Expand all Collapse all | Results 1 - 5 of 5 |
1. CJM 2010 (vol 62 pp. 758)
General Preservers of Quasi-Commutativity Let ${ M}_n$ be the algebra of all $n \times n$ matrices over $\mathbb{C}$. We say that $A, B \in { M}_n$ quasi-commute if there exists a nonzero $\xi \in \mathbb{C}$ such that $AB = \xi BA$. In the paper we classify bijective not necessarily linear maps $\Phi \colon M_n \to M_n$ which preserve quasi-commutativity in both directions.
Keywords:general preservers, matrix algebra, quasi-commutativity Categories:15A04, 15A27, 06A99 |
2. CJM 2003 (vol 55 pp. 3)
An Exactly Solved Model for Mutation, Recombination and Selection It is well known that rather general mutation-recombination models can be
solved algorithmically (though not in closed form) by means of Haldane
linearization. The price to be paid is that one has to work with a
multiple tensor product of the state space one started from.
Here, we present a relevant subclass of such models, in continuous time,
with independent mutation events at the sites, and crossover events
between them. It admits a closed solution of the corresponding
differential equation on the basis of the original state space, and
also closed expressions for the linkage disequilibria, derived by means
of M\"obius inversion. As an extra benefit, the approach can be extended
to a model with selection of additive type across sites. We also derive
a necessary and sufficient criterion for the mean fitness to be a Lyapunov
function and determine the asymptotic behaviour of the solutions.
Keywords:population genetics, recombination, nonlinear $\ODE$s, measure-valued dynamical systems, MÃ¶bius inversion Categories:92D10, 34L30, 37N30, 06A07, 60J25 |
3. CJM 2002 (vol 54 pp. 757)
Strongly Projective Graphs We introduce the notion of strongly projective graph, and characterise
these graphs in terms of their neighbourhood poset. We describe certain
exponential graphs associated to complete graphs and odd cycles. We
extend and generalise a result of Greenwell and Lov\'asz \cite{GreLov}:
if a connected graph $G$ does not admit a homomorphism to $K$, where $K$
is an odd cycle or a complete graph on at least 3 vertices, then the
graph $G \times K^s$ admits, up to automorphisms of $K$, exactly $s$
homomorphisms to $K$.
Categories:05C15, 06A99 |
4. CJM 2001 (vol 53 pp. 592)
Ideal Structure of Multiplier Algebras of Simple $C^*$-algebras With Real Rank Zero We give a description of the monoid of Murray-von Neumann equivalence
classes of projections for multiplier algebras of a wide class of
$\sigma$-unital simple $C^\ast$-algebras $A$ with real rank zero and stable
rank one. The lattice of ideals of this monoid, which is known to be
crucial for understanding the ideal structure of the multiplier
algebra $\mul$, is therefore analyzed. In important cases it is shown
that, if $A$ has finite scale then the quotient of $\mul$ modulo any
closed ideal $I$ that properly contains $A$ has stable rank one. The
intricacy of the ideal structure of $\mul$ is reflected in the fact
that $\mul$ can have uncountably many different quotients, each one
having uncountably many closed ideals forming a chain with respect to
inclusion.
Keywords:$C^\ast$-algebra, multiplier algebra, real rank zero, stable rank, refinement monoid Categories:46L05, 46L80, 06F05 |
5. CJM 1999 (vol 51 pp. 792)
Tensor Products and Transferability of Semilattices In general, the tensor product, $A \otimes B$, of the lattices $A$ and
$B$ with zero is not a lattice (it is only a join-semilattice with
zero). If $A\otimes B$ is a {\it capped\/} tensor product, then
$A\otimes B$ is a lattice (the converse is not known). In this paper, we
investigate lattices $A$ with zero enjoying the property that $A\otimes
B$ is a capped tensor product, for {\it every\/} lattice $B$ with zero;
we shall call such lattices {\it amenable}.
The first author introduced in 1966 the concept of a {\it sharply
transferable lattice}. In 1972, H.~Gaskill defined,
similarly, sharply transferable semilattices, and characterized them
by a very effective condition (T).
We prove that {\it a finite lattice $A$ is\/} amenable {\it if{}f it is\/}
sharply transferable {\it as a join-semilattice}.
For a general lattice $A$ with zero, we obtain the result: {\it $A$ is
amenable if{}f $A$ is locally finite and every finite sublattice of $A$
is transferable as a join-semilattice}.
This yields, for example, that a finite lattice $A$ is amenable
if{}f $A\otimes\FL(3)$ is a lattice if{}f $A$ satisfies (T), with
respect to join. In particular, $M_3\otimes\FL(3)$ is not a lattice.
This solves a problem raised by R.~W.~Quackenbush in 1985 whether
the tensor product of lattices with zero is always a lattice.
Keywords:tensor product, semilattice, lattice, transferability, minimal pair, capped Categories:06B05, 06B15 |