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The purpose of this paper is to investigate the real quadratic number fields $Q(sqrt{d})$ which contain the specific form of the continued fractions expansions of integral basis element  where $dequiv 2,3( mod  4)$ is a square free positive integer. Besides, the present paper deals with determining the fundamental unit$$epsilon _{d}=left(t_d+u_dsqrt{d}right) 2left.right > 1$$and  $n_d$ and $m_d...

Let $n_q(k,d)$ denote the smallest value of $n$ for which there exists a linear $[n,k,d]$-code over the Galois field $GF(q)$. An $[n,k,d]$-code whose length is equal to $n_q(k,d)$ is called {em optimal}. In this paper we present some matrix generators for the family of optimal $[n,3,d]$ codes over $GF(7)$ and $GF(11)$. Most of our given codes in $GF(7)$ are non-isomorphic with the codes pre...

Let S be a fixed finite symmetric subset of SLd(Z), and assume that it generates a Zariski-dense subgroup G. We show that the Cayley graphs of πq(G) with respect to the generating set πq(S) form a family of expanders, where πq is the projection map Z → Z/qZ.

Quantitative measurements of fluorescein in the eye are widely used in studying the function of the bloodretinal barrier, but they also can be of value in measuring the flow rate of the tears and aqueous humor and the barriers offered by the corneal layers and the other anterior structures of the eye. Concentrations down to 10~ g/ml can be determined in the ocular humors, and this allows measur...

Abstract We consider a quiver with potential (QP) $(Q(D),W(D))$ and an iced (IQP) $(\overline {Q}(D), F(D), \overline {W}(D))$ associated Postnikov Diagram D prove that their mutations are compatible the geometric exchanges of . This ensures we may define QP $(Q,W)$ IQP {Q},F,\overline {W})$ for Grassmannian cluster algebra up to mutation equivalence. It shows is always rigid (thus nondegenerat...