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Combinatorics of linear iterated function systems with overlaps
No full textLet p_0, ..., p_{m-1} be points in {\mathbb{R}}^d , and let \{f_j\}_{j=0}^{m-1} be a one-parameter family of similitudes of {\mathbb{R}}^d :
\begin{eqnarray*}
f_j({\hbox{\bit x}}) = {\lambda}{\hbox{\bit x}} + (1-{\lambda}){\hbox{\bit p}}_j,\tqs j=0,\dots,m-1,
\end{eqnarray*}
where λ ∈ (0, 1) is our parameter. Then, as is well known, there exists a unique self-similar attractor S_λ satisfying S_λ =\bigcup_{j=0}^{m-1} f_j(S_λ) . Each x ∈ S_λ has at least one address (i_1,i_2,\dots)\in\prod_1^\infty\{0,1,\dots,m-1\} , i.e. \lim_n f_{i_1}f_{i_2}\dots f_{i_n}({\bf 0})=\x .
We show that for λ sufficiently close to 1, each x ∈ S_λ setmn {p0, ..., pm-1} has 2^{\aleph_0} different addresses. If λ is not too close to 1, then we can still have an overlap, but there exist xs which have a unique address. However, we prove that almost every x ∈ Sλ has 2^{\aleph_0} addresses, provided S_λ contains no holes and at least one proper overlap. We apply these results to the case of expansions with deleted digits.
Furthermore, we give sharp sufficient conditions for the open set condition to fail and for the attractor to have no holes.
These results are generalizations of the corresponding one-dimensional results, however most proofs are different
A class of noncommutative projective surfaces
Get PDFLet A=k+A_1+A_2.... be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A)=k(Y)[t,t^{-1},sigma], where sigma is an automorphism of the integral projective surface Y. Then we prove that A can be written as a naive blowup algebra of a projective surface X birational to Y. This enables one to obtain a deep understanding of the structure of these algebras; for example, generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme. This is despite the fact that the simple objects in the quotient category qgr A will always be in (1-1) correspondence with the closed points of the scheme X
Weak, strong and detached oblique shocks in gravity driven granular free-surface flows
No full textHazardous natural flows such as snow-slab avalanches, debris flows, pyroclastic flows and lahars are part of a much wider class of dense gravity-driven granular free-surface flows that frequently occur in industrial processes as well as in foodstuffs in our kitchens! This paper investigates the formation of oblique granular shocks, when the oncoming flow is deflected by a wall or obstacle in such a way as to cause a rapid change in the flow height and velocity. The theory for non-accelerative slopes is qualitatively similar to that of gasdynamics. For a given deflection angle there are three possibilities: a weak shock may form close to the wall; a strong shock may extend across the chute; or the shock may detach from the tip. Weak shocks have been observed in both dense granular free-surface flows and granular gases. This paper shows how strong shocks can be triggered in chute experiments by careful control of the downstream boundary conditions. The resulting downstream flow height is much thicker than that of weak shocks and there is a marked decrease in the downstream velocity. Strong shocks therefore dissipate much more energy than weak shocks. An exact solution for the angle at which the flow detaches from the wedge is derived and this is shown to be in excellent agreement with experiment. It therefore provides a very useful criterion for determining whether the flow will detach. In experimental, industrial and geophysical flows the avalanche is usually accelerated, or decelerated, by the net effect of the gravitational acceleration and basal sliding friction as the slope inclination angle changes. The presence of these source terms necessarily leads to gradual changes in the flow height and velocity away from the shocks, and this in turn modifies the local Froude number of the flow. A shock-capturing non-oscillating central method is used to compute numerical solutions to the full problem. This shows that the experiments can be matched very closely when the source terms are included and explains the deviations away from the classical oblique-shock theory. We show that weak shocks bend towards the wedge on accelerative slopes and away from it on decelerative slopes. In both cases the presence of the source terms leads to a gradual increase in the downstream flow thickness along the wedge, which suggests that defensive dams should increase in height further down the slope, contrary to current design criteria but in accordance with field observations of snow-avalanche deposits from a defensive dam in Northwestern Iceland. Movies are available with the online version of the paper
Geometric Brownian Motion with delay: mean square characterisation
Get PDFA geometric Brownian motion with delay is the solution of a stochastic differential equation where the drift
and diffusion coefficient depend linearly on the past of the solution, i.e. a linear stochastic functional
differential equation. In this work the asymptotic behavior in mean square of a geometric Brownian motion
with delay is completely characterized by a sufficient and necessary condition in terms of the drift and
diffusion coefficients
Permutation complexes for profinite groups
No full textAn important tool in the analysis of discrete groups of finite virtual cohomological dimension is the existence of a finite dimensional contractible CW-complex on which the group acts with finite stabilizers. We develop a purely algebraic analogue for profinite groups. This enables us to reveal the connection between finiteness conditions on the cohomology of the group and those on the normalizers of the finite p-subgroups
Cubic structures, equivariant Euler characteristics and lattices of modular form
No full textWe use the theory of cubic structures to give a fixed point Riemann-Roch
formula for the equivariant Euler characteristics of coherent sheaves on projective
at
schemes over with a tame action of a finite abelian group. This formula supports a
conjecture concerning the extent to which such equivariant Euler characteristics may be
determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed
point locus. Our results are applied to study the module structure of modular forms
having Fourier coeficients in a ring of algebraic integers, as well as the action of diamond
Hecke operators on the Mordell-Weil groups and Tate-Shafarevich groups of Jacobians of
modular curves
Optimal Scaling of Random Walk Metropolis algorithms with Discontinuous target densities
No full textWe consider the optimal scaling problem for high-dimensional
Random walk Metropolis (RWM) algorithms where the target
distribution has a discontinuous probability density function. All
previous analysis has focused upon continuous target densities.
The main result is a weak convergence result as the dimensionality
of the target densities converges to . In particular,
when the proposal variance is scaled by , the sequence of
stochastic processes formed by the first component of each Markov
chain converges to an appropriate Langevin diffusion process.
Therefore optimising the efficiency of the RWM algorithm is
equivalent to maximising the speed of the limiting diffusion. This
leads to an asymptotic optimal acceptance rate of under quite general conditions. The results have major
practical implications for the implementation of RWM algorithms by
highlighting the detrimental effect of choosing RWM algorithms
over Metropolis-within-Gibbs algorithms
Recovery Patterns for Iterative Methods in a Parallel Unstable Environment
Get PDFSeveral recovery techniques for parallel iterative methods are presented.
First, the implementation of checkpoints in parallel iterative
methods is described and analyzed. Then, a simple checkpoint-free fault tolerant
scheme for parallel iterative methods, the lossy approach, is presented.
When one processor fails and all its data is lost, the system is
recovered by computing a new approximate solution using the data of the
non-failed processors. The iterative method is then restarted with this
new vector. The main advantage of the lossy approach over standard
checkpoint algorithms is that it does not increase the computational cost
of the iterative solver, when no failure occurs. Experiments are presented
that compare the different techniques. The fault tolerant FT-MPI library
is used. Both iterative linear solvers and eigensolvers are considered