On the nilpotent * - Fourier transform

LetG be a nilpotent Lie group. The adapted nilpotent Fourier transform was introduced by D. Arnal and J. C. Cortet,:L(G) C (V,L(^2d )), whereL(G) is the Schwartz space ofG andV × ^2k is aG-invariant Zariski open set ing ^* the dual of the Lie algebra ofG. We prove the surjectivity of this transformation, which allows us to extend it to distribution spaces.     

On the extremality of the disordered state for the Ising model on the Bethe lattice

We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k4387j842664r8lt/xxlarge8805.gif" alt="ge" align="MIDDLE" BORDER="0">2 is extremal if and only if 4387j842664r8lt/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0"> is less or equal to the spin glass transition value, given by tanh(4387j842664r8lt/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0"> _c ^SG = 1/4387j842664r8lt/xxlarge8730.gif" alt="radic" align="MIDDLE" BORDER="0">k.The work was partially supported by the NSF grant DMS 9504513.     

Four-dimensional BF theory as a topological quantum field theory

Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that four-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G=GL(4,) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of four-manifolds.     

Some results about the spectrum of commutative Banach algebras under the weak topology and applications

LetA be a commutative Banach algebra with a nonempty spectrum 43pu1167l4/xxlarge931.gif" alt="Sgr" align="BASELINE" BORDER="0">A. By 43pu1167l4/xxlarge8220.gif" alt="ldquo" align="MIDDLE" BORDER="0">weak43pu1167l4/xxlarge8221.gif" alt="rdquo" align="MIDDLE" BORDER="0"> we denote the relative weak topology induced on 43pu1167l4/xxlarge931.gif" alt="Sgr" align="BASELINE" BORDER="0">A by 43pu1167l4/xxlarge963.gif" alt="sgr" align="BASELINE" BORDER="0">(A ^*,A ^**). In this note we study some properties of the topological space (43pu1167l4/xxlarge931.gif" alt="Sgr" align="BASELINE" BORDER="0">A, weak) and present some applications of the results obtained and tools used to amenability, weakly compact homomorphisms, weakly compact subsets of the spectrum of the uniform algebras and to a characterization of the synthesizable ideals of the algebraA.     

Harmonic analysis in infinite dimension

We study the Hermite transform onL ²() where is a Gaussian measure on a Lusin locally convex spaceE. We are then lead to a Hilbert space () of analytic functions onE which is also a natural range for the Laplace transform. LetB be a convenient Hilbert-Schmidt operator on the Cameron-Martin spaceH of . There exists a natural sequence Cap _n of capacities onE associated toB. This implies the Kondratev-Yokoi theorem about positive linear forms on the Hida test-functions space.     

Random walk on the infinite cluster of the percolation model

Summary We consider random walk on the infinite cluster of bond percolation on ^d . We show that, in the supercritical regime whend3, this random walk is a.s. transient. This conclusion is achieved by considering the infinite percolation cluster as a random electrical network in which each open edge has unit resistance. It is proved that the effective resistance of this network between a nominated point and the points at infinity is almost surely finite.G.R.G. acknowledges support from Cornell University, and also partial support by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell UniversityH.K. was supported in part by the N.S.F. through a grant to Cornell University     

About a minimax theorem of Matthies,Strang and Christiansen

We give a new minisup theorem for noncompact strategy sets. Our result is of the type of the Matthies-Strang-Christiansen minimax theorem where the hyperplane should be replaced by any closed convex set. As an application, we derive a slight generalization of the Matthies-Strang-Christiansen minimax theorem.     

A geometric construction of local representations of local Lie groups

Local transformation groups acting on a manifold X define a natural action of on a space D(X), of functions on X. The natural action induces a local representation of on a Hilbert subspace of the space of distributions on D(X).     

A remark on compact matrix quantum groups

A simpler set of axioms of the theory of compact matrix quantum groups (pseudogroups) is found.Supported by Japan Society for Promoting Science. On leave from the Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoa 74, 00-682 Warsaw, Poland.     

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

We study the well‐posedness of the fractional differential equations with infinite delay on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying ,  and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of 43383a-b926-4a5f-96b8-0077569008f9/mana201800104-math-0007.png"> in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.