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The Edmund Landau Minerva Center for Research in Mathematical Analysis and Related Areas


List of Publications supported by the Landau Center for research in Mathematical Analysis and Related Areas

1 October 1990 -- 30 September 1991

  1. 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 (Lg)(x) = ΓRy=x g(y)/[R1(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/z-x and, in particular, for the Fredholm determinant D(λ) =det(IλL). 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.

  2. Y. Kifer: Equilibrium states for random expanding transformations.

  3. A. Eizenberg, M. Freidlin: Averaging principle for perturbed random evolution equations and corresponding Dirichlet problems.

  4. 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 + qoz)t + f(u)x = 0, zt + kφ(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".

  5. A. Eizenberg: Divergence of invariant measure under small tandom perturbations.

  6. Y. Kifer: Averaging in dynamical systems and large deviations.

    Abstract: The paper treats ordinary differential equations of the form dze(t)/dt = εB(ze(t),f ty) 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.

  7. Y. Kupitz: Spanned k-sided hyperplanes of finite sets in Rd.

    Abstract: It is shown that the convex hull of a spanning set V of ⌊1/2(d+1)k⌋ + 1 points in Rd 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).

  8. J. Rodenmüller, B. Peleg: The least-core, 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 constant-sum 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 least-core, the nucleolus, and the kernel and show their close relationship (coincidence) with the minimal reprecentation.

  9. I. Benjamimi: An inequality for the heat kernel.

  10. I. Benjamimi, Y. Peres: A correlation inequality for tree-index Markov chains.

  11. Z. Sela: The isomorphism problem for torsion-free 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.

  12. 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.

  13. I. Benjamimi, Y. Peres: Markov chain indexed by trees.

  14. 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.

  15. 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 Z-N-D traveling wave solution. In the appendices, a second order numerical scheme for the model is described, and some numerical results are presented.

  16. A. Kalma: Computation of reacting flows with high activation energy.

    Abstract: A GRP-scheme 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 second-order scheme generalizing the first-order Godunov scheme, having the property of high resolution of shocks and other discontinuities. For a γ-law gas we give explicit expressions of the solutions fot any "Arrhenius-type" reaction rate function.

  17. H. Furstenberg, Y. Peres, B. Weiss: Perfect filtering and double disjointness.

    Abstract: Suppose a stationary process {Un} is used to select from several stationary processes, i.e., if Un=i then we observe Yn which is the n'th variable in the i'th process. When can we recover {Un} from {Yn}?

  18. 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.

  19. 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) = -Σγ∈Γμ(γ)∫Ω log dγv/dv dv. We show that for countable groups with property
    T : α(Y,v) ≥ α0(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 R-rank one that ergodically w.r.t. Lebesgue measure class on the maximal boundary G/P gives rise to an amenable factor group Γ/N.

  20. Z. Sela: The conjugacy problem for knot groups.

    Abstract: Combining Thurston's description of a knot complement, his Dehn-Surgery theorem and results from the theory of (Gromov) hyperbolic groups, we solve the conjugacy problem for knot groups.

  21. 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 Rn+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 d-characteristic which depends on the radius of the sphere.

  22. A. Eizenberg: Elliptic perturbations for a class of Hamilton-Jacobi equations.

  23. G.M. Levin, M.L. Sodin: Polynomials with disconnected Julia sets and Green maps.

  24. I. Benjamimi: λ1(M) > 0 and Liouville property.

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