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"الكيمياء غير العضوية الحيوية"
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الكيمياء الشيقة
التفاعلات الكيميائية | الصناعات الكيميائية | التركيب الكيميائي | الكيميائيون | العناصر | المعادن.
eBook
A Note on E-Self-Projective Modules
E-self-projectivity is a generalization of self-projectivity. In this paper, we give several properties of e-self-projective modules, and discuss the question of when an e-self-projective module is projective or self-projective .
Journal Article
When GP - injective Rings are P - injective
The aim of this paper is to investigate principally and general principally injective rings satisfying additional conditions. A ring R is called left P -injective if for every a R, aR = rl(a) where L(.) and r(.) denote left and right annihilators respectively. The ring R is called left GP-injective if for any 0≠ a R, there exists n > 0 such that an ≠ 0 and anR = rl(an). As a response to an open question on GP-injective rings, an example of a lef GP-injective ring which is not left P -injective is given. Various results are developed, many extending known results.
Journal Article
On The Excess of Hadamard Matrix
A (-1, 1) -matrix is a matrix whose only entries are the numbers -1 or 1. In this paper for the most part we will be interested in special (-1,1) -matrices called Hadamard matrices. A Hadamard matrix of order n is an n x n (-1,1) - matrix H, satisfying H'H = H'H = nIn, where H' denotes the transpose of H and In is the identity matrix of order n. If H is a Hadamard matrix of order n, let w(H) = number of plus ones in H and let σ (H) = sum of all the entries of H. The numbers w(H) and σ (H) are called the weight of H and the excess of H, respectively. Further let w(n)= max { w(H) :H EΩ (n) } and σ (n)= max { σ (H) : H EΩ(n) } where Ω(n) is the class of all Hadamard matrices of order n. We call w(n) and σ (n) the maximum weight and the maximum excess of the class Q(n), respectively. The functions w and a were first introduced by Schmidt (1973) and subsequently studied by Schmidt and Wang (1977), Best (1977), Enomoto and Miyamoto (1980), Hammer, Levingston, Seberry (1978), and many other authors. The purpose of this paper is to report on what progress has been made on the maximum excess problem or equivalently the problem of maximum weight. In this paper, we first derive the relationship between w(H) and σ (H) as well as between w(n) and σ (n). The paper then proceeds to elaborate on the papers by Schmidt and Wang (1977) and Best (1977). Perhaps a key and most useful result in this paper is the inequality σ (n) ≤ n√n obtained by Best (1977). We conclude this paper by giving some results and examples of Hadamard matrices with maximum excess.
Journal Article