Topological entropy for set valued maps

Any continuous map T on a compact metric space X induces in a natural way a continuous map on the space K(X) of all non-empty compact subsets of X. Let T be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map is zero or infinity. Moreover, the topological entropy of is zero, where C(X) denotes the space of all non-empty compact and connected subsets of X. For general continuous maps on compact metric spaces these results are not valid.     

Invariant manifolds of partial functional differential equations

This paper is concerned with the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces when the nonlinear perturbation has a small global Lipschitz constant and locally C^k-smooth near the trivial solution. Such a nonlinear perturbation arises in many applications through the usual cut-off procedure, but the requirement in the existing literature that the nonlinear perturbation is globally C^k-smooth and has a globally small Lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut-off function. Our general results are illustrated by and applied to partial functional differential equations for which the phase space (where r>0 and being a Banach space) has no smooth inner product structure and for which the validity of variation-of-constants formula is still an interesting open problem.     

The transitivity of induced maps

For a metric continuum X, we consider the hyperspaces X² and C(X) of the closed and nonempty subsets of X and of subcontinua of X, respectively, both with the Hausdorff metric. For a given map we investigate the transitivity of the induced maps and . Among other results, we show that if X is a dendrite or a continuum of type λ and is a map, then C(f) is not transitive. However, if X is the Hilbert cube, then there exists a transitive map such that f² and C(f) are transitive.     

Uniqueness properties of solutions of Schrödinger equations

Under suitable assumptions on the potentials V and a, we prove that if u∈C([0,1],H¹) is a solution of the linear Schrödinger equation

     

Some properties of sharp Hessenberg pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider an ordered pair of F-linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable on V; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V₀+V₁+?+V_i+1 for 0?i?d, where V_-1:=0 and V_d+1:=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and . We call such a pair a Hessenberg pair on V. It is known that if the Hessenberg pair A,A^∗ on V is irreducible then d=δ and for 0?i?d the dimensions of V_i and coincide. We say a Hessenberg pair A,A^∗ on V is sharp whenever it is irreducible and .In this paper, we give the definitions of a Hessenberg system and a sharp Hessenberg system. We discuss the connection between a Hessenberg pair and a Hessenberg system. We also define a finite sequence of scalars called the parameter array for a sharp Hessenberg system, which consists of the eigenvalue sequence, the dual eigenvalue sequence and the split sequence. We calculate the split sequence of a sharp Hessenberg system. We show that a sharp Hessenberg pair is a tridiagonal pair if and only if there exists a nonzero nondegenerate bilinear form on V that satisfies 〈Au,v〉=〈u,Av〉 and 〈A^∗u,v〉=〈u,A^∗v〉 for all u,v∈V.     

A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations

In dimension n?3, we define a generalization of the classical two-dimensional partial Legendre transform, that reduces interior regularity of the generalized Monge-Ampère equation to regularity of a divergence form quasilinear system of special form. This is then used to obtain smoothness of C^2,1 solutions, having n-1 nonvanishing principal curvatures, to certain subelliptic Monge-Ampère equations in dimension n?3. A corollary is that if k?0 vanishes only at nondegenerate critical points, then a C^2,1 convex solution u is smooth if and only if the symmetric function of degree n-1 of the principal curvatures of u is positive, and moreover, u fails to be when not smooth.     

Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces

First, the existence and structure of uniform attractors in H is proved for nonautonomous 2D Navier-Stokes equations on bounded domain with a new class of distribution forces, termed normal in (see Definition 3.1), which are translation bounded but not translation compact in . Then, the properties of the kernel section are investigated. Last, the fractal dimension is estimated for the kernel sections of the uniform attractors obtained.     

The density of 0's in recurrence double sequences

Let be a double sequence over a finite field satisfying a linear recurrence with constant coefficients, with at most finitely many nonzero elements on each row. Given a nonzero element g of , we show how to obtain an explicit formula for the number of g's in the first qⁿ rows of A. We also characterize the cases when the density of 0's is 1.     

Subword complexity and projection bodies

A polytope P⊆[0,1d⁾ and an induce a so-called Hartman sequence which is by definition 1 at the kth position if and 0 otherwise, k∈Z. We prove an asymptotic formula for the subword complexity of such a Hartman sequence. This result establishes a connection between symbolic dynamics and convex geometry: If the polytope P is convex then the subword complexity of asymptotically equals the volume of the projection body ΠP of P for almost all .     

Center manifolds for difference equations—smooth parameter dependence

For sufficiently small C¹ perturbations of (nonautonomous) linear difference equations with a nonuniform exponential trichotomy, we establish the existence of center manifolds with the optimal C¹ regularity. We also consider the case of parameter-dependent perturbations and we obtain the C¹ dependence of the center manifolds on the parameter. In addition, we consider arbitrary growth rates with the usual exponential estimates of the form in the notion of exponential trichotomy replaced by where ρ is now an arbitrary function. The proof of the regularity, both of the center manifold and of its dependence, on the parameter is based on the fiber contraction principle. The most technical part of the argument concerns the continuity of the fiber contraction that essentially needs a direct argument.     

Category and compact leaves

We prove that if F is a C¹-foliation of a compact manifold M with finite transverse saturated LS category, , then F has a compact leaf. In contrast, we show that if F is expansive on some non-trivial minimal set of F, then . Examples of foliations are given to illustrate the main results of the paper.     

Blow-up of solutions to a degenerate parabolic equation not in divergence form

We study nonglobal positive solutions to the Dirichlet problem for u_t=u^p(Δu+u) in bounded domains, where 0<p<2. It is proved that the set of points at which u blows up has positive measure and the blow-up rate is exactly . If either the space dimension is one or p<1, the ω-limit set of consists of continuous functions solving . In one space dimension it is shown that actually as t→T, where w coincides with an element of a one-parameter family of functions inside each component of its positivity set; furthermore, we study the size of the components of {w>0} with the result that this size is uniquely determined by Ω in the case p<1, while for p>1, the positivity set can have the maximum possible size for certain initial data, but it may also be arbitrarily close to the minimal length π.     

Quadratic algebras with Ext algebras generated in two degrees

Green and Marcos (2005) [2] call a graded k-algebra δ-Koszul if the corresponding Yoneda algebra is finitely generated and there exists a function δ:N→N such that is zero if j≠δ(i). For any integer m≥3 we exhibit a noncommutative quadratic δ-Koszul algebra for which the Yoneda algebra is generated in degrees (1,1) and (m,m+1). These examples answer a question of Green and Marcos. These algebras are not Koszul but m-Koszul (in the sense of Backelin).     

A further three critical points theorem

If X is a real Banach space, we denote by W_X the class of all functionals possessing the following property: if {u_n} is a sequence in X converging weakly to u∈X and lim inf_n→∞Φ(u_n)≤Φ(u), then {u_n} has a subsequence converging strongly to u.In this paper, we prove the following result:Let X be a separable and reflexive real Banach space; an interval; a sequentially weakly lower semicontinuous C¹ functional, belonging to W_X, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X^∗; a C¹ functional with compact derivative. Assume that, for each λ∈I, the functional Φ−λJ is coercive and has a strict local, not global minimum, say .Then, for each compact interval [a,b]⊆I for which , there exists r>0 with the following property: for every λ∈[a,b] and every C¹ functional with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation

Φ^′(x)=λJ^′(x)+μΨ^′(x)     

A compactly supported solution to a three-dimensional uniformly elliptic equation without zero-order term

A second order, nonvariational, elliptic operator L and a function V are constructed in with the following properties: the operator L is uniformly elliptic, without zero-order term and smooth almost everywhere in ; the function (1<p<3) solves the equation LV=0 in , it has compact support but it is not identically zero.