A t ? (v; m; k;) covering design is a pair (X; A), where X is a set of v elements (called points) and A is a multiset of k-subsets of X (called blocks), such that every m-subset of X intersects at least members of A in at least t points. It is required that v k t and v m t. Such a covering design gives an upper bound on C (v; m; k; t), the number of blocks in a minimal t ? (v; m; k;) covering d...

We notice that there is a 2-cover P of S 1 SO(3) (we denote this group D 1/2 ) by the group U(2): P sends a matrix z into a pair (det z,v). The matrix v here is the image of u under a standard covering map p from SU(2) onto SO(3), see (3.4) below. Finally, u (being a matrix from SU(2) ) is determined (up to a sign) from the decomposition z = du, here d 2 = det z. Both P and p are group homomorp...

For a graph G with vertex set V (G) = {v1, v2, . . . , vn}, let S be the covering set of G having the maximum degree over all the minimum covering sets of G. Let NS[v] = {u ∈ S : uv ∈ E(G)} ∪ {v} be the closed neighbourhood of the vertex v with respect to S. We define a square matrix AS(G) = (aij), by aij = 1, if |NS[vi] ∩NS[vj]| ≥ 1, i 6= j and 0, otherwise. The graph G associated with the mat...

Heat kernels are used in this paper to express the analytic index of projectively invariant Dirac type operators on Γ covering spaces of compact manifolds, as elements in the K-theory of certain unconditional completions of the twisted group algebra of Γ. This is combined with V. Lafforgue’s results in the untwisted case, to compute the range of the trace on the K-theory of these algebras, unde...

For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action i...

The minimum number of k-subsets out of a v-set such that each t-set is contained in at least one k-set is denoted by C (v; k; t). In this paper, a computer search for nding good such covering designs, leading to new upper bounds on C (v; k; t), is considered. The search is facilitated by predetermining automorphisms of desired covering designs. A stochastic heuristic search (embedded in the gen...

Let U, g, k and A be positive integers with u :::: k. A (k, A)-grOUp divisible covering design ((k, A)-GDCD) with type gU is a A-cover of pairs by k-tuples of a gu-set X with u holes of size g, which are disjoint and spanning. The covering number, C(k, A; gil), is the minimum number of blocks in a (k, A)-GDCD of type gUo In this paper, the detennination ofllie fimction C(3, A; gil) begun by [6]...

We pursue an analogy of the Schur-Weyl reciprocity for the spinor groups and pick up the irreducible spin representations in the tensor space ∆ ⊗⊗ k V . Here ∆ is the fundamental representation of Pin(N) and V is the natural (vector) representation of the orthogonal group O(N). We consider the centralizer algebra CPk = EndPin(N)(∆ ⊗⊗k V ) for Pin(N), the double covering group of O(N) and define...

These notes, from a first course in algebraic topology, introduce the fundamental group and the fundamental groupoid of a topological space and use them to classify covering spaces.