Inner Functions and Cyclic Composition Operators on H(Bn)

The present paper shows that the algebra generated by {C|  Aut(B_n)} is cyclic on H²(B_n), and any nonconstant function f  H²(B_n) is a cyclic vector of . In addition, the hypercyclic and cyclic composition operators will be discussed.     

Every 4-connected line graph of a quasi claw-free graph is hamiltonian connected

Let G be a graph. For u,vV(G) with dist_G(u,v)=2, denote J_G(u,v)={wN_G(u)∩N_G(v)|N_G(w)N_G(u)N_G(v){u,v}}. A graph G is called quasi claw-free if J_G(u,v)≠ for any u,vV(G) with dist_G(u,v)=2. In 1986, Thomassen conjectured that every 4-connected line graph is hamiltonian. In this paper we show that every 4-connected line graph of a quasi claw-free graph is hamiltonian connected.     

Universal Polynomial Majorants on Convex Bodies

Let K be a convex body in ^d (d2), and denote by B_n(K) the set of all polynomials p_n in ^d of total degree n such that |p_n|1 on K. In this paper we consider the following question: does there exist a p*_nB_n(K) which majorates every element of B_n(K) outside of K? In other words can we find a minimal γ1 and p*_nB_n(K) so that |p_n(x)|γ |p*_n(x)| for every p_nB_n(K) and x ^d\K? We discuss the magnitude of γ and construct the universal majorants p*_n for evenn. It is shown that γ can be 1 only on ellipsoids. Moreover, γ=O(1) on polytopes and has at most polynomial growth with respect to n, in general, for every convex body K.     

Optimal Sobolev Imbeddings Involving Rearrangement-Invariant Quasinorms

Let m and n be positive integers with n2 and 1mn−1. We study rearrangement-invariant quasinorms _R and _D on functions f: (0, 1)→ such that to each bounded domain Ω in ⁿ, with Lebesgue measure |Ω|, there corresponds C=C(|Ω|)>0 for which one has the Sobolev imbedding inequality _R(u*(|Ω| t))C_D(|^mu|* (|Ω| t)), uC^m₀(Ω), involving the nonincreasing rearrangements of u and a certain mth order gradient of u. When m=1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which _D need not be rearrangement-invariant, _R(u*(|Ω| t))C_D((d/dt) ∫_{{x ⁿ : |u(x)|>u*(|Ω| t)}} |(u)(x)| dx), uC¹₀(Ω). In both cases we are especially interested in when the quasinorms are optimal, in the sense that _R cannot be replaced by an essentially larger quasinorm and _D cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of Trudinger, Strichartz, Hansson, Brézis, and Wainger.     

Long time behaviour for generalized complex Ginzburg–Landau equation

In this paper, the two-dimensional generalized complex Ginzburg–Landau equation (CGL)

u_t=ρu−Δφ(u)−(1+iγ)Δu−νΔ²u−(1+iμ)|u|^2σu+αλ₁(|u|²u)+β(λ₂)|u|²

is studied. The existence of global attractor for this equation with periodic boundary condition is established and upper bounds of Hausdorff and fractal dimensions of attractor are obtained.     

LOCAL TIME ANALYSIS OF ADDITIVE LEVY PROCESSES WITH DIFFERENT L(E)VY EXPONENTS

Let X1 XN be independent, classical Levy processes on R^d with Levy exponents ψ1,…, ψN, respectively. The corresponding additive Levy process is defined as the following N-parameter random field on R^d, X（t） △= X1（t1） ＋ ... ＋ XN（tN）, At∈N. Under mild regularity conditions on the ψi＇s, we derive estimate for the local and uniform moduli of continuity of local times of X = {X（t）; t ∈R^N}.     

Maximal Inequalities and Lebesgue's Differentiation Theorem for Best Approximant by Constant over Balls

For fL_p( ⁿ), with 1p<∞, >0 and x ⁿ we denote by T(f)(x) the set of every best constant approximant to f in the ball B(x, ). In this paper we extend the operators T_p to the space L_p−1( ⁿ)+L_∞( ⁿ), where L₀ is the set of every measurable functions finite almost everywhere. Moreover we consider the maximal operators associated to the operators T_p and we prove maximal inequalities for them. As a consequence of these inequalities we obtain a generalization of Lebesgue's Differentiation Theorem.     

Strong and uniform mean stability of cosine and sine operator functions

It is first observed that a uniformly bounded cosine operator function C() and the associated sine function S() are totally non-stable. Then, using a zero-one law for the Abel limit of a closed linear operator, we prove some results concerning strong mean stability and uniform mean stability of C(). Among them are: (1) C() is strongly (C,1)-mean stable (or (C,2)-mean stable, or Abel-mean stable) if and only if 0ρ(A)σ_c(A); (2) C() is uniformly (C,2)-mean stable if and only if S() is uniformly (C,1)-mean stable, if and only if , if and only if , if and only if C() is uniformly Abel-mean stable, if and only if S() is uniformly Abel-mean stable, if and only if 0ρ(A).     

A vanishing theorem on Kaehler Finsler manifolds

Let M be a connected compact complex manifold endowed with a strongly pseudoconvex complex Finsler metric F. In this paper, we first define the complex horizontal Laplacian □_h and complex vertical Laplacian □_v on the holomorphic tangent bundle T^1,0M of M, and then we obtain a precise relationship among □_h,□_v and the Hodge–Laplace operator on (T^1,0M,,), where , is the induced Hermitian metric on T^1,0M by F. As an application, we prove a vanishing theorem of holomorphic p-forms on M under the condition that F is a Kaehler Finsler metric on M.     

Lipschitz regularity for scalar minimizers of autonomous simple integrals

We prove Lipschitz regularity for a minimizer of the integral , defined on the class of the AC functions having x(a)=A and x(b)=B. The Lagrangian may have L(s,) nonconvex (except at ξ=0), while may be non-lsc, measurability sufficing for ξ≠0 provided, e.g., L^**() is lsc at (s,0) s. The essential hypothesis (to yield Lipschitz minimizers) turns out to be local boundedness of the quotient φ/ρ() (and not of L^**() itself, as usual), where φ(s)+ρ(s)h(ξ) approximates the bipolar L^**(s,ξ) in an adequate sense. Moreover, an example of infinite Lavrentiev gap with a scalar 1-dim autonomous (but locally unbounded) lsc Lagrangian is presented.     

On constants in some inequalities for intermediate derivatives on a finite interval

Let L_q (1q<∞) be the space of functions f measurable on I=[−1,1] and integrable to the power q, with normL_∞ is the space of functions measurable on I with normWe denote by AC the set of all functions absolutely continuous on I. For nN, q[1,∞] we setW_n,q={f:f⁽ⁿ⁻¹⁾AC, f⁽ⁿ⁾L_q}.In this paper, we consider the problem of accuracy of constants A, B in the inequalities (1)|| f^(m)||_qA|| f||_p+B|| f^(m+k+1)||_r, mN, kW; p,q,r[1,∞], fW_m+k+1,r.     

Turán type reverse Markov inequalities for compact convex sets

For a compact convex set the well-known general Markov inequality holds asserting that a polynomial p of degree n must have p^′c(K)n²p. On the other hand for polynomials in general, p^′ can be arbitrarily small as compared to p.The situation changes when we assume that the polynomials in question have all their zeroes in the convex set K. This was first investigated by Turán, who showed the lower bounds p^′(n/2)p for the unit disk D and for the unit interval I[-1,1]. Although partial results provided general lower estimates of order , as well as certain classes of domains with lower bounds of order n, it was not clear what order of magnitude the general convex domains may admit here.Here we show that for all bounded and convex domains K with nonempty interior and polynomials p with all their zeroes lying in K p^′c(K)np holds true, while p^′C(K)np occurs for any K. Actually, we determine c(K) and C(K) within a factor of absolute numerical constant.     

The Approximation Functional and Interpolation with Perturbed Continuity

The continuity conditions at the endpoints of interpolation theorems, Ta_BjM_j a_Aj for j=0, 1, can be written with the help of the approximation functional: E(t, Ta; B₁, B₀)_L^∞M₀ a_A0 and E(t, Ta; B₀, B₁)_L^∞M₁ a_A1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L^∞ norms can be replaced by BMO( ₊) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition fL⁰, i.e., f*L^∞, where f*(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(p, q) spaces by the weaker f*BMO( ₊).     

Theorems about the attractor for incompressible non-Newtonian flow driven by external forces that are rapidly oscillating in time but have a smooth average

This paper discusses the incompressible non-Newtonian fluid with rapidly oscillating external forces g(x,t)=g(x,t,t/) possessing the average g⁰(x,t) as →0⁺, where 0<₀<1. Firstly, with assumptions (A₁)–(A₅) on the functions g(x,t,ξ) and g⁰(x,t), we prove that the Hausdorff distance between the uniform attractors and in space H, corresponding to the oscillating equations and the averaged equation, respectively, is less than O() as →0⁺. Then we establish that the Hausdorff distance between the uniform attractors and in space V is also less than O() as →0⁺. Finally, we show for each [0,₀].     

Conditions for a class of dimensional dual hyperovals to be of polar type

A d-dimensional dual hyperoval with monomial is of polar type if and only if d is even, Gal(GF(2^d+1)/GF(2)) and σ²=id_GF(2^d+1).     

On some d-dimensional dual hyperovals in

Let d≥3. Let H be a d+1-dimensional vector space over GF(2) and {e₀,…,e_d} be a specified basis of H. We define Supp(t){e_t1,…,e_tl}, a subset of a specified base for a non-zero vector t=e_t1++e_tl of H, and Supp(0)0/. We also define J(t)Supp(t) if |Supp(t)| is odd, and J(t)Supp(t){0} if |Supp(t)| is even.For s,tH, let {a(s,t)} be elements of H(HH) which satisfy the following conditions: (1) a(s,s)=(0,0), (2) a(s,t)=a(t,s), (3) a(s,t)≠(0,0) if s≠t, (4) a(s,t)=a(s^′,t^′) if and only if {s,t}={s^′,t^′}, (5) {a(s,t)|tH} is a vector space over GF(2), (6) {a(s,t)|s,tH} generate H(HH). Then, it is known that S{X(s)|sH}, where X(s){a(s,t)|tH{s}}, is a dual hyperoval in PG(d(d+3)/2,2)=(H(HH)){(0,0)}.In this note, we assume that, for s,tH, there exists some x_s,t in GF(2) such that a(s,t) satisfies the following equation: Then, we prove that the dual hyperoval constructed by {a(s,t)} is isomorphic to either the Huybrechts’ dual hyperoval, or the Buratti and Del Fra’s dual hyperoval.     

Traveling fronts in space–time periodic media

This paper is concerned with the existence of pulsating traveling fronts for the equation:

(1)

∂_tu−(A(t,x)u)+q(t,x)u=f(t,x,u),