Let S be closed, m-convex subset of R d S locally a full ddimensional, with Q the corresponding set of Inc points of S If q is an essential inc point of order k then for some neighborhood U of q Q u is expressible as a union of k or fewer (d2)-dimenslonal manifolds, each containing q For S compact, if to every q E Q there corresponds a k > 0 such that q is an essential inc point of order k then...

The qk (full) factorial design with replication λ is the multi-set consisting of occurrences each element q-ary vector length k; we denote this by λ×[q]k. An m×n row-column strength t an arrangement elements λ×[q]k into array (which say type Ik(m,n,q,t)) such that for row (column), set vectors therein are rows orthogonal degree k, size n (respectively, m), q levels and t. Such arrays used in ex...

A graph G is a (Kq , k) vertex stable graph (q ≥ 3) if it contains a Kq after deleting any subset of k vertices (k ≥ 0). We are interested by the (Kq , κ(q)) vertex stable graphs of minimum size where κ(q) is the maximum value for which for every nonnegative integer k < κ(q) the only (Kq , k) vertex stable graph of minimum size is Kq+k.

Let X and Y be finite-type CW-spaces (X connected, Y simply connected), such that the ring H∗(Y,Q) is a k-rescaling of H∗(X,Q). If H∗(X,Q) is a Koszul algebra, then the graded Lie algebra π∗(ΩY )⊗Q is the k-rescaling of gr∗(π1X)⊗Q. If Y is a formal space, then the converse holds, and Y is coformal. Furthermore, if X is formal, with Koszul cohomology algebra, there exist filtered group isomorphi...

Chen, W.Y.C. and R.P. Stanley, Derangements on the n-cube, Discrete Mathematics 115 (1993) 65-15. Let Q. be the n-dimensional cube represented by a graph whose vertices are sequences of O’s and l’s of length n, where two vertices are adjacent if and only if they differ only at one position. A k-dimensional subcube or a k-face of Q. is a subgraph of Q. spanned by all the vertices u1 u2 u, with c...

0 //K ⊂ //E q //Q //1. Here K stands for kernel and Q stands for quotient. She assumed that K is abelian (= commutative), so she wrote its product as + with identity element 0. She did not assume that Q is abelian, so she wrote its product as · with identity element 1. That explains the joke notation with 0 at the left and 1 at the right. The sequence is exact, which means that K is a normal su...

We prove the exceptional zero conjecture is true for semistable elliptic curves E/Q over number fields of the form F ( e2πi/q n ,∆ 1/q 1 , . . . ,∆ 1/q d ) where F is a totally real field, and the split multiplicative prime p 6= 2 is inert in F (e2πi/q n ) ∩ R. In 1986 Mazur, Tate and Teitelbaum [9] attached a p-adic L-function to an elliptic curve E/Q with split multiplicative reduction at p. ...

Given three nonnegative integers $p,q,r$ and a finite field $F$, how many Hankel matrices $\left( x_{i+j}\right) _{0\leq i\leq p,\ 0\leq j\leq q}$ over $F$ have rank $\leq r$ ? This question is classical, the answer ($q^{2r}$ when $r\leq\min\left\{ p,q\right\} $) has been obtained independently by various authors using different tools (Daykin, Elkies, Garcia Armas, Ghorpade Ram). In this note, ...