Existence of Equilibria for Multivalued Mappings and Its Application to Vectorial Equilibria

In this paper, we apply a new fixed-point theorem and use various monotonicity and some coercivity conditions to establish equilibrium theorems for multimaps. As a simple consequence, we give a unified approach to vectorial equilibria for multimaps. We show that, from our results, some well-known classical results, such as the Ky Fan minimax inequality theorem and the Browder and Hartman-Stampacchia theorems concerning the existence for variational inequalities, can be derived easily.     

Revisiting Landauʼs density theorems for Paley–Wiener spaces

We present a surprisingly simple approach to Landau?s density theorems for sampling and interpolation, which provides stronger versions of these results. In particular, we extend the interpolation theorem to unbounded spectra.     

Simple proof of the Lagrange multiplier rule for the fixed endpoint problem in the calculus of variations

A new simple proof of the Lagrange multiplier rule is presented in this paper. The approach used involves simple analytical techniques that are very easy to follow and does not involve theorems on imbeddability in a one-parameter family of curves or matrix-rank analysis as do most of the existing techniques. The proof is here developed for the fixed-endpoint problem in a three-dimensional space.     

Semi-Discrete Ingham-Type Inequalities

One of the general methods in linear control theory is based on harmonic and non-harmonic Fourier series. The key of this approach is the establishment of various suitable adaptations and generalizations of the classical Parseval equality. A new and systematic approach was begun in our papers [1]-[4] in collaboration with Baiocchi. Many recent results of this kind, obtained through various Ingham-type theorems, were exposed recently in [9]. Although this work concentrated on continuous models, in connection with numerical simulations a natural question is whether these results also admit useful discrete versions. The purpose of this paper is to establish discrete versions of various Ingham-type theorems by using our approach. They imply the earlier continuous results by a simple limit process.     

On the (N+1)-dimensional local fractional reduced differential transform method and its applications

In this paper, we generalize the (N+1)-dimensional local fractional reduced differential transform method (LFRDTM) within the local fractional derivative sense. First, we show some new properties, lemmas, theorems and corollariesfor the (N+1)-dimensional LFRDTM. Second, these new properties, lemmas and theorems can be proved immediately after. Thirdly, we used two examples to state that this approach is efficient and simple to find numerical solutions to higher dimensional local fractional partial differential equations. Finally, we can be seen that this work can be looked as an extension of the prior work.     

Multivalent preference structures

This paper presents a valence approach for assessing multiattribute utility functions. Unlike the decomposition approach which uses independence axioms on whole attributes to obtain utility representations, the valence approach partitions the elements of each attribute into classes on the basis of equivalent conditional preference orders. These partitions generate multivalent utility independence axioms that lead to additive-multiplicative and quasi-additive representation theorems for multiaatribute utility functions defined over product sets of equivalence classes. Preference interdependencies are thereby reflected in these classes, so attribute interactions are readily interpreted and the functional forms of the representations are kept simple.     

Slowly oscillating functions in semigroup compactifications and convolution algebras

The paper proposes a unified approach to many key theorems proved in the last twenty years in different areas of abstract harmonic analysis. This approach is based on the so-called slowly oscillating functions which were introduced in coarse geometry. In addition to this method being the most natural and simple, it also leads to the generalisation of some of the results and to the achievement of some new results. Several of these results concern the topological centres of convolution algebras and semigroup compactifications.     

Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings

We develop a systematic matrix-analytic approach, based on intertwinings of Markov semigroups, for proving theorems about hitting-time distributions for finite-state Markov chains—an approach that (sometimes) deepens understanding of the theorems by providing corresponding sample-path-by-sample-path stochastic constructions. We employ our approach to give new proofs and constructions for two theorems due to Mark Brown, theorems giving two quite different representations of hitting-time distributions for finite-state Markov chains started in stationarity. The proof, and corresponding construction, for one of the two theorems elucidates an intriguing connection between hitting-time distributions and the interlacing eigenvalues theorem for bordered symmetric matrices.     

Sampling theorems on locally compact groups from oscillation estimates

We present a general approach to derive sampling theorems on locally compact groups from oscillation estimates. We focus on the L ²-stability of the sampling operator by using notions from frame theory. This approach yields particularly simple and transparent reconstruction procedures. We then apply these methods to the discretization of discrete series representations and to Paley–Wiener spaces on stratified Lie groups.     

Stochastic integration without tears

Stochastic integration theory is developed by axiomatizing the concept of semi-martingale in terms of a continuity property of integrals of simple functions. Using this approach, stochastic integration for left-continuous integrands, the change of variables formula and properties of the quadratic variation process are established in an elementary way. Submartingale decomposition theorems are introduced at a late stage in order to extend the results to general predictable integrands.     

Dichotomic maps for delayed equations

The aim of this work is to apply the method of “dichotomic maps” to study stability of the null solutions of delayed functional differential equations. Preliminaries are presented followed by proofs of main theorems on simple and asymptotic stability. Some particular cases which have been studied by others authors with more restrictive hypotheses and more elaborated methods are showed as illustrative examples enhancing the advantages of the dichotomic maps approach.     

Remarks on ordinary differential equations in ordered Banach spaces

This paper contains a unified simple approach to the main theorems on ordinary differential equations in ordered Banach spaces when the positive cone has a nonempty interior. The main features are (i) the inclusion of the case of absolutely continuous functions where the differential inequality holds only a.e.; (ii) the different situation corresponding to weak or strong inequalities in the initial conditions. An application to matrix differential equations yields generalizations of known theorems for matrix Riccati equations, among others.Dedicated to professor E. Hlawka on the occasion of his seventieth birthdayGuest professor at the Mathematisches Institut I, University of Karlsruhe, under auspices of the U.S. Special Program, Alexander von Humboldt-Stiftung.     

Projective background of the infinitesimal rigidity of frameworks

We present proofs of two classical theorems. The first one, due to Darboux and Sauer, states that infinitesimal rigidity is a projective invariant; the second one establishes relations (infinitesimal Pogorelov maps) between the infinitesimal motions of a Euclidean framework and of its hyperbolic and spherical images. The arguments use the static formulation of infinitesimal rigidity. The duality between statics and kinematics is established through the principles of virtual work. A geometric approach to statics, due essentially to Grassmann, makes both theorems straightforward. Besides, it provides a simple derivation of the formulas both for the Darboux-Sauer correspondence and for the infinitesimal Pogorelov maps. The research for this article was supported by the DFG Research Unit “Polyhedral Surfaces”.     

Operator Ergodic Theorems for Actions of Free Semigroups and Groups

New ergodic theorems are obtained for measure-preserving actions of free semigroups and groups. These theorems are derived from ergodic theorems for Markov operators. This approach also allows one to obtain ergodic theorems for some classes of Markov semigroups. Results of the paper generalize classical ergodic theorems of Kakutani, Oseledets, and Guivarc'h, and recent ergodic theorems of Grigorchuk, Nevo, and Nevo and Stein.     

Moudafi's viscosity approximations with Meir-Keeler contractions

In this paper, we discuss Moudafi's viscosity approximations with Meir-Keeler contractions. We first present very simple proofs of Xu's theorems concerning Moudafi's approximations. We next prove that Browder's and Halpern's type convergence theorems imply Moudafi's viscosity approximations. Using them, we finally state several deduced theorems.     

Multiplication maps and vanishing theorems for Toric varieties

We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.     

Derivation of the limit equations of elasticity theory on thin nets

A new approach is proposed to the study of plane problems of elasticity on rod junctures. This approach is based on so-called structural theorems and yields a fairly simple method of passing to the limit in the integral identity and thus obtaining the limit problem. In the case of nonlinear problems, the structural theorems make it possible to use variational methods like that of Γ-convergence of energy functionals. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 55–97, 2005.     

Actions of Noncompact Semisimple Groups on Lorentz Manifolds

The above title is the same, but with “semisimple” instead of “simple,” as that of a notice by Nadine Kowalsky. There, she announced many theorems on the subject of actions of simple Lie groups preserving a Lorentz structure. Unfortunately, she published proofs for essentially only half of the announced results before her premature death. Here, using a different, geometric approach, we generalize her results to the semisimple case, and give proofs of all her announced results. Received: May 2006, Revision: February 2007, Accepted: March 2007     

Optimal aggregation of fuzzy preference relations with an application to broadband internet service selection

This paper investigates the aggregation of multiple fuzzy preference relations into a collective fuzzy preference relation in fuzzy group decision analysis and proposes an optimization based aggregation approach to assess the relative importance weights of the multiple fuzzy preference relations. The proposed approach that is analytical in nature assesses the weights by minimizing the sum of squared distances between any two weighted fuzzy preference relations. Relevant theorems are offered in support of the proposed approach. Multiplicative preference relations are also incorporated into the approach using an appropriate transformation technique. An eigenvector method is introduced to derive the priorities from the collective fuzzy preference relation. The proposed aggregation approach is tested using two numerical examples. A third example involving broadband internet service selection is offered to illustrate that the proposed aggregation approach provides a simple, effective and practical way of aggregating multiple fuzzy preference relations in real-life situations.