Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

The Bergman kernel: Explicit formulas,deflation, Lu Qi-Keng problem and Jacobi polynomials

We investigate the Bergman kernel function for the intersection of two complex ellipsoids {(z,w ₁,w ₂) ∈ C ⁿ⁺²: |z ₁|²+...+|z _n |²+|w ₁| ^q < 1, |z ₁|²+...+|z _n |²+|w ₂| ^r < 1}. We also compute the kernel function for {(z ₁,w ₁,w ₂) ∈ C³: |z ₁|^2/n + |w ₁| ^q < 1, |z ₁|^2/n + |w ₂| ^r < 1} and show deflation type identity between these two domains. Moreover in the case that q = r = 2 we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem.     

The local variational principle of topological pressure for sub-additive potentials

Let (X,T) be a topological dynamical system and be a sub-additive potential on C(X,R). Let U be an open cover of X. Then for any T-invariant measure μ, let . The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle:

     

On the Nitsche conjecture for harmonic mappings in ℝ<Superscript>2</Superscript> and ℝ<Superscript>3</Superscript>

We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. LetA(r, 1)={z:r<|z|<1}. Nitsche's conjecture states that if there exists a univalent harmonic mapping from an annulusA(r, 1), to an annulusA(s, 1), thens is at most 2r/(r ²+1).Lyzzaik's result states thats<t wheret is the length of the Grötzsch's ring domain associated withA(r, 1) (see [5]). Weitsman's result states thats≤1/(1+1/2(r logr)²) (see [8]).Our result for two-dimensional space states thats≤1/(1+1/2 log² r) which improves Weitsman's bound for allr, and Lyzzaik's bound forr close to 1. For three-dimensional space the result states thats≤1/(r?logr).     

A restricted sum formula among multiple zeta values

For positive integers α₁,α₂,…,αr with αr?2, the multiple zeta value or r-fold Euler sum is defined as

     

Distribution of exponential functions with squarefull exponent in residue rings

For a real x ≥ 1 we denote by S[x] the set of squarefull integers n ≤x, that is, the set of positive integers n ≤ such that l²|n for any prime divisor l|n. We estimate exponential sums of the form

     

A short note on Lq theory for Stokes problem with a pressure-dependent viscosity

We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on p and on the symmetric part of a gradient of u, namely, it is represented by a stress tensor T (Du, p):= v(p, |D|²)D which satisfies r-growth condition with r ∈ (1, 2]. In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for example in the paper Caffarelli, Peral (1998).     

Some variations on Tverberg’s theorem

Define T(d, r) = (d + 1)(r - 1) + 1. A well known theorem of Tverberg states that if n ≥ T(d, r), then one can partition any set of n points in R^d into r pairwise disjoint subsets whose convex hulls have a common point. The numbers T(d, r) are known as Tverberg numbers. Reay added another parameter k (2 ≤ k ≤ r) and asked: what is the smallest number n, such that every set of n points in R^d admits an r-partition, in such a way that each k of the convex hulls of the r parts meet. Call this number T(d, r, k). Reay conjectured that T(d, r, k) = T(d, r) for all d, r and k. In this paper we prove Reay’s conjecture in the following cases: when k ≥ [d+3/2], and also when d < rk/r-k - 1. The conjecture also holds for the specific values d = 3, r = 4, k = 2 and d = 5, r = 3, k = 2.     

An Example of an Unbounded Maximal Singular Operator

Suppose that an even integrable function Ω on the unit sphere S ¹ in R ² with mean value zero satisfies

$\mathop{\mathrm{essup}}\limits_{\xi\in \mathbf{S}^{1}}\biggl|\int_{\mathbf{S}^{1}}\Omega(\theta)\log\frac{1}{|\theta\cdot\xi|}\,d\theta\biggr|<+\infty,$

then the singular integral operator T _Ω given by convolution with the distribution p.v.?Ω(x/|x|)|x|^?2, initially defined on Schwartz functions, extends to an L ²-bounded operator. We construct examples of a function Ω satisfying the above conditions and of a continuous bounded integrable function f such that

$\limsup_{\epsilon\to 0^+}\biggl|\int_{\epsilon<|y|}\Omega(y/|y|)|y|^{-2}f(x-y)dy\biggr|=\infty\quad \hbox{a. e.}$

     

The iterated Aluthge transforms of a matrix converge

Given an r×r complex matrix T, if T=U|T| is the polar decomposition of T, then, the Aluthge transform is defined byΔ(T)=|T|^1/2U|T|^1/2. Let Δn(T) denote the n-times iterated Aluthge transform of T, i.e., Δ⁰(T)=T and Δn(T)=Δ(Δn−1(T)), n∈N. We prove that the sequence {Δn(T)}_n∈N converges for every r×r matrix T. This result was conjectured by Jung, Ko and Pearcy in 2003. We also analyze the regularity of the limit function.     

Regularity results for a class of obstacle problems under nonstandard growth conditions

We prove regularity results for minimizers of functionals in the class , where is a fixed function and f is quasiconvex and fulfills a growth condition of the type

L⁻¹|z|^p(x)?f(x,ξ,z)?L(1+|z|^p(x)),     

New upper bounds on the spectral radius of unicyclic graphs

Let G=(V(G),E(G)) be a unicyclic simple undirected graph with largest vertex degree Δ. Let C_r be the unique cycle of G. The graph G-E(C_r) is a forest of r rooted trees T₁,T₂,…,T_r with root vertices v₁,v₂,…,v_r, respectively. Let

     

The Schröder-Bernstein property for weakly minimal theories

For a countable, weakly minimal theory T, we show that the Schröder-Bernstein property (any two elementarily bi-embeddable models are isomorphic) is equivalent to each of the following:

1. 1.
   For any U-rank-1 type q ∈ S(acl ^eq (?)) and any automorphism f of the monster model C, there is some n < ω such that f ⁿ (q) is not almost orthogonal to q ? f(q) ? … ? f ^n?1(q)
    
2. 2.
   T has no infinite collection of models which are pairwise elementarily bi-embeddable but pairwise nonisomorphic.
    

We conclude that for countable, weakly minimal theories, the Schröder-Bernstein property is absolute between transitve models of ZFC.     

Monotone weak Lindelöfness

The definition of monotone weak Lindelöfness is similar to monotone versions of other covering properties: X is monotonically weakly Lindelöf if there is an operator r that assigns to every open cover U a family of open sets r(U) so that (1) ∪r(U) is dense in X, (2) r(U) refines U, and (3) r(U) refines r(V) whenever U refines V. Some examples and counterexamples of monotonically weakly Lindelöf spaces are given and some basic properties such as the behavior with respect to products and subspaces are discussed.     

Generalizations of Bohr inequality for Hilbert space operators

Let B(H) be the space of all bounded linear operators on a complex separable Hilbert space H. Bohr inequality for Hilbert space operators asserts that for A,B∈B(H) and p,q>1 real numbers such that 1/p+1/q=1,

²|A+B|?p²|A|+q²|B|     

A class number free criterion for catalan's conjecture

The Catalan conjecture asserts that the equation X^U−Y^V=1 with U,V>1 has no other solution in integers but 3²−2³=1 (Catalan's Conjecture, Academic Press, New York, 1994). We prove that, for primes U=p and V=q yielding a solution to the Catalan equation, the simultaneous conditions

     

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

We study spectral properties for HK,Ω, the Krein-von Neumann extension of the perturbed Laplacian −Δ+V defined on , where V is measurable, bounded and nonnegative, in a bounded open set Ω⊂Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r>1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula

     

Hyperbolic Chevalley groups on ℂ<Superscript>2</Superscript>

Let Γ ? U (1, 1) be the subgroup generated by the complex reflections. Suppose that Γ acts discretely on the domain K = {(z ₁, z ₂) ∈ ?² ||z ₁|² ? |z ₂|² < 0} and that the projective group PΓ acts on the unit disk B = {|z ₁/z ₂| < 1} as a Fuchsian group of signature (n ₁, ..., n _s ), s ? 3, n _i ? 2. For such groups, we prove a Chevalley type theorem, i.e., find a necessary and sufficient condition for the quotient space K/Γ to be isomorphic to ?² ? {0}.     

Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.