Spectral Distribution of Transport Operator Arising in Growing Cell Populations

Transport equation with partly smooth boundary conditions arising in growing cell populations is studied inLp (1<p<+∞)space. It is to prove that the transport operatorAHgenerates aC0semigroup and the ninth-order remainder termR9(t)of the Dyson-Phillips expansion of the semigroup is compact, and the spectrum of transport operatorAHconsists of only finite isolated eigenvalues with finite algebraic multiplicities in a tripΓω. The main methods rely on theory of linear operators, comparison operators, and resolvent operators approach.

Nonexistence of Almost Moore Digraphs of Diameter Three

Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound $M(d,k)=1+d+\cdots +d^k$, where $d>1$ and $k>1$ denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when $d=2,3$ or $k=2$. In this paper, we prove that almost Moore digraphs of diameter $k=3$ do not exist for any degree $d$. The enumeration of almost Moore digraphs of degree $d$ and diameter $k=3$ turns out to be equivalent to the search of binary matrices $A$ fulfilling that $AJ=dJ$ and $I+A+A^2+A^3=J+P$, where $J$ denotes the all-one matrix and $P$ is a permutation matrix. We use spectral techniques in order to show that such equation has no $(0,1)$-matrix solutions. More precisely, we obtain the factorization in ${\Bbb Q}[x]$ of the characteristic polynomial of $A$, in terms of the cycle structure of $P$, we compute the trace of $A$ and we derive a contradiction on some algebraic multiplicities of the eigenvalues of $A$. In order to get the factorization of $\det(xI-A)$ we determine when the polynomials $F_n(x)=\Phi_n(1+x+x^2+x^3)$ are irreducible in ${\Bbb Q}[x]$, where $\Phi_n(x)$ denotes the $n$-th cyclotomic polynomial, since in such case they become 'big pieces' of $\det(xI-A)$. By using concepts and techniques from algebraic number theory, we prove that $F_n(x)$ is always irreducible in ${\Bbb Q}[x]$, unless $n=1,10$. So, by combining tools from matrix and number theory we have been able to solve a problem of graph theory.

Reductions of ideals in local rings

This paper contains some contributions to the analytic theory of ideals. The central concept is that of a reduction which is defined as follows: if and are ideals and ⊆ , then is called a reduction of if n = n+1 for all large values of n. The usefulness of the concept depends mainly on two facts. First, it defines a relationship between two ideals which is preserved under homomorphisms and ring extensions; secondly, what we may term the reduction process gets rid of superfluous elements of an ideal without disturbing the algebraic multiplicities associated with it. For example, the process when applied to a primary ideal belonging to the maximal ideal of a local ring gives rise to a system of parameters having the same multiplicity; but the methods work almost equally well for an arbitrary ideal and bring to light some interesting facts which are rather obscured in the special case. The concept seems to be suitable for a variety of applications. The present paper contains one instance which is a generalized form of the associative law for multiplicities (see § 8), and the authors hope to give other illustrations in a separate paper.