Abstract Chung, Diaconis, and Graham considered random processes of the form Xn+1 = 2Xn + bn (mod p) where X0 = 0, p is odd, and bn for n = 0, 1, 2, . . . are i.i.d. random variables on {−1, 0, 1}. If Pr(bn = −1) = Pr(bn = 1) = β and Pr(bn = 0) = 1− 2β, they asked which value of β makes Xn get close to uniformly distributed on the integers mod p the slowest. In this paper, we extend the results...

The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson’s group F and some of its generalizations interpret the Arithmetic without parameters.

Let M ⊂ C and M ′ ⊂ C ′ be two smooth (C) generic submanifolds with p0 ∈ M and p ′ 0 ∈ M . We shall consider holomorphic mappings H : (C , p0) → (C ′ , p0), defined in a neighborhood of p0 ∈ C N , such that H(M) ⊂ M ′ (and, more generally, smooth CR mappings (M, p0) → (M , p0); see below). We shall always work under the assumption that M is of finite type at p0 in the sense of Kohn and Bloom–Gr...

For every fixed graph H and every fixed 0 < α < 1, we show that if a graph G has the property that all subsets of size αn contain the “correct” number of copies of H one would expect to find in the random graph G(n, p) then G behaves like the random graph G(n, p); that is, it is p-quasi-random in the sense of Chung, Graham, and Wilson [4]. This solves a conjecture raised by Shapira [8] and solv...

In this note, we show how the determinant of the q-distance matrix Dq(T ) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85–88]. Further, by means of the result, w...

The distance matrix of a tree is extensively investigated in the literature. The classical result concerns the determinant of the matrix D (see Graham and Pollak [7]), which asserts that if T is any tree on n vertices then det(D) = (−1)(n− 1)2. Thus, det(D) is a function dependent only on n, the number of vertices of the tree. The formula for the inverse of the matrix D was obtained in a subseq...

Let χ1(n) denote the maximum possible absolute value of an entry of the inverse of an n by n invertible matrix with 0, 1 entries. It is proved that χ1(n) = n 1 2. This solves a problem of Graham and Sloane. Let m(n) denote the maximum possible number m such that given a set of m coins out of a collection of coins of two unknown distinct weights, one can decide if all the coins have the same wei...

The analysis of chessboard pebbling by Andrew Odlyzko is strengthened and generalized, rst to higher dimension and then to arbitrary posets. 1 The pebbling game The pebbling game of Kontsevich is played on the grid points of the rst quadrant. One starts with a single pebble on the origin and a move consists of replacing any pebble with two pebbles, one above and one to the right of the vanishin...

The heart of the matter is written by Graham Green who was born on 2nd October the year 1904 in England. Graham Green went to a school where his father was the principal. His father being the principal of the school that Graham green attended, it resulted to him being mercilessly teased by his classmates. While in his teenage, Green underwent a mental crisis. So bad was the situation that he tr...

A theorem now known as Sperner’s Lemma [5] states that a largest collection of subsets of an n-element set such that no subset contains another is obtained by taking the collection of all the subsets with cardinal bn=2c. (We denote by bxc, resp. dxe, the largest integer less than or equal to x, resp. the smallest integer greater than or equal to x.) In other words, the density of a largest anti...