Let $ mathcal{H}(mathbb{D}) $ denote the space of analytic functions on the open unit disc $mathbb{D}$. For a weight $mu$ and a nonnegative integer $n$, the $n$'th weighted type space $ mathcal{W}_mu ^{(n)} $ is the space of all $fin mathcal{H}(mathbb{D}) $ such that $sup_{zin mathbb{D}}mu(z)left|f^{(n)}(z)right|begin{align*}left|f right|_{mathcal{W}_...

for every $1leq s< n$, the $s^{th}$ derivative of a polynomial $p(z)$ of degree $n$ is a polynomial $p^{(s)}(z)$ whose degree is $(n-s)$. this paper presents a result which gives generalizations of some inequalities regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle. besides, our result gives interesting refinements of some well-known results.

We present a short way of proving the inequalities obtained by Wright in [Journal of Graph Theory, 4: 393 – 407 (1980)] concerning the number of connected graphs with l edges more than vertices. 1. Preliminaries For n ≥ 0 and −1 ≤ l ≤ ( n 2 ) − n, let c(n, n + l) be the number of connected graphs with n vertices and n+ l edges. Quantifying c(n, n+ l) represents one of the fundamental tasks in t...

Abstract C 8 H 10 N 2 O , orthorhombic, Pbca (no. 61), a = 11.0571(2) Å, b 8.1172(2) c 17.6080(4) V 1580.37(6) Å 3 Z 8, R gt ( F ) 0.0324, wR ref 0.0872, T 100(2) K.

Abstract C 9 H 8 N 5 O CuI, orthorhombic, Pbca (no. 61), a = 11.0801(4) Å, b 6.8489(3) c 35.3975(13) Z 8, V 2686.20(18) Å 3 , R gt ( F ) 0.0325, wR ref 2 0.0707, T 292 K.

In this paper, upper bounds on the minimum Euclidean weights of Type I codes over Z6 , and self-dual codes over Z8 and Z10 , are derived for modest lengths. The notion of extremality for Euclidean weights is also introduced. We construct new extremal self-dual codes over these rings. Most of these codes are obtained via the double circulant and quasitwisted constructions. New extremal odd unimo...

1. Let Zk~ be the manifold of all nondegenerate complex k × n-matrices, k < n , let Gkn be the Grassmannian of k-dimensional subspaces in C~', and let ~r: Zkn -~ Gk, be the na tura l projection (7~(z) for z E Zk~ is the k-dimensional subspace of C ~ generated by the rows of z). For a matr ix z E Zkn denote by px(z) = pil...ik(z) the minor of z composed by the columns with indices from the set I...

Let Sw+2(Γ0(N)) be the vector space of cusp forms of weight w+2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2(Γ0(2)), and extend the Eichler-Shimura-Manin isomorphism theorem to Γ0(2). This implies that there are natu...

The Jacobi symbol satisfies many formulas that the Legendre symbol does, such as these: for a, b ∈ Z and odd m,n ∈ Z+, (1) a ≡ b mod n⇒ ( a n) = ( b n), (2) ( n ) = ( a n)( b n), (3) (−1 n ) = (−1) (n−1)/2 and ( 2 n) = (−1) (n2−1)/8, (4) ( n m) = (−1) (n ). But there is one basic rule about Legendre symbols that is not listed above for the Jacobi symbol: an analogue of Euler’s congruence a(p−1)...