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On the reconstruction of conductivity of bordered two-dimensional surface in R^3 from electrical currents measurements on its boundary
An electrical potential U on bordered surface X (in Euclidien
three-dimensional space) with isotropic conductivity function sigma>0 satisfies
equation d(sigma d^cU)=0, where d^c is real operator associated with complex
(conforme) structure on X induced by Euclidien metric of three-dimensional
space. This paper gives exact reconstruction of conductivity function sigma on
X from Dirichlet-to-Neumann mapping (for aforementioned conductivity equation)
on the boundary of X. This paper extends to the case of the Riemann surfaces
the reconstruction schemes of R.Novikov (1988) and of A.Bukhgeim (2008) given
for the case of domains in two-dimensional Euclidien space. The paper extends
and corrects the statements of Henkin-Michel (2008), where the inverse boundary
value problem on the Riemann surfaces was firstly considered
A vector equilibrium problem for the two-matrix model in the quartic/quadratic case
We consider the two sequences of biorthogonal polynomials (p_{k,n})_k and
(q_{k,n})_k related to the Hermitian two-matrix model with potentials V(x) =
x^2/2 and W(y) = y^4/4 + ty^2. From an asymptotic analysis of the coefficients
in the recurrence relation satisfied by these polynomials, we obtain the
limiting distribution of the zeros of the polynomials p_{n,n} as n tends to
infinity. The limiting zero distribution is characterized as the first measure
of the minimizer in a vector equilibrium problem involving three measures which
for the case t=0 reduces to the vector equilibrium problem that was given
recently by two of us. A novel feature is that for t < 0 an external field is
active on the third measure which introduces a new type of critical behavior
for a certain negative value of t. We also prove a general result about the
interlacing of zeros of biorthogonal polynomials.Comment: 60 pages, 9 figure
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