133 research outputs found
Metric trees of generalized roundness one
Every finite metric tree has generalized roundness strictly greater than one.
On the other hand, some countable metric trees have generalized roundness
precisely one. The purpose of this paper is to identify some large classes of
countable metric trees that have generalized roundness precisely one.
At the outset we consider spherically symmetric trees endowed with the usual
combinatorial metric (SSTs). Using a simple geometric argument we show how to
determine decent upper bounds on the generalized roundness of finite SSTs that
depend only on the downward degree sequence of the tree in question. By
considering limits it follows that if the downward degree sequence of a SST satisfies , then has generalized roundness one. Included among the
trees that satisfy this condition are all complete -ary trees of depth
(), all -regular trees () and inductive limits
of Cantor trees.
The remainder of the paper deals with two classes of countable metric trees
of generalized roundness one whose members are not, in general, spherically
symmetric. The first such class of trees are merely required to spread out at a
sufficient rate (with a restriction on the number of leaves) and the second
such class of trees resemble infinite combs.Comment: 14 pages, 2 figures, 2 table
Group-invariant solutions of a nonlinear acoustics model
Based on a recent classification of subalgebras of the symmetry algebra of
the Zabolotskaya-Khokhlov equation, all similarity reductions of this equation
into ordinary differential equations are obtained. Large classes of
group-invariant solutions of the equation are also determined, and some
properties of the reduced equations and exact solutions are discussed.Comment: 14 page
Positive definite metric spaces
Magnitude is a numerical invariant of finite metric spaces, recently
introduced by T. Leinster, which is analogous in precise senses to the
cardinality of finite sets or the Euler characteristic of topological spaces.
It has been extended to infinite metric spaces in several a priori distinct
ways. This paper develops the theory of a class of metric spaces, positive
definite metric spaces, for which magnitude is more tractable than in general.
Positive definiteness is a generalization of the classical property of negative
type for a metric space, which is known to hold for many interesting classes of
spaces. It is proved that all the proposed definitions of magnitude coincide
for compact positive definite metric spaces and further results are proved
about the behavior of magnitude as a function of such spaces. Finally, some
facts about the magnitude of compact subsets of l_p^n for p \le 2 are proved,
generalizing results of Leinster for p=1,2, using properties of these spaces
which are somewhat stronger than positive definiteness.Comment: v5: Corrected some misstatements in the last few paragraphs. Updated
reference
Resonant properties of a nonlinear dissipative layer excited by a vibrating boundary: Q-factor and frequency response
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