Global existence and collisions for symmetric configurations of nearly parallel vortex filaments

We consider the Schrödinger system with Newton-type interactions that was derived by R. Klein, A. Majda and K. Damodaran (1995) [17] to modelize the dynamics of N nearly parallel vortex filaments in a 3-dimensional homogeneous incompressible fluid. The known large time existence results are due to C. Kenig, G. Ponce and L. Vega (2003) [16] and concern the interaction of two filaments and particular configurations of three filaments. In this article we prove large time existence results for particular configurations of four nearly parallel filaments and for a class of configurations of N   nearly parallel filaments for any N?2$N?2$. We also show the existence of travelling wave type dynamics. Finally we describe configurations leading to collision.     

A parallel nearly implicit time-stepping scheme

Across-the-space parallelism still remains the most mature, convenient and natural way to parallelize large scale problems. One of the major problems here is that implicit time stepping is often difficult to parallelize due to the structure of the system. Approximate implicit schemes have been suggested to circumvent the problem (M.A. Botchev et al., Appl. Numer. Math. 31 (3) (1999) 239). These schemes have attractive stability properties and they are also very well parallelizable.The purpose of this article is to give an overall assessment of the parallelism of the method.     

Renormalized energy of interacting Ginzburg-Landau vortex filaments

We consider the Ginzbug–Landau energy in a cylinder in³, and a canonical approximation for critical points with anassembly of n2 periodic vortex lines near the axis of the cylinder.We find a formula for the energy which, up to a large additiveconstant and to leading order, is the action functional of then-body problem with a logarithmic potential in ², the axis variableplaying the role of time. A special family of rotating helicoidalcritical points of the functional is found to be non-degenerateup to the invariances of the problem, and therefore persistentunder small perturbations. Our analysis suggests the presenceof very complex stationary configurations for vortex filaments,potentially also involving intersecting filaments.     

A completeness theorem for the linear stability problem of nearly parallel flows

A completeness/expansion theorem, analogous to that of DiPrima and Habetler (D-H), is proved for the equation governing the linear stability of nearly parallel flows, to which the D-H theorem does not apply. It is also proved that only a finite number of eigenvalues with negative real parts can occur. Both results are based on a theorem of Gohberg and Kreǐn.     

Eddington's statistical theory

The theory of earlier papers is continued to give a new discussion of § 5 ofFundamental Theory, which has been shown to be unsatisfactory.     

On the existence of leapfrogging pair of circular vortex filaments

We propose and analyze a system of nonlinear partial differential equations describing the motion of a pair of vortex filaments. As a preliminary analysis, we first consider the case when the filaments are arranged as straight and parallel lines, and explicitly solve the system to show that the motion of the lines resemble that of point vortices moving in a plane. Then, we consider the motion of a pair of coaxial circular vortex filaments. We show that in this case, the system can be reduced to a two‐dimensional Hamiltonian system. Based on this formulation, we give a condition for the initial configuration and parameters of the filaments for leapfrogging to occur, and prove that the condition is in fact necessary and sufficient.     

Asymptotic distribution theory for general statistical functionals

Summary In this paper an asymptotic theory is developed for a general statistical functional which includesL-estimators,M-estimators, andR-estimators as special cases. It is shown that a proof of the asymptotic normality ofL-estimators,M-estimators, andR-estimators can be based on one important fact, namely, that the inverse c.d.f. is compactly differentiable.     

Complexity,parallel computation and statistical physics

The intuition that a long history is required for the emergence of complexity in natural systems is formalized using the notion of depth. The depth of a system is defined in terms of the number of parallel computational steps needed to simulate it. Depth provides an objective, irreducible measure of history that is applicable to systems of the kind studied in statistical physics. It is argued that physical complexity cannot occur in the absence of substantial depth and that depth is a useful proxy for physical complexity. The ideas are illustrated for a variety of systems in statistical physics. © 2006 Wiley Periodicals, Inc. Complexity 11: 46–64, 2006     

An algebraic theory for statistical information based on the theory of hints

Statistical problems were at the origin of the mathematical theory of evidence, or Dempster–Shafer theory. It was also one of the major concerns of Philippe Smets, starting with his PhD dissertation. This subject is reconsidered here, starting with functional models, describing how data is generated in statistical experiments. Inference is based on these models, using probabilistic assumption-based reasoning. It results in posterior belief functions on the unknown parameters. Formally, the information used in the process of inference can be represented by hints. Basic operations on hints are combination, corresponding to Dempster’s rule, and focussing. This leads to an algebra of hints. Applied to functional models, this introduces an algebraic flavor into statistical inference. It emphasizes the view that in statistical inference different pieces of information have to be combined and then focussed onto the question of interest. This theory covers Bayesian and Fisher type inference as two extreme cases of a more general theory of inference.     

Double vortex condensates in the Chern-Simons-Higgs theory

For a selfdual model introduced by Hong-Kim-Pac [18] and Jackiw-Weinberg [19] we study the existence of double vortex-condensates“bifurcating” from the symmetric vacuum state as the Chern-Simons coupling parameter k tends to zero. Surprisingly, we show a connection between the asymptotic behavior of the given double vortex as with the existence of extremal functions for a Sobolev inequality of the Moser-Trudinger's type on the flat 2-torus ([22], [1] and [15]). In fact, our construction yields to a “best” minimizing sequence for the (non-coercive) associated extremal problem, in the sense that, the infimum is attained if and only if the given minimizing sequence admits a convergent subsequence. Received: March 3, 1998 / Accepted October 23, 1998     

Measure theory of statistical convergence

The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the studies of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper, in terms of subdifferential, first shows a representation theorem for all finitely additive probability measures defined on the σ-algebra of all subsets of N, and proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measure μ with μ(k) = 0 for all singletons {k}). This paper also shows that classical statistical measures have many nice properties, such as: The set of all such measures endowed with the topology of point-wise convergence on forms a compact convex Hausdorff space; every classical statistical measure is of continuity type (hence, atomless), and every specific class of statistical measures fits a complementation minimax rule for every subset in N. Finally, this paper shows that every kind of statistical convergence can be unified in convergence of statistical measures. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10771175, 10471114)     

General theory of statistical decisions

Author's abstract of the dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Physical-Mathematical Sciences. The dissertation was defended on January 21, 1969 at the Scientific Council of the Institute of Applied Mathematics of the Academy of Sciences of the SSSR. The official opponents of the thesis were: Academician Yu. V. Linnik, Corresponding Member of the Academy of Sciences SSSR A. A. Lyapunov, Academician of the Academy of Science USSR B. V. Gnedenko, and Doctor of Physical-Mathematical Sciences Professor F. I. Karpelovich.Translated from Matematicheskie Zametki, Vol. 5, No. 5, pp. 635–648, May, 1969.     

The boundary layer on a wall nearly parallel to a magnetic field

This paper calculates the structure of the boundary layer which occurs at the wall of a nonconducting duct in fully-developed laminar flow of a conducting fluid when the wall is inclined at a small angle to the uniform transverse magnetic field and when the Hartmann numberM is high. When is large compared withM ^–1/2 the layer splits into a Hartmann layer on the inclined wall and a weak shear layer emanating from the duct corner along field lines. These two layers cause equal reductions in the total flow rate.

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Zusammenfassung Diese Arbeit behandelt die Struktur der Grenzschicht, die an der Wand eines nicht leitfähigen Kanals entsteht die eine vollentwickelte Laminarströmung einer leitfähigen Flüssigkeit führt, wobei die Wand einen kleinen Winkel mit dem gleichförmigen transversalen Magnetfeld bildet und die Hartmannsche ZahlM groß ist. Für große Werte von im Vergleich zuM ^–1/2 teilt sich die Grenzschicht in einen Hartmannsche Schicht an der geneigten Wand und eine schwache Scherschicht, die von der Ecke des Kanals entlang der Feldlinien ausgeht. Die beiden Schichten verursachen die gleichen Verminderungen in der Durchflußmenge.

     

On the filaments in a typical laser with plane parallel mirrors

Zusammenfassung Ein Ausdruck für die Intensitätsverteilung der in einem Fabry-Pérot-Interferometer mit einer grossen Fresnel-Zahl resonierenden Wellen wird angegeben. Auf dieses theoretische Ergebnis aufbauend, wird ein Modell für den fadenförmigen Laser-Mechanismus entwickelt, um die experimentellen Beobachtungen zu interpretieren.     

Measurements and infinite-dimensional statistical inverse theory

The most important ingredient of the statistical inverse theory is the indirect and noisy measurement of the unknown. Without the measurement, the formula for the posterior distribution becomes useless. However, inserting the measurement into the posterior distribution is not always simple. In the general setting, the posterior distribution is defined as a regular conditional probability. Hence it is known up to almost all measurements, which is inconvenient when we are given a single measurement. This shortage is covered in the finite-dimensional statistical inverse theory by fixing versions of probability density functions. A usual choice is to consider continuous probability density functions. Unfortunately, infinite-dimensional probability measures lack density functions which prohibits us from using the same method in the general setting. In this work, other possibilities for fixing the posterior distributions are discussed in the Gaussian framework. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New approaches to statistical learning theory

We present new tools from probability theory that can be applied to the analysis of learning algorithms. These tools allow to derive new bounds on the generalization performance of learning algorithms and to propose alternative measures of the complexity of the learning task, which in turn can be used to derive new learning algorithms.