864 research outputs found
A note on the divergence-free Jacobian Conjecture in R^2
We give a shorter proof to a recent result by Neuberger, in the real case.
Our result is essentially an application of the global asymptotic stability
Jacobian Conjecture. We also extend some of the results presented in Neuberger
Stationary layered solutions for a system of Allen-Cahn type equations
We consider a class of semilinear elliptic system of the form where is a
double well non negative symmetric potential. We show, via variational methods,
that if the set of solutions to the one dimensional system , which connect the two minima of as has
a discrete structure, then the given system has infinitely many layered
solutions
Average resistance of toroidal graphs
The average effective resistance of a graph is a relevant performance index
in many applications, including distributed estimation and control of network
systems. In this paper, we study how the average resistance depends on the
graph topology and specifically on the dimension of the graph. We concentrate
on -dimensional toroidal grids and we exploit the connection between
resistance and Laplacian eigenvalues. Our analysis provides tight estimates of
the average resistance, which are key to study its asymptotic behavior when the
number of nodes grows to infinity. In dimension two, the average resistance
diverges: in this case, we are able to capture its rate of growth when the
sides of the grid grow at different rates. In higher dimensions, the average
resistance is bounded uniformly in the number of nodes: in this case, we
conjecture that its value is of order for large . We prove this fact
for hypercubes and when the side lengths go to infinity.Comment: 24 pages, 6 figures, to appear in SIAM Journal on Control and
Optimization (SICON
A (short) survey on Dominated Splitting
We present here the concept of Dominated Splitting and give an account of
some important results on its dynamics.Comment: 19 page
Combustion dynamics in steady compressible flows
We study the evolution of a reactive field advected by a one-dimensional
compressible velocity field and subject to an ignition-type nonlinearity. In
the limit of small molecular diffusivity the problem can be described by a
spatially discretized system, and this allows for an efficient numerical
simulation. If the initial field profile is supported in a region of size l <
lc one has quenching, i.e., flame extinction, where lc is a characteristic
length-scale depending on the system parameters (reacting time, molecular
diffusivity and velocity field). We derive an expression for lc in terms of
these parameters and relate our results to those obtained by other authors for
different flow settings.Comment: 6 pages, 5 figure
Nivat's conjecture holds for sums of two periodic configurations
Nivat's conjecture is a long-standing open combinatorial problem. It concerns
two-dimensional configurations, that is, maps where is a finite set of symbols. Such configurations are often
understood as colorings of a two-dimensional square grid. Let denote
the number of distinct block patterns occurring in a configuration
. Configurations satisfying for some
are said to have low rectangular complexity. Nivat conjectured that such
configurations are necessarily periodic.
Recently, Kari and the author showed that low complexity configurations can
be decomposed into a sum of periodic configurations. In this paper we show that
if there are at most two components, Nivat's conjecture holds. As a corollary
we obtain an alternative proof of a result of Cyr and Kra: If there exist such that , then is periodic. The
technique used in this paper combines the algebraic approach of Kari and the
author with balanced sets of Cyr and Kra.Comment: Accepted for SOFSEM 2018. This version includes an appendix with
proofs. 12 pages + references + appendi
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