No abstract uploaded!

No abstract uploaded!

The main purpose of this paper is to study the asymptotic property of one class of number-theoretic functions and give a sharp hybrid mean-value formula by using the generalized Bernoulli numbers, Gauss sums and the mean-value theorems of Dirichlet$L$-functions.

In this paper, a random graph process {$G$($t$)}_{$t$≥1}is studied and its degree sequence is analyzed. Let {$W$_{$t$}}_{$t$≥1}be an i.i.d. sequence. The graph process is defined so that, at each integer time$t$, a new vertex with$W$_{$t$}edges attached to it, is added to the graph. The new edges added at time$t$are then preferentially connected to older vertices, i.e., conditionally on$G$($t$-1), the probability that a given edge of vertex$t$is connected to vertex$i$is proportional to$d$_{$i$}($t$-1)+δ, where$d$_{$i$}($t$-1) is the degree of vertex$i$at time$t$-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ=min{τ_{W},τ_{P}}, where τ_{W}is the power-law exponent of the initial degrees {$W$_{$t$}}_{$t$≥1}and τ_{P}the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.

The face numbers of simplicial complexes without missing faces of dimension larger than$i$are studied. It is shown that among all such ($d$−1)-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal$f$-vector; and moreover, among all such 2-Cohen–Macaulay (2-CM) complexes, the same sphere has the componentwise minimal$h$-vector. It is also verified that the$l$-skeleton of a flag ($d$−1)-dimensional 2-CM complex is 2($d$−$l$)-CM, while the$l$-skeleton of a flag piecewise linear ($d$−1)-sphere is 2($d$−$l$)-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension and the number of vertices are established.