Applications of nonstandard analysis to spin models

Methods of nonstandard analysis are used to construct a Markov semigroup representing the stochastic evolution of an infinite spin system with finite range interaction by means of a hyperfinite spin system. The hyperfinite spin system is then used to derive classical results about phase transitions of the stochastic Ising model without the use of thermodynamic limits.     

Planning the unusual: Applications of control theory to nonstandard problems

This paper presents a survey on intertemporal optimization problems arising in different areas such as literature, theology, ethics, sports, health care, criminology, vampirology, and history. Although some of the models considered seem to lack seriousness, they are undoubtedly valuable for real-life economic decisions.     

理想单子在非标准饱和模型下的若干应用

通过利用非标准分析中的饱和模型,对其中的理想进行了讨论,从而得到了理想的非标准特征,并进一步利用这一特征证明了单子论中的一些相关定理．     

Resolutive ideal boundaries of nonlinear resistive networks

In this paper, we deal with nonlinear resistive networks in the framework of modular sequence spaces, introduced by De Michele and Soardi. We consider ideal boundaries of a network and investigate Dirichlet boundary value problems for solutions of Poisson equations.

     

Some remarks on nonstandard theory of classes

In the paper, we discuss several hitherto open problems concerning classes in nonstandard set theory. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 233–255, 2005.     

Spectral theory of group representations and their nonstandard hull

We construct the nonstandard hull of a not necessarily bounded strongly continuous representationU of the locally compact semigroupS on a Banach spaceE. Then we apply our results to the theory of the spectrum σ (U) ofU, mainly in cases whereS is an abelian group, e.g.S=R. First of all we obtain generalizations to the unbounded case of results known for the bounded one. Secondly we introduce the notion of the Riesz part R σ(U) of σ(U) and characterize those representations satisfying σ(U)=R σ(U). We illustrate the theory developed so far by applications to representations on Banach lattices. Dedicated to Prof. Dr. H. H. Schaefer on the occasion of his 60th birthday     

A nonstandard set theory in the \displaystyle\in-language

Abstract. We demonstrate that a comprehensive nonstandard set theory can be developed in the standard -language. As an illustration, a nonstandard is obtained. Received: 17 March 1998 / Revised version: 30 October 1998     

Finite dimensional potential theory applied to numerical analysis of linear systems

It is shown how to use finite dimensional potential theory in the numerical analysis of certain iterative methods for solving differential equations.     

Harmonic analysis in (UMD)-spaces: Applications to the theory of bases

In the paper, a general method for the construction of bases and unconditional finite-dimensional basis decompositions for spaces with the property of unconditional martingale differences is proposed. The construction makes use of a certain strongly continuous representation of Cantor's group in these spaces. The results are applied to vector function spaces and symmetric spaces of measurable operators associated with factors of type II.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 890–905, December, 1995.     

A nonstandard proof of a lemma from constructive measure theory

Suppose that f_n is a sequence of nonnegative functions with compact support on a locally compact metric space, that T is a nonnegative linear functional, and that T f_n < T f₀. A result of Bishop, foundational to a constructive theory of functional analysis, asserts the existence of a point x such that f_n (x) < f ₀(x). This paper extends this result to arbitrary Hausdorff spaces, and gives short proofs using nonstandard analysis. While such arguments used are not themselves constructive, they can illuminate where the difficulty lies in finding the point x. An algorithm for constructing x is then given, with a nonstandard proof that the algorithm converges to a correct value. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Applications of the theory of semi-embeddings to Banach space theory

Let X and Y be Banach spaces and T:X → Y an injective bounded linear operator. T is called a semi-embedding if T maps the closed unit ball of X to a closed subset of Y. (This concept was introduced by Lotz, Peck, and Porta, Proc. Edinburgh Math. Soc.22 (1979), 233–240.) It is proved that if X semi-embeds in Y, and X is separable, then X has the Radon-Nikodym property provided Y does. It is shown that if L¹ semi-embeds in Y, then Y fails the Schur property and contains a subspace isomorphic to l¹. As a consequence of the proof, it is shown that if X is a subspace of L¹, either L¹ embeds in X or l¹ embeds in $\text{L}{}^1\text{X}$. The simpler result that L¹ does not semi-embed in c₀ is treated separately. This result is used to deduce the classic result of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients vanishing at infinity. Some generalizations of the notion of semi-embedding are given, and several complements and open questions are discussed.     

Applications of the theory of modular forms to number theory

The survey is devoted to arithmetic questions in the theory of modular forms and, in particular, to arithmetic applications of modular functions; mainly only elementary and analytic aspects of this topic, as a rule, for the case of a single variable are presented.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 15, pp. 5–91, 1977.