Incomplete Character Sums and Polynomial Interpolation of the Discrete Logarithm

In the first part of the paper, certain incomplete character sums over a finite field F_p^r are considered which in the case of finite prime fields F_p are of the form ∑^A+N−1_n=Aχ(g(n))ψ(f(n)), where A and N are integers with 1≤N<p, g and f are polynomials over F_p, and χ denotes a multiplicative and ψ an additive character of F_p. Excluding trivial cases, it is shown that the above sums are at most of the order of magnitude N^1/2p^r/4. Recently, Shparlinski showed that a polynomial f over the integers which coincides with the discrete logarithm of the finite prime field F_p for N consecutive elements of F_p must have a degree at least of the order of magnitude Np^−1/2. In this paper this result is extended to arbitrary F_p^r. The proof is based on the above new bound for incomplete hybrid character sums.     

Compositions of analytic functions of the form Fn(z) = Fn−1(fn(z)), fn(z) → f(z)

Frequently, in applications, a function is iterated in order to determine its fixed point, which represents the solution of some problem. In the variation of iteration presented in this paper fixed points serve a different purpose. The sequence {F_n(z)} is studied, where F₁(z) = f₁(z) and F_n(z) = F_n−1(f_n(z)), with f_n → f. Many infinite arithmetic expansions exhibit this form, and the fixed point, α, of f may be used as a modifying factor (z = α) to influence the convergence behaviour of these expansions. Thus one employs, rather than seeks the fixed point of the function f.     

On some isometries on the Bergman-Privalov class on the unit ball

Bergman-Privalov class AN_α(B) consists of all holomorphic functions on the unit ball B⊂Cⁿ such that

‖f‖_ANα:=∫_Bln(1+∣f(z)∣)dV_α(z)<∞,     

Embedding cycles of various lengths into star graphs with both edge and vertex faults

The n-dimensional star graph S_n is an attractive alternative to the hypercube graph and is a bipartite graph with two partite sets of equal size. Let F_v and F_e be the sets of faulty vertices and faulty edges of S_n, respectively. We prove that S_n − F_v − F_e contains a fault-free cycle of every even length from 6 to n! − 2∣F_v∣ with ∣F_v∣ + ∣F_e∣ ? n − 3 for every n ? 4. We also show that S_n − F_v − F_e contains a fault-free path of length n! − 2∣F_v∣ − 1 (respectively, n! − 2∣F_v∣ − 2) between two arbitrary vertices of S_n in different partite sets (respectively, the same partite set) with ∣F_v∣ + ∣F_e∣ ? n − 3 for every n ? 4.     

A characterization of semicontinuity-preserving multifunctions

We give a necessary and sufficient condition on a multifunction Γ that the function inf_u?Γ(x)f(u) be lower semicontinuous for all f from an arbitrary class F. The condition is then concretized for important special classes F.     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

The number of polynomial functions which permute the matrices over a finite field

Let F = GF(q) denote the finite field of order q, and let F_n×n denote the algebra of n × n matrices over F. A function f:F_n×n → F_n×n is called a scalar polynomial function if there exists a polynomial f(x) ?F[x] which represents f when considered as a matrix function under substitution. In this paper a formula is obtained for the number of permutations of F_n×n which are scalar polynomial functions.     

On pointwise approximation of arbitrary functions by countable families of continuous functions

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyf _n :n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F?X and every real number ε>0 one can choosen∈ω such that ∥f(x)?f_n(x)∥<ε for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A?X, there exists a continuous mapf _A:X→R ^ω such that A=f _A ^?1 (A). Splitting spaces will be studied systematically.     

Scalar polynomial functions on the n×n matrices over a finite field

Let F=GF(q) denote the finite field of order q, and let $\text{?(x)?F[x]}$. Then f(x) defines, via substitution, a function from F_n×n, the n×n matrices over F, to itself. Any function $\text{?:F}{}_{\mathrm{n\times n}}\text{→ F}{}_{\mathrm{n\times n}}$ which can be represented by a polynomialf(x)?F[x] is called a scalar polynomial function on F_n×n. After first determining the number of scalar polynomial functions on F_n×n, the authors find necessary and sufficient conditions on a polynomial $\text{?(x) ? F[x]}$ in order that it defines a permutation of (i) $\text{D}$_n, the diagonalizable matrices in F_n×n, (ii)$\text{R}$_n, the matrices in F_n×n all of whose roots are in F, and (iii) the matric ring F_n×n itself. The results for (i) and (ii) are valid for an arbitrary field F.     

Certain sufficient conditions of uniformity for systems of functions of many-valued logic

For any finite system A of functions of the k-valued logic taking values in the set E _s = {0,1,…, s ? 1}, k ≥ s ≥ 2, such that the closed class generated by restriction of functions from A on the set E _s contains a near-unanimity function, it is proved that there exist constants c and d such that for an arbitrary function f ∈ [A] the depth D _A (f) and the complexity L _A (f) of f in the class of formulas over A satisfy the relation D _A (f) ≤ clog₂ L _A (f) + d.     

Several spectrally arbitrary ray patterns

A ray pattern A of order n is said to be spectrally arbitrary if given any monic nth degree polynomial f(x) with coefficients from ?, there exists a matrix realization of A such that its characteristic polynomial is f(x). An n?×?n ray pattern A is said to be minimally spectrally arbitrary if replacing any nonzero entry of A by zero destroys this property. In this article, several families of ray patterns are presented and proved to be minimally spectrally arbitrary. We also show that for n?≥?5, when A _n is spectrally arbitrary, then it is minimally spectrally arbitrary.     

Spectrally arbitrary ray patterns

An n×n ray pattern A is said to be spectrally arbitrary if for every monic nth degree polynomial f(x) with coefficients from C, there is a matrix in the pattern class of A such that its characteristic polynomial is f(x). In this article the authors extend the nilpotent-Jacobi method for sign patterns to ray patterns, establishing a means to show that an irreducible ray pattern and all its superpatterns are spectrally arbitrary. They use this method to establish that a particular family of n×n irreducible ray patterns with exactly 3n nonzeros is spectrally arbitrary. They then show that every n×n irreducible, spectrally arbitrary ray pattern has at least 3n-1 nonzeros.     

On systems of finite sets with constraints on their unions and intersections

Let F be a family of subsets of an n-element set. F is said to be of type (n, r, s) if A ∈ F, B ∈ F implies that |A ∪ B| ? n ? r, and |A ∩ B| ? s. Let f(n, r, s) = max {|F| : F is of type (n, r, s)}. We prove that f(n, r, s) ? f(n ? 1, r ? 1, s) + f(n ? 1, r + 1, s) if r > 0, n > s. And this result is used to give simple and unified proofs of Katona's and Frankl's results on f(n, r, s) when s = 0 and s = 1.     

Comparison of the best uniform approximations of analytic functions in the disk and on its boundary

Denote by C _A the set of functions that are analytic in the disk |z| < 1 and continuous on its closure |z| ≤ 1; let ? _n , n = 0, 1, 2, ..., be the set of rational functions of degree at most n. Denote by R _n (f) (R _n (f) _A ) the best uniform approximation of a function f ∈ C _A on the circle |z| = 1 (in the disk |z| ≤ 1) by the set ? _n . The following equality is proved for any n ≥ 1: sup{R _n (f) _A /R _n (f): f ∈ C _A ? ? _n } = 2. We also consider a similar problem of comparing the best approximations of functions in C _A by polynomials and trigonometric polynomials.     

On superpositions of continuous functions

We show that if Φ is an arbitrary countable set of continuous functions of n variables, then there exists a continuous, and even infinitely smooth, function ψ(x₁,...,x_n) such that ψ(x ₁, ...,x _n ) ?g [? (f ₁(x ₁, ... ,f _f (x _n ))] for any function ? from Φ and arbitrary continuous functions g and f_i, depending on a single variable.     

On the distribution of values of certain divisor functions

Let {?_d} be a sequence of nonnegative numbers and f(n) = Σ?_d, the sum being over divisors d of n. We say that f has the distribution function F if for all c ≥ 0, the number of integers n ≤ x for which f(n) > c is asymptotic to xF(c), and we investigate when F exists and when it is continuous.     

Inequalities between ∥f(A + B)∥ and ∥f(A) + f(B)∥

The conjecture posed by Aujla and Silva [J.S. Aujla, F.C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003) 217-233] is proved. It is shown that for any m-tuple of positive-semidefinite n × n complex matrices A_j and for any non-negative convex function f on [0, ∞) with f(0) = 0 the inequality ?f(A₁) + f(A₂) + ? + f(A_m)? ? ? f(A₁ + A₂ + ? + A_m)? holds for any unitarily invariant norm ? · ?. It is also proved that ?f(A₁) + f(A₂) + ? + f(A_m)? ? f(?A₁ + A₂ + ? + A_m?), where f is a non-negative concave function on [0, ∞) and ? · ? is normalized.     

Asymptotic estimates of sums involving the Moebius function

Let p(n) denote the smallest prime factor of an integer n>1 and let p(1)=∞. We study the asymptotic behavior of the sum M(x,y)=Σ_{1≤n≤x,p(n)>y}μ(n) and use this to estimate the size of A(x)=max_|f|≤1|Σ_2≤n<xμ(n)f(p(n))|, where μ(n) is the Moebius function. Applications of bounds for A(x), M(x,y) and similar quantities are discussed.     

Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

Let F_q be the finite field of q elements with characteristic p and F_q^m its extension of degree m. Fix a nontrivial additive character Ψ of F_p. If f(x₁,…, x_n)∈F_q[x₁,…, x_n] is a polynomial, then one forms the exponential sum S_m(f)=∑_{(x1,…,xn)∈(Fq^m)ⁿ}Ψ(Tr_Fq^m/Fp(f(x₁,…,x_n))). The corresponding L functions are defined by L(f, t)=exp(∑^∞_m=0S_m(f)t^m/m). In this paper, we apply Dwork's method to determine the Newton polygon for the L function L(f(x), t) associated with one variable polynomial f(x) when deg f(x)=4. As an application, we also give an affirmative answer to Wan's conjecture for the case deg f(x)=4.