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In the first section the Weil conjectures for non-singular primals are stated and several examples are given. Particularities for curves are described in section two. The remaining sections are devoted to elliptic cubic curves. In particular, the number of points that a cubic can have is precisely given, as well as the number of inequivalent curves with a fixed number of points.
Let M, N be any disjoint subsets of the vertex set of the graph G with ‖M‖=m and ‖N‖=n. We say that G ε C(m, n) if there is a cycle K in G such that M $$\subseteq$$ VK and N∩VK=φ. If g Is k-connected, then it is an old result of Dirac that G ε C(k, 0). It is easy to produce k-connected graphs which are not C(k+1, 0). Hence the best we can hope of an arbitrary k-connected graph is that it is...
Recently, a number of new connections between complete mappings, sequencings of groups, and the construction of neofields and block designs have come to light. Also, some progress has been made in determining classes of groups which are sequenceable or R-sequenceable. We survey these results, point out their inter-connections and indicate some unsolved problems.
The problem of covering and partitioning the edge set of a simple graph with a minimum number of complete subgraphs has been studied by several writers over the years. This paper surveys some of the progress made so far and presents a number of open problems.
If a Room square of side s contains a Room subsquare of side t, then s≥3t+2. For t=3 or 5, there is no Room square of side t, yet one can construct (incomplete) Room squares of side s "missing" subsquares of side 3 or 5 (the same bound s≥3t+2 holds). It has been conjectured that if s and t are odd, s≥3t+2 and (s,t) ⊄ (5,1), then there exists a Room square of side s containing (or...
We survey recent results on the embedding of generalized quadrangles, partial geometries, semi partial geometries, and (0,α)-geometries in the finite projective space PG(n,q).
We refine the Paige-Wexler canonical form for incidence matrices of finite projective planes and thus obtain a simple relationship between the incidence matrix and a corresponding planar ternary ring. We also demonstrate a simple relationship between an incidence matrix and a corresponding set of mutually orthogonal latin squares.
The clique covering number of a graph is the smallest number of complete subgraphs needed to cover its edge-set. For each n, we determine the set of those integers which are clique covering numbers of connected, cubic graphs on n vertices. The analogous result for 4-regular graphs is stated.
The relative merits of teams involved in a competition can be expressed by a partial ordering on the teams. The result of a round of the competition can be expressed by a multiplication of partial orderings, satisfying certain axioms. Several possible multiplications are described, and some properties of these multiplications displayed.
Deciding whether a triple system with λ=3 can be decomposed, or partitioned, into triple systems with smaller λ is NP-complete; this contrasts with the polynomial time algorithm for decomposing triple systems with λ=2. Two extensions of this result are examined here — to triple systems without repeated blocks, and to triple systems with λ=4.
For j=1, ..., r, let hj and kj be integers such that 0≤hj≤kj−1 and ωj=exp [2πi / kj]. Then, the number of ways of placing n≥0 different balls into r distinct cells such that, for j=1, ..., r, the number of balls in the jth cell is congruent to hj modulo kj, is $$\left( {\begin{array}{*{20}c}r \\{\Pi k_j } \\{j = 1} \\\end{array}...
We define a group Hadamard matrix to be a generalised Hadamard matrix whose rows form a group. We show that for abelian groups G, only group Hadamard matrices of type ps for Cp×...×Cp exist. We also show that the matrices for Cp of each possible order are unique up to equivalence. We indicate a connection between strongly independent sets and row group Hadamard matrices. We show that...
A labelled coloured bipartite graph, of LCBG, is a bipartite (simple) graph whose vertices have been 2-coloured and the vertices of each colour labelled independently. It is shown that for fixed r⩾3 the proportion of r-regular LCBGs on 2n vertices which are r-connected approaches 1 as n → ∞. Also, fix r⩾3 and q>0; let g=max(4,2{q/(2(r−2))}). Then the numbers of the following types of r-regular...
The concept of a system of orthogonal designs enables many of the construction techniques of orthogonal design theory to be unified and generalized by one theorem.
We generalize the results of paper [1] on systems of orthogonal designs to a theory of systems of general higher dimensional designs and present several new constructions for such designs.
According to the definition of the famous Italian geometer, Beniamino Segre, an oval of a finite projective plane is a maximal sized set of points, no three of which are collinear. One of the most influential theorems of finite geometry has been Segre's 1954 result that every oval of a finite Desarguesian plane of odd order is an irreducible conic. In 1957 and 1962 he showed that the even order case...
Despite the existence of such salient successes as the use of ergodic theory to establish Szemerédi's theorem, dynamical systems and stochastic processes have not been widely employed in the derivation of combinatoric results. This paper offers instances of what the authors believe should be a more widespread tool in the routine analysis of combinatoric problems.
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