Robust tori-like Lagrangian coherent structures

In general the term "Lagrangian coherent structure" (LCS) is used to make reference about structures whose properties are similar to a time-dependent analog of stable and unstable manifolds from a hyperbolic fixed point in Hamiltonian systems. Recently, the term LCS was used to describe a different type of structure, whose properties are similar to those of invariant tori in certain classes of two-dimensional incompressible flows. A new kind of LCS was obtained. It consists of barriers, called robust tori that block the trajectories in certain regions of the phase space. We used the Double-Gyre Flow system as the model. In this system, the robust tori play the role of a skeleton for the dynamics and block, horizontally, vortices that come from different parts of the phase space. (C) 2012 Elsevier B.V. All rights reserved.

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

On bifurcation of solutions of the Yamabe problem in product manifolds

We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non-homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds. (C) 2011 Elsevier Masson SAS. All rights reserved.

A NOTE ON OPEN 3-MANIFOLDS SUPPORTING FOLIATIONS BY PLANES

We show that if N, an open connected n-manifold with finitely generated fundamental group, is C-2 foliated by closed planes, then pi(1)(N) is a free group. This implies that if pi(1)(N) has an abelian subgroup of rank greater than one, then F has at least a nonclosed leaf. Next, we show that if N is three dimensional with fundamental group abelian of rank greater than one, then N is homeomorphic to T-2 x R. Furthermore, in this case we give a complete description of the foliation.

Invariant Solutions of Richards' Equation for Water Movement in Dissimilar Soils

Scaling methods allow a single solution to Richards' equation (RE) to suffice for numerous specific cases of water flow in unsaturated soils. During the past half-century, many such methods were developed for similar soils. In this paper, a new method is proposed for scaling RE for a wide range of dissimilar soils. Exponential-power (EP) functions are used to reduce the dependence of the scaled RE on the soil hydraulic properties. To evaluate the proposed method, the scaled RE was solved numerically considering two test cases: infiltration into relatively dry soils having initially uniform water content distributions, and gravity-dominant drainage occurring from initially wet soil profiles. Although the results for four texturally different soils ranging from sand to heavy clay (adopted from the UNSODA database) showed that the scaled solution were invariant for a wide range of flow conditions, slight deviations were observed when the soil profile was initially wet in the infiltration case or deeply wet in the drainage case. The invariance of the scaled RE makes it possible to generalize a single solution of RE to many dissimilar soils and conditions. Such a procedure reduces the numerical calculations and provides additional opportunities for solving the highly nonlinear RE for unsaturated water flow in soils.

An estimate on the fractal dimension of attractors of gradient-like dynamical systems

This paper is dedicated to estimate the fractal dimension of exponential global attractors of some generalized gradient-like semigroups in a general Banach space in terms of the maximum of the dimension of the local unstable manifolds of the isolated invariant sets, Lipschitz properties of the semigroup and the rate of exponential attraction. We also generalize this result for some special evolution processes, introducing a concept of Morse decomposition with pullback attractivity. Under suitable assumptions, if (A, A*) is an attractor-repeller pair for the attractor A of a semigroup {T(t) : t >= 0}, then the fractal dimension of A can be estimated in terms of the fractal dimension of the local unstable manifold of A*, the fractal dimension of A, the Lipschitz properties of the semigroup and the rate of the exponential attraction. The ingredients of the proof are the notion of generalized gradient-like semigroups and their regular attractors, Morse decomposition and a fine analysis of the structure of the attractors. As we said previously, we generalize this result for some evolution processes using the same basic ideas. (C) 2012 Elsevier Ltd. All rights reserved.

Gauge and integrable theories in loop spaces

We propose an integral formulation of the equations of motion of a large class of field theories which leads in a quite natural and direct way to the construction of conservation laws. The approach is based on generalized non-abelian Stokes theorems for p-form connections, and its appropriate mathematical language is that of loop spaces. The equations of motion are written as the equality of a hyper-volume ordered integral to a hyper-surface ordered integral on the border of that hyper-volume. The approach applies to integrable field theories in (1 + 1) dimensions, Chern-Simons theories in (2 + 1) dimensions, and non-abelian gauge theories in (2 + 1) and (3 + 1) dimensions. The results presented in this paper are relevant for the understanding of global properties of those theories. As a special byproduct we solve a long standing problem in (3 + 1)-dimensional Yang-Mills theory, namely the construction of conserved charges, valid for any solution, which are invariant under arbitrary gauge transformations. (C) 2012 Elsevier B.V. All rights reserved.

Axioms for the coincidence index of maps between manifolds of the same dimension

We study the coincidence theory of maps between two manifolds of the same dimension from an axiomatic viewpoint. First we look at coincidences of maps between manifolds where one of the maps is orientation true, and give a set of axioms such that characterizes the local index (which is an integer valued function). Then we consider coincidence theory for arbitrary pairs of maps between two manifolds. Similarly we provide a set of axioms which characterize the local index, which in this case is a function with values in Z circle plus Z(2). We also show in each setting that the group of values for the index (either Z or Z circle plus Z(2)) is determined by the axioms. Finally, for the general case of coincidence theory for arbitrary pairs of maps between two manifolds we provide a set of axioms which characterize the local Reidemeister trace which is an element of an abelian group which depends on the pair of functions. These results extend known results for coincidences between orientable differentiable manifolds. (C) 2012 Elsevier B.V. All rights reserved.

Integral form of Yang-Mills equations and its gauge invariant conserved charges

Despite the fact that the integral form of the equations of classical electrodynamics is well known, the same is not true for non-Abelian gauge theories. The aim of the present paper is threefold. First, we present the integral form of the classical Yang-Mills equations in the presence of sources and then use it to solve the long-standing problem of constructing conserved charges, for any field configuration, which are invariant under general gauge transformations and not only under transformations that go to a constant at spatial infinity. The construction is based on concepts in loop spaces and on a generalization of the non-Abelian Stokes theorem for two-form connections. The third goal of the paper is to present the integral form of the self-dual Yang-Mills equations and calculate the conserved charges associated with them. The charges are explicitly evaluated for the cases of monopoles, dyons, instantons and merons, and we show that in many cases those charges must be quantized. Our results are important in the understanding of global properties of non-Abelian gauge theories.

Invariant Natural Killer T Cell Agonist Modulates Experimental Focal and Segmental Glomerulosclerosis

A growing body of evidence demonstrates a correlation between Th2 cytokines and the development of focal and segmental glomerulosclerosis ( FSGS). Therefore, we hypothesized that GSL-1, a monoglycosylceramide from Sphingomonas ssp. with pro-Th1 activity on invariant Natural Killer T ( iNKT) lymphocytes, could counterbalance the Th2 profile and modulate glomerulosclerosis. Using an adriamycin( ADM)-based model of FSGS, we found that BALB/c mice presented albuminuria and glomerular degeneration in association with a Th2-like pro-fibrogenic profile; these mice also expressed a combination of inflammatory cytokines, such as IL-4, IL-1 alpha, IL-1 beta, IL-17, TNF-alpha, and chemokines, such as RANTES and eotaxin. In addition, we observed a decrease in the mRNA levels of GD3 synthase, the enzyme responsible for GD3 metabolism, a glycolipid associated with podocyte physiology. GSL-1 treatment inhibited ADM-induced renal dysfunction and preserved kidney architecture, a phenomenon associated with the induction of a Th1-like response, increased levels of GD3 synthase transcripts and inhibition of pro-fibrotic transcripts and inflammatory cytokines. TGF-beta analysis revealed increased levels of circulating protein and tissue transcripts in both ADM- and GSL-1-treated mice, suggesting that TGF-beta could be associated with both FSGS pathology and iNKT-mediated immunosuppression; therefore, we analyzed the kidney expression of phosphorylated SMAD2/3 and SMAD7 proteins, molecules associated with the deleterious and protective effects of TGF-beta, respectively. We found high levels of phosphoSMAD2/3 in ADM mice in contrast to the GSL-1 treated group in which SMAD7 expression increased. These data suggest that GSL-1 treatment modulates the downstream signaling of TGF-beta through a renoprotective pathway. Finally, GSL-1 treatment at day 4, a period when proteinuria was already established, was still able to improve renal function, preserve renal structure and inhibit fibrogenic transcripts. In conclusion, our work demonstrates that the iNKT agonist GSL-1 modulates the pathogenesis of ADM-induced glomerulosclerosis and may provide an alternative approach to disease management.

CID: an efficient complexity-invariant distance for time series

The ubiquity of time series data across almost all human endeavors has produced a great interest in time series data mining in the last decade. While dozens of classification algorithms have been applied to time series, recent empirical evidence strongly suggests that simple nearest neighbor classification is exceptionally difficult to beat. The choice of distance measure used by the nearest neighbor algorithm is important, and depends on the invariances required by the domain. For example, motion capture data typically requires invariance to warping, and cardiology data requires invariance to the baseline (the mean value). Similarly, recent work suggests that for time series clustering, the choice of clustering algorithm is much less important than the choice of distance measure used.In this work we make a somewhat surprising claim. There is an invariance that the community seems to have missed, complexity invariance. Intuitively, the problem is that in many domains the different classes may have different complexities, and pairs of complex objects, even those which subjectively may seem very similar to the human eye, tend to be further apart under current distance measures than pairs of simple objects. This fact introduces errors in nearest neighbor classification, where some complex objects may be incorrectly assigned to a simpler class. Similarly, for clustering this effect can introduce errors by “suggesting” to the clustering algorithm that subjectively similar, but complex objects belong in a sparser and larger diameter cluster than is truly warranted.We introduce the first complexity-invariant distance measure for time series, and show that it generally produces significant improvements in classification and clustering accuracy. We further show that this improvement does not compromise efficiency, since we can lower bound the measure and use a modification of triangular inequality, thus making use of most existing indexing and data mining algorithms. We evaluate our ideas with the largest and most comprehensive set of time series mining experiments ever attempted in a single work, and show that complexity-invariant distance measures can produce improvements in classification and clustering in the vast majority of cases.