共查询到20条相似文献,搜索用时 46 毫秒
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Surinder K. Sehgal 《Journal of Number Theory》1974,6(2):124-127
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Marvin I. Knopp 《Journal of Number Theory》1980,12(1):2-9
If h, k ∈ Z, k > 0, the Dedekind sum is given by , with , . The Hecke operators Tn for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities (n ∈ Z+) , where (h, k) = 1, k > 0 and σ(n) is the sum of the positive divisors of n. Petersson had earlier proved (1) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (1) when n is prime. 相似文献
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This paper deals with the class of Q-matrices, that is, the real n × n matrices M such that for every q ∈ n×1, the linear complementarity problem , , has a solution. In general, the results are of two types. First, sufficient conditions are given on a matrix M so that M ∈ Q. Second, conditions are given so that M ? Q. 相似文献
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Jennifer Seberry Wallis 《Linear and Multilinear Algebra》2013,61(3):197-207
Recent advances in the construction of Hadamard matrices have depeaded on the existence of Baumert-Hall arrays and four (1, ?1) matrices A B C Dof order m which are of Williamson type, that is they pair-wise satisfy i) MNT = NMT , ∈ {A B C D} and ii) AAT + BBT + CCT + DDT = 4mIm . It is shown that Williamson type matrices exist for the orders m = s(4 ? 1)m = s(4s + 3) for s∈ {1, 3, 5, …, 25} and also for m = 93. This gives Williamson matrices for several new orders including 33, 95,189. These results mean there are Hadamard matrices of order i) 4s(4s ?1)t, 20s(4s ? 1)t,s ∈ {1, 3, 5, …, 25}; ii) 4s(4:s + 3)t, 20s(4s + 3)t s ∈ {1, 3, 5, …, 25}; iii) 4.93t, 20.93t for t ∈ {1, 3, 5, … , 61} ∪ {1 + 2 a 10 b 26 c a b c nonnegative integers}, which are new infinite families. Also, it is shown by considering eight-Williamson-type matrices, that there exist Hadamard matrices of order 4(p + 1)(2p + l)r and 4(p + l)(2p + 5)r when p ≡ 1 (mod 4) is a prime power, 8ris the order of a Plotkin array, and, in the second case 2p + 6 is the order of a symmetric Hadamard matrix. These classes are new. 相似文献
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Edward Spence 《Discrete Mathematics》1977,18(1):79-85
Szekeres has established the existence of a skew-Hadamard matrix of order 2(q + 1) in the case q ≡ 5 (mod 8), a prime power. His method utilized complementary difference sets in the elementary abelian group of order q. The main result of this paper is to show that, for the same q, there exist skew-Hadamard matrices of order 2(q + 1) that are of the Goethals-Seidel type. This is achieved by using a cyclic relative difference set with parameters . 相似文献
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Jennifer Seberry 《Graphs and Combinatorics》1986,2(1):81-87
It is shown that ifq is a prime power then there are Williamson-type matrices of order
- 1/2q 2(q + 1) whenq ≡ 1 (mod 4).
- 1/4q 2(q + 1) whenq ≡ 3 (mod 4) and there are Williamson-type matrices of order 1/4(q + 1).