Topologically inseparable functions II: Infinitary case

Given a set A and a function A: A → A, we study the set of all functions g: A → A that are continuous for all topologies for which f continuous. We prove that in a sense to be made precise in the text, for any essentially infinitary function f, any non-constant such g equals f ⁿ , for some n∈ ?. We also prove a similar result for the clone of n-ary functions from A ⁿ → A.     

Generalizations of ?wierczkowski’s lemma and the arity gap of finite functions

?wierczkowski’s lemma-as it is usually formulated-asserts that if f:Aⁿ→A is an operation on a finite set A, n≥4, and every operation obtained from f by identifying a pair of variables is a projection, then f is a semiprojection. We generalize this lemma in various ways. First, it is extended to B-valued functions on A instead of operations on A and to essentially at most unary functions instead of projections. Then we characterize the arity gap of functions of small arities in terms of quasi-arity, which in turn provides a further generalization of ?wierczkowski’s lemma. Moreover, we explicitly classify all pseudo-Boolean functions according to their arity gap. Finally, we present a general characterization of the arity gaps of B-valued functions on arbitrary finite sets A.     

Bourgain algebras and tightness in some algebras of analytic functions

We prove that the Bourgain algebra of the polydisk algebra A(Δn) is A(Δn) itself and disprove the tightness of some algebras of analytic functions; in particular that of H^∞(BE).     

Partitions of the set of natural numbers and their representation functions

For a given set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. In this paper we give a simple proof to two results by Sándor.     

Improving eigenvalue bounds using extra bounds

Bounds for various functions of the eigenvalues of a Hermitian matrix A, based on the traces of A and A², are improved. A technique is presented whereby these bounds can be improved by combining them with other bounds. In particular, the diagonal of A, in conjunction with majorization, is used to improve the bounds. These bounds all require O(n²) multiplications.     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.     

Norms of powers of matrices with constrained spectra

If a matrix A of unit norm on n-dimensional Hilbert space has eigenvalues close to zero, then 6Aⁿ6 must be small. It is proved here that 6Aⁿ6 ?nr+O(r²), where r is the spectral radius of A, and the rate at which 6Aⁿ6 can approach 1 as r→1 is also ascertained [1?6Aⁿ6? O((1?r)^2n?1)]. More precise bounds for 6Aⁿ6 are obtained, some in terms of r and others in terms of all the eigenvalues of A.     

A subclass of parabolic starlike and uniformly convex functions

Let A be the class of analytic functions in the open unit disk U. A function f in A satisfying the normalization is said to be in the class SP_n if Dⁿf is a parabolic starlike function, where Dⁿ is a notation of the Salagean operator. In this paper, several basic properties and characteristics of the class SP_n are investigated. These include subordination, convolution properties, class-preserving integral operators, and Fekete-Szegö problems.     

On the logarithmic coefficients of Bazilevic? functions

The objective of the present paper is to study the logarithmic coefficients of Bazilevic? functions. We obtain the inequality ∣γ_n∣ ? An⁻¹logn (A is an absolute constant) which holds for Bazilevic? functions.     

On fixed point sets of homeomorphisms of then-ball

Conditions are investigated under which a subsetA can be the fixed point set of a homeomorphism ofB ⁿ . If eitherA ∩ ?B ⁿ ≠ Ø andn arbitrary orA ∩ ?B ⁿ =Ø andn even it is necessary and sufficient thatA is non-empty and closed. IfA ∩ ?B ⁿ =Ø andn odd, conditions which are either necessary or sufficient (but not both) are given.     

Defect indices of powers of a contraction

Let A be a contraction on a Hilbert space H. The defect index d_A of A is, by definition, the dimension of the closure of the range of I-A^∗A. We prove that (1) d_Aⁿ?nd_A for all n?0, (2) if, in addition, Aⁿ converges to 0 in the strong operator topology and d_A=1, then d_Aⁿ=n for all finite n,0?n?dimH, and (3) d_A=d_A^∗ implies d_Aⁿ=d_A^n∗ for all n?0. The norm-one index k_A of A is defined as sup{n?0:‖Aⁿ‖=1}. When dimH=m<∞, a lower bound for k_A was obtained before: k_A?(m/d_A)-1. We show that the equality holds if and only if either A is unitary or the eigenvalues of A are all in the open unit disc, d_A divides m and d_Aⁿ=nd_A for all n, 1?n?m/d_A. We also consider the defect index of f(A) for a finite Blaschke product f and show that d_f(A)=d_Aⁿ, where n is the number of zeros of f.     

A semilattice structure for Arens products

Let (A,?) be a Banach algebra. Then for n∈?, A ⁽²ⁿ⁾ has 2 ⁿ Arens products. In this paper we study the relations between the Arens products on A ⁽²ⁿ⁾. Moreover, if P _n (A) denotes the set of all Arens products on A ⁽²ⁿ⁾, for n∈?, we show that $P(A)=\bigcup_{n=1}^{\infty} P_{n}(A)$ is a ∧-semilattice. Also, we study P(A) as an infinite commutative semigroup and P(A)?{?} as a free semigroup generated by two elements. Then we investigate amenability and weak amenability for their semigroup Banach algebras.     

On Singular Points of Meromorphic Functions Determined by Continued Fractions

It is shown that Leighton’s conjecture about singular points of meromorphic functions represented by C-fractions K^∞_n=1(a _n z ^αn /1) with exponents α₁, α₂,... tending to infinity, which was proved by Gonchar for a nondecreasing sequence of exponents, holds also for meromorphic functions represented by continued fractions K^∞_n=1(a _n A _n (z)/1), where A₁,A₂,... is a sequence of polynomials with limit distribution of zeros whose degrees tend to infinity.     

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.     

On some questions concerning permanents of (1,−1)-matrices

Let Ω _n denote the set of alln×n-(1,?1)-matrices. E.T.H. Wang has posed the following problem: For eachn≧4, can one always find nonsingularA∈Ω _n such that |perA|=|detA| (*)? We present a solution forn≦6 and, more generally, we show that (*) does not hold ifn=2 ^k ?1,k≧2, even for singularA∈Ω _n . Moreover, we prove that perA≠0 ifA∈Ω _n ,n=2 ^k ?1, and we derive new results concerning the divisibility of the permanent in Ω _n by powers of 2.     

Lower bound inequalities for norms of symmetrized tensor powers

Let V be a complex inner product space of positive dimension m with inner product 〈·,·〉, and let T_n(V) denote the set of all n-linear complex-valued functions defined on V×V×?×V (n-copies). By S_n(V) we mean the set of all symmetric members of T_n(V). We extend the inner product, 〈·,·〉, on V to T_n(V) in the usual way, and we define multiple tensor products A₁⊗A₂⊗?⊗A_n and symmetric products A₁·A₂?A_n, where q₁,q₂,…,q_n are positive integers and A_i∈T_qi(V) for each i, as expected. If A∈S_n(V), then A^k denotes the symmetric product A·A?A where there are k copies of A. We are concerned with producing the best lower bounds for ‖A^k‖², particularly when n=2. In this case we are able to show that ‖A^k‖² is a symmetric polynomial in the eigenvalues of a positive semi-definite Hermitian matrix, M_A, that is closely related to A. From this we are able to obtain many lower bounds for ‖A^k‖². In particular, we are able to show that if ω denotes 1/r where r is the rank of M_A, and , then

     

Upper and lower bounds on norms of functions of matrices

Given an n by n matrix A, we look for a set S in the complex plane and positive scalars m and M such that for all functions p bounded and analytic on S and throughout a neighborhood of each eigenvalue of A, the inequalities

m·inf{‖f‖_L^∞(S):f(A)=p(A)}?‖p(A)‖?M·inf{‖f‖_L^∞(S):f(A)=p(A)}     

Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions

We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if Ω is any open set in Cd, and L is a suitable transfer operator acting on Bergman space A²(Ω), its eigenvalue sequence {λn(L)} is bounded by |λn(L)|?Aexp(−an1/d), where a,A>0 are explicitly given.