6,956 research outputs found
A H\"older-type inequality on a regular rooted tree
We establish an inequality which involves a non-negative function defined on
the vertices of a finite -ary regular rooted tree. The inequality may be
thought of as relating an interaction energy defined on the free vertices of
the tree summed over automorphisms of the tree, to a product of sums of powers
of the function over vertices at certain levels of the tree. Conjugate powers
arise naturally in the inequality, indeed, H\"{o}lder's inequality is a key
tool in the proof which uses induction on subgroups of the automorphism group
of the tree
Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure
For directed graph iterated function systems (IFSs) defined on R, we prove
that a class of 2-vertex directed graph IFSs have attractors that cannot be the
attractors of standard (1-vertex directed graph) IFSs, with or without
separation conditions. We also calculate their exact Hausdorff measure. Thus we
are able to identify a new class of attractors for which the exact Hausdorff
measure is known
Generalised dimensions of measures on almost self-affine sets
We establish a generic formula for the generalised q-dimensions of measures
supported by almost self-affine sets, for all q>1. These q-dimensions may
exhibit phase transitions as q varies. We first consider general measures and
then specialise to Bernoulli and Gibbs measures. Our method involves estimating
expectations of moment expressions in terms of `multienergy' integrals which we
then bound using induction on families of trees
Inhomogeneous parabolic equations on unbounded metric measure spaces
We study inhomogeneous semilinear parabolic equations with source term f
independent of time u_{t}={\Delta}u+u^{p}+f(x) on a metric measure space,
subject to the conditions that f(x)\geq 0 and u(0,x)=\phi(x)\geq 0. By
establishing Harnack-type inequalities in time t and some powerful estimates,
we give sufficient conditions for non-existence, local existence, and global
existence of weak solutions. This paper generalizes previous results on
Euclidean spaces to general metric measure spaces
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