Remarks on some basic concepts in the KKM theory

In the KKM theory, some authors adopt the concepts of the compact closure (ccl), compact interior (cint), transfer compactly closed-valued multimap, transfer compactly l.s.c. multimap, and transfer compactly local intersection property, respectively, instead of the closure, interior, closed-valued multimap, l.s.c. multimap, and possession of a finite open cover property. In this paper, we show that such adoption is inappropriate and artificial. In fact, any theorem with a term with “transfer” attached is equivalent to the corresponding one without “transfer”. Moreover, we can invalidate terms with “compactly” attached by giving a finer topology on the underlying space. In such ways, we obtain simpler formulations of KKM type theorems, Fan-Browder type fixed point theorems, and other results in the KKM theory on abstract convex spaces.     

弱广义向量拟似变分不等式解的存在性

研究拓扑向量空间上弱广义向量拟似变分不等式解的存在性问题.与Khaliq和Rashid等在单调或伪单调假设条件下,利用KKM定理证明解的存在性不同,在集值映射是非空凸值映射并且有开下截口的条件下,利用Fan-Browder不动点原理证明解的存在性.     

局部FC-空间上的几乎不动点和不动点

利用古典的KKM原理的开[闭]形式得到FC-空间上的KKM型定理并给出局部FC-空间上的上[下]半连续映射的几乎不动点定理,最后给出在局部FC-空间上具有闭值且其值为FC-子空间的上半连续映射的不动点定理.     

Locally covering maps in metric spaces and coincidence points

We study the notion of α-covering map with respect to certain subsets in metric spaces. Generalizing results from [1] we use this notion to give some coincidence theorems for pairs of single-valued and multivalued maps one of which is relatively α-covering while the other satisfies the Lipschitz condition. These assertions extend some classical contraction map principles. We define the notion of α-covering multimap at a point and give conditions under which the covering property of a multimap at each interior point of a set implies that it is covering on the whole set. As applications we consider the solvability of a system of inclusions and the existence of a positive trajectory for a semilinear feedback control system. This paper is dedicated to Professor Felix Browder on the occasion of his jubilee     

Fixed Point Theorems for New Classes of Multimaps

A closed multimap T : X ? X having the KKM property has a fixed point whenever X is a compact convex set satisfying a certain topological condition. Our new result properly generalizes a known one due to W. K. Kim [4].     

On the transitivity of multifunctions and density of orbits in generalized topological spaces

The connection between transitivity and existence of a dense orbit for multifunctions $\phi \: X\multimap X$ in generalized topological spaces is studied. Moreover strongly transitive multifunctions and functions in generalized topological spaces are investigated.     

Best approximation, coincidence and fixed point theorems for quasi-lower semicontinuous set-valued maps in hyperconvex metric spaces

Suppose X is a compact admissible subset of a hyperconvex metric spaces M, and suppose F:XM is a quasi-lower semicontinuous set-valued map whose values are nonempty admissible. Suppose also G:XX is a continuous, onto quasi-convex set-valued map with compact, admissible values. Then there exists an x₀X such that

As applications, we give some coincidence and fixed point results for weakly inward set-valued maps. Our results, generalize some well-known results in literature.     

Multi-Model Communication and Data Assimilation for Mitigating Model Error and Improving Forecasts

Models for weather and climate prediction are complex, and each model typi-cally has at least a small number of phenomena that are poorly represented, such as perhaps the Madden-Julian Oscillation (MJO for short) or El Ni\~{n}o-Southern Oscillation (ENSO for short) or sea ice. Furthermore, it is often a very challenging task to modify and improve a complex model without creating new deficiencies. On the other hand, it is sometimes possible to design a low-dimensional model for a particular phenomenon, such as the MJO or ENSO, with significant skill, although the model may not represent the dynamics of the full weather-climate system. Here a strategy is proposed to mitigate these model errors by taking advantage of each model''s strengths. The strategy involves inter-model data assimilation, during a forecast simulation, whereby models can exchange information in order to obtain more faithful representations of the full weather-climate system. As an initial investigation, the method is examined here using a simplified scenario of linear models, involving a system of stochastic partial differential equations (SPDEs for short) as an imperfect tropical climate model and stochastic differential equations (SDEs for short) as a low-dimensional model for the MJO. It is shown that the MJO prediction skill of the imperfect climate model can be enhanced to equal the predictive skill of the low-dimensional model. Such an approach could provide a route to improving global model forecasts in a minimally invasive way, with modifications to the prediction system but without modifying the complex global physical model itself.     

Fixed point theorems for better admissible multimaps on abstract convex spaces

In this paper, a better admissible class B+ is introduced and a new fixed point theorem for better admissible multimap is proved on abstract convex spaces. As a consequence, we deduce a new fixed point theorem on abstract convex Φ-spaces. Our main results generalize some recent work due to Lassonde, Kakutani, Browder, and Park.     

非紧的一般化凸空间上不动点定理和supinfsup不等式

利用一般化凸空间上的KKM型定理得到有限交定理,然后作为应用讨论了在没有紧性限制的一般化凸空间上集值映射的不动点的存在问题以及Von Neumann-Fan型supinfsup不等式(等式)问题,最后给出了极大极小等式.     

Period two implies all periods for a class of ODEs: A multivalued map approach

We present an elementary proof that, for a multivalued map with nonempty connected values and monotone margins, the existence of a periodic orbit of any order implies the existence of periodic orbits of all orders. This generalizes a very recent result of this type in terms of scalar ordinary differential equations without uniqueness, due to F. Obersnel and P. Omari, obtained by means of lower and upper solutions techniques.

     

Optimal income taxation with discrete skill distribution

This paper studies the pattern of the optimal marginal income tax rates in a discrete model allowing all forms of individual skill distribution. It derives an explicit solution to the optimal marginal income tax rates in terms of the parameters of the model, and then rigorously shows the optimal marginal tax rate can be U-shaped, inverse U-shaped, strictly increasing, or strictly decreasing in the interior of skill levels, depending crucially on skill distribution. The numerical examples indicate that the optimal marginal tax rates can be W-shaped and inverse W-shaped in the interior of skill levels. The explicit solution to the optimal marginal income tax rate derived in this discrete model can be used to find optimal marginal income tax rates for an economy with any empirical skill distribution, without the need to estimate the density function of skill.     

Viscosity approximation methods for nonexpansive multimaps in Banach spaces

We prove strong convergence of the viscosity approximation process for nonexpansive nonself multimaps. Furthermore, an explicit iteration process which converges strongly to a fixed point of multimap T is constructed. It is worth mentioning that, unlike other authors, we do not impose the condition ＂Tz = {z}＂ on the map T.     

Compact Browder maps and equilibria of abstract economies

We obtain a partial resolution of a conjecture raised by Ben-El-Mechaiekh; that is, for a convex subset X of a Hausdorff t.v.s., any compact Browder map T:X ? X (a multimap with nonempty convex values and open fibers) has a fixed point. From this new result, we deduce a collectively fixed point theorem with applications to existences of equilibrium points and maximal elements of an abstract economy. Consequently, some known results are extended.     

Semantic hashing

We show how to learn a deep graphical model of the word-count vectors obtained from a large set of documents. The values of the latent variables in the deepest layer are easy to infer and give a much better representation of each document than Latent Semantic Analysis. When the deepest layer is forced to use a small number of binary variables (e.g. 32), the graphical model performs “semantic hashing”: Documents are mapped to memory addresses in such a way that semantically similar documents are located at nearby addresses. Documents similar to a query document can then be found by simply accessing all the addresses that differ by only a few bits from the address of the query document. This way of extending the efficiency of hash-coding to approximate matching is much faster than locality sensitive hashing, which is the fastest current method. By using semantic hashing to filter the documents given to TF-IDF, we achieve higher accuracy than applying TF-IDF to the entire document set.     

Modeling the span of control of leaders with different skill sets

In leading a team doing routine engineering design, two orthogonal skill sets can be distinguished: domain-specific or technical skills, and interpersonal “managerial” skills which are more general. This paper presents a computational model of team performance that relates these two types of skills to the amount of managerial and communication work generated given a certain team size (i.e. span of control). This model can be used to derive the optimal managerial profile for any team size, or the optimal team size for the skill set of a specific manager, provided the nature of the work remains fixed. The analysis of the model reveals several interesting insights. First, managerial skills are found to increase team performance up to a point after which it starts to decrease again. Second, a manager needs to compensate for low domain knowledge with high people skills, so optimal managerial skill level increases with lower domain knowledge. Third, both abilities have a significant impact on the manager’s allocated time for his/her group; however, more influence is noticed for managerial skills. Finally, the manager was found to be more essential to large teams.     

Assigning cross-trained workers to departments: A two-stage optimization model to maximize utility and skill improvement

We develop a general framework that is applicable in both manufacturing and service settings for assigning cross-trained workers across departments. The framework consists of a two-stage optimization model where two objective functions, departmental utility and skill improvement, are considered sequentially. Departmental utility is a function of departmental labor shortage and the first-stage optimization model maximizes total departmental utility subject to typical assignment constraints. The second stage model seeks to maximize total skill improvement, which is quantified by a hyperbolic learning curve, while trying not to deviate from the utility level obtained during the first stage optimization. Our computational experiments suggest that incorporating the skill improvement function explicitly in the model results in significant improvement in the total skill level of the workforce and thus leads to more effective worker assignments.     

数据处理服务外包企业多层复合技能人力资源调配问题研究

服务外包是推进我国产业结构调整的重要方式。数据处理作为服务外包中基础业务之一,对人力依赖程度很高。企业只有合理有效安排员工生产才能及时处理并以低成本交付订单。文章以数据处理业务为研究背景,考虑订单加工整个流程和员工技能种类,建立两步多层复合技能人力调配分段模型。选取实地调研企业数据运用模型进行求解。结果表明了模型的有效性,对于有效提升企业接包能力,促进我国服务外包发展有重要意义。     

Teaching mathematical modelling in microeconomics

Three modelling excercises from microeconomics are presented, both as examples of a methodology for teaching economics to mathematically inclined students, and as a vehicle for teaching modelling as a general skill. The first exercise, a price adjustment model for perfect competition, is given in detail within a formulation/solution/generalization/validation scheme with suggestions for an interactive teaching strategy. The results of giving the exercise to a second‐year BSc class in Mathematics and Computing are reported. Outlines of two further exercises relating to oligopoly and related models are given, and an Appendix provides a series of graduated problems for the first of these exercises.     

Impact of computer animations in cognitive learning: differentiation

In mathematic courses, construction of some concepts by the students in a meaningful way may be complicated. In such circumstances, to embody the concepts application of the required technologies may reinforce learning process. Onset of learning process over daily life events of the student's environment may lure their attention and may enable them to gain from the preliminary knowledge. Therefore, a good initiation may be realized in the course of meaningful learning. The underlying meaning of the abstract concepts by computer animations may be accomplished in class environments. That study is conducted searching out to discover the effects of animations over the learning process in mathematic courses. The study was performed over the 58 university freshman students selected randomly. Thirty-two students constituted the experiment group and 26 students constituted the control group. Computer animations-aided instruction model in constructive form were applied on the experiment group and non-computer-aided instruction model in constructive form were implemented on the control group. Student academic success via a test method developed by explored group with confidence rate .819 (Cronbach's alpha) revealed that data were evaluated by two-way variance analyses. The findings provided from the final test shows that the experiment group students were significantly higher according to the control group students in terms of academic success average scores. Computer animations were observed to be significant to assimilate the derivative concept in a discrete way over the students, to appeal their attention, animations of real life events observed to transform the abstract meanings in the events to a concrete manner. Students of whom the concrete stage is constructed meaningfully found to be tactful in reaching to semi-abstract and abstract stages.