Proceedings of the London Mathematical Society - Advance Access

All but finitely many non-trivial zeros of the approximations of the Epstein zeta function are simple and on the critical line

Ki, H. — 2006-01-01

Some errors, which include Proposition 4.1(2), and typographical mistakes are corrected.

Universal finitary codes with exponential tails

Harvey, N., Holroyd, A. E., Peres, Y., Romik, D. — 2006-01-01

In 1977, Keane and Smorodinsky showed that there exists a finitary homomorphism from any finite-alphabet Bernoulli process to any other finite-alphabet Bernoulli process of strictly lower entropy. In 1996, Serafin proved the existence of a finitary homomorphism with finite expected coding length. In this paper, we construct such a homomorphism in which the coding length has exponential tails. Our construction is source-universal, in the sense that it does not use any information on the source distribution other than the alphabet size and a bound on the entropy gap between the source and target distributions. We also indicate how our methods can be extended to prove a source-specific version of the result for Markov chains.

Finite-order meromorphic solutions and the discrete Painleve equations

Halburd, R. G., Korhonen, R. J. — 2006-01-01

Let w(z) be an admissible finite-order meromorphic solution of the second-order difference equation\[ w(z+1)+w(z-1) = R(z,w(z)) \]where R(z, w(z)) is rational in w(z) with coefficients that are meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else the above equation can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painlevé equations of the above form, together with their autonomous versions. This suggests that the existence of finite-order meromorphic solutions is a good detector of integrable difference equations.

Tightness for the interfaces of one-dimensional voter models

Belhaouari, S., Mountford, T., Valle, G. — 2006-01-01

We show that for the voter model on {0, 1}^Z corresponding to a random walk with kernel p(·) and starting from unanimity to the right and opposing unanimity to the left, a tight interface between zeros and ones exists if p(·) has finite second moment but does not if p(·) fails to have finite moment of order for some <2.

Finiteness of integrals of functions of Levy processes

Erickson, K. B., Maller, R. A. — 2006-01-01

We prove necessary and sufficient conditions for the almost sure convergence of the integrals$$\int_1^\infty g(a(t)+M_t)\,df(t)\quad\hbox{and}\quad \int_0^1 g(a(t)+M_t)\,df(t),$$and thus of ₀g(a(t) + M_t)df(t), where M_t=sup{|X_s| : s≤t} is the two-sided maximum process corresponding to a Lévy process (X_t)_t≥0, a(·) is a non-decreasing function on [0, ) with a(0)=0, g(·) is a positive non-increasing function on (0, ), possibly with g(0+)=, and f(·) is a positive non-decreasing function on [0, ) with f(0)=0. The conditions are expressed in terms of the canonical measure, (·), of the process X_t. The special case when a(x)=0, f(x)=x and g(·) is equivalent to the tail of (at zero or infinity) leads to an interesting comparison of M_t with the largest jump of X_t in (0, t].

Some results concerning the convergence at zero and infinity of integrals like g(a(t)+|X_t|) dt, g(S_t) dt, and g(R_t) dt, where S_t is the supremum process and R_t=S_t–X_t is the process reflected in its supremum, are also given. We also consider the convergence of integrals such as ₀Eg(a(t)+M_t)df(t), etc.

Tenth order mock theta functions in Ramanujan's lost notebook III

Choi, Y.-S. — 2006-01-01

Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously, the author proved six of the eight tenth order mock theta function identities. It is the purpose of this paper to prove the fifth and sixth identities of Ramanujan's tenth order mock theta functions. The properties of modular forms are used for the proofs of theta function identities.

Cohomology of Lie superalgebras slm|n and osp2|2n

Su, Y., Zhang, R. B. — 2006-01-01

We explicitly compute the first and second cohomology groups of the classical Lie superalgebras sl_m|n and osp_2|2n with coefficients in the finite-dimensional irreducible modules and the Kac modules. We also show that the second cohomology groups of these Lie superalgebras with coefficients in the respective universal enveloping algebras (under the adjoint action) vanish. The latter result, in particular, implies that the universal enveloping algebras sl_m|n and osp_2|2n do not admit any non-trivial formal deformations of Gerstenhaber type.

Scattering theory on SL(3)/SO(3): connections with quantum 3-body scattering

Mazzeo, R., Vasy, A. — 2006-01-01

In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than 1 and their geometric perturbations. Our goal here is to explain how analysis of the Laplacian on the globally symmetric space SL(3, R)/SO(3, R) is very closely related to quantum three-body scattering. In particular, we adapt geometric constructions from recent advances in that field by one of us (A.V.), as well as from a previous paper of ours concerning resolvents for product spaces, to give a precise description of the resolvent and the spherical functions on this space. Amongst the many technical advantages, these methods give results which are uniform up to the walls of the Weyl chambers.

On the helicity in 3D-periodic Navier-Stokes equations I: The non-statistical case

Foias, C., Hoang, L., Nicolaenko, B. — 2006-01-01

We consider the three-dimensional Navier-Stokes equations with potential forces and study the helicity of the regular solutions which are periodic in the space variables. We will give a detailed description of the behavior of the helicity for large times. In particular, the following asymptotic dichotomy of the helicity will be established: the helicity either is identically zero or is eventually non-zero and converges to zero as t^de^–2h₀t for time t -> . The relation between the helicity and the energy is also investigated in connection with that between the energy and enstrophy. Our study relies on the theory of the asymptotic expansion of the regular solutions of the Navier-Stokes equations and its associated normalization map as well as a Phragmen-Linderl öf principle. The application of this principle is possible due to our proof that the domain of analyticity (in complexified time) of the regular solutions contains (up to a logarithmic correction) a right half plane.

The spine of a Fourier-Stieltjes algebra

Ilie, M., Spronk, N. — 2006-01-01

We define the spine A ^*(G) of the Fourier–Stieltjes algebra B (G) of a locally compact group G. This algebra encodes information about much of the fine structure of B (G), particularly information about certain homomorphisms and idempotents. We show that A ^*(G) is graded over a certain semi-lattice, that of non-quotient locally precompact topologies on G. We compute the spine's spectrum G^*, which admits a semi-group structure. We discuss homomorphisms from A ^*(G) to B (H) where H is another locally compact group; and we show that A ^*(H) contains the image of every completely bounded homomorphism from the Fourier algebra A (H) of any amenable group G. We also show that A ^*(G) contains all of the idempotents in B (G). Finally, we compute examples for vector groups, abelian lattices, minimally almost periodic groups and the (ax+b)-group; and we explore the complexity of A ^*(G) for the discrete rational numbers and free groups.

Computations in non-commutative Iwasawa theory

Dokchitser, T., Dokchitser, V., Coates, J., Sujatha, R. — 2006-01-01

We study special values of L-functions of elliptic curves over Q twisted by Artin representations that factor through a false Tate curve extension $\Q\big(\mu_{p^{\infty}},\sqrt[p^{\infty}]m\,\big)/\Q$. In this setting, we explain how to compute L-functions and the corresponding Iwasawa-theoretic invariants of non-abelian twists of elliptic curves. Our results provide both theoretical and computational evidence for the main conjecture of non-commutative Iwasawa theory.

Invariant Gaussian measures for operators on Banach spaces and linear dynamics

Bayart, F., Grivaux, S. — 2006-01-01

We give conditions for an operator T on a complex separable Banach space X with sufficiently many eigenvectors associated to eigenvalues of modulus 1 to admit a non-degenerate invariant Gaussian measure with respect to which it is weak-mixing. The existence of such a measure depends on the geometry of the Banach space and on the possibility of parametrizing the T-eigenvector fields of T in a regular way. We also investigate the connection with frequent hypercyclicity.

Witt vectors and equivariant ring spectra applied to cobordism

Brun, M. — 2006-01-01

Given a finite group G we show that Dress and Siebeneicher's ring of G-typical Witt vectors on the Lazard ring, that is, on the polynomial ring on countably many indeterminates over the integers, embeds as a subring of the unitary cobordism ring of G-manifolds. We also show that the ring of G-typical Witt vectors on the Lazard ring embeds as a subring of the ring of homotopy groups of the G-fixed point spectrum of the spectrum MU representing cobordism. The above results are derived by exploiting the interaction between restriction, additive transfer and multiplicative transfer. This interaction is described by two Mackey functors satisfying a distributivity relation encoded in a formalism developed by Tambara.

Good grading polytopes

Brundan, J., Goodwin, S. M. — 2006-01-01

Let g be a finite-dimensional semisimple Lie algebra over C and eg a nilpotent element. Elashvili and Kac have recently classified all good Z-gradings for e. We instead consider good R-gradings, which are naturally parameterized by an open convex polytope in a Euclidean space arising from the reductive part of the centralizer of e in g. As an application, we prove that the isomorphism type of the finite W-algebra attached to a good R-grading for e is independent of the particular choice of good grading.

Diophantine analysis and torsion on elliptic curves

Ingram, P. — 2006-01-01

In a recent paper of Bennett and the author, it was shown that the elliptic curve defined by y²=x³+Ax+B, where A and B are integers, has no rational points of finite order if A is sufficiently large relative to B (at least if one assumes the abc Conjecture of Masser and Oesterlé). In the present article we show, perhaps surprisingly, that the rational torsion on the above curve is also quite restricted if B is sufficiently large relative to A. In particular, we demonstrate that for any >0 there is a constant c such that if A and B are integers satisfying |B|>c |A|⁶⁺, then the elliptic curve defined above has no rational torsion points, other than a possible point of order 2 (again making use of the abc Conjecture in some cases). We then extend this by proving similar results for elliptic curves admitting non-trivial Q-isogenies, elliptic curves written in other forms, and elliptic curves over certain number fields. Curiously, the results on isogenies lead to two unexpected irrationality measures for certain algebraic numbers.

Turbulence, amalgamation, and generic automorphisms of homogeneous structures

Kechris, A. S., Rosendal, C. — 2006-01-01

We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin, Hurley and Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample generics). We show that Polish groups with ample generics have the small index property (generalizing results of Hodges, Hodkinson, Lascar and Shelah) and arbitrary homomorphisms from such groups into separable groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of many -stable, N₀-categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group S has a unique non-trivial separable group topology. For several interesting groups we also establish Serre's properties (FH) and (FA).

A bifurcation problem governed by the boundary condition II

Garcia-Melian, J., Rossi, J. D., Sabina de Lis, J. C. — 2006-01-01

In this work we consider the problem u=a(x) u^p in ${\partial u\over\partial \nu}=\lambda u$ on , where is a smooth bounded domain, is the outward unit normal to , is regarded as a parameter and 0<p<1. We consider both cases where a(x)>0 in or a(x) is allowed to vanish in a whole subdomain ₀ of . Our main results include existence of non-negative non-trivial solutions in the range 0<<₁, where ₁ is characterized by means of an eigenvalue problem, uniqueness and bifurcation from infinity of such solutions for small , and the appearance of dead cores for large enough .