A numerical approach to some basic theorems in singularity theory

In this paper, we give the explicit bounds for the data of objects involved in some basic theorems of singularity theory: the inverse, implicit and rank theorems for Lipschitz mappings, the splitting lemma and the Morse lemma, the density and openness of Morse functions. We expect that the results will make singularities more applicable and will be useful for numerical analysis and some fields of computing.     

A characterization of orthogonal duality in matroid theory

An operation on matroids is a function defined from the collection of all matroids on finite sets to itself which preserves isomorphism of matroids and sends a matroid on a set S to a matroid on the same set S. We show that orthogonal duality is the only non-trivial operation on matroids which interchanges contraction and deletion.     

A new approach to the Morse-Conley theory and some applications

Summary We present a new approach to the Morse theory which is based on a generalization of the Conley index to non locally compact spaces. The variant of the Morse theory which we obtain seems suitable for the applications to nonlinear functionals analysis. Some applications are given here; they mainly concern the study of periodic solutions of second order Hamiltonian systems. Other applications are in some quoted papers.     

A new look at some basic concepts in arbitrage pricing theory

The notion of No Free Lunch with Vanishing Risk (or NFLVR in short) w.r.t. admissible strategies depends on the choice of numeraire. Yan introduced the notion of allowable strategy and showed that condition of NFLVR w.r.t. allowable strategies is independent of the choice of numeraire and is equivalent to the existence of an equivalent martingale measure for the deflated price process. In this paper we establish a version of the Kramkov's optional decomposition theorem in the setting of equivalent martingale measures. Based on this theorem, we have a new look at some basic concepts in arbitrage pricing theory: superhedging, fair price, attainable contingent claims, complete markets and etc.     

Unified approach to some geometric results in variational analysis

Based on a study of a minimization problem, we present the following results applicable to possibly nonconvex sets in a Banach space: an approximate projection result, an extended extremal principle, a nonconvex separation theorem, a generalized Bishop-Phelps theorem and a separable point result. The classical result of Dieudonné (on separation of two convex sets in a finite-dimensional space) is also extended to a nonconvex setting.     

The basic stratum of some simple Shimura varieties

Under simplifying hypotheses we prove a relation between the $\ell $ -adic cohomology of the basic stratum of a Shimura variety of PEL-type modulo a prime of good reduction of the reflex field and the cohomology of the complex Shimura variety. In particular we obtain explicit formulas for the number of points in the basic stratum over finite fields. We obtain our results using the trace formula and truncation of the formula of Kottwitz for the number of points on a Shimura variety over a finite fields.     

Symmetric spaces approach to some fixed point results

In this paper we present a simple and unified approach to the fixed point results on cone symmetric spaces and metric type spaces based on symmetric spaces fixed point theory. We also give a new characterization of semi-metric spaces with open balls.     

Extensions of some factorization results from simple graphs to multigraphs

It was shown in a recent paper that an rs-regular multigraph G with maximum multiplicity μ(G) ≤ r can be factored into r regular simple graphs if first we allow the deletion of a relatively small number of hamilton cycles from G. In this paper, we use this theorem to obtain extensions of some factorization results on simple graphs to new results on multigraphs. © 1997 John Wiley & Sons, Inc. Inc.     

An economic approach to some classical theorems in optimization theory

This short article offers economically intuitive proofs of the Euler equation and the maximum principle based on one of the best known results in economics, namely that the marginal utility of one extra dollar spent on each consumption goods is the same for all the consumption goods as required by budget-constrained utility maximization.     

Euclideaness and final polynomials in oriented matroid theory

This paper deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial [3], [5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel [6], [7] concerning non-degenerate cycling of linear programs in non-euclidean oriented matroids.     

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.     

A simple approach to Hardy inequalities

We describe a simple method of proving Hardy-type inequalities of second and higher order with weights for functions defined in ℝ ⁿ . It is shown that we can obtain such inequalities with sharp constants by applying the divergence theorem to specially chosen vector fields. Another approach to Hardy inequalities based on the application of identities of Rellich-Pokhozhaev type is also proposed. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 563–572, April, 2000.     

A relation-algebraic approach to simple games

Simple games are a powerful tool to analyze decision-making and coalition formation in social and political life. In this paper, we present relation-algebraic models of simple games and develop relational specifications for solving some basic problems of them. In particular, we test certain fundamental properties of simple games and compute specific players and coalitions. We also apply relation algebra to determine power indices. This leads to relation-algebraic specifications, which can be evaluated with the help of the BDD-based tool Rel View after a simple translation into the tool’s programming language. In order to demonstrate the visualization facilities of Rel View we consider an example of the Catalonian Parliament after the 2003 election.     

Using the power of ideal solutions: simple proofs of some old and new results in location theory

4OR - When all objectives have a common minimum the existence of this ideal solution directly yields all efficient solutions of the corresponding multi-objective problem, as well as all minimising...     

A simple perturbation approach to Blasius equation

In this paper, we couple the iteration method with the perturbation method to solve the well-known Blasius equation. The obtained approximate analytic solutions are valid for the whole solution domain. Comparison with Howarth’s numerical solution reveals that the proposed method is of high accuracy, the first iteration step leads to 6.8% accuracy, and the second iteration step yields the 0.73% accuracy of initial slop.