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Let v be a positive integer and let K be a set of positive integers. A ( v,K, 1)-Mendelsohn design, which we denote briefly by (itv,K}, 1)-MD, is a pair ( X,B) where X is a υ-set (of points) and B is a collection of cyclically ordered subsets of X (called blocks) with sizes in the set K such that every ordered pair of points of X are consecutive in exactly one block of B. If for all t = 1, 2, ..., r}, every ordered pair of points of X are t-apart in exactly one block of B, then the ( v,K, 1)-MD is called an r-fold perfect design and denoted briefly by an r-fold perfect ( v,K, 1)-MD. If K = “ k” and r = k − 1, then an r-fold perfect ( v, “k”, 1)-MD is essentially the more familiar ( v, k, 1)- perfect Mendelsohn design, which is briefly denoted by ( v, k, 1)-PMD. In this paper, we investigate the existence of 4-fold perfect ( v, “5, 8”, 1)-Mendelsohn designs. [ABSTRACT FROM AUTHOR]

Let σ( k, n) be the smallest even integer such that each n–term positive graphic sequence with term sum at least σ( k, n) can be realized by a graph containing a clique of k + 1 vertices. Erdős et al. (Graph Theory, 1991, 439–449) conjectured that σ( k, n) = ( k −1)(2 n− k)+2. Li et al. (Science in China, 1998, 510–520) proved that the conjecture is true for k ≥ 5 and $$ n \geqslant {\left( {\begin{array}{*{20}c} {k} \\ {2} \\ \end{array} } \right)} + 3 $$ , and raised the problem of determining the smallest integer N( k) such that the conjecture holds for n ≥ N( k). They also determined the values of N( k) for 2 ≤ k ≤ 7, and proved that $$ {\left\lceil {\frac{{5k - 1}} {2}} \right\rceil } \leqslant N{\left( k \right)} \leqslant {\left( {\begin{array}{*{20}c} {k} \\ {2} \\ \end{array} } \right)} + 3 $$ for k ≥ 8. In this paper, we determine the exact values of σ( k, n) for n ≥ 2 k+3 and k ≥ 6. Therefore, the problem of determining σ( k, n) is completely solved. In addition, we prove as a corollary that $$ N{\left( k \right)} = {\left\lceil {\frac{{5k - 1}} {2}} \right\rceil } $$ for k ≥ 6. [ABSTRACT FROM AUTHOR]

Assume that m ⩾ 2, p is a prime number, (m,p(p - 1)) = 1, -1 ∉ (p) ⊂ (∤/m∤)* and [(∤/m∤)* : (p)] = 4. In this paper, we calculate the value of Gauss sum G(X) = Σx∈F*qX(x)ζT(x)P over ...q, where q = pf, f = φ(m)/4, X is a multiplicative character of ...q and T is the trace map from ...q to ...p. Under our assumptions, G(X) belongs to the decomposition field K of p in ℚ(ζm) and K is an imaginary quartic abelian number field. When the Galois group Gal(K/ℚ) is cyclic, we have studied this cyclic case in another paper: "Gauss sums of index four: (1) cyclic case" (accepted by Acta Mathematica Sinica, 2003). In this paper we deal with the non-cyclic case. [ABSTRACT FROM AUTHOR]

Finite dimensional modules over Weil algebras are investigated and corresponding gauge bundle functors, from the category of vector bundles into the category of fibered manifolds, are determined. The equivalence of the two definitions of gauge Weil functors is proved and a number of geometric examples is presented, including a new description of vertical Weil bundles. [ABSTRACT FROM AUTHOR]