格中依次最短无关组与Minkowski约化基等价的充分条件

本文讨论了格中基子集、依次最短无关组及Minkowski约化基之间的向量长度关系,利用无关组与基之间的一些制约性质,给出了Minkowski约化基达到依次最短长度,以及依次最短无关组成为Minkowski约化基的一些充分条件.     

Inequalities between successive minima and intrinsic volumes of a convex body

The second theorem of Minkowski establishes a relation between the successive minima and the volume of a 0-symmetric convex body. Based on this theorem we will prove a series of inequalities connecting the product of certain successive minima with certain intrinsic volumes.     

Notes on lattice points of zonotopes and lattice-face polytopes

Minkowski’s second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski’s bound by replacing the volume by the lattice point enumerator of a convex body. In this context we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our results for lattice zonotopes and lattice-face polytopes imply, in particular, that for 0-symmetric lattice-face polytopes and lattice parallelepipeds the volume can be replaced by the lattice point enumerator.     

Successive-minima-type inequalities

We show analogues of Minkowski's theorem on successive minima, where the volume is replaced by the lattice point enumerator. We further give analogous results to some recent theorems by Kannan and Lovász on covering minima.     

An Optimization Problem Related to Minkowski’s Successive Minima

The purpose of this paper is to establish an inequality connecting the lattice point enumerator of a 0-symmetric convex body with its successive minima. To this end, we introduce an optimization problem whose solution refines former methods, thus producing a better upper bound. In particular, we show that an analogue of Minkowski’s second theorem on successive minima with the volume replaced by lattice point enumerator is true up to an exponential factor, whose base is approximately 1.64.     

Successive minima and slopes of hermitian vector bundles over number fields

The purpose of this paper is to clarify the relationship between the successive minima and the slopes of a hermitian vector bundle on the spectrum of the ring of integers of an algebraic number field. The main result is a lower and an upper bound for each successive minimum in terms of the corresponding slope.     

A new transfer principle in the geometry of numbers

In my paper (Proc. Roy. Soc. Edinburgh Sect. A 64 (1956), 223–238), I gave a general transfer principle in the geometry of numbers which consisted of inequalities linking the successive minima of a convex body in n dimensions with those of a convex body in N dimensions where in general N is greater than n. This result contained in particular my earlier theorem on compound convex bodies (Proc. London Math. Soc. (3) 5 (1955), 358–379). In the present paper I apply essentially the same method to prove a new transfer principle which connects the successive minima of a convex body in m dimensions and those of a convex body in n dimensions with the successive minima of a convex body in mn dimensions.     

Minkowski’s Second Theorem over a Simple Algebra

We generalize the notion of successive minima, Minkowski’s second theorem and Siegel’s lemma to a free module over a simple algebra whose center is a global field.     

Minima, pentes et algèbre tensorielle

Slopes of an adelic vector bundle exhibit a behaviour akin to successive minima. Comparisons between the two amount to a Siegel lemma. Here we use Zhang’s version for absolute minima over the algebraic numbers. We prove a Minkowski-Hlawka theorem in this context. We also study the tensor product of two hermitian bundles bounding both its absolute minimum and maximal slope, thus improving an estimate of Chen. We further include similar inequalities for exterior and symmetric powers, in terms of some lcm of multinomial coefficients.     

A successive descent algorithm over a system of local minima

A successive descent algorithm over a system of local minima has been developed to find the global minimum of a function of many variables defined on a simply connected compact set. If the number of local minima is finite and a bound on the global minimum is given, the algorithm finds the global minimum in finitely many steps. Test examples are presented. Translated from Prikladnaya Matematika i Informatika, No. 30, pp. 46–54, 2008.     

Lattice-point enumerators of ellipsoids

Minkowski’s second theorem on successive minima asserts that the volume of a 0-symmetric convex body K over the covolume of a lattice Λ can be bounded above by a quantity involving all the successive minima of K with respect to Λ. We will prove here that the number of lattice points inside K can also accept an upper bound of roughly the same size, in the special case where K is an ellipsoid. Whether this is also true for all K unconditionally is an open problem, but there is reasonable hope that the inductive approach used for ellipsoids could be extended to all cases.     

Convergence of the solution for a domain decomposed,heterogeneously modeled flow past a concave profile problem

The free streamline problem investigated is that of fluid flow past a symmetric truncated concave‐shaped profile between walls. An open wake or cavity is formed behind the profile. Conformal mapping techniques are used to solve this problem. The problem formulated in the hodograph plane is decomposed into two nonoverlapping domains. Heterogeneous modeling is then used to describe the problems, i.e., a different governing differential equation in each domain. In one of these domains, a Baiocchi‐type transformation is used to obtain a fixed domain formulation for the part of the transformed problem containing an unknown boundary. In the other domain, the Baiocchi‐type transformation is extended across the boundary between the two domains, thus yielding a different problem formulation. This also assures that the dependent variables and their normal derivatives are continuous along this common boundary. The numerical solution scheme, a successive over‐relaxation approach, is applied over the whole problem domain with the use of a projection‐operation over only the fixed domain formulated part. Numerical results are obtained for the case of a truncated circular profile. These results are found to be in good agreement with another published result. The existence and uniqueness of the solution to the problem as a variational inequality is shown, and the convergence of the numerical solution using a domain decomposition method scheme is demonstrated by assuming some convergence property on the common interface of the two subdomains. © 2000 John Wiley & Sons, Inc. Numeer Methods Partial Differential Eq 16: 459–479, 2000     

Lattice Points in Large Borel Sets and Successive Minima

Let B be a Borel set in E^d with volume V(B) = ∞. It is shown that almost all lattices L in E^d contain infinitely many pairwise disjoint d-tuples, that is sets of d linearly independent points in B. A consequence of this result is the following: let S be a star body in E^d with V(S ) = ∞. Then for almost all lattices L in E^d the successive minima λ₁(S,L),..., λ_d(S,L) of S with respect to L are 0. A corresponding result holds for most lattices in the Baire category sense. A tool for the latter result is the semi-continuity of the successive minima.     

Minkowski’s Second Theorem over a Simple Algebra

We generalize the notion of successive minima, Minkowski’s second theorem and Siegel’s lemma to a free module over a simple algebra whose center is a global field. The author was partly supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science.     

On Optimal Constants for Best Two-Dimensional Simultaneous Diophantine Approximations

 The main results of this paper state optimal constants for estimates of so-called successive minima in two dimensions under a constraint on the denominator. While these inequalities are known for every dimension, best possible constants within these estimates are, of course, notknown for any dimension larger than one and remain unknown for all dimensions larger than two. (Received 29 April 1998; in revised form 23 November 1998)     

Variational Sets of Perturbation Maps and Applications to Sensitivity Analysis for Constrained Vector Optimization

We consider sensitivity analysis in terms of variational sets for nonsmooth vector optimization. First, relations between variational sets, or their minima/weak minima, of a set-valued map and that of its profile map are obtained. Second, given an objective map, relationships between the above sets of this objective map and that of the perturbation map and weak perturbation map are established. Finally, applications to constrained vector optimization are given. Many examples are provided to illustrate the essentialness of the imposed assumptions and some advantages of our results.     

Voronoi's algorithm in purely cubic congruence function fields of unit rank 1

The first part of this paper classifies all purely cubic function fields over a finite field of characteristic not equal to 3. In the remainder, we describe a method for computing the fundamental unit and regulator of a purely cubic congruence function field of unit rank 1 and characteristic at least 5. The technique is based on Voronoi's algorithm for generating a chain of successive minima in a multiplicative cubic lattice, which is used for calculating the fundamental unit and regulator of a purely cubic number field.

     

Minimal zonotopes containing the crosspolytope

Motivated by the problem to improve Minkowski’s lower bound on the successive minima for the class of zonotopes we determine the minimal volume of a zonotope containing the standard crosspolytope. It turns out that this volume can be expressed via the maximal determinant of a ±1-matrix, and that in each dimension the set of minimal zonotopes contains a parallelepiped. Based on that link to ±1- matrices, we characterize all zonotopes attaining the minimal volume in dimension 3 and present related results in higher dimensions.     

The penalized profile sampler

The penalized profile sampler for semiparametric inference is an extension of the profile sampler method [B.L. Lee, M.R. Kosorok, J.P. Fine, The profile sampler, Journal of the American Statistical Association 100 (2005) 960-969] obtained by profiling a penalized log-likelihood. The idea is to base inference on the posterior distribution obtained by multiplying a profiled penalized log-likelihood by a prior for the parametric component, where the profiling and penalization are applied to the nuisance parameter. Because the prior is not applied to the full likelihood, the method is not strictly Bayesian. A benefit of this approximately Bayesian method is that it circumvents the need to put a prior on the possibly infinite-dimensional nuisance components of the model. We investigate the first and second order frequentist performance of the penalized profile sampler, and demonstrate that the accuracy of the procedure can be adjusted by the size of the assigned smoothing parameter. The theoretical validity of the procedure is illustrated for two examples: a partly linear model with normal error for current status data and a semiparametric logistic regression model. Simulation studies are used to verify the theoretical results.     

Computationally tractable pairwise complexity profile

Quantifying the complexity of systems consisting of many interacting parts has been an important challenge in the field of complex systems in both abstract and applied contexts. One approach, the complexity profile, is a measure of the information to describe a system as a function of the scale at which it is observed. We present a new formulation of the complexity profile, which expands its possible application to high‐dimensional real‐world and mathematically defined systems. The new method is constructed from the pairwise dependencies between components of the system. The pairwise approach may serve as both a formulation in its own right and a computationally feasible approximation to the original complexity profile. We compare it to the original complexity profile by giving cases where they are equivalent, proving properties common to both methods, and demonstrating where they differ. Both formulations satisfy linear superposition for unrelated systems and conservation of total degrees of freedom (sum rule). The new pairwise formulation is also a monotonically nonincreasing function of scale. Furthermore, we show that the new formulation defines a class of related complexity profile functions for a given system, demonstrating the generality of the formalism. © 2013 Wiley Periodicals, Inc. Complexity 18:20–27, 2013