Abundance of elliptic dynamics on conservative three-flows

We consider a compact three-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C ¹-residual (dense G _δ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M. This is the flow-setting counterpart of Newhouse's Theorem 1.3 (S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 (1977), pp. 1061–1087). Our result follows from two theorems, the first one says that if Λ is a hyperbolic invariant set for some class C ¹ zero divergence vector field X on M, then either X is Anosov, or else Λ has empty interior. The second one says that, if X is not Anosov, then for any open set U ? M there exists Y arbitrarily close to X such that Y ^t has an elliptical closed orbit through U.     

On the volume of singular-hyperbolic sets

An attractor Λ for a 3-vector field X is singular-hyperbolic if all its singularities are hyperbolic and it is partially hyperbolic with volume expanding central direction. We prove that C ^1?+?α singular-hyperbolic attractors, for any α?> 0, always have zero volume, extending an analogous result for uniformly hyperbolic attractors. The same result holds for a class of higher dimensional singular attractors. Moreover, we prove that if Λ is a singular-hyperbolic attractor for X then either it has zero volume or X is an Anosov flow. We also present examples of C ¹ singular-hyperbolic attractors with positive volume. In addition, we show that C ¹ generically we have volume zero for C ¹ robust classes of singular-hyperbolic attractors.     

Inverse shadowing for structurally stable flows

We introduce the notion of inverse shadowing for continuous flows, and show that a structurally stable flow generated by a C ¹ vector field on a compact smooth manifold has the inverse shadowing property with respect to the classes of continuous methods.     

The real analytic Feigenbaum–Coullet–Tresser attractor in the disc

We consider a real analytic diffeomorphism ψ₀ on an n-dimensional disc 𝒟, n ≥ 2, exhibiting a Feigenbaum–Coullet–Tresser (FCT) attractor. We assume that in the C ^ω(𝒟) topology it is far from the standard FCT map φ₀ fixed by the double renormalization. We prove that ψ₀ persists along a codimension-one manifold ? ? C ^ω(𝒟), and that it is the bifurcating map along any one-parameter family in C ^ω(𝒟) transversal to ?, from diffeomorphisms exhibiting sinks to those which exhibit chaos, filling a gap in the usually accepted proof of this assertion. The main tool in the proofs is a theorem of functional analysis, which we state and prove in this article, characterizing the existence of codimension-one submanifolds in any abstract functional Banach space.     

Chaotic double cycling

We study the dynamics of a generic vector field in the neighbourhood of a heteroclinic cycle of non-trivial periodic solutions whose invariant manifolds meet transversely. The main result is the existence of chaotic double cycling: there are trajectories that follow the cycle making any prescribed number of turns near the periodic solutions, for any given bi-infinite sequence of turns. Using symbolic dynamics, arbitrarily close to the cycle, we find a robust and transitive set of initial conditions whose trajectories follow the cycle for all time and that is conjugate to a Markov shift over a finite alphabet. This conjugacy allows us to prove the existence of infinitely many heteroclinic and homoclinic subsidiary connections, which give rise to a heteroclinic network with infinitely many cycles and chaotic dynamics near them, exhibiting themselves switching and cycling. We construct an example of a vector field with Z ₃ symmetry in a five-dimensional sphere with a heteroclinic cycle having this property.     

Conchoidal transform of two plane curves

The conchoid of a plane curve C is constructed using a fixed circle B in the affine plane. We generalize the classical definition so that we obtain a conchoid from any pair of curves B and C in the projective plane. We present two definitions, one purely algebraic through resultants and a more geometric one using an incidence correspondence in P ² × P ². We prove, among other things, that the conchoid of a generic curve of fixed degree is irreducible, we determine its singularities and give a formula for its degree and genus. In the final section we return to the classical case: for any given curve C we give a criterion for its conchoid to be irreducible and we give a procedure to determine when a curve is the conchoid of another.     

Normal Lyapunov exponents and asymptotically stable attractors

Suppose f is a C ^1+α map and leaves a lower-dimensional compact attractor A. In this article, we show that if for every f-ergodic probability measure supported on A, the normal Lyapunov exponents are negative, then this attractor could be a high-dimensional attractor. Moreover, we prove that the supremum of the normal Lyapunov exponents on the set of all ergodic measures can be achieved.     

Ergodic measure-expansive diffeomorphisms

In this paper, we consider the set of diffeomorphisms which are measure-expansive for any ergodic measure, and study the set from the viewpoint of geometric theory of dynamical systems. It is proved that (1) there exists a non-empty C¹-open set of robustly non-hyperbolic and transitive diffeomorphisms such that each element of the set is measure-expansive for any ergodic measure, and that (2) C¹-generically, a diffeomorphism is measure-expansive for any generic ergodic measure.     

A Quasi-Boundary Semi-Analytical Method for Backward in Time Advection-Dispersion Equation

In this paper, we take the advantage of an analytical method to solve the advection-dispersion equation (ADE) for identifying the contamination problems. First, the Fourier series expansion technique is employed to calculate the concentration field C(x, t) at any time t< T. Then, we consider a direct regularization by adding an extra term αC(x,0) on the final condition to carry off a second kind Fredholm integral equation. The termwise separable property of the kernel function permits us to transform itinto a two-point boundary value problem. The uniform convergence and error estimate of the regularized solution C^α(x,t) are provided and a strategy to select the regularized parameter is suggested. The solver used in this work can recover the spatial distribution of the groundwater contaminant concentration. Several numerical examples are examined to show that the new approach can retrieve all past data very well and is good enough to cope with heterogeneous parameters’ problems, even though the final data are noised seriously.     

1-Perfect Uniform and Distance Invariant Partitions

Let F ⁿ be the n-dimensional vector space over ℤ₂. A (binary) 1-perfect partition of F ⁿ is a partition of F ⁿ into (binary) perfect single error-correcting codes or 1-perfect codes. We define two metric properties for 1-perfect partitions: uniformity and distance invariance. Then we prove the equivalence between these properties and algebraic properties of the code (the class containing the zero vector). In this way, we characterize 1-perfect partitions obtained using 1-perfect translation invariant and not translation invariant propelinear codes. The search for examples of 1-perfect uniform but not distance invariant partitions enabled us to deduce a non-Abelian propelinear group structure for any Hamming code of length greater than 7. Received: March 6, 2000; revised version: November 30, 2000     

Diffeomorphisms with the s-limit shadowing property

Let Diff(M) be the space of diffeomorphisms of a closed C ^∞ manifold M endowed with the C ¹-topology, and denote by ?(f) the chain recurrent set of f?∈?Diff(M). In this article, we show that C ¹-generically f _|?(f) has the s-limit shadowing property if and only if f satisfies both Axiom A and no-cycle condition.     

C 1-stably ℒ p shadowable homoclinic classes

Let p be a hyperbolic saddle of diffeomorphism f on closed manifold M and H(p, f) be the homoclinic class associated with it. In this article, we introduce the notion of C ¹-stably ? _p shadowing and prove that if f is C ¹-stably ? _p shadowable on a homoclinic class H(p, f) then, H(p, f) has a dominated splitting. Moreover, we prove that if f is C ¹-stably Lipschitz ? _p shadowable on H(p, f) and H(p, f)-germ of f is expansive then the homoclinic class is hyperbolic.     

On Systems of Linear Equations with Nonnegative Coefficients

We consider a system of linear equations with positive coefficients, where the entries of the nonnegative irreducible coefficient matrix depend on a parameter vector. We say that the parameter vector is feasible if there exists a positive solution to this system. A set of all feasible parameter vectors is called the feasibility set. If all the positive entries are log-convex functions, the paper shows that the associated Perron root is log-convex on the parameter set and the l¹-norm of the solution is log-convex on the feasibility set. These results imply that the feasibility set is a convex set regardless whether the l¹-norm of the solution is bounded by some positive real number or not. Finally, we show important applications of these results to wireless communication networks and prove some other interesting results for this special case.This work was partly supported by the German Federal Ministry of Education and Research (BMBF) under grant BU150This work was supported by the German Research Foundation (DFG) under grant BO1734/1-2     

Anosov diffeomorphisms on nilmanifolds modelled on a free nilpotent Lie group

In this article we give an elementary proof, using standard arguments from algebraic number theory, of the fact that a nilmanifold modelled on a free c-step nilpotent Lie group on n generators admits an Anosov diffeomorphism if and only if n > c. In fact, we need to show that for any integer n > 1, there exists a matrix A ∈ GL(n, ?), such that any product of less than n eigenvalues of A is of modulus ≠ 1.     

Entanglement between two atoms interacting with a stochastic field

The entanglement property between two two-level atoms interacting with a stochastic field is studied. We analytically prove that when the correlation time κ ^?1 of the stochastic field is very short compared to the radiative lifetime γ ^?1 of the atom, the steady-state entanglement between the two atoms can appear. The concurrence characterizing the entanglement degree has also been calculated by using Monte Carlo simulation. It is shown that the correlation time has a strong effect on the entanglement. We also discuss the influence of strength of the stochastic process on the entanglement. The validity of the decorrelation approximation is also investigated.     

The Minimal Generator Matrix of a Vector Space

The atomic vectors of a finitely generated vector space C over a field F are characterized for C a subspace of the product vector space ? = ∏ _{i =1} ⁿ ? _i over F. For finite fields, the minimal trellis diagram for mixed-codes is determined, and this provides the L-section minimal trellis diagram for linear codes. As an example, an extremely simple yet comprehensive analysis of the trellis structure of Reed-Muller codes is given. In particular, a trellis oriented generator matrix for the 2 ^l -section minimal trellis diagram of a Reed-Muller code is presented. Received: February 27, 1997; revised version: May 6, 1999     

A three‐dimensional C1 finite element for gradient elasticity

In gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C¹ continuous interpolation in finite element discretizations of the displacement field only. Employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement, due to the scarcity of C¹ finite elements and their perceived computational cost. In this context, the lack of three‐dimensional C¹ elements is of particular concern. In this paper we present a new C¹ hexahedral element which, to the best of our knowledge, is the first three‐dimensional C¹ element ever constructed. It is shown to pass the single element and patch tests, and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity. It is further shown that C¹ elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good‐quality solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

Quasi-linearization of parameter-depending germs of vector fields

In this paper, we study normalizationand quasi-linearization of a family of germs of hyperbolic vector fields at the origin. We show that, when the eigenvalues of these systems satisfy the so-called strongly 1-resonant condition, i.e. there exists a relation of the form (r, λ) = 0, then they can be simplified, in the context of orbital equivalence, to a normal form which can be integrated in an explicit and convenient way. More precisely, given a family of germs of sufficiently smooth vector fields having a generic nonlinear part, with strongly 1-resonant, then for any integer k, there is a neighbourhood at the origin of the parameter space such that for any , is C ^k equivalent to a system having only one resonant term on each component.     

Use of normalized porosity in models for the porosity dependence of mechanical properties

The use of normalized porosity in models for the porosity dependence of mechanical properties is addressed first for the frequently used power law expression for such dependence, i.e., E/E₀ = (1−P)ⁿ where E is the property of interest at any volume fraction porosity (P) and E₀ is the value of E at P = 0. Normalizing P by P_C, the value of P at which the property of interest inherently goes to zero, giving E/E₀ = (1−P/P_C)ⁿ, clearly calls attention to the importance of P_C values < 1 (e.g., potentially as low as ∼ 0.2), a fact long known but inadequately recognized. Serious problems from the arbitrary use of both n and P_C as fitting parameters with little or no guidance as to the dependence that n and P_C (which is microstructurally sensitive) have on the type of porosity are shown. Further, porosity normalization of the power law model indicates at best limited compression of different porosity dependences into a single universal porosity dependence function and little distinguishing of property dependences as a function of the type of porosity. However, normalized porosity of minimum solid area (MSA) models gives a single universal porosity dependence. The difference in responses to P normalization of the two modeling approaches is attributed to their being based respectfully on little or no pore character and on detailed pore character. Thus, P normalization may be a valuable tool for evaluating porosity models, but must be applied in a more rigorous fashion, i.e., P_C determined primarily by measurement and correlation with the type of porosity (as with MSA models) and not as an arbitrary fitting parameter as used in the evaluations of the power law model.     

On piecewise smoothness of conjugacy of class P circle homeomorphisms to diffeomorphisms and rotations

We give a characterization of piecewise C ¹ class P-homeomorphism f of the circle with irrational rotation number and finitely many break points which is piecewise C ¹ conjugate to a C ¹-diffeomorphism. The following properties are equivalent:

?(i)?f is conjugate to a C ¹-diffeomorphism of the circle by a piecewise C ¹-homeomorphism. (ii)?The product of jumps of f in the break points contained in a same orbit is trivial. (iii)?f is conjugate to a C ¹-diffeomorphism of the circle by a piecewise linear (PL)-homeomorphism or a piecewise quadratic homeomorphism.

For a PL-homeomorphism f having the property (ii): f is conjugate to a rotation either by a PL-homeomorphism or by a piecewise analytic homeomorphism.