Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices

Using the variational method, it is shown that the set of all strong peak functions in a closed algebra A of Cb(K) is dense if and only if the set of all strong peak points is a norming subset of A. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space X is a norming subset of , then the set of all strongly norm attaining elements in is dense. In particular, the set of all points at which the norm of is Fréchet differentiable is a dense Gδ subset. In the last part, using Reisner's graph-theoretic approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space X with an absolute norm, its polynomial numerical indices are one if and only if X is isometric to . Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm.     

Well-posedness and ill-posedness for a fifth-order shallow water wave equation

In this paper we establish a new bilinear estimate in suitable Bourgain spaces by using a fundamental estimate on dyadic blocks for the Kawahara equation which was obtained by the [k;Z] multiplier norm method of Tao (2001) [2]; then the local well-posedness of the Cauchy problem for a fifth-order shallow water wave equation in with is obtained by the Fourier restriction norm method. And some ill-posedness in with is derived from a general principle of Bejenaru and Tao.     

The Chase radical and reduced products

Let κ be a regular uncountable cardinal. We shall give a criterion for certain reduced products of torsion-free abelian groups to be ℵ₁-free. As an application we shall show that the norm of the Chase radical is ℵ₁ in ZFC, a result which was previously known only under the assumption of the continuum hypothesis 2^ℵ₀=ℵ₁.     

Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B^--XF^--F⁺X+XB⁺X=0, where , and with a nonnegative matrix P, positive diagonal matrices D^±, and nonnegative parameters f, and . We prove the existence of the minimal nonnegative solution X^∗ under the physically reasonable assumption , and study its numerical computation by fixed-point iteration, Newton’s method and doubling. We shall also study several special cases; e.g. when and P is low-ranked, then is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.     

Eventually norm continuous semigroups on Hilbert space and perturbations

The eventually norm continuous semigroups on Hilbert space and perturbation are studied in this paper. By resolvent of infinitesimal generator, the sufficient and necessary conditions for eventually norm continuous semigroups are given. Using the result obtained, it is proved that if is infinitesimal generator of an eventually norm continuous semigroup T(t), then there is a subspace Ξ_A of such that, for any , the semigroup S(t) generated by preserves the property of T(t).     

Norm or numerical radius attaining polynomials on C(K)

Let be the Banach space of all complex-valued continuous functions on a compact Hausdorff space K. We study when the following statement holds: every norm attaining n-homogeneous complex polynomial on attains its norm at extreme points. We prove that this property is true whenever K is a compact Hausdorff space of dimension less than or equal to one. In the case of a compact metric space a characterization is obtained. As a consequence we show that, for a scattered compact Hausdorff space K, every continuous n-homogeneous complex polynomial on can be approximated by norm attaining ones at extreme points and also that the set of all extreme points of the unit ball of is a norming set for every continuous complex polynomial. Similar results can be obtained if “norm” is replaced by “numerical radius.”     

Pressure representation and boundary regularity of the Navier-Stokes equations with slip boundary condition

We first represent the pressure in terms of the velocity in . Using this representation we prove that a solution to the Navier-Stokes equations is in under the critical assumption that , with r?3, while for r=3 the smallness is required. In [H.J. Choe, Boundary regularity of weak solutions of the Navier-Stokes equations, J. Differential Equations 149 (2) (1998) 211-247], a boundary L^∞ estimate for the solution is derived if the pressure on the boundary is bounded. In our work, we remove the boundedness assumption of the pressure. Here, our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate for L^∞-norm of u.     

On strong convergence to 3D steady vortex sheets

In this paper, we first establish a strong convergence criterion of approximate solutions for the 3D steady incompressible Euler equations. For axisymmetric flows, under the assumption that the vorticity is of one sign and uniformly bounded in L¹ space, we obtain a sufficient and necessary condition for the strong convergence in of approximate solutions. Furthermore, for one-sign and L¹-bounded vorticity, it is shown that if a sequence of approximate solutions concentrates at an isolated point in (r,z)-plane, then the concentration point can appear neither in the region near the axis (including the symmetry axis itself) nor in the region far away from the axis. Finally, we present an example of approximates solutions which converge strongly in by using Hill's spherical vortex.     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.     

Invariant functionals and the uniqueness of invariant norms

Let τ be a representation of a compact group G on a Banach space (X,||·||). The question we address is whether X carries a unique invariant norm in the sense that ||·|| is the unique norm on X for which τ is a representation. We characterize the uniqueness of norm in terms of the automatic continuity of the invariant functionals in the case when X is a dual Banach space and τ is a -continuous representation of G on X such that τ(G) consists of -continuous operators. We illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces Lp(Ω), where Ω is a locally compact Hausdorff space equipped with a positive Radon measure and G acts on Ω as a group of continuous invertible measure-preserving transformations.     

Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to N-waves

The goal of this article is to develop a new technique to obtain better asymptotic estimates for scalar conservation laws. General convex flux, f″(u)?0, is considered with an assumption . We show that, under suitable conditions on the initial value, its solution converges to an N-wave in L¹ norm with the optimal convergence order of O(1/t). The technique we use in this article is to enclose the solution with two rarefaction waves. We also show a uniform convergence order in the sense of graphs. A numerical example of this phenomenon is included.     

The asymptotic representations on the norm of the Fourier operators

In this paper by a new method we first get . Based on it, we then obtain a new asymptotic representation on the norm Ln of the Fourier operators as follows: .     

Polynomially based multi-projection methods for Fredholm integral equations of the second kind

In this paper, we propose a multi-projection and iterated multi-projection methods for Fredholm integral equations of the second kind with a smooth kernel using polynomial bases. We obtain super-convergence rates for the approximate solutions, more precisely, we prove that in M-Galerkin and M-collocation methods not only iterative solution approximates the exact solution u in the supremum norm with the order of convergence n^-4k, but also the derivatives of approximate the corresponding derivatives of u in the supremum norm with the same order of convergence, n being the degree of polynomial approximation and k being the smoothness of the kernel.     

Bounded Kähler class rigidity of actions on Hermitian symmetric spaces

We prove that the conjugacy class of a Zariski dense representation , q>p?1, of a finitely generated group Γ is completely determined by the pull-back via π of a bounded cohomology class in defined in terms of the Kähler form on the associated symmetric space. Under the assumption that is finite dimensional, we show that, up to equivalence, there is only a finite number of such representations for fixed q>p?1; moreover, under the hypothesis that injects into , we estimate the total number of such representations (for all q>p?1) to be bounded above by .     

An explicit formula for the action of a finite group on a commutative ring

Let G be a finite group, k a commutative ring upon which G acts. For every subgroup H of G, the trace (or norm) map is defined. is onto if and only if there exists an element x_H such that . We will show that the existence of x_P for every subgroup P of prime order determines the existence of x_G by exhibiting an explicit formula for x_G in terms of the x_P, where P varies over prime order subgroups. Since is onto if and only if is, where g∈G is an arbitrary element, we need to take only one P from each conjugacy class. We will also show why a formula with less factors does not exist, and show that the existence or non-existence of some of the x_P’s (where we consider only one P from each conjugacy class) does not affect the existence or non-existence of the others.     

Mixed means over balls and annuli and lower bounds for operator norms of maximal functions

In this paper we prove mixed-means inequalities for integral power means of an arbitrary real order, where one of the means is taken over the ball , centered at and of radius , δ>0. Therefrom we deduce the corresponding Hardy-type inequality, that is, the operator norm of the operator S_δ which averages over , introduced by Christ and Grafakos in Proc. Amer. Math. Soc. 123 (1995) 1687-1693. We also obtain the operator norm of the related limiting geometric mean operator, that is, Carleman or Levin-Cochran-Lee-type inequality. Moreover, we indicate analogous results for annuli and discuss estimations related to the Hardy-Littlewood and spherical maximal functions.     

An extension of a Bourgain-Lindenstrauss-Milman inequality

Let ‖⋅‖ be a norm on Rn. Averaging ‖(ε₁x₁,…,εnxn)‖ over all the n² choices of , we obtain an expression |||x||| which is an unconditional norm on Rn. Bourgain, Lindenstrauss and Milman [J. Bourgain, J. Lindenstrauss, V.D. Milman, Minkowski sums and symmetrizations, in: Geometric Aspects of Functional Analysis (1986/1987), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 44-66] showed that, for a certain (large) constant η>1, one may average over ηn (random) choices of and obtain a norm that is isomorphic to |||⋅|||. We show that this is the case for any η>1.