Measures and forking

Shelah's theory of forking (or stability theory) is generalized in a way which deals with measures instead of complete types. This allows us to extend the method of forking from the class of stable theories to the larger class of theories which do not have the independence property. When restricted to the special case of stable theories, this paper reduces to a reformulation of the classical approach. However, it goes beyond the classical approach in the case of unstable theories. Methods from ordinary forking theory and the Loeb measure construction from nonstandard analysis are used.     

Very simple theories without forking

We prove Vaught's conjecture for minimal trivial simple theories satisfying the generalized independence theorem. Research supported by KBN grant 2 P03A 006 09     

The stable forking conjecture and generic structures

 We prove that for any simple theory which is constructed via Fr?issé-Hrushovski method, if the forking independence is the same as the d-independence then the stable forking property holds. Received: 22 January 2001 / Published online: 19 December 2002 This article is part of the author's D-Phil thesis, written at the University of Oxford and supported by the Ministry of Higher Education of Iran. The author would like to thank the Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran, for its financial support whilst working on this article. Mathematics Subject Classification (2000): 03C45 Key words or phrases: Generic structures – Fr?issé-Hrushovski method – Predimension – Simple theories – Stable theories – Stable forking conjecture     

Stable theories without dense forking chains

Summary We define a generalized notion of rank for stable theories without dense forking chains, and use it to derive that every type is domination-equivalent to a finite product of regular types. We apply this to show that in a small theory admitting finite coding, no realisation of a nonforking extension of some strong type can be algebraic over some realisation of a forking extension.Research supported by the Deutsche ForschungsgemeinschaftThanks as always to Alistair. Research supported by NSERC and FCARPartially supported by NSF grant DMS90 06628     

On the uniqueness property of forking in abstract elementary classes

In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for μ‐forking, a certain independence notion arising from splitting. This had been a longstanding technical difficulty when constructing forking‐like notions in this setup. As an application, we show that the two versions of forking symmetry appearing in the literature (the one defined by Shelah for good frames and the one defined by VanDieren for splitting) are equivalent.     

Fundamentals of Computational Conformal Geometry

Computational conformal geometry is an inter-disciplinary field between mathematics and computer science. This work introduces the fundamentals of computational conformal geometry, including theoretic foundation, computational algorithms, and engineering applications. Two computational methodologies are emphasized, one is the holomorphic differentials based on Riemann surface theory and the other is surface Ricci flow from geometric analysis.     

Fundamentalsätze der metrischen Geometrie

For arbitrary quadratic forms, including the cases of characteristic 2 and of infinite dimensions, several affine-metric and projective-metric structures are considered, and the corresponding isomorphisms are determined. As an application, a general fundamental theorem of the miquelian circle geometry is proved.

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Martin Barner zum 65. Geburtstag gewidmet     

Fundamentals of a mathematical theory of genetic codes

This paper continues the authors’ long-time studies concerning biomathematics (see [1]). One direction of research was related to the development of a new area of modern biomathematics, namely, a mathematical theory of genetic codes. The foundations of this theory are described below. To the author’s knowledge, the results presented have not been found in other researchers’ publications.     

Fundamentals of the theory of non-fabric glass-reinforced plastics

Starting from the known mechanical properties of the components (glass reinforcement and resin) the stress-strain relations for an anisotropic viscoelastic material (glass-reinforced plastic) are determined. Using Volterra's principle [2] of replacing the elastic constants in the solution of the problem of the theory of elasticity [5] by integral operators, relations for the given material are obtained and found to be in fairly good agreement with the experimental data. In a separate study the high-temperature behavior of GRP is investigated by calculating the integral operators, which are functions of the viscous properties of the resin and the temperature.Mekhanika polimerov, Vol. 1, No. 1, pp. 151–158, 1965     

Fundamentals of the statistical mechanics of composite systems

Conclusions The proposed variant of the statistical theory of composite media makes it possible to derive relations between the effective parameters of the medium and the dispersion characteristics of the structure, and also to account for the effect of specimen shape and variable structural heterogeneity on these parameters. In the limiting case of an infinitely large specimen, all relationships comply with the results of the traditional theory of composite systems.Presented at the Sixth All-Union Conference on the Mechanics of Polymer and Composite Materials (Riga, November, 1986).Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 21–30, January–February, 1988.     

Fundamentals of Quaternionic Kinematics in Euclidean 4-Space

Based on a kinematic mapping for the group SE(4) of displacements of Euclidean 4-space, we show that the mapping of basic elements (points, oriented lines, oriented planes, oriented hyperplanes, instantaneous screws) can be written compactly in terms of 2 × 2 quaternionic matrices. Moreover we discuss the kinematics on the velocity level by investigating instantaneous screws and their geometric parameters.     

Gradient-constrained minimum networks. I. Fundamentals

In three-dimensional space an embedded network is called gradient-constrained if the absolute gradient of any differentiable point on the edges in the network is no more than a given value m. A gradient-constrained minimum Steiner tree T is a minimum gradient-constrained network interconnecting a given set of points. In this paper we investigate some of the fundamental properties of these minimum networks. We first introduce a new metric, the gradient metric, which incorporates a new definition of distance for edges with gradient greater than m. We then discuss the variational argument in the gradient metric, and use it to prove that the degree of Steiner points in T is either three or four. If the edges in T are labelled to indicate whether the gradients between their endpoints are greater than, less than, or equal to m, then we show that, up to symmetry, there are only five possible labellings for degree 3 Steiner points in T. Moreover, we prove that all four edges incident with a degree 4 Steiner point in T must have gradient m if m is less than 0.38. Finally, we use the variational argument to locate the Steiner points in T in terms of the positions of the neighbouring vertices.     

Fundamentals of the theory of varieties of nilpotent MR-groups

We expose fundamentals of the theory of varieties of nilpotent MR-groups and compare various definitions of nilpotency in this category.     

Fundamentals of the statistical theory of equilibrium spatial clusters

The paper is devoted to the problems of the statistical thermodynamics of spatially bounded clusters of many particles. The dependence between the temperature of a cluster and its size at finite energy is established.Institute of Geology and Prospecting, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 7, pp. 75–81, July, 1994.