Inverse Additive Problems for Minkowski Sumsets II

The Brunn–Minkowski Theorem asserts that μ _d (A+B)^1/d ≥μ _d (A)^1/d +μ _d (B)^1/d for convex bodies A,B?? ^d , where μ _d denotes the d-dimensional Lebesgue measure. It is well known that equality holds if and only if A and B are homothetic, but few characterizations of equality in other related bounds are known. Let H be a hyperplane. Bonnesen later strengthened this bound by showing $$\mu_d(A+B)\geq (M^{1/(d-1)}+N^{1/(d-1)} )^{d-1}\biggl(\frac{\mu_d(A)}{M}+\frac {\mu_d(B)}{N} \biggr),$$ where M=sup?{μ _d?1((x+H)∩A)∣x∈? ^d } and $N=\sup\{\mu_{d-1}((\mathbf{y}+H)\cap B)\mid \mathbf{y}\in \mathbb {R}^{d}\}$ . Standard compression arguments show that the above bound also holds when M=μ _d?1(π(A)) and N=μ _d?1(π(B)), where π denotes a projection of ? ^d onto H, which gives an alternative generalization of the Brunn–Minkowski bound. In this paper, we characterize the cases of equality in this latter bound, showing that equality holds if and only if A and B are obtained from a pair of homothetic convex bodies by ‘stretching’ along the direction of the projection, which is made formal in the paper. When d=2, we characterize the case of equality in the former bound as well.     

Hamiltonian Operators with Zero-Divergence Constraints

Theoretical and Mathematical Physics - Using previously proposed techniques, we derive the defining system for a differential algebra associated with zero-divergence constraints. We study this...     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

Optimal Market Dealing Under Constraints

We consider a market dealer acting as a liquidity provider by continuously setting bid and ask prices for an illiquid asset in a quote-driven market. The market dealer may benefit from the bid–ask spread, but has the obligation to permanently quote both prices while satisfying some liquidity and inventory constraints. The objective is to maximize the expected utility from terminal liquidation value over a finite horizon and subject to the above constraints. We characterize the value function as the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation, and further enrich our study with numerical results. The contributions of our study concern both the modelling aspects and the dynamic structure of the control strategies. Important features and constraints characterizing market making problems are no longer ignored.     

Additive Maps Preserving Similarity of Operators on Banach Spaces

Let X be an infinite-dimensional complex Banach space and denote by B（X） the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B（X） onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B（X） such that either Φ（T） = cATA^-1 ＋ ψ（T）I for all T, or Φ（T） = cAT＊A^-1 ＋ ψ（T）I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero.     

Special Additive and Multiplicative Commutator Representations for Diagonal Operators

Let ?? be a dense linear subspace of a separable Hilbert space and let ??⁺(??) be the maximal Op*-Algebra on ??. The paper deals with a class of diagonal operators of ??⁺(??), for which we can prove some results concerning special commutator representations. For this end ideas of [1] are generalized.     

Additive Results for Moore-Penrose Inverse of Lambert Conditional Operators

Analysis Mathematica - In this paper, we discuss measure theoretic characterizations for Moore-Penrose inverse of Lambert conditional operators, denoted by (MwEMu)?, in some operator classes...     

市场约束条件下的非寿险费率厘定

在非寿险费率厘定中,经常遇到的一个实际问题是某些风险类别的费率不能过高或不能过低。在这种约束条件下,传统的广义线性模型将不能直接用于费率厘定。本文给出了一种在一般线性约束条件下,如何应用迭代算法对常用的广义线性模型进行调整,从而得到满足特定约束条件的费率厘定结果。本文的实证研究结果表明,该方法具有灵活性和现实可行性,能够解决非寿险费率厘定中常见的市场约束问题。     

Additive Results of the Drazin Inverse for Bounded Linear Operators

Given two bounded linear operators $P$ and $Q$ on a Banach space the formula for the Drazin inverse of $P+Q$ is given, under the assumptions $P^2 Q+PQ^2=0$ and $P^3 Q=PQ^3=0$ . In particular, some recent results arising in Drazin (Am Math Mon 65:506–514, 1958), Hartwig et al. (Linear Algebra Appl 322:207–217, 2001) and Castro-González et al. (J Math Anal Appl 350:207–215, 2009) are extended.     

Bounds for f-Divergences Under Likelihood Ratio Constraints

In this paper we establish an upper and a lower bound for the f-divergence of two discrete random variables under likelihood ratio constraints in terms of the Kullback-Leibler distance. Some particular cases for Hellinger and triangular discrimination, ²-distance and Rényi's divergences, etc. are also considered.     

Quantile Portfolio Optimization Under Risk Measure Constraints

This paper analyzes the problem of optimal portfolio choice with budget and risk constraints. The problem is formulated in terms of quantile functions and the risk is quantified through a large family of coherent risk measures. The solution is obtained analyzing the problem without constraints using Lagrange multipliers, getting a unique solution to the optimization problem.     

Generalized Variational Inequalities with Pseudomonotone Operators Under Perturbations

Variational inequalities and generalized variational inequalities with perturbed operators and constraints are considered and convergence of solutions to such problems is proved under an assumption of pseudomonotonicity. The paper extends previous results given by the authors proved in the setting of monotone operators.     

Improved Bounds on Families Under k-wise Set-Intersection Constraints

Let p be a prime, and let L be a set of s congruence classes modulo p. Let be a family of subsets of [n] such that the size modulo p of each member of is not in L, but the size modulo p of every intersection of k distinct members of is in L. We prove that , improving the bound due to Grolmusz and generalizing results proved for k = 2 by Snevily. Work supported in part by the NSA under Award No. MDA904-03-1-0037.     

Sequencing a Set of Alternatives Under Time Constraints

We consider a problem of sequencing a set of alternatives (i.e. manufacturing methods, job applicants or target journals) available for selection to complete a project. Associated with each alternative are the probability of successful completion, the completion time, and the reward obtained upon successfully completing the alternative. The optimal sequencing strategy that maximizes the expected present value of total rewards, is derived based on a simple ordering parameter. We further consider an extension in which one of the alternatives will not be available for selection if not selected by a certain time, and another extension in which the selection process is allowed only for a limited period of time. We propose solution strategies to the selection and sequencing problem under time constraints.     

二阶随机占优约束下考虑偏度的投资组合模型

在DentchevaRuszczynski(2006)模型的基础上,考虑偏度对构建投资组合的影响,建立了二阶随机占优约束下最大化组合收益率偏度的投资组合优化模型,并应用分段线性近似方法将模型转化为一个非线性混合整数规划问题.利用中国股票市场的历史数据对所建模型进行了实证分析,结果表明,所建新模型比均值-方差-偏度模型和市场指数具有更稳健的表现.