正弦定理与欧氏平行公理的等价性

本人在数学通报(1991年第11期)上发表一篇短文(见[1])证明了正弦定理与余弦定理是等价的,但在从正弦定理导出余弦定理时,用了一个条件,即△ABC的内角和为π。因此有些读者就问这个条件能否去掉,若不能去掉就不能说正弦定理与余弦定理是等价的。问题提得好。首先我们要明确什么叫命题的等价性?所谓命题等价     

Verlinde模性范畴上的Casimir数及其应用

本文计算了秩为n+1的一类特殊的Verlinde模性范畴L的Casimir数,计算结果表明该Casimir数为2n+4.作为应用,由Higman定理知域K上的Grothendieck代数Gr(L)_Z K是半单代数当且仅当2n+4在域K中不为零.这也给出了第二类型n+1次Dickson多项式E_(n+1)(X)在K[X]中无重因式的一个等价刻画.如果2n+4在域K中为零,借助于Dickson多项式的有关因式分解定理,本文完全给出了Grothendieck代数Gr(L)_Z K的Jacobson根.     

有限p-群的半p-交换性和正则性

本文定义了所谓“半p-交换p-群”(定义3),例1和例2说明半p-交换性与正则性(定义1)并不等价.但定理1证明了:有限p-群是正则的与它的每个子群(包括自身)的任一商群都是半P-交换的这两个条件等价.这就给出了有限正则P-群的一个充分必要条件.     

两类树的独立集多项式的单峰性

研究了图的独立集多项式的单峰性,给出具有爪图结构的几类图的独立集多项式等价的无爪图,并在此基础上证明了两类具有爪图结构的树T(n,n+1,m)和T(I,i+1,k,j,j+1)的独立集多项式具有单峰性,从而为具有爪图结构的其它树的单峰性提供了一个证明方法.     

非可加测度空间上Egoroff定理的伪条件

定义了伪条件(M)和伪Egoroff条件,证明了在非可加测度理论上伪Egoroff条件与Egoroff定理的一种伪形式等价,伪条件(M)是伪Egoroff条件的一个必要条件,以及Egoroff定理的上述伪形式是伪条件(M)的一个必要条件,从而得到了在非可加测度空间上Egoroff定理四种形式的等价条件。     

中心对称与中心交错矩阵几何

应用对称(交错)矩阵几何基本定理,本文证明了域上中心对称(交错)矩阵几何基本定理.     

资产定价第一基本定理的拓扑描述

资产定价的第一基本定理是数量金融学中核心的定理之一 ,本文证明了在 L∞ 的弱 * 拓扑 σ(L∞ ,L1)中的凸集分离定理 ,并在此定理的基础上给出了没有无风险免费午餐的拓扑描述 ,证明了市场公平性与没有无风险免费午餐条件的等价性 ,从而重新证明了资产定价的第一基本定理 .     

弱偏好序下带容量房屋匹配混合模型的机制设计

研究弱偏好序下, 带容量房屋市场混合模型(CHMTeT)的机制设计问题, 并针对该模型提出了一类算法机制, 该机制是TTC算法机制的推广, 称之为剔除筛选算法(简记为CTTC)机制. 此外, 证明了CHMTeT模型应用CTTC算法得到的这一类机制(即CTTC机制)满足个人理性、帕累托有效性和防策略操纵性, 并得出CTTC算法的时间复杂度为O(n_{1}^{2}\\(n_{1}n_{2}+n_{2}^{2})), 其中n_{1}为参与人数, n_{2}为房子数.     

关于资产定价的第一基本定理

资产定价的第一基本定理是数理金融学中最核心的定理之一 ,本文证明了在 L∞的弱 *拓扑 σ(L∞ ,L1)中的凸集分离定理 ,并在此定理的基础上给出了没有无风险免费午餐的拓扑描述 ,证明了市场公平性与没有无风险免费午餐条件的等价性 ,从而重新证明了资产定价的第一基本定理 .     

具有风险偏好的模糊网络最短路算法

运用结构元理论来求解具有风险偏好的、带有模糊权值的网络最短路问题.首先,简要介绍模糊结构元及相关定理.之后,提出了组合序,证明组合序是全序.组合序含有参数θ,随着θ的取值范围不同,序反映风险偏好的类型不同.在组合序和相关定理的基础上,证明了模糊网络最短路的判定定理,定理表明:求模糊网络最短路等价求一经典网络最短路,且风险偏好大小由θ的取值来确定.最后,通过一个例子来说明求解过程.     

社会福利函数独裁的特征

在这篇短文中,给出了关于社会福利函数F的半严格正向响应的概念,并且证明了如果备选对象至少有三个,则弱帕累托性质与半严格正向响应性质是独裁的充分必要条件.作为应用,我们给出了社会选择函数防止策略性操纵的一个等价描述,并对社会福利函数引进了防止局部策略性操纵的概念,得到了一个类似于Gibbard—Satterthwaite定理的结论.     

Hedonic coalition formation games with variable populations: core characterizations and (im)possibilities

We consider hedonic coalition formation games with variable sets of agents and extend the properties competition sensitivity and resource sensitivity (introduced by Klaus, Games Econ Behav 72:172–186, 2011, for roommate markets) to hedonic coalition formation games. Then, we show that on the domain of solvable hedonic coalition formation games, the Core is characterized by coalitional unanimity and Maskin monotonicity (see also Takamiya, Maskin monotonic coalition formation rules respecting group rights. Niigata University, Mimeo, 2010, Theorem 1). Next, we characterize the Core for solvable hedonic coalition formation games by unanimity, Maskin monotonicity, and either competition sensitivity or resource sensitivity (Corollary 2). Finally, and in contrast to roommate markets, we show that on the domain of solvable hedonic coalition formation games, there exists a solution not equal to the Core that satisfies coalitional unanimity, consistency, competition sensitivity, and resource sensitivity (Example 2).     

Monotonicity and Consistency in Matching Markets

Objective: To obtain axiomatic characterizations of the core of one-to-one and one-to-many matching markets. Methods: The axioms recently applied to characterize the core of assignment games were adapted to the models of this paper. Results: The core of one-to-one matching markets is characterized by two different lists of axioms. The first one consists of weak unanimity, population monotonicity, and Maskin monotonicity. The second consists of weak unanimity, population monotonicity, and consistency. If we allow for weak preferences, the core is characterized by weak unanimity, population monotonicity, Maskin monotonicity, and consistency. For one-to-many matchings, the same lists as for the case of strict preferences characterize the core. Conclusions: The cores of the discrete matching markets are characterized by axioms that almost overlap with the axioms characterizing the core of the continuous matching markets. This provides an axiomatic explanation for the observations in the literature that almost parallel properties are obtained for the core of the two models. We observe that Maskin monotonicity is closely related to consistency in matching marketsThis research is financially supported by Waseda University Grant for Special Research Projects #2000A−887, 21COE-GLOPE, and Grant-in-Aid for Scientific Research #15530125, JSPS. This paper was presented at the 7th. International Meeting of the Society for Social Choice and Welfare held in Osaka, Japan. The comments of the participants are gratefully acknowledged. The author thanks Professors William Thomson, Eiichi Miyagawa and anonymous referees for their valuable comments and suggestions. Any remaining errors are independent     

Strategy-proofness, solidarity, and consistency for multiple assignment problems

We consider a problem of allocating indivisible objects when agents may desire to consume more than one object and no monetary transfers are allowed. We are interested in allocation rules that satisfy desirable properties from an economic and social point of view. In addition to strategy-proofness and Pareto efficiency, we consider consistency and two solidarity properties (replacement-domination and population-monotonicity). In most of the cases, these properties are satisfied only by serially dictatorial rules. Received: November 1999/Final version: December 2001     

An alternative characterization of the deferred acceptance algorithm

Kojima and Manea (Econometrica 78(2):633–653, 2010) present two characterizations of when an allocation rule corresponds to the agent-proposing deferred acceptance (DA) algorithm for some substitutable priority rule of the objects being assigned. Building on their results we characterize when an allocation rule is outcome equivalent to the DA algorithm for every substitutable priority rule. In particular, an assignment rule satisfies mutual best (MB) if an agent is always assigned her most preferred object whenever she has the highest priority for it. This mild requirement is a necessary but far from sufficient condition for an assignment rule to be stable. We demonstrate that any allocation mechanism that satisfies MB along with non-wastefulness, population monotonicity and either individually rational monotonicity or weak Maskin monotonicity not only is a stable assignment mechanism but is equivalent to the agent proposing DA algorithm.     

Strategy-proofness and “median voters”

We consider the problem of choosing a level of the public good for an economy in which agents have continuous and single-peaked preferences (Black, 1948). We show that a solution satisfyingstrategy-proofness andcontinuity if and only if it is an augmented median-voter solution. Anaugmented median-voter solution is described in terms of 2ⁿ parameters (which satisfy an antimonotonicity condition) as follows:n+1 of them are selected according to an increasing order of the peaks; the outcome is the median of thesen+1 parameters and then peaks. This result establishes a formal connection between strategy-proofness and a generalized notion of median voter. (Similar median formulas were used by Moulin (1980) to describe smaller classes of solutions.) We provide an interpretation of these 2 ⁿ parameters in terms of the following properties:anonymity, voter sovereignty, unanimity, andPareto efficiency.I wish to thank William Thomson for his enormous effort in supervision. I am grateful to Jeff Banks, Marcus Berliant, Tarik Kara, Hideo Konishi, Shige Serizawa, John Weymark, two anonymous referees of this journal, and the participants of the Theory Workshop at the University of Rochester, the Econometric Society Meeting, Winter 1994, and the International Game Theory Conference at SUNY, Stony Brook, July 1993 for their useful comments. All remaining errors are my own responsibility.     

Monotonic core solutions: beyond Young’s theorem

Young’s theorem implies that every core concept violates monotonicity. In this paper, we investigate when such a violation of monotonicity by a given core concept is justified. We introduce a new monotonicity property for core concepts. We pose several open questions for this new property. The open questions arise because the most important core concepts (the nucleolus and the per capita nucleolus) do not satisfy the property even in the class of convex games.     

Sturm-Liouville问题的特征值对势函数的依赖性

研究了定义在[0,1]上的Sturm-Liouville问题的特征值对势函数的连续依赖性.应用比较定理和定义区间单调性证明了:当部分区间[x0,1]上的势函数趋于无穷大时,[0,1]区间上的特征值渐进趋近于[0,x0]区间上的某个特征值.推广了一些作者对Sturm-Liouville问题研究的相应结果,并为其相应问题的研究提供了一个新的视角.     

Banach analytic sets and a non-linear version of the Levi extension theorem

We prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite-dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite-dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic family.