Matching Theorem Research Articles

A Dynamic Generalization of Becker's Assortative Matching Result

This paper considers a dynamic partnership model in which agents' future productivity depends in part on their current match. Efficient rankings over human capital distributions are characterized in several common stochastic orders. These rankings are used to characterize the returns to human capital for individual agents in the associated market equilibrium. Assortative matching theorems are developed using the planner's characterization results. While positive assortative matching is efficient given sufficient complementarity in static production and dynamic transitions, curvature assumptions on static production are necessary for a robust sorting theory in this dynamic environment.

The matching theorems in G-convex spaces with applications

The purpose of this paper is to establish some new matching theorems in G-convex spaces and, as applications, to obtain some new fixed point theorems, section theorems and a minimax theorem in G-convex spaces. The results presented in this paper improve and generalize the corresponding results in [1–5,7–12].

MATCHING THEOREMS AND SIMULTANEOUS RELATION PROBLEMS

In this paper we give two matching theorems of Ky Fan type concerning open or closed coverings of nonempty convex sets in a topological vector space. One of them will permit us to put in evidence, when X and Y are convex sets in topological vector spaces, a new subclass of KKM(X, Y) different by any admissible class <TEX>$\mathfrak{u}_c$</TEX>(X, Y). For this class of set-valued mappings we establish a KKM-type theorem which will be then used for obtaining existence theorems for the solutions of two types of simultaneous relation problems.

Alternative principles and their applications

In this paper we establish two alternative principles of the following type: If X and Y are convex subsets of two locally convex Hausdorff topological vector spaces and $${F,S:X \multimap Y}$$ are two set-valued mappings satisfying certain conditions, then either there exists $${x_0 \in X}$$ such that $${F(x_0) = \emptyset}$$ or $${\bigcap_{x \in X}S(x) \neq \emptyset}$$ . As first applications of the alternative principles we obtain two matching theorems of Ky Fan type. Next, are given several analytic alternatives and minimax inequalities. Finally we establish two very general alternative theorems concerning existence of solutions of a vector equilibrium problem.

Balanced list edge-colourings of bipartite graphs

Galvin solved the Dinitz conjecture by proving that bipartite graphs are edge-choosable. We improve Galvin’s method and deduce from any colouring of the edges of bipartite graph G some further list edge-colouring properties of G. In particular, for bipartite graphs, it follows from the existence of balanced bipartite edge-colourings that balanced list edge-colourings exist as well. While the key to Galvin’s proof is the stable marriage theorem of Gale and Shapley, our result is based on the well-known “many-to-many” version of the stable matching theorem.

Smith and Rawls share a room: stability and medians

We consider one-to-one, one-sided matching (roommate) problems in which agents can either be matched as pairs or remain single. We introduce a so-called bi-choice graph for each pair of stable matchings and characterize its structure. Exploiting this structure we obtain as a corollary the “lone wolf” theorem and a decomposability result. The latter result together with transitivity of blocking leads to an elementary proof of the so-called stable median matching theorem, showing how the often incompatible concepts of stability (represented by the political economist Adam Smith) and fairness (represented by the political philosopher John Rawls) can be reconciled for roommate problems. Finally, we extend our results to two-sided matching problems.

Generalized [formula omitted] property in a hyperconvex metric space and its applications

In this paper, we first define the family 2 - g K K M ( X , Y ) in a hyperconvex metric space, and then we get a 2 - g K K M theorem and a fixed point theorem without compactness assumption. Next, by using the 2 - g K K M theorem, we get the matching theorems, coincidence theorems, variational inequality theorems and minimax inequality theorems.

The Stable Roommates Problem with Choice Functions

The stable marriage theorem of Gale and Shapley states that for n men and n women there always exists a stable marriage scheme, that is, a set of marriages such that no man and woman mutually prefer one another to their partners. The stable marriage theorem was generalized in two directions: the stable roommates problem is the “one-sided” version, where any two agents on the market can form a partnership. The generalization by Kelso and Crawford is in the “two-sided” model, but on one side of the market agents have a so-called substitutable choice function, and stability is interpreted in a natural way. It turned out that even if both sides of the market have substitutable choice functions, there still exists a stable assignment. The latter version contains the “many-to-many” model where up to a personal quota, polygamy is allowed for both men and women in the two-sided market. The goal of this work is to solve the stable partnership problem, a generalization of the one-sided model with substitutable choice functions. We do not quite reach that: besides substitutability, we also need for the result the so called increasing property (or, in other terminology, the law of aggregate demand). Luckily, choice functions in well-known stable matching theorems comply with this property. The main result is a generalization of Irving’s algorithm, the first efficient method to solve the stable roommates problem. We deduce from the algorithm a generalization of Tan’s result on characterizing the existence of a stable matching and prove a generalization of the rural hospital theorem and the so-called splitting property of many-to-many stable matchings. We show that our algorithm is linear-time in some sense and indicate that for general (i.e. not necessarily increasing) substitutable choice functions the stable partnership problem is NP-complete.

Short containers in Cayley graphs

The star diameter of a graph measures the minimum distance from any source node to several other target nodes in the graph. For a class of Cayley graphs from abelian groups, a good upper bound for their star diameters is given in terms of the usual diameters and the orders of elements in the generating subsets. This bound is tight for several classes of graphs including hypercubes and directed n -dimensional tori. The technique used is the so-called disjoint ordering for a system of subsets, due to Gao, Novick and Qiu [S. Gao, B. Novick, K. Qiu, From Hall’s matching theorem to optimal routing on hypercubes, J. Comb. Theory B 74 (1998) 291–301].

Smith and Rawls Share a Room: Stability and Medians

We consider one-to-one, one-sided matching (roommate) problems in which agents can either be matched as pairs or remain single. We introduce a so-called bi-choice graph for each pair of stable matchings and characterize its structure. Exploiting this structure we obtain as a corollary the lone wolf theorem and a decomposability result. The latter result together with transitivity of blocking leads to an elementary proof of the so-called stable median matching theorem, showing how the often incompatible concepts of stability (represented by the political economist Adam Smith) and fairness (represented by the political philosopher John Rawls) can be reconciled for roommate problems. Finally, we extend our results to two-sided matching problems.

GENERALIZED KKM THEOREMS ON HYPERCONVEX METRIC SPACES WITH APPLICATIONS

In this work, we establish a generalized KKM theorem in a hyperconvex metric space, and then we use this theorem to get a fixed point theorem, the matching theorem, the coincidence theorem, minimax inequality theorems and the variational inequality theorems.

A new KKM theorem in L-convex spaces and some applications

In this paper, a new KKM theorem is established in L-convex spaces. As applications, a Ky Fan matching theorem for compactly open covers, a Fan–Browder coincidence theorem, a Fan–Browder fixed point theorem and a maximal element theorem are obtained in L-convex spaces. These theorems unify, improve and generalize some recent results in the references therein. Finally, the equilibrium existence theorems for abstract economies and qualitative games in L-convex spaces are yielded.

Matching theorems for systems of a finitely generated Coxeter group

In this paper we prove a series of matching theorems for two sets of Coxeter generators of a finitely generated Coxeter group that identify common features of the two sets of generators. As an application, we describe an algorithm for finding a set of Coxeter generators of maximum rank for a finitely generated Coxeter group.

New results related to a conjecture of Manickam and Singhi

In 1988 Manickam and Singhi conjectured that for every positive integer d and every n ≥ 4 d , every set of n real numbers whose sum is non-negative contains at least ( n − 1 d − 1 ) subsets of size d whose sums are non-negative. In this paper we make use of Hall’s matching theorem in order to study some numbers related to this conjecture.

W- G- F- KKM mapping, intersection theorems and minimax inequalities in FC-space

By using basic KKM theorem, a new matching theorem and some minimax inequalities for set-valued mappings defined on the FC-spaces are proved under very weak assumptions. These results generalized many known results from the recent literature.

Some results for the family [formula omitted] and the Φ-mapping

In this paper, we first establish a fixed point theorem for a k-set contraction map on the family KKM ( X , X ) , which not needs to be a compact map. Next, we establish the matching theorems, coincidence theorems and minimax theorems on the family KKM ( X , Y ) and the Φ-mapping.

The matching theorems and coincidence theorems for generalized R-KKM mapping in topological spaces

In this paper we present some new matching theorems with open cover and closed cover by using the generalized R-KKM theorems [L. Deng, X. Xia, Generalized R-KKM theorem in topological space and their applications, J. Math. Anal. Appl. 285 (2003) 679–690] in the topological spaces with property (H). As applications, some coincidence theorems are established in topological spaces. Our results extend and generalize some known results.

KKM mappings in metric spaces

We introduce the class of KKM-type mappings on metric spaces and establish some fixed point theorems for this class. We also obtain a generalized Fan's matching theorem, a generalized Fan–Browder's type theorem, and a new version of Fan's best approximation theorem.

Photosynthetic models with maximum entropy production in irreversible charge transfer steps

Steady-state bacterial photosynthesis is modelled as cyclic chemical reaction and is examined with respect to overall efficiency, power transfer efficiency, and entropy production. A nonlinear flux–force relationship is assumed. The simplest two-state kinetic model bears complete analogy with the performance of an ideal (zero ohmic resistance of the P–N junction) solar cell. In both cases power transfer to external load is much higher than the 50% allowed by the impedance matching theorem for the linear flux–force relationship. When maximum entropy production is required in the transition with a load, one obtains high optimal photochemical yield of 97% and power transfer efficiency of 91%. In more complex photosynthetic models, entropy production is maximized in all irreversible electron/proton (non-slip) transitions in an iterative procedure. The resulting steady-state is stable with respect to an extremely wide range of initial values for forward rate constants. Optimal proton current increases proportionally to light intensity and decreases with an increase in the proton-motive force (the backpressure effect). Optimal affinity transfer efficiency is very high and nearly perfectly constant for different light absorption rates and for different electrochemical proton gradients. Optimal overall efficiency (of solar into proton-motive power) ranges from 10% (bacteriorhodopsin) to 19% (chlorophyll-based bacterial photosynthesis). Optimal time constants in a photocycle span a wide range from nanoseconds to milliseconds, just as corresponding experimental constants do. We conclude that photosynthetic proton pumps operate close to the maximum entropy production mode, connecting biological to thermodynamic evolution in a coupled self-amplifying process.

Discrete anomalies and the null cone of SYM theories

A stronger version of an anomaly matching theorem (AMT) is proven that allows to anticipate the matching of continuous as well as discrete global anomalies. The AMT shows a connection between anomaly matching and the geometry of the null cone of SYM theories. Discrete symmetries are shown to be broken at the origin of the moduli space in Seiberg–Witten theories.