Some relations between additive functions-I

Aequationes mathematicae -     

On integral-valued additive functions,II

A pair [₁,f₂] of integral-valued arithmetical functions is said to be uniformly distributed modulo [q₁, q₂], where q₁ and q₂ are integers > 1, if, for every pair [r₁, r₂] of integers, the set of those positive integers n which satisfy

$\text{f}{}_1\text{(n)≡(}\text{mod}\text{q}{}_1\text{)}\text{and}\text{f}{}_2\text{(n)≡r}{}_2\text{(}\text{mod}\text{q}{}_2$

has density $\text{1}\text{(q}{}_1\text{q}{}_2\text{)}$.We give necessary and sufficient conditions for a pair of integral-valued additive functions to be uniformly distributed modulo [q₁, q₂] in the case when (q₁, q₂) > 1.     

Some relations between certain inequalities concerning analytic and univalent functions

By making use of the well-known assertions in [S.S. Miller, P.T. Mocanu, Differential Subordinations, Theory and Applications, Marcel Dekker, New York-Basel, 2000] and [M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A Math. Sci. 69 (7) (1993) 234–237], several useful results of certain inequalities dealing with analytic and univalent functions are obtained and then certain consequences of them are also indicated.     

On the monotonicity properties of additive representation functions, II

For a set A of positive integers and any positive integer n, let R₁(A,n), R₂(A,n) and R₃(A,n) denote the number of solutions of a+a^′=n with the additional restriction a,a^′∈A; a,a^′∈A,a<a^′ and a,a^′∈A,a≤a^′ respectively. In this paper, we specially focus on the monotonicity of R₃(A,n). Moreover, we show that there does not exist any set A⊂N such that R₂(A,n) or R₃(A,n) is eventually strictly increasing.     

Some classes of completely monotonic functions, II

A function is said to be completely monotonic if for all x > 0 and n = 0,1,2,.... In this paper we present several new classes of completely monotonic functions. Our functions have in common that they are defined in terms of the classical gamma, digamma, and polygamma functions. Moreover, we apply one of our monotonicity theorems to prove a new inequality for prime numbers. Some of the given results extend and complement theorems due to Bustoz & Ismail, Clark & Ismail, and other researchers. 2000 Mathematics Subject Classification Primary—11A41, 26A48, 33B15; Secondary—26D15     

Some continuous functions related to corner polyhedra,II

The group problem on the unit interval is developed, with and without continuous variables. The connection with cutting planes, or valid inequalities, is reviewed. Certain desirable properties of valid inequalities, such as minimality and extremality are developed, and the connection between valid inequalities for P(I, u ₀) and P _- ⁺ (I, u ₀) is developed. A class of functions is shown to give extreme valid inequalities for P _- ⁺ (I, u ₀) and for certain subsetsU ofI. A method is used to generate such functions. These functions give faces of certain corner polyhedra. Other functions which do not immediately give extreme valid inequalities are altered to construct a class of faces for certain corner polyhedra. This class of faces grows exponentially as the size of the group grows.     

Some asymptotic formulas on generalized divisor functions,II

Let A be an infinite sequence of positive integers a₁ < a₂ <… and put $\text{f}{}_{\mathrm{A}}\text{(x) =}\text{Σ}{}_{\mathrm{a\in A,\; a\le x}}\text{(}\text{1}\text{a}\text{)}$, $\text{D}{}_{\mathrm{A}}\text{(x) =}\text{max}{}_{\mathrm{1\le n\le x}}\text{Σ}{}_{\mathrm{<mtext>a\in A,</mtext><mtext>a</mtext><mtext>n</mtext>}}\text{1}$. In Part I, it was proved that $\text{lim}{}_{\mathrm{x\to +\infty }}\text{sup}\text{D}{}_{\mathrm{A}}\text{(x)}\text{f}{}_{\mathrm{A}}\text{(x)}\text{= +∞}$. In this paper, this theorem is sharpened by estimating D_A(x) in terms of f_A(x). It is shown that lim_x→+∞sup D_A(x) exp(?c₁(logf_A(x))²) = +∞ and that this assertion is not true if c₁ is replaced by a large constant c₂.     

Some diophantine relations involving circular functions of rational angles

The eigenvalues of the 3 off-diagonal matrices of rank n with elements $\text{1}\text{+i}\text{cot}\text{[(j?k)}\text{π}\text{n}\text{],}\text{sin}{}^{-2}\text{[(j?k)}\text{π}\text{n}\text{]}$and $\text{sin}{}^{-4}\text{[(j?k)}\text{π}\text{n}\text{] (j=}\text{1,2,…,}\text{n,k=}\text{1,2,…,}\text{n,j≠k)}$, are computed. The sums over k from 1 to n?1 of $\text{cot(}\text{kπ}\text{n}\text{)sin(}\text{2skπ}\text{n}\text{)}$ and$\text{sin}{}^{\mathrm{-p}}\text{(}\text{kπ}\text{n}\text{)cos(}\text{2skπ}\text{n}\text{)}$ are moreover computed for s integer and p=2 and 4. The results are given by simple formulae in terms of integers.     

Some relations between Hausdorff-dimensions and entropies

Some relations between the Hausdorff-dimensions and the entropies for expansive homeomorphisms on compact metric spaces are obtained. Project partially supported by the National Natural Science Foundation of China.