Theorems of the Alternative and Duality

This paper investigates the relations between theorems of the alternative and the minimum norm duality theorem. A typical theorem of the alternative is associated with two systems of linear inequalities and/or equalities, a primal system and a dual one, asserting that either the primal system has a solution, or the dual system has a solution, but never both. On the other hand, the minimum norm duality theorem says that the minimum distance from a given point z to a convex set is equal to the maximum of the distances from z to the hyperplanes separating z and . We consider the theorems of Farkas, Gale, Gordan, and Motzkin, as well as new theorems that characterize the optimality conditions of discrete l ₁-approximation problems and multifacility location problems. It is shown that, with proper choices of , each of these theorems can be recast as a pair of dual problems: a primal steepest descent problem that resembles the original primal system, and a dual least–norm problem that resembles the original dual system. The norm that defines the least-norm problem is the dual norm with respect to that which defines the steepest descent problem. Moreover, let y solve the least norm problem and let r denote the corresponding residual vector. If r=0, which means that z , then y solves the dual system. Otherwise, when r0 and z , any dual vector of r solves both the steepest descent problem and the primal system. In other words, let x solve the steepest descent problem; then, r and x are aligned. These results hold for any norm on . If the norm is smooth and strictly convex, then there are explicit rules for retrieving x from r and vice versa.     

Necessary and sufficient well-posedness conditions for a convolution equation of the second kind with even kernel on a finite interval

We derive some necessary and sufficient conditions for the well-posedness of a convolution equation of the second kind with even kernel on a finite interval. In order to check these conditions it suffices to compute a one-dimensional integral (of a given function) with precision less than 0.5. As an intermediate result we give a strengthening of the Fredholm alternative for the equation in question with an arbitrary kernel in L ₁.     

假设检验中的三个问题及其思考

讨论了假设检验的教学和实践中可能会遇到的三个问题.分析得到三个结论:理论上原假设与备择假设可以任意设定,但实践中常常按照某种规则来设定;假设检验的两个常用的决策准则中,p值准则更精确,为决策者接受原假设提供了一个"风险"的度量;假设检验理论在处理利用样本推断总体的问题中是普遍有效的.     

Heterogeneous Electrocatalytic Oxygen Evolution Reaction by a Sol-Gel Electrode with Entrapped Na3[Ru2(μ-CO3)4]: The Effect of NaHCO3

The Na₃[Ru₂(μ-CO₃)₄] complex is acting as a water oxidation catalyst in a homogeneous system. Due to the significance of heterogeneous systems and the effect of bicarbonate on the kinetic, we studied the bicarbonate effect on the heterogeneous electrocatalyst by entrapping the Na₃[Ru₂(μ-CO₃)₄] complex in a sol-gel matrix. We have developed two types of sol-gel electrodes, which differ by the precursor, and are demonstrating their stability over a minimum of 200 electrochemical cycles. The pH increases affected the currents and k_cat for both types of electrodes, and their hydrophobicity, which was obtained from the precursor type, influenced the electrocatalytic process rate. The results indicate that NaHCO₃ has an important role in the catalytic activity of the presented heterogeneous systems; without NaHCO₃, the diffusing species is probably OH⁻, which undergoes diffusion via the Grotthuss mechanism. To the best of our knowledge, this is the first study to present a simple and fast one-step entrapment process for the Na₃[Ru₂(μ-CO₃)₄] complex by the sol-gel method under standard laboratory conditions. The results contribute to optimizing the WSP, ultimately helping expand the usage of hydrogen as a green and more readily available energy source.     

考虑牛鞭效应与反牛鞭效应联合作用在不完美信息下供应链决策者的合作博弈与演进

理性假设下,供应链运作过程中出现牛鞭效应也会出现反牛鞭效应,在考虑牛鞭效应与反牛鞭效应联合作用下分别构建二级供应链合作博弈模型与n级供应链合作博弈模型,得到Nash均衡策略,构建合作Nash均衡下供应链上订货模型,并通过数值算例验证订货模型及牛鞭效应特性与反牛鞭效应特性交替作用的最优化局势优于传统视角的消除牛鞭效应带来的供应链优化.研究表明:供应链上下游在周期内表现出牛鞭效应特性与反牛鞭效应特性的交替作用,会使得供应链整体上市场需求偏差减少、运作成本降低.贡献在于给出消除牛鞭效应的新思路,从整体上消除牛鞭效应或反牛鞭效应,使得整条供应链不管是短期的还是长期都是最优的,在长期还会给供应链企业带来显著的战略优势.     

Exceptional sets related to Hayman's alternative

Let E be a subset of the complex plane C consisting of a countable set of points tending to ∞ and let k?1 be an integer. We derive a spacing condition (dependent on k) on the points of E which ensures that, if f is a function meromorphic in C with sufficiently large Nevanlinna deficiency at the poles, then either f takes every complex value infinitely often, or the kth derivative f(k) takes every non-zero complex value infinitely often, in C−E. This improves a previous result of Langley.     

First Order System of Equations of Mixed Type and Second Order Equation of Mixed Type

A system of first order equations of mixed type, which may be reduced to a general second order equation of mixed type, is considered. Uniqueness of solution to the generalized Tricomi problem is proved by the method of auxiliary function. Existence of H¹ strong solution is based on a characteristic problem and is proved by the Fredholm's alternative properties.     

天然气吸附剂的开发及其储气性能的研究——Ⅰ-吸附剂制备与体积法评定吸附剂的储气性能

报道了天然气吸附剂的基本制备技术,并采用体积法评定了吸附剂的储气性能。结果表明,采用作者开发的吸附剂制备工艺可获得性能优良的天然气吸附剂。以木质素为原料制备的粉状吸附剂,其比表面积可达２９１２ｍ２／ｇ,微孔体积可达１－４８ｃｍ３／ｇ,平均孔径为１－４８ｎｍ ,堆密度为０－３０ｇ／ｃｍ３ 。在６－０ＭＰａ、２５ ℃下,天然气的吸附储存量可达到１４０Ｖ／Ｖ。天然气吸附剂的储气性能与其比表面、微孔数量、填充密度等参数有重要关系     

“Alternative medical therapy” use among singers: Prevalence and implications for the medical care of the singer

Singers are extremely conscious of health problems that affect their voices and well-being and often take an active role in seeking care for these problems. They frequently seek treatment from providers or with modalities considered “alternative” to traditional medical care. A survey of singers was completed to elucidate their attitudes and practices regarding “alternative modalities” of medical care. Frequently singers will self-medicate or take advice from people not well versed in the special needs of a professional voice user. They will fail to share this information with the physician when seeking “traditional” medical care. These practices may predispose the singer to suboptimal or even dangerous care. These results are discussed, as well as the implications for the medical physician treating the singer. The possible detrimental pharmacologic effects of “natural therapies” widely used by singers are presented, with special attention to the particular concerns for the professional singer