In the context of the degree/diameter problem, the ‘defect’ of a graph represents the difference between the corresponding Moore bound and its order. Thus, a graph with maximum degree d and diameter two has defect two if its order is n = d − 1. Only four extremal graphs of this type, referred to as (d, 2, 2)-graphs, are known at present: two of degree d = 3 and one of degree d = 4 and 5, respec...

A vanishing sum a0 + a1ζn + . . .+ an−1ζ n−1 n = 0 where ζn is a primitive n-th root of unity and the ai’s are nonnegative integers is called minimal if the coefficient vector (a0, . . . , an−1) does not properly dominate the coefficient vector of any other such nonzero sum. We show that for every c ∈ N there is a minimal vanishing sum of n-th roots of unity whose greatest coefficient is equal ...

We consider the problem of computing square root a perturbation scaled identity matrix, , where and are matrices with . This arises in various applications, including computer vision optimization methods for machine learning. derive new formula th that involves weighted sum powers matrix is particularly attractive root, since has just one term when also class Newton iterations exploit low-rank ...

In 1999, for (0≤k<∞), the concept of conic domain by defining k-uniform convex functions were introduced Kanas and Wisniowska then in 2000, they defined related k-starlike denoted k−UCV k−ST respectively. Motivated their studies, our work, we define class k-parabolic starlike functions, k−SHm,q, using quasi-subordination m-fold symmetric analytic making use Ωk. We determine coefficient bound...

Let G = GLn(q) be the general linear group of degree n ≥ 2 defined over a finite field Fq of characteristic p. We fix a prime l 6= p and let stand R for a local principal ideal domain having characteristic 0, maximal ideal lR, and containing a primitive p-th root of unity. Then the residue field K = R/lR has characteristic l and a primitive p-th root of unity. By a Steinberg lattice of G over R...

then σ is a singular value of A and u and v are corresponding left and right singular vectors, respectively. (For generality it is assumed that the matrices here are complex, although given these results, the analogs for real matrices are obvious.) If, for a given positive singular value, there are exactly t linearly independent corresponding right singular vectors and t linearly independent co...

Let E be an elliptic curve, and let Ln be the Kummer extension generated by a primitive pnth root of unity and a pn-th root of a for a fixed a ∈ Q − {±1}. A detailed case study by Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led these authors to predict unbounded and strikingly regular growth for the rank of E over Ln in certain cases. The aim of this note is to explain how some of th...

In this paper we give a complete classiication of the minimal cyclic M q (n)-modules and construct them explicitly. Also, we give a complete classiica-tion of the minimal cyclic modules of the so-called Dipper-Donkin quantum matrix algebra as well as of two other natural quantized matrix algebras. In the last part of the paper we relate the results to the De Concini { Procesi conjecture. 1. int...

We present a new cube root algorithm in finite field Fq with q a power of prime, which extends Cipolla-Lehmer type algorithms and has lower complexity than Tonelli-Shanks type algorithms. Efficient computation of r-th root in Fq has many applications in computational number theory and many other related areas. There are two standard algorithms for computing rth root in finite field. One is Adle...

Let Fq be the finite field with q elements and let ! be a primitive n-th root of unity in an extension eld Fqd of Fq. Given a polynomial P 2 Fq[x] of degree less than n, we will show that its discrete Fourier transform (P (1); P (!); :::; P (!n¡1)) 2Fqd n can be computed essentially d times faster than the discrete Fourier transform of a polynomial Q 2 Fqd[x] of degree less than n, in many case...