Long binary sequences related to cyclic difference sets are investigated. Among all known constructions of cyclic difference sets we show that only sequences constructed from Hadamard difference sets can have an asymptotic nonzero merit factor. Maximal length shift register sequences, Legendre, and twin-prime sequences are all constructed from Hadamard difference sets. We prove that the asympto...

Abstract We prove that an n-component fibered link L in S3 is strongly quasi-positive if and only $\tau(L)=g_3(L)+n-1$, where $g_3(L)$ denotes the Seifert genus τ Ozsváth–Szabó concordance invariant. also provide a table which contains list of some prime links with at most 9 crossings; we explicitly determine ones are their maximal self-linking number.

A frequency square is a matrix in which each row and column permutation of the same multiset symbols. type $$(n;\lambda )$$ if it contains $$n/\lambda $$ symbols, occurs $$\lambda times per column. In case when =n/2$$ we refer to as binary. set k-MOFS k squares such that any two are superimposed, possible ordered pair equally often. k-maxMOFS not contained $$(k+1)$$ -MOFS . For even n, let $$\m...

Given a connected graph G = ( V , E ) $G=(V,E)$ and length function ℓ : → R $\ell :E\to {\mathbb{R}}$ we let d v w ${d}_{v,w}$ denote the shortest distance between vertex $v$ $w$ . A t $t$ -spanner is subset ′ ⊆ $E^{\prime} \subseteq E$ such that if ${d}_{v,w}^{^{\prime} }$ denotes distances in subgraph $G^{\prime} =(V,E^{\prime} )$ then ≤ }\le t{d}_{v,w}$ for all ∈ $v,w\in V$ We study size of ...

Fix α > 0, and sample N integers uniformly at random from { 1, 2, . . . , ⌊ eαN ⌋} . Given η > 0, the probability that the maximum of the pairwise GCDs lies between N2−η and N2+η converges to 1 as N → ∞. More precise estimates are obtained. This is a Birthday Problem: two of the random integers are likely to share some prime factor of order N2 / log(N). The proof generalizes to any arithmetical...

Theorem 1: Let un, n > 0, be the general term of a given sequence of integers and define the transformation T^yk)(un) as T^xyJc){un) = xun+lc +yun for every n > 0, A: being a positive integer. Then, if x mdy are nonzero integers and there exists a positive prime number/? which divides T(x,y,k)(n) f° every n>0 and is relatively prime to x, the distribution of the residues of (un) modulo p is eit...

In this paper, we define a subclass of analytic functions by denote $T_{\beta}H\left( z,C_{n}^{\left( \lambda \right) }\left( t\right) $ satisfying the following subordinate condition \begin{equation*} \left( 1-\beta \frac{zf^{^{\prime }}\left( z\right) }{f\left( }\right) +\beta 1+\frac{zf^{^{\prime \prime }{f^{^{\prime \prec \frac{1}{\left( 1-2tz+z^{2}\right) ^{\lambda }}, \end{equation*} wher...

Proof. Using that (Z/p)× is a cyclic group of order p − 1 (i.e. the existence of primitive roots), we see that there is a square root of −1 (that is, a non-trivial fourth root of 1) in (Z/p)× if and only if p ≡ 1 mod 4. Suppose now that p ≡ −1 mod 4, and suppose that α and β are two elements of Z[i] such that p|αβ. Then p = N(p)|N(α)N(β), and so (after relabelling if necessary) we may assume th...

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.