On the independence of expansions of algebraic numbers in an integer base

Let b 2 be an integer. According to a conjecture of ÉmileBorel, the b-adic expansion of any irrational algebraic numberbehaves in some respects ‘like a random sequence’.We give a contribution to the following related problem: let and ' be irrational algebraic numbers, then prove that theirb-adic expansions either have the same tail, or behave in somerespects ‘like independent random sequences’.     

On the stability of some integer programming algorithms

In the present paper we develop our approach for studying the stability of integer programming problems. We prove that the L-class enumeration method is stable on integer linear programming problems in the case of bounded relaxation sets [9]. The stability of some cutting plane algorithms is discussed.     

An expansion for the number of partitions of an integer

The Ramanujan Journal - We consider p(n) the number of partitions of a natural number n, starting from an expression derived by Báez-Duarte in (Adv Math 125(1):114–120, 1997) by relating...     

On the metric theory of the nearest integer continued fraction expansion

Supposek _n denotes either (n) or (p _n) (n=1,2,...) where the polynomial maps the natural numbers to themselves andp _k denotes thek ^th rationals prime. Also let denote the sequence of convergents to a real numberx and letc _n(x)) _n=1 be the corresponding sequence of partial quotients for the nearest integer continued fraction expansion. Define the sequence of approximation constants _n(x)) _n=1 by

On an exponential sum

Let p be a prime number, n be a positive integer, and ƒ(x) = ax^k + bx. We put

Rationality of exponential functions at integer arguments

Whereas exponential equations have been widely studied, we will deal with integer arguments where an exponential function assumes rational values.     

Existence and optimality of w-non-adjacent forms with an algebraic integer base

We consider digit expansions in lattices with endomorphisms acting as base. We focus on the w-non-adjacent form (w-NAF), where each block of w consecutive digits contains at most one non-zero digit. We prove that for sufficiently large w and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a w-NAF. If the eigenvalues of the endomorphism are large enough and w is sufficiently large, then the w-NAF is shown to minimise the weight among all possible expansions of the same lattice element using the same digit system.     

On some congruence with application to exponential sums

We will study the solution of a congruence,x ≡g ^1/2)ωg(2 ⁿ ) mod 2 ⁿ , depending on the integersg andn, where ω _g (2 ⁿ ) denotes the order ofg modulo 2 ⁿ . Moreover, we introduce an application of the above result to the study of an estimation of exponential sums.     

On some semilinear elliptic equation involving exponential growth

In this paper, we establish the existence of at least two nontrivial solutions for some semilinear elliptic equation involving a nonlinearity term having a critical exponential growth condition. Our main argument is critical point theory.     

On the number of hypercubic bipartitions of an integer

For n≤2^k$n\le 2^k$ we study the maximum number of edges of an induced subgraph on n$n$ vertices of the k$k$-dimensional hypercube _Qk$Q_k$. In the process we revisit a well-known divide-and-conquer maximin recurrence f(n)=max(min(_n1,_n2)+f(_n1)+f(_n2))$f\left(n\right)=max(min(n_1,n_2)+f\left(n_1\right)+f\left(n_2\right))$ where the maximum is taken over all proper bipartitions n=_n1+_n2$n=n_1+n_2$. We first use known results to present a characterization of those bipartitions n=_n1+_n2$n=n_1+n_2$ that yield the maximum f(n)=min(_n1,_n2)+f(_n1)+f(_n2)$f\left(n\right)=min(n_1,n_2)+f\left(n_1\right)+f\left(n_2\right)$. Then we use this characterization to present the main result of this article, namely, for a given n∈N$n\in \mathbb{N}$, the determination of the number h(n)$h\left(n\right)$ of these bipartitions that yield the said maximum f(n)$f\left(n\right)$. We present recursive formulae for h(n)$h\left(n\right)$, a generating function h(x)$h\left(x\right)$, and an explicit formula for h(n)$h\left(n\right)$ in terms of a special representation of n$n$.     

On the correlation structure of exponential order statistics and some extensions

In this paper, we carry out a theoretical study of the correlation coefficients between exponential order statistics and their monotonicity properties. Then, aided by a Monte Carlo simulation study, we make some conjectures about the correlation structure of order statistics from a larger class of distributions.     

On an integral transformation for integer programming

It is shown that the transformation of Oguz, of an integer programming problem into an equivalent problem with free variables, is more simply achieved through generalized inverses. Further, generalized inverses offer tests of consistency and feasibility.     

On digit sums of multiples of an integer

Let g>1 be an integer and sg(m) be the sum of digits in base g of the positive integer m. In this paper, we study the positive integers n such that sg(n) and sg(kn) satisfy certain relations for a fixed, or arbitrary positive integer k. In the first part of the paper, we prove that if n is not a power of g, then there exists a nontrivial multiple of n say kn such that sg(n)=sg(kn). In the second part of the paper, we show that for any K>0 the set of the integers n satisfying sg(n)?Ksg(kn) for all k∈N is of asymptotic density 0. This gives an affirmative answer to a question of W.M. Schmidt.     

On the construction of irredundant coverings of an integer matrix

New estimates are derived for the computational complexity of the problem of constructing irredundant coverings of an integer matrix (search for maximal conjunctions of a special logical function).     

On the integer points on some special hyper-elliptic curves over a finite field

If $\text{l}$_r(p) is the least positive integral value of x for which y² ≡ x(x + 1) ? (x + r ? 1)(modp) has a solution, we conjecture that $\text{l}$_r(p) ≤ r² ? r + 1 with equality for infinitely many primes p. A proof is sketched for r = 5. A further generalization to y² ≡ (x + a₁) ? (x + a_r) is suggested, where the a's are fixed positive integers.     

On the obstruction to the existence of an exponential basis

An obstruction is constructed to the existence of an exponential basis for an integral extension of a Dedekind ring. This obstruction provides various 3 examples of integral extensions of the ring k[x] that do not have an exponential basis over k[x], where k is an arbitrary field. Translated from Zapiski Nauchykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 198, pp. 89–91, 1991.