A Kähler structure on the moduli space of isometric maps of a circle into Euclidean space

We study the spaces and and _Lip of smooth (resp. non-degenerate Lipschitz) isometric maps of a circle into Euclidean space modulo orientation preserving Euclidean motions. We prove that and _Lip are infinite dimensional Kähler manifolds. In particular, they are complex Fréchet (resp. Banach) manifolds. This is proved by an infinite dimensional version of the Kirwan, Kempf-Ness Theorem [Kir84], [KN78], [Nes84] relating symplectic quotients to holomorphic quotients, applied to the action ofPSL ₂() on the free loop space ofS ².Oblatum 15-X-1994 & 5-VII-1995This research was supported in part by NSF grant DMS-92-05154.This research was partially supported by AFOSR grant F49620-92-J-0093.     

Existence of Cesàro limit of bounded solution of evolution equation in banach space

An existence criterion for the Cesàro limit of a bounded solution y(t) of the problem dy(t)/dt = Ay(t), y(0)=y₀, t [O, ), where A is a closed linear operator with dense domain of definition D(A) in a reflexive Banach space E, is obtained under the condition that there exists a sufficiently small interval (O, ) belonging to the set of the regular points (A) of the operator A.Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1279–1280, September, 1992.     

Partitions of the phase space of a measure preserving ℤ d -action into towers

Alpern proved that the phase space of an aperiodic measure preserving automorphismT can be decomposed into Rokhlin-Halmos towers of given heightsh _i and weightsm _i whenever the numbersh _i are relatively prime. In this paper an extension of the Alpern theorem to the case of free ℤ ^d -actions is given. Namely, we investigate the decomposability of the action phase space into towers of rectangular, form and present conditions on the configuration (the set of tower forms) sufficient for the existence of such a decomposition. The proof of the main result uses the technique of tilings. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 712–725, May, 1999.     

Generalized lipschitz conditions and riesz derivatives on the space of bessel potentials Lp α

In the foregoing Note (this Journal Vol.I.p. 75-99) the space of n-dimensional Bessel potentials L^p _x was deseribed in terms of generalized Lipschitz conditions of f or its Riesz transform for 0<∝≦2 The still open case ∝>1 is treated in the first half of this paper, firstly by introducing appropriate iterates of the cited conditions, secondly by using derivatives of f and its Riesz transform, in particular the Laplacian △ and the gradient of the Riesz transformation(▽,R and by applying the former results In Section 6 a definition of a Riesz derivative of order ∝ is given and based upon the concept: Integrate f(m-α)-times in the sense of Riesz and then differentiate [d]m-times (by considering the limit of suitable difference quotients of f). Necessary and sufficient conditions for the existence of these Riesz derivatives are obtained All results also hold in the non-reflexive spaces[d]     

Existence and regularity of solutions for evolution equations with Riemann–Liouville fractional derivatives

In this paper, we derive the existence and uniqueness of mild solutions for inhomogeneous fractional evolution equations in Banach spaces by means of the method of fractional resolvent. Furthermore, we give the necessary and sufficient conditions for the existence of strong solutions. An example of the fractional diffusion equation is also presented to illustrate our theory.     

Global solutions of abstract quasi-linear evolution equations of “hyperbolic” type

The problem of existence and uniqueness of global classical solutions of abstract quasi-linear evolution equations is considered in a general Banach space. The results obtained here are applied to the initial value problems for hyperbolic partial differential equations.     

A criterion for embedding of anisotropic Sobolev–Morrey spaces into the space of uniformly continuous functions

We prove the embedding theorems of the Sobolev–Morrey spaces into the space of uniformly continuous functions so extending the classical Sobolev Theorems.     

Generalized notion of “liquid” and phase transition to turbulence

We try to explain the physical meaning of the notion of liquid “without interaction” and its characteristic property of having a small number of degrees of freedom. We show the relationship between opalescence and turbulence.     

The isometric extension of “into” mappings on unit spheres of AL-spaces

In this paper, we show that if V0 is an isometric mapping from the unit sphere of an AL-space onto the unit sphere of a Banach space E, then V0 can be extended to a linear isometry defined on the whole space.     

Method of isomonodromic deformations for “degenerate” third Painlevé equation

To study the solutions of the equation (u)=e^u–eu which is a version of the degenerate third Painlevé equation we consider a linear ordinary differential equation in 3×3 matrices. By means of the monodromy data of this linear equation we parametrize the asymptotics of all solutions as 0, as well as the asymptotics of regular solutions of the nonlinear equation studied as ±.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 161, pp. 45–53, 1987.The author thanks A. R. Its for posing the problem.     

The evolution of metanorms: quis custodiet ipsos custodes?

How are norms maintained? Axelrod (in Am. Political Sci. Rev. 80(4): 1095–1111, 1986) used an evolutionary computational model to proffer a solution: the metanorm (norm to enforce norm enforcement). Although often discussed, this model has neither been sufficiently replicated nor explored. In this paper we replicate and extend that model. Results were generally supportive of the original. Speculations in the original regarding the requirement to link sanctions underlying the metanorm structure were not supported, as differentiating punishment likelihoods against defectors from punishment likelihoods against shirkers (non-enforcers of the norm against defection) lead to more efficient and effective sanctioning structures that allowed norm emergence. Replications of the Groups game (two groups differing in numbers and power) generally supported the original reports, but true norms against defection emerged only if sanctioning structures were differentiated, resulting in the Strong group developing a dominant norm against others defecting (Metavengeance). That is, when groups are involved with differential power, Metanorms fail unless a more sophisticated sanctioning structure (Metavengeance) is supported.     

On the paper “Asymptotic behavior of meromorphic functions of completely regular growth and of their logarithmic derivatives”

Translated fromSibirski Matematicheski Zhurnal, Vol. 33, No. 4, pp. 219–219, July–August, 1992.     

Natural selection: A concept in need of some evolution?

In some respects natural selection is a quite simple theory, arrived at through the logical integration of three propositions (the presence of variation within natural populations, an absolutely limited resources base, and procreation capacities exceeding mere replacement numbers) whose individual truths can hardly be denied. Its relation to the larger subject of evolution, however, remains problematic. It is suggested here thata scaling‐down of the meaning of natural selection to “the elimination of the unfit,” as originally intended by Alfred Russel Wallace (1823–1913), might ultimately prove a more effective means of relating it to larger‐scale, longer‐term, evolutionary processes. © 2011 Wiley Periodicals, Inc. Complexity, 2011