Preprints
List of Publications supported by the Landau Center for research in
Mathematical Analysis and Related Areas
1 October 1990  30 September 1991
 G.M. Levin, M.L. Sodin, P.M. Yuditski: A Ruelle operator for a Julia
set.
 Abstract:
Let R be an expanding rational function with a real bounded Julia set,
and let
(L_{g})(x) = ΓR_{y}=x
g(y)/[R^{1}(y)]^{2}
be a Ruelle operator acting
in a space of functions analytic in a neighbourhood of the Julia set. We obtain
explicit expressions for the resolvent function E(x,z;λ) =
(IλE)1 1/zx and, in particular, for the Fredholm determinant
D(λ) =det(I_{λ}L).
It gives us an equation for calculating the escape rate. We relate our results
to orthogonal polynomials with respect to the balanced measure of R.
Two examples are considered.
 Y. Kifer: Equilibrium states for random expanding
transformations.
 A. Eizenberg, M. Freidlin: Averaging principle for perturbed random
evolution equations and corresponding Dirichlet problems.
 A. Levy: Existence & Uniqueness for Majda's model of dynamic
combustion.
 Abstract:
In the paper we outline our solution, of the Cauchy problem, for Majda's model
of dynamic combustion, i.e. the sustem:
(u + q_{o}z)t + f(u)_{x} = 0, z_{t} +
k_{φ}(u)_{z} = 0,
in the class of bounded measurable functions. We
define a weak entropy solution for this system, state the uniqueness
theorem in this class, and outline the existence proof under the assumption
that the initial data is of bounded variation. The existence proves via the
"vanishing viscosity method".
 A. Eizenberg: Divergence of invariant measure under small tandom
perturbations.
 Y. Kifer: Averaging in dynamical systems and large deviations.
 Abstract:
The paper treats ordinary differential equations of the form
dz^{e}(t)/dt = εB(z^{e}(t),f ^{t}y)
where f ^{t} is a hyperbolic flow. Large
deviations bounds for the averaging principle are obtained here in the
form appeared in [F1],[F2] for the case when the flow f ^{t}
is replaced by a Markov process.
 Y. Kupitz: Spanned ksided hyperplanes of finite sets in
R^{d}.
 Abstract:
It is shown that the convex hull of a spanning set V of
⌊1/2(d+1)k⌋ + 1 points in R^{d}
has a supporting spanned hyperplane (equivalently: a facet of
[V]) which
misses at least k points of V. Spanning sets of cardinality
⌊1/2(d+1)k⌋ not having such a supporting hyperplane are
fully described. The related problem where the spanned hyperplane has not to
be supporting, but still leaves at least k points of V on one
side is discussed and solved in some instances (e.g., d odd or
k≤9, k≠8).
 J. Rodenmüller, B. Peleg: The leastcore, nucleolus & kernel
of homogeneous weighted majority games.
 Abstract:
Homogeneous weighted majority games were already introduced by von Neumann
and Morgenstern. They discussed uniqueness of the representation and the
"main simple solution" for constantsum games. For the same class of games,
Peleg studied the kernel and the nucleolus. Ostermann and Rosenmüller
described the nature of representations of general homogeneous weighted
majority games. The present paper starts out to close the gap: for the
general weighted majority game "without steps", we discuss the leastcore,
the nucleolus, and the kernel and show their close relationship (coincidence)
with the minimal reprecentation.
 I. Benjamimi: An inequality for the heat kernel.
 I. Benjamimi, Y. Peres: A correlation inequality for treeindex
Markov chains.
 Z. Sela: The isomorphism problem for torsionfree hyperbolic groups
with no small action on a real tree.
 Abstract:
Using canonical representatives in hyperbolic groups and the decidability of
the universal theory in free groups, we solve the isomorphism problem for the
groups in the title. The solution implies the solvability of the homeomorphism
problem for closed hyperbolic manifolds and for negatively curved ones of
dimension ≥ 5.
 S. Mozes: Mixing of all orders of Lie groups actions.
 Abstract:
We show that for Lie groups whose adjoint representation reflects their
topology, mixing implies of all orders. In particular, we prove that mixing
action of a semisimple Lie group are mixing of all orders, answering a
conjecture of B. Marcus.
 I. Benjamimi, Y. Peres: Markov chain indexed by trees.
 B. Rubin: Fractional integrals and weakly singular integral
equations of the first kind in the n'th dimensional ball.
 Abstract:
The purpose of the paper is to introduce and to investigate a new class of
fractional integrals connected with balls in Rn. A Riesz
potential IαΩφ over a ball Ω is represented by a
composition of such integrals. Using this representation we obtain necessary
and sufficient solvability conditions for the equation
IαΩφ = f in the space
Lp(Ω;ω) with a power weight ωx and solve the
equation in a closed form. The investigation is based on a special Fourier
analysis adopted for operators commuting with rotations and dilations in
Rn.
 A. Levy: On Majda's model of dynamic combustion.
 Abstract:
Madjda's model of dynamic combustion, consists of the system,
(u + q0z)t + f(u)x = 0, zt + Kφ(u)z = 0
In the paper the Cauchy problem is considered. A weak entropy solution
for this system is defined, existence, uniqueness and contunuos dependence on
initial date are proved. as well as finie propagation speed, for initial date
in L∞. The existence is proved via the "vanishing viscosity
method". Furthermore it is proved that the solution to the Riemann problem
converges as t → ∞ to ZND traveling wave solution. In the
appendices, a second order numerical scheme for the model is described, and
some numerical results are presented.
 A. Kalma: Computation of reacting flows with high activation
energy.
 Abstract:
A GRPscheme is introduced for the numerical integration of the Euler system
of equations of compressible flow in a duct of variable cross section, subject
to external potential. The GRP (Generalized Riemann Problem) scheme is based
on an analytic solution of the GRP at jump discontinuities. It is secondorder
scheme generalizing the firstorder Godunov scheme, having the property of high
resolution of shocks and other discontinuities. For a γlaw gas we give
explicit expressions of the solutions fot any "Arrheniustype" reaction rate
function.
 H. Furstenberg, Y. Peres, B. Weiss: Perfect filtering and double
disjointness.
 Abstract:
Suppose a stationary process {U_{n}} is used to select from
several stationary processes, i.e., if U_{n}=i then we observe
Y_{n} which is
the n'th variable in the i'th process. When can we recover
{U_{n}} from {Y_{n}}?
 M.S. Goldstein, G.Ya. Grabarnik: Almost sure convergence theorem
in von Neumann algebras (some new results).
 Abstract:
The subadditive sequences of operators which belong to a von Neumann algebra
with a faithful normal state and a given positive linear kernel are considered.
We prove the almost sure convergence in Egorov's sense for such sequences.
 A. Nevo: Factors of Poisson boundaries.
 Abstract:
The differential entropy of a (Γ,μ) space (Y,v) with a
stationary measure is defined by:
α(Y,v) =
Σ_{γ}∈Γμ(γ)∫_{Ω}
log dγv/dv dv.
We show that for countable groups with property
T : α(Y,v) ≥ α_{0}(Γ,μ) > 0,
if μ is
irreducible and α(Y,v) ≠ 0. We then show that the factor of the
Poisson boundary B(Γ,μ) obtained as the space of ergodic
components of the action of a normal subgroup N of Γ is
the Poisson boundary B(Γ,μ) of the factor group. A corollary
is that a normal subgroup N of a lattice Γ i a simple Lie
group G of Rrank one that ergodically w.r.t. Lebesgue measure
class on the maximal boundary G/P gives rise to an amenable factor group
Γ/N.
 Z. Sela: The conjugacy problem for knot groups.
 Abstract:
Combining Thurston's description of a knot complement, his DehnSurgery
theorem and results from the theory of (Gromov) hyperbolic groups, we solve
the conjugacy problem for knot groups.
 B. Rubin: The inversion of fractional integrals on a sphere.
[ps] [pdf]
 Abstract:
The purpose of the paper is to invert Riesz potentials and some other
fractional integrals on a spherical surface in R^{n+1}
in the closed form.
New descriptions of spaces of the fractional smoothness on a sphere are
obtained in terms of the orders n,N+2,N+4,... on a sphere may be Noether
operators with a dcharacteristic which depends on the radius of the
sphere.
 A. Eizenberg: Elliptic perturbations for a class of
HamiltonJacobi equations.
 G.M. Levin, M.L. Sodin: Polynomials with disconnected Julia sets
and Green maps.
 I. Benjamimi: λ_{1}(M) > 0 and Liouville
property.
