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1.
We consider Hardy-Rellich inequalities and discuss their possible improvement. The procedure is based on decomposition into spherical harmonics, where in addition various new inequalities are obtained (e.g. Rellich-Sobolev inequalities). We discuss also the optimality of these inequalities in the sense that we establish (in most cases) that the constants appearing there are the best ones. Next, we investigate the polyharmonic operator (Rellich and higher order Rellich inequalities); the difficulties arising in this case come from the fact that (generally) minimizing sequences are no longer expected to consist of radial functions. Finally, the successively use of the Rellich inequalities lead to various new higher order Rellich inequalities. 相似文献
2.
We examine linear inequalities satisfied by the
flag $f$-vectors of polytopes. One source of these inequalities
is the toric $g$-vector; convolutions of its entries are non-negative
for rational polytopes. We prove a conjecture of Meisinger about a
redundancy in these inequalities. Another source of inequalities is
the {\bf cd}-index; among all $d$-polytopes, each {\bf cd}-index coefficient
is minimized on the $d$-simplex. We show that not all of the {\bf cd}-index
inequalities are implied by the toric $g$-vector inequalities, and that not all of the toric $g$-vector inequalities are implied by the {\bf cd}-index
inequalities.
Finally, we show that some inequalities from
convolutions of {\bf cd}-index coefficients are implied by other
{\bf cd}-index inequalities. 相似文献
3.
Karen Aardal 《Mathematical Programming》1998,81(2):149-175
We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack cover, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities. The separation problem based on the family of submodular inequalities is NP-hard in general. For the well known subclass of flow cover inequalities, however, we show that if the client set is fixed, and if all capacities are equal, then the separation problem can be solved in polynomial time. For the flow cover inequalities based on an arbitrary client set and general capacities, and for the effective capacity and single depot inequalities we develop separation heuristics. An important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining. The way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities. The family of combinatorial inequalities was originally developed for the uncapacitated facility location problem, but is also valid for the capacitated problem. No computational experience using the combinatorial inequalities has been reported so far. Here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities. We report on computational results from solving 60 medium size problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. 相似文献
4.
Alysson M. Costa Jean-François Cordeau Bernard Gendron 《Computational Optimization and Applications》2009,42(3):371-392
Solving multicommodity capacitated network design problems is a hard task that requires the use of several strategies like
relaxing some constraints and strengthening the model with valid inequalities. In this paper, we compare three sets of inequalities
that have been widely used in this context: Benders, metric and cutset inequalities. We show that Benders inequalities associated
to extreme rays are metric inequalities. We also show how to strengthen Benders inequalities associated to non-extreme rays
to obtain metric inequalities. We show that cutset inequalities are Benders inequalities, but not necessarily metric inequalities.
We give a necessary and sufficient condition for a cutset inequality to be a metric inequality. Computational experiments
show the effectiveness of strengthening Benders and cutset inequalities to obtain metric inequalities. 相似文献
5.
E. I. Galakhov 《Proceedings of the Steklov Institute of Mathematics》2008,260(1):112-122
We prove the nonexistence of solutions for some nonlinear ordinary differential equations and inequalities, for quasilinear partial differential equations and inequalities in bounded domains with singular points on the boundary, and for systems of such equations and inequalities. The proofs are based on the method of nonlinear capacity. We also give examples demonstrating that the conditions obtained are sharp in the class of problems under consideration. 相似文献
6.
We establish refined Sobolev inequalities between the Lorentz spaces and homogeneous Besov spaces. The sharpness of these
inequalities is illustrated on several examples, in particular based on non-uniformly oscillating functions known as chirps.
These results are also used to derive refined Hardy inequalities. 相似文献
7.
We derive new trace inequalities for NURBS-mapped domains. In addition to Sobolev-type inequalities, we derive discrete trace inequalities for use in NURBS-based isogeometric analysis. All dependencies on shape, size, polynomial degree, and the NURBS weighting function are precisely specified in our analysis, and explicit values are provided for all bounding constants appearing in our estimates. As hexahedral finite elements are special cases of NURBS, our results specialize to parametric hexahedral finite elements, and our analysis also generalizes to T-spline-based isogeometric analysis. We compare the bounding constants appearing in our explicit trace inequalities with numerically computed optimal bounding constants, and we discuss application of our results to a Laplace problem. We finish this paper with a brief exploration of so-called patch-wise trace inequalities for isogeometric analysis. 相似文献
8.
We develop a method for generating valid convex quadratic inequalities for mixed0–1 convex programs. We also show how these inequalities can be generated in the linear case by defining cut generation problems using a projection cone. The basic results for quadratic inequalities are extended to generate convex polynomial inequalities. 相似文献
9.
Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces
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In this paper,we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces.The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type.We obtain the first order Poincare inequalities for vector fields satisfying Hrmander's condition in variable non-isotropic Sobolev spaces.We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups.Moreover,we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups.These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian.Our results are only stated and proved for vector fields satisfying Hrmander's condition,but they also hold for Grushin vector fields as well with obvious modifications. 相似文献
10.
We investigate some variational inequalities associated with the equations of the stationary motion of granulated media with a constant density. These inequalities replace the usual equations of motion in the case when some additional constraints are imposed on the flow. We prove the existence of solutions of the inequalities, study their regularity, uniqueness, dependence on data and relations to solutions of the equations of motion. 相似文献
11.
George A. Anastassiou 《Applicable analysis》2013,92(5):607-624
Here we present univariate Sobolev-type fractional inequalities involving fractional derivatives of Canavati, Riemann–Liouville and Caputo types. The results are general L p inequalities forward and converse on a closed interval. We give an application to a fractional ODE. We present also the mean Sobolev-type fractional inequalities. 相似文献
12.
Q. H. Ansari Z. Khan A. H. Siddiqi 《Journal of Optimization Theory and Applications》2005,127(2):263-283
In this paper, we introduce weighted variational inequalities over product of sets and system of weighted variational inequalities.
It is noted that the weighted variational inequality problem over product of sets and the problem of system of weighted variational
inequalities are equivalent. We give a relationship between system of weighted variational inequalities and systems of vector
variational inequalities. We define several kinds of weighted monotonicities and establish several existence results for the
solution of the above-mentioned problems under these weighted monotonicities. We introduce also the weighted generalized variational
inequalities over product of sets, that is, weighted variational inequalities for multivalued maps and systems of weighted
generalized variational inequalities. Extensions of weighted monotonicities for multivalued maps are also considered. The
existence of a solution of weighted generalized variational inequalities over product of sets is also studied. The existence
results for a solution of weighted generalized variational inequality problem give also the existence of solutions of systems
of generalized vector variational inequalities.
The first and third author express their thanks to the Department of Mathematical Sciences, King Fahd University of Petroleum
and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The authors are also grateful to the referees
for comments and suggestions improving the final draft of this paper. 相似文献
13.
We study 0-1 reformulations of the multicommodity capacitated network design problem, which is usually modeled with general integer variables to represent design decisions on the number of facilities to install on each arc of the network. The reformulations are based on the multiple choice model, a generic approach to represent piecewise linear costs using 0-1 variables. This model is improved by the addition of extended linking inequalities, derived from variable disaggregation techniques. We show that these extended linking inequalities for the 0-1 model are equivalent to the residual capacity inequalities, a class of valid inequalities derived for the model with general integer variables. In this paper, we compare two cutting-plane algorithms to compute the same lower bound on the optimal value of the problem: one based on the generation of residual capacity inequalities within the model with general integer variables, and the other based on the addition of extended linking inequalities to the 0-1 reformulation. To further improve the computational results of the latter approach, we develop a column-and-row generation approach; the resulting algorithm is shown to be competitive with the approach relying on residual capacity inequalities. 相似文献
14.
We study valid inequalities for optimization models that contain both binary indicator variables and separable concave constraints. These models reduce to a mixed-integer linear program (MILP) when the concave constraints are ignored, or to a nonconvex global optimization problem when the binary restrictions are ignored. In algorithms designed to solve these problems to global optimality, cutting planes to strengthen the relaxation are traditionally obtained using valid inequalities for the MILP only. We propose a technique to obtain valid inequalities that are based on both the MILP constraints and the concave constraints. We begin by characterizing the convex hull of a four-dimensional set consisting of a single binary indicator variable, a single concave constraint, and two linear inequalities. Using this analysis, we demonstrate how valid inequalities for the single node flow set and for the lot-sizing polyhedron can be “tilted” to give valid inequalities that also account for separable concave functions of the arc flows. We present computational results demonstrating the utility of the new inequalities for nonlinear transportation problems and for lot-sizing problems with concave costs. To our knowledge, this is one of the first works that simultaneously convexifies both nonconvex functions and binary variables to strengthen the relaxations of practical mixed-integer nonlinear programs. 相似文献
15.
We investigate strong inequalities for mixed 0-1 integer programs derived from flow cover inequalities. Flow cover inequalities
are usually not facet defining and need to be lifted to obtain stronger inequalities. However, because of the sequential nature
of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting
coefficients, lifting of flow cover inequalities is computationally very demanding. We present a computationally efficient
way to lift flow cover inequalities based on sequence independent lifting techniques and give computational results that show
the effectiveness of our lifting procedures.
Received May 15, 1996 / Revised version received August 7, 1998
Published online June 28, 1999 相似文献
16.
Bernd Carl 《Journal of Functional Analysis》1981,41(3):290-306
We establish inequalities between entropy numbers and approximation numbers for operators acting between Banach spaces. Furthermore we derive inequalities between eigenvalues and entropy numbers for operators acting on a Banach space. The results are compared with the classical inequalities of Bernstein and Jackson. 相似文献
17.
Hüseyin Budak Samet Erden Muhammad Aamir Ali 《Mathematical Methods in the Applied Sciences》2021,44(1):378-390
We first establish two new identities, based on the kernel functions with either two section or three sections, involving quantum integrals by using new definition of quantum derivative. Then, some new inequalities related to Simpson's 1/3 formula for convex mappings are provided. In addition, Newton type inequalities, for functions whose quantum derivatives in modulus or their powers are convex, are deduced. We also mention that the results in this work generalize inequalities given in earlier study. 相似文献
18.
The main results of this paper concern sharp constants for the Moser‐Trudinger inequalities on spheres in complex space ?n. We derive Moser‐Trudinger inequalities for smooth functions and holomorphic functions with different sharp constants (see Theorem 1.1). The sharp Moser‐Trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the CR setting. Though there is a close connection in spirit between inequalities proven here on complex spheres and those on the Heisenberg group for functions with compact support in any finite domain proven earlier by the same authors [17], derivation of the sharp constants for Moser‐Trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the Heisenberg group. Variants of Moser‐Onofri‐type inequalities are also given on complex spheres as applications of our sharp inequalities (see Theorems 1.2 and 1.3). One of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients (see Theorem 1.4). © 2004 Wiley Periodicals, Inc. 相似文献
19.
Yu. P. Virchenko 《Ukrainian Mathematical Journal》1998,50(6):870-878
We consider models of statistical mechanics of the type of lattice gas with attractive interaction of general kind. We propose
a method for obtaining inequalities that connect multipoint correlation functions of different order. This method allows one,
on the one hand, to strengthen similar inequalities, which can be obtained within the framework of the FKG method, and on
the other hand, to obtain new inequalities. We introduce the notion of duality for models of lattice gas. We show that if,
under the transformation p ⇒ 1 - p, the correlation inequalities for a model with attraction turn into the corresponding inequalities that are also satisfied,
then the correlation functions of the dual model also satisfy the latter inequalities.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 765–773, June, 1998. 相似文献
20.
We obtain weak type (1,q) inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces.The proofs use some properties of maximal operators.Our results are closely related to the strong type inequalities in [13,14,15]. 相似文献