The scheme LambdaCD deals with a non empty set A, a unary functor F yielding a set, a unary functor G yielding a set, and a unary predicate P, and states that: There exists a function f such that dom f = A and for every element x of A holds if P[x], then f(x) = F(x) and if not P[x], then f(x) = G(x) for all values of the parameters. The following propositions are true: (1) Let L be a non empty ...

Let G be a k-edge-connected graph and let L denote the subset of all vertices having odd degree in G. For every subset K = {u1, u2, . . . , uk} of L with |K| ≤ |L| 2 , and for every function h defined on K having the property that h(ui) ∈ ‰

A primitive multiple scheme is a Cohen-Macaulay $Y$ such that the associated reduced $X=Y_{red}$ smooth, irreducible, and can be locally embedded in smooth variety of dimension $\dim(X)+1$. If $n$ multiplicity $Y$, there canonical filtration $X=X_1\subset X_2\subset\cdots\subset X_n=Y$, $X_i$ $i$. The simplest example trivial to line bundle $L$ on $X$: it $n$-th infinitesimal neighborhood $X$, ...

Let K be a locally compact hypergroup. In this paper we initiate the concept of fundamental domain in locally compact hypergroups and then we introduce the Borel section mapping. In fact, a fundamental domain is a subset of a hypergroup K including a unique element from each cosets, and the Borel section mapping is a function which corresponds to any coset, the related unique element in the fun...

Let G be a transitive permutation group on a set ? and let m be a positive integer. If no element of G moves any subset of ? by more than m points, then |? | [2mp I (p-1)] wherep is the least odd primedividing |G |. When the bound is attained, we show that | ? | = 2 p q ….. q where ? is a non-negative integer with 2 < p, r 1 and q is a prime satisfying p < q < 2p, ? = 0 or 1, I i n....

Given positive integers a1, . . . , an, t, the fixed weight subset sum problem is to find a subset of the ai that sum to t, where the subset has a prescribed number of elements. It is this problem that underlies the security of modern knapsack cryptosystems, and solving the problem results directly in a message attack. We present new exponential algorithms that do not rely on lattices, and henc...

David Brydges and Thomas Spencer's Lace Expansion is abstracted, and it is shown how it sometimes gives rise to sieves. LACES Definition: Let P be a finite set of properties. A mapping l that assigns to any subset S ⊂ P another subset l(S), is called a lace-map, A set L for which l(L) = L is called a lace. By applying (ii) to G = l(S), it is seen that l(l(S)) = l(S), for any set of properties S...

let g be a transitive permutation group on a set ? and let m be a positive integer. if no element of g moves any subset of ? by more than m points, then |? | [2mp i (p-1)] wherep is the least odd primedividing |g |. when the bound is attained, we show that | ? | = 2 p q ….. q where ? is a non-negative integer with 2 < p, r 1 and q is a prime satisfying p < q < 2p, ? = 0 or 1, i i n. furthermore...

Given an infinite group G and a subset A of G we let ∆(A) = {g ∈ G : |gA ∩ A| = ∞} (this is sometimes called the combinatorial derivation of A). A subset A of G is called: large if there exists a finite subset F of G such that FA = G; ∆-large if ∆(A) is large and small if for every large subset L of G, (G \ A) ∩ L is large. In this note we show that every nonsmall set is ∆-large, answering a qu...