A characterization of infimum functions

An abstract characterization of multi-place infimum functions on meet-semilattices is given.     

本质下确界的最优性条件及其相对熵算法实现

为了研究带约束的本质下确界优化问题,介绍了m阶偏差积分并研究了它的性质,给出了其最优性条件和概念算法.基于极小化相对熵的技术,提出了一种实现算法,并有效地解决该优化问题.数值算例验证了算法的有效性.     

On the infimum problem of Hilbert space effects

The quantum effects for a physical system can be described by the set ε(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. The infimum problem of Hilbert space effects is to find under what condition the infimum A∧B exists for two quantum effects A and B∈ε(H). The problem has been studied in different contexts by R. Kadison, S. Gudder, M. Moreland, and T. Ando. In this note, using the method of the spectral theory of operators, we give a complete answer of the infimum problem. The characterizations of the existence of infimum A∧B for two effects A. B∈ε(H) are established.     

计算多项式函数的全局下确界和全局最小值的有效算法

通过捕获所谓的严格临界点, 本文提出了一个计算实多项式函数的全局下确界和全局最小值的有效方法. 对于实数域R 上一个n 元多项式f, 该方法可用来判定f 在Rⁿ 上是否具有有限的全局下确界. 在f 具有有限的全局下确界的情况下, f 的下确界可严格地表示为码(h; a, b), 其中h 是一个实单元多项式, a 和b 是使得a < b 的两个有理数, 而(h; a, b) 代表h(z) 在开区间]a, b[ 中仅有的实根.此外, 当f 具有有限下确界时, 本文的方法可进一步判定f 的下确界能否达到. 在我们的算法设计中,著名的吴方法起着重要作用.     

Algorithms for computing the global infimum and minimum of a polynomial function

By catching the so-called strictly critical points,this paper presents an effective algorithm for computing the global infimum of a polynomial function.For a multivariate real polynomial f ,the algorithm in this paper is able to decide whether or not the global infimum of f is finite.In the case of f having a finite infimum,the global infimum of f can be accurately coded in the Interval Representation.Another usage of our algorithm to decide whether or not the infimum of f is attained when the global infimum of f is finite.In the design of our algorithm,Wu’s well-known method plays an important role.     

两个闭子空间之间夹角余弦最大最小问题的谱刻画

本文通过算子分块技巧给出了Hilbert 空间中两个闭子空间之间夹角的最大最小余弦的谱刻画, 同时给出了一些相关的结论.     

Spectral-like duality for distributive Hilbert algebras with infimum

Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and \({\wedge}\)-semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.     

Global infimum of strictly convex quadratic functions with bounded perturbations

The problem of minimizing [(f)\tilde]=f+p{\tilde f=f+p} over some convex subset of a Euclidean space is investigated, where f(x) = x ^T Ax + b ^T x is strictly convex and |p| is only assumed to be bounded by some positive number s. It is shown that the function [(f)\tilde]{\tilde f} is strictly outer γ-convex for any γ > γ*, where γ* is determined by s and the smallest eigenvalue of A. As consequence, a γ*-local minimal solution of [(f)\tilde]{\tilde f} is its global minimal solution and the diameter of the set of global minimal solutions of [(f)\tilde]{\tilde f} is less than or equal to γ*. Especially, the distance between the global minimal solution of f and any global minimal solution of [(f)\tilde]{\tilde f} is less than or equal to γ*/2. This property is used to prove a roughly generalized support property of [(f)\tilde]{\tilde f} and some generalized optimality conditions.     

On the infimum of a quasiconvex vector function over an intersection

We give sufficient conditions for the infimum of a quasiconvex vector function f over an intersection \(\bigcap_{i\in I}R_{i}\) to agree with the supremum of the infima of f over the R _i ’s.     

On the infimum of the local time of a Wiener process

Let {W(t), t0} be a standard Wiener process, and let L(x, t) be its jointly continuous local time. Define

Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum

A mapping is called isotone if it is monotone increasing with respect to the order defined by a pointed closed convex cone. Finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in [1, 2, 3, 4, 5]. Such cones are called isotone projection cones. In particular it was shown that any isotone projection cone is latticial [2]. This problem is extended by replacing the projection mapping with a continuous isotone retraction onto the cone. By introducing the notion of sharp mappings, it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous isotone retraction onto the cone whose complement is sharp. This result is used for characterizing a subdual latticial cone by the isotonicity of a generalization of the positive part mapping x ↦ x ⁺. This generalization is achieved by generalizing the infimum for subdual cones. The theoretical results of this paper exhibit fundamental properties of the lattice structure of the space which were not analysed before.     

Parallel summation,Maxwell's principle and the infimum of projections

In this paper we apply Maxwell's principle to give simple proofs of the properties of R: S, the parallel sum of two positive semi-definite linear operators. The parallel sum has been studied by Anderson, Ando, Duffin, Fillmore, Mitra, Williams, and others. In particular we give a short, elementary, and geometric proof of the result of Anderson and Duffin that gives the infimum of two orthogonal projections as twice their parallel sum.     

On the infimum attained by the reflected fractional Brownian motion

Let {B _H (t):t≥0} be a fractional Brownian motion with Hurst parameter \(H\in (\frac {1}{2},1)\) . For the storage process \(Q_{B_{H}}(t)=\sup _{-\infty \le s\le t}\) \(\left (B_{H}(t)-B_{H}(s)-c(t-s)\right )\) we show that, for any T(u)>0 such that \(T(u)=o(u^{\frac {2H-1}{H}})\) , $$\mathbb P (\inf_{s\in[0,T(u)]} Q_{B_{H}}(s)>u)\sim\mathbb P(Q_{B_{H}}(0)>u),$$ as \(u\to \infty \) . This finding, known in the literature as the strong Piterbarg property, goes in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.     

On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation

This paper concerns the Monge's transport problem in a general Polish space. We find optimal conditions to establish the equality between the infimum of Monge's problem and the minimum of the Kantorovich's relaxed version of the problem. A preliminary version of the results of this paper is contained in the Ph.D. thesis [A. Pratelli, Existence of optimal transport maps and regularity of the transport density in mass transportation problems, Ph.D. Thesis, Scuola Normale Superiore, Pisa, Italy, 2003. Available on http://cvgmt.sns.it/].     

On resolvents of integral and partially integral operators

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 2, pp. 88–91, March–April, 1993.