Problems on Polytopes, Their Groups, and Realizations

The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete Geometry, to appear

Families of sequences with good family complexity and cross-correlation measure

In this paper we study pseudorandomness of a family of sequences in terms of two measures, the family complexity ($f$-complexity) and the cross-correlation measure of order $\ell$. We consider sequences not only on binary alphabet but also on $k$-symbols ($k$-ary) alphabet. We first generalize some known methods on construction of the family of binary pseudorandom sequences. We prove a bound on the $f$-complexity of a large family of binary sequences of Legendre-symbols of certain irreducible polynomials. We show that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order. Next, we present another family of binary sequences having high $f$-complexity and low cross-correlation measure. Then we extend the results to the family of sequences on $k$-symbols alphabet.Comment: 13 pages. Comments are welcome

Hyperbolic summation for the divisor function involving the integer part function

Let $\tau(n)$ be the function number of divisors of a positive integer $n\geq 1$ and let $\lfloor{.}\rfloor$ denotes the greatest integer function. In this paper, we use the process B of the Van der Corput method to give an asymptotic formula of the sum $$\sum_{n_1n_2\leq x}\tau(\lfloor\frac{x}{n_1n_2}\rfloor).$$Comment: All remarks are welcom

On χs-Orthogonal Matrices

In this paper we, introduced the concept of χs-orthogonal matrices and extended some results of Abaraetal, [3] in the context of secondary transpose