A variational approach to the regularity of minimal surfaces of annulus type in Riemannian manifolds

Given two Jordan curves in a Riemannian manifold, a minimal surface of annulus type bounded by these curves is described as the harmonic extension of a critical point of some functional (the Dirichlet integral) in a certain space of boundary parametrizations. The H2,2-regularity of the minimal surface of annulus type will be proved by applying the critical points theory and Morrey's growth condition.     

Moser’s Quadratic,Symplectic Map

In 1994, Jürgen Moser generalized Hénon’s area-preserving quadratic map to obtain a normal form for the family of four-dimensional, quadratic, symplectic maps. This map has at most four isolated fixed points. We show that the bounded dynamics of Moser’s six parameter family is organized by a codimension-three bifurcation, which we call a quadfurcation, that can create all four fixed points from none.The bounded dynamics is typically associated with Cantor families of invariant tori around fixed points that are doubly elliptic. For Moser’s map there can be two such fixed points: this structure is not what one would expect from dynamics near the cross product of a pair of uncoupled Hénon maps, where there is at most one doubly elliptic point. We visualize the dynamics by escape time plots on 2d planes through the phase space and by 3d slices through the tori.     

Non-autonomous Julia sets with escaping critical points

We consider non-autonomous iteration which is a generalization of standard polynomial iteration where we deal with Julia sets arising from composition sequences for arbitrarily chosen polynomials with uniformly bounded degrees and coefficients. In this paper, we look at examples where all the critical points escape to infinity. In the classical case, any example of this type must be hyperbolic and there can be only one Fatou component, namely the basin at infinity. This result remains true in the non-autonomous case if we also require that the dynamics on the Julia set be hyperbolic or semi-hyperbolic. However, in general it fails and we exhibit three counterexamples of sequences of quadratic polynomials all of whose critical points escape but which have bounded Fatou components.     

CLASSIFICATION OF CUBIC KOLMOGOROV TYPE SYSTEM AND ITS APPLICATION

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Fractional dynamics of populations

Nature often presents complex dynamics, which cannot be explained by means of ordinary models. In this paper, we establish an approach to certain fractional dynamic systems using only deterministic arguments. The behavior of the trajectories of fractional non-linear autonomous systems around the corresponding critical points in the phase space is studied. In this work we arrive to several interesting conclusions; for example, we conclude that the order of fractional derivation is an excellent controller of the velocity how the mentioned trajectories approach to (or away from) the critical point. Such property could contribute to faithfully represent the anomalous reality of the competition among some species (in cellular populations as Cancer or HIV). We use classical models, which describe dynamics of certain populations in competition, to give a justification of the possible interest of the corresponding fractional models in biological areas of research.     

Simulation of phase-transformations based on numerical minimization of intersecting Gibbs energy potentials

We present a novel approach for the simulation of solid to solid phase-transformations in polycrystalline materials. To facilitate the utilization of a non-affine micro-sphere formulation with volumetric-deviatoric split, we introduce Helmholtz free energy functions depending on volumetric and deviatoric strain measures for the underlying scalar-valued phase-transformation model. As an extension of affine micro-sphere models [5], the non-affine micro-sphere formulation with volumetric-deviatoric split allows to capture different Young's moduli and Poisson's ratios on the macro-scale [1]. As a consequence, the temperature-dependent free energy assigned to each individual phase takes the form of an elliptic paraboloid in volumetric-deviatoric strain space, where the energy landscape of the overall material is obtained from the contributions of the individual constituents. For the evolution of volume fractions, we use an approach based on statistical physics–taking into account actual Gibbs energy barriers and transformation probabilities [2]. The computation of individual energy barriers between the phases considered is enabled by numerical minimization of parametric intersection curves of elliptic Gibbs energy paraboloids. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spatial functional normal mixed effect approach for curve classification

This paper proposes a spatial functional formulation of the normal mixed effect model for the statistical classification of spatially dependent Gaussian curves, in both parametric and state space model frameworks. Fixed effect parameters are represented in terms of a functional multiple regression model whose regression operators can change in space. Local spatial homogeneity of these operators is measured in terms of their Hilbert–Schmidt distances, leading to the classification of fixed effect curves in different groups. Assuming that the Gaussian random effect curves obey a spatial autoregressive dynamics of order one [SARH(1) dynamics], a second functional classification criterion is proposed in order to detect local spatially homogeneous patterns in the mean quadratic functional variation of Gaussian random effect curve increments. Finally, the two criteria are combined to detect local spatially homogeneous patterns in the regression operators and in the functional mean quadratic variation, under a state space approach. A real data example in the financial context is analyzed as an illustration.     

Where to place a hole to achieve a maximal escape rate

A natural question of how the survival probability depends upon a position of a hole was seemingly never addressed in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period) which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the faster is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole.     

Multiple separatrix crossing in multi-degree-of-freedom Hamiltonian flows

Summary We study separatrix crossing in near-integrablek-degree-of-freedom Hamiltonian flows, 2 <k < , whose unperturbed phase portraits contain separatrices inn degrees of freedom, 1 <n <k. Each of the unperturbed separatrices can be recast as a codimension-one separatrix in the 2k-dimensional phase space, and the collection of these separatrices takes on a variety of geometrical possibilities in the reduced representation of a Poincaré section on the energy surface. In general 0 l n of the separatrices will be available to the Poincaré section, and each separatrix may be completely isolated from all other separatrices or intersect transversely with one or more of the other available separatrices. For completely isolated separatrices, transitions across broken separatrices are described for each separatrix by the single-separatrix crossing theory of Wiggins, as modified by Beigie. For intersecting separatrices, a possible violation of a normal hyperbolicity condition complicates the analysis by preventing the use of a persistence and smoothness theory for compact normally hyperbolic invariant manifolds and their local stable and unstable manifolds. For certain classes of multi-degree-of-freedom flows, however, a local persistence and smoothness result is straightforward, and we study the global implications of such a local result. In particular, we find codimension-one partial barriers and turnstile boundaries associated with each partially destroyed separatrix. From the collection of partial barriers and turnstiles follows a rich phase space partitioning and transport formalism to describe the dynamics amongst the various degrees of freedom. A generalization of Wiggins' higher-dimensional Melnikov theory to codimension-one surfaces in the multi-separatrix case allows one to uncover invariant manifold geometry. In the context of this perturbative analysis and detailed numerical computations, we study invariant manifold geometry, phase space partitioning, and phase space transport, with particular attention payed to the role of a vanishing frequency in the limit approaching the intersection of the partially destroyed separatrices. The class of flows under consideration includes flows of basic physical relevance, such as those describing scattering phenomena. The analysis is illustrated in the context of a detailed study of a 3-degree-of-freedom scattering problem.     

Local expansion concepts for detecting transport barriers in dynamical systems

In the last two decades, the mathematical analysis of material transport has received considerable interest in many scientific fields such as ocean dynamics and astrodynamics. In this contribution we focus on the numerical detection and approximation of transport barriers in dynamical systems. Starting from a set-oriented approximation of the dynamics we combine discrete concepts from graph theory with established geometric ideas from dynamical systems theory. We derive the global transport barriers by computing the local expansion properties of the system. For the demonstration of our results we consider two different systems. First we explore a simple flow map inspired by the dynamics of the global ocean. The second example is the planar circular restricted three body problem with Sun and Jupiter as primaries, which allows us to analyze particle transport in the solar system.     

Exit time moments, boundary value problems, and the geometry of domains in Euclidean space

Let X _t be a diffusion in Euclidean space. We initiate a study of the geometry of smoothly bounded domains in Euclidean space using the moments of the exit time for particles driven by X _t , as functionals on the space of smoothly bounded domains. We provide a characterization of critical points for each functional in terms of an overdetermined boundary value problem. For Brownian motion we prove that, for each functional, the boundary value problem which characterizes critical points admits solutions if and only if the critical point is a ball, and that all critical points are maxima. Received: 23 January 1997 / Revised version: 21 January 1998     

Polynomial Hurwitz Numbers and Intersections on \overline M _{0,k}

We express Hurwitz numbers of polynomials of arbitrary topological type in terms of intersection numbers on the moduli space of curves of genus zero with marked points.     

The Finite Dean Problem: Nonlinear Theory

The Dean problem models flows in curved pipes of square cross-sectionby considering pressure-driven axisymmetric flow between concentriccylinders. The inclusion of ends on these cylinders introducesa range of flows not seen in an infinite formulation. Weaklynonlinear analyses are made possible by considering end conditionssuch as those of Blennerhassett & Hall (1979). This paperextends to Dean flow this nonlinear work of Hall (1980). A newderivation of the amplitude equations will be given, which alsouses a perturbation from the intersection points of the oddand even curves of critical Taylor number against cylinder heightgiven by linear theory. Equilibrium solutions of the amplitudeequations are then found and their stability examined. Unlikethe Taylor-vortex results of Hall, the properties of these solutionsdepend on the particular intersection point under consideration.A comparison is given with previous work on the finite Deanproblem.     

Converting homotopies to isotopies and dividing homotopies in half in an effective way

We prove two theorems about homotopies of curves on two-dimensional Riemannian manifolds. We show that, for any \({\epsilon > 0}\) , if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by \({L + \epsilon}\) . If the manifold is orientable, then for any \({\epsilon > 0}\) we show that, if we can contract a curve \({\gamma}\) traversed twice through curves of length bounded by L, then we can also contract \({\gamma}\) through curves bounded in length by \({L + \epsilon}\) . Our method involves cutting curves at their self-intersection points and reconnecting them in a prescribed way. We consider the space of all curves obtained in this way from the original homotopy, and use a novel approach to show that this space contains a path which yields the desired homotopy.     

Kontsevich's formula and the WDVV equations in tropical geometry

Using Gromov-Witten theory the numbers of complex plane rational curves of degree d through 3d−1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we establish the same results entirely in the language of tropical geometry. In particular this shows how the concepts of moduli spaces of stable curves and maps, (evaluation and forgetful) morphisms, intersection multiplicities and their invariance under deformations can be carried over to the tropical world.     

New algorithms for product positioning

Two new algorithms are proposed for the problem of positioning a new product in attribute space in order to attract the maximum number of consumers. Each consumer is assumed to choose the existing or new product closest to his ideal point according to the Euclidean norm. The first algorithm is based on finding a finite number of intersection points of indifference surfaces. The second algorithm proceeds by considering sets of balls bounded by indifference surfaces and finding points belonging to the largest weighted number of them. Problems with up to 500 consumers groups, 40 existing products and 20 attributes are solved exactly.     

Computing genus 2 curves from invariants on the Hilbert moduli space

We give a new method for generating genus 2 curves over a finite field with a given number of points on the Jacobian of the curve. We define two new invariants for genus 2 curves as values of modular functions on the Hilbert moduli space and show how to compute them. We relate them to the usual three Igusa invariants on the Siegel moduli space and give an algorithm to construct curves using these new invariants. Our approach simplifies the complex analytic method for computing genus 2 curves for cryptography and reduces the amount of computation required.