The diameter of random regular graphs

We give asymptotic upper and lower bounds for the diameter of almost everyr-regular graph onn vertices (n → ∞).     

The Spearman footrule and a Markov chain property

An equivalent representation of the Spearman footrule is considered and a characterization in terms of a Markov chain is established. A martingale approach is thereby incorporated in the study of the asymptotic normality of the statistics.     

Examples of cell-like decompositions of the infinite-dimensional manifolds σ and Σ

Denote by σ the subspace of Hilbert space {(x_i)?l₂:x_i=0 for all but finitely many i}. Examples of cell-like decompositions of σ are constructed that have decomposition spaces that are not homeomorphic to σ. At one extreme is a cell-like decomposition G of σ produced using ghastly finite dimensional examples such that the decomposition space σ?G contains no embedded 2-cell but (σ?G)×$\text{R}$ is homeomorphic to σ. At the other extreme is a cell-like decomposition G of σ satisfying: (a) the nondegeneracy set N_G={g?G:g≠point} consists of countably many arcs (necessarily tame); (b) the nondegeneracy set N_G is a closed subset of the decomposition space σ?G; (c) each map f:B²→σ?G of a 2-cell into σ?G can be approximated arbitrarily closely by an embedding; (d) σ?G is not homeomorphic to σ but (σ?G)×$\text{R}$ is homeomorphic to σ. The fact that both conditions (a) and (b) can be satisfied (and have (d) hold) is directly attributable to σ’s incompleteness as a topological space.     

Varieties of pairs of nilpotent matrices annihilating each other

We classify the irreducible components of the varieties \[ \V(n,a,b) = \{ (A,B) \in \M_n(\field) \times \M_n(\field) \mid AB = BA = A^a = B^b = 0 \}. \]     

The singular locus of the secant varieties of a smooth projective curve

Let X be a smooth irreducible non-degenerated projective curve in some projective space P^N. Let r be a positive integer such that 2r + 1 < N and let S_r(X) be the r-th secant variety of X. It is a variety of dimension 2r + 1. In this paper we prove that the singular locus is the (r - 1)-th secant variety S_{r- 1}(X) if X does not have any (2r + 2)-secant 2r-space divisor. Received: 26 November 2002     

A characterization of $$ \cal N $$ -constrained groups

We prove that a finite group G is -constrained if and only if it contains a nilpotent subgroup I satisfying for all .Received: 22 July 2002     

An alternate proof of Cohn's four squares theorem

While various techniques have been used to demonstrate the classical four squares theorem for the rational integers, the method of modular forms of two variables has been the standard way of dealing with sums of squares problems for integers in quadratic fields. The case of representations by sums of four squares in was resolved by Götzky, while those of and were resolved by Cohn. These efforts utilized modular forms. In previous work, the author was able to demonstrate Götzky's theorem by means of the geometry of numbers. Here Cohn's theorem on representation by the sum of four squares for is proven by a combination of geometry of numbers and quaternionic techniques.     

On Differences of Semicubical Powers

For X,Y,>0, let and define I ₈(X,Y,) to be the cardinality of the set. In this paper it is shown that, for >0, Y ²/X ³=O(), =O(Y ³/X ³) and X=O (Y ²), one has I ₈(X,Y,)=O(X ² Y ²+X min (X ^{3/2} Y ³, X ^{11/2} Y ^{–1})+X min (^{1/3} X ² Y ³, X ^{14/3} Y ^{1/3})), with the implicit constant depending only on . There is a brief report on an application of this that leads, by way of the Bombieri-Iwaniec method for exponential sums, to some improvement of results on the mean squared modulus of a Dirichlet L-function along a short interval of its critical line.