On a problem regarding coefficients of cyclotomic polynomials

Let an(k) be the coefficient of tk in the nth cyclotomic polynomial

     

Isomorphic properties of intersection bodies

We study isomorphic properties of two generalizations of intersection bodies - the class of k-intersection bodies in Rn and the class of generalized k-intersection bodies in Rn. In particular, we show that all convex bodies can be in a certain sense approximated by intersection bodies, namely, if K is any symmetric convex body in Rn and 1≤k≤n−1 then the outer volume ratio distance from K to the class can be estimated by

     

High dimensional random sections of isotropic convex bodies

We study two properties of random high dimensional sections of convex bodies. In the first part of the paper we estimate the central section function for random F∈Gn,k and K⊂Rn a centrally symmetric isotropic convex body. This partially answers a question raised by V.D. Milman and A. Pajor (see [V.D. Milman, A. Pajor, Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: Lecture Notes in Math., vol. 1376, Springer, 1989, p. 88]). In the second part we show that every symmetric convex body has random high dimensional sections F∈Gn,k with outer volume ratio bounded by

     

Some identities involving theta functions

Let (|q|<1). For k∈N it is shown that there exist k rational numbers A(k,0),…,A(k,k−1) such that

     

On the number of representations of certain integers as sums of 11 or 13 squares

Let r_k(n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p,

     

The boundary of higher rank numerical ranges

The rank-k-numerical range of an n×n matrix A is defined as

     

Mean value of some exponential sums and applications to Kloosterman sums

Let q, m, n, k be integers with q?3 and k?1, define the exponential sum

     

The generalized Roper-Suffridge extension operator for some biholomorphic mappings

Suppose f is a spirallike function of type β (or starlike function of order α) on the unit disk D in C. Let , where 1?p₁?2 (or 0<p₁?2), pj?1, j=2,…,n, are real numbers. In this paper, we prove that

     

Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings

Suppose that K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E. Let be two nonself asymptotically nonexpansive mappings with sequences {kn},{ln}⊂[1,∞), limn→∞kn=1, limn→∞ln=1, , respectively. Suppose {xn} is generated iteratively by

     

Precise rates in the law of logarithm for the moment convergence of i.i.d. random variables

Let be a sequence of i.i.d. random variables with EX=0 and EX²=σ²<∞. Set , Mn=maxk?n|Sk|, n?1. Let r>1, then we obtain

     

Beta-expansion and continued fraction expansion

For any real number β>1, let ε(1,β)=(ε₁(1),ε₂(1),…,εn(1),…) be the infinite β-expansion of 1. Define . Let x∈[0,1) be an irrational number. We denote by kn(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n digits in the β-expansion of x. If is bounded, we obtain that for all x∈[0,1)?Q,

     

On the Turán number for the hexagon

A long-standing conjecture of Erd?s and Simonovits is that ex(n,C_2k), the maximum number of edges in an n-vertex graph without a 2k-gon is asymptotically as n tends to infinity. This was known almost 40 years ago in the case of quadrilaterals. In this paper, we construct a counterexample to the conjecture in the case of hexagons. For infinitely many n, we prove that

     

The products on the unit sphere and even-dimension spaces

The distribution δ(k)(r−1) focused on the unit sphere Ω of Rm is defined by

     

Log-concavity and LC-positivity

A triangle {a(n,k)}_0?k?n of nonnegative numbers is LC-positive if for each r, the sequence of polynomials is q-log-concave. It is double LC-positive if both triangles {a(n,k)} and {a(n,n−k)} are LC-positive. We show that if {a(n,k)} is LC-positive then the log-concavity of the sequence {xk} implies that of the sequence {zn} defined by , and if {a(n,k)} is double LC-positive then the log-concavity of sequences {xk} and {yk} implies that of the sequence {zn} defined by . Examples of double LC-positive triangles include the constant triangle and the Pascal triangle. We also give a generalization of a result of Liggett that is used to prove a conjecture of Pemantle on characteristics of negative dependence.     

A note on the kth eigenvalue of trees

In [Linear Algebra Appl. 149 (1991) 19-34], Shao proved that for a tree T on n vertices, the kth eigenvalue

     

Approximation numbers for the finite sections of Toeplitz operators with piecewise continuous symbols

In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator T(a)∈L(?p), 1<p<∞, where a is a piecewise continuous function on the unit circle. We prove that the behavior of the approximation numbers of the finite sections Tn(a)=PnT(a)Pn depends heavily on the Fredholm properties of the operators T(a) and . In particular, if the operators T(a) and are Fredholm on ?p, then the approximation numbers of Tn(a) have the so-called k-splitting property. But, in contrast with the case of continuous symbols, the splitting number k is in general larger than .     

Partition identities II. The results of Bateman and Erd?s

This paper shows that the natural setting for the Bateman and Erd?s study of monotonicity of the kth difference of partition functions a(n) is the class of partition identities