95 research outputs found
Asymptotic enumeration of Eulerian circuits for graphs with strong mixing properties
We prove an asymptotic formula for the number of Eulerian circuits for graphs
with strong mixing properties and with vertices having even degrees. The exact
value is determined up to the multiplicative error ,
where is the number of vertice
Instability in the Gel'fand inverse problem at high energies
We give an instability estimate for the Gel'fand inverse boundary value
problem at high energies. Our instability estimate shows an optimality of
several important preceeding stability results on inverse problems of such a
type
Reconstruction of a potential from the impedance boundary map
We give formulas and equations for finding generalized scattering data for
the Schr\"odinger equation in open bounded domain at fixed energy from the
impedance boundary map (or Robin-to-Robin map). Combining these results with
results of the inverse scattering theory we obtain efficient methods for
reconstructing potential from the impedance boundary map
On the class of graphs with strong mixing properties
We study three mixing properties of a graph: large algebraic connectivity,
large Cheeger constant (isoperimetric number) and large spectral gap from 1 for
the second largest eigenvalue of the transition probability matrix of the
random walk on the graph. We prove equivalence of this properties (in some
sense). We give estimates for the probability for a random graph to satisfy
these properties. In addition, we present asymptotic formulas for the numbers
of Eulerian orientations and Eulerian circuits in an undirected simple graph
Exponential instability in the inverse scattering problem on the energy interval
International audienceWe consider the inverse scattering problem on the energy interval in three dimensions. We are focused on stability and instability questions for this problem. In particular, we prove an exponential instability estimate which shows optimality of the logarithmic stability result of [Stefanov, 1990] (up to the value of the exponent)
Exponential instability in the Gel'fand inverse problem on the energy intervals
We consider the Gel'fand inverse problem and continue studies of [Mandache,2001]. We show that the Mandache-type instability remains valid even in the case of Dirichlet-to-Neumann map given on the energy intervals. These instability results show, in particular, that the logarithmic stability estimates of [Alessandrini,1988], [Novikov,Santacesaria,2010] and especially of [Novikov,2010] are optimal (up to the value of the exponent)
Asymptotic behaviour of the number of the Eulerian circuits
We determine the asymptotic behaviour of the number of the Eulerian circuits
in undirected simple graphs with large algebraic connectivity (the
second-smallest eigenvalue of the Laplacian matrix). We also prove some new
properties of the Laplacian matrix
Stability estimates for determination of potential from the impedance boundary map
International audienceWe study the impedance boundary map (or Robin-to-Robin map) for the Schrodinger equation in open bounded demain at fixed energy in multidimensions. We give global stability estimates for determining potential from these boundary data and, as corollary, from the Cauchy data set. Our results include also, in particular, an extension of the Alessandrini identity to the case of the impedance boundary map
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