

Chernoff bounds for discriminating between two markov processes

We study a statistical hypothesis testing problem, where a sample function of a Markov process with one of two sets of known parameters is observed over a finite time interval. When a log likelihood ratio test is used to discriminate between the two sets of parameters, we give bounds on the probability of choosing an incorrect hypothesis, and on the total probability of error, for both discrete and continuous time parameter, and discrete and continuous state space. The asymptotic behavior of the bounds is examined as the observation interval becomes infinite.     

Stochastic bounds on the zero range processes with superlinear jump rates

We present some results concerning the regularity of dynamics of the zero-range process. These results are achieved by using coupling based on attractivity of the models. The essential novelty is that these methods are carried out without any restriction on the increase of the jump rates depending on the local configurations, in contrast to the usually required sub-linear growth conditions. Based on the bounds presented here, the dynamics of these models may be constructed.     

Exact upper bounds for the number of the polynomials with given discriminants

We generalize the result of Davenport on the sum of absolute values of discriminants of integer polynomials of degree three. For the first time, we find the exact upper bound for the number of polynomials with given discriminant in the class of cubic polynomials of bounded height for the whole interval of values of discriminants.     

Remarks on the paper ‘Exact bounds on average granular pressures in a general bin model’

In the paper referred to in the title to this communication, Shahinpoor argued that increased density due to material consolidation of a bulk material in a vertical bin reduces the stresses the bulk material exerts on the bin wall below those that would be present if the material were unconsolidated. In this paper it is shown that increased density due to material consolidation actually increases rather than reduces these stresses. Shahinpoor's conclusions are incorrect because he misinterprets his analytical results.