<Emphasis Type="Italic">p</Emphasis>-adic properties of division polynomials and elliptic divisibility sequences

For a fixed rational point P E (K) on an elliptic curve, we consider the sequence of values (F_n (P))_n1 of the division polynomials of E at P. For a finite field we prove that the sequence is periodic. For a local field we prove (under certain hypotheses) that there is a power q=p^e so that for all m1, the limit of exists in K and is algebraic over We apply this result to prove an analogous p-adic limit and algebraicity result for elliptic divisibility sequences.Mathematics Subject Classification (1991): 11G07, 11D61, 14G20, 14H52The authors research supported by NSA grant H98230-04-1-0064.     

Hyponormality of Toeplitz operators with rational symbols

In this article we introduce a notion of `division' for rational functions and then give a criterion for hyponormality of (f, g are rational functions) in the cases where g divides f. Furthermore we show that we may assume, without loss of generality, that g divides f when we consider the hyponormality of . Supported in part by a grant from Faculty Research Fund, Sungkyunkwan University, 2004. Supported in part by a grant (R14-2003-006-01000-0) from the Korea Research Foundation.     

An immersion theorem for Vaisman manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold admitting a Kähler covering , with monodromy acting on by Kähler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on . We prove that any compact Vaisman manifold admits a natural holomorphic immersion to a Hopf manifold (ⁿ0). As an application, we obtain that any Sasakian manifold has a contact immersion to an odd-dimensional sphere.Mathematics Subject Classification (2000): 53C55, 14E25, 53C25Liviu Ornea is member of EDGE, Research Training Network HRPN-CT-2000-00101, supported by the European Human Potential Programme.Misha Verbitsky is an EPSRC advanced fellow supported by CRDF grant RM1-2354-MO02 and EPSRC grant GR/R77773/01.Both authors acknowledge financial support from Ecole Polytechnique (Palaiseau).     

On the zero-distribution of Epstein zeta-functions

We prove that the mean value of the real parts of the nontrivial zeros of the Epstein zeta-function associated with a positive definite quadratic form in n variables is equal to . Furthermore, we show that Epstein zeta-functions in general have an asymmetric zero-distribution with respect to the critical line Re .     

Asymptotic laws for regenerative compositions: gamma subordinators and the like

For a random closed set obtained by exponential transformation of the closed range of a subordinator, a regenerative composition of generic positive integer n is defined by recording the sizes of clusters of n uniform random points as they are separated by the points of . We focus on the number of parts K_n of the composition when is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for K_n and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the Lévy measure is regularly varying at 0+. Research supported in part by N.S.F. Grant DMS-0071448     

On topological Kadec norms

We present a topological analogue of the classic Kadec Renorming Theorem, as follows. Let be two separable metric topologies on the same set X. We prove that every point in X has an -neighbourhood basis consisting of sets that are -closed if and only if there exists a function φ: X→ℝ that is -lower semi-continuous and such that is the weakest topology on X that contains and that makes φ continuous. An immediate corollary is that the class of almost n-dimensional spaces consists precisely of the graphs of lower semi-continuous functions with at most n-dimensional domains.     

Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m5) of positive sectional curvature on which an (m–1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By [Wi], these results imply a homeomorphism classification for positively curved n-manifolds (n8) of almost maximal symmetry rank Supported by CNPq of Brazil, NSFC Grant 19741002, RFDP and Qiu-Shi Foundation of China.Supported partially by NSF Grant DMS 0203164 and a research grant from Capital normal university.     

Topological classification of holomorphic, semi-hyperbolic germs, in ``Quasi-Absence' of resonances

We give the classification, under topological conjugacy, of invertible holomorphic germs f:, with λ₁, . . . ,λ_n eigenvalues of d f₀, and |λ_i|≠1 for i=2, . . . ,n while λ₁ is a root of the unity, in the suitable hypothesis of ``quasi-absence' of resonances (i.e., assuming that for r_i≥0 and i=2, . . . ,n, with ).     

On the elliptic genus of generalised Kummer varieties

Borisov and Libgober ([2]) recently proved a conjecture of Dijkgraaf, Moore, Verlinde, and Verlinde (see [6]) on the elliptic genus of a Hilbert scheme of points on a surface. We show how their result can be used together with our work on complex genera of generalised Kummer varieties [17] to deduce the following formula, conjectured by Kawai and Yoshioka ([15]), on the elliptic genus of a generalised Kummer variety A^[[n>^]] of dimension 2(n–1): Here is the weak Jacobi form of weight –1 and index and V(n) is the Hecke operator sending Jacobi forms of index r to Jacobi forms of index nr (see [7]).The author was supported by the Deutsche Forschungsgemeinschaft.Revised version: 14 November 2003     

Isomorphisms of algebras associated with directed graphs

Given countable directed graphs G and G, we show that the associated tensor algebras (G) and (G) are isomorphic as Banach algebras if and only if the graphs G are G are isomorphic. For tensor algebras associated with graphs having no sinks or no sources, the graph forms an invariant for algebraic isomorphisms. We also show that given countable directed graphs G, G, the free semigroupoid algebras and are isomorphic as dual algebras if and only if the graphs G are G are isomorphic. In particular, spatially isomorphic free semigroupoid algebras are unitarily isomorphic. For free semigroupoid algebras associated with locally finite directed graphs with no sinks, the graph forms an invariant for algebraic isomorphisms as well.Mathematics Subject Classification (2000): 47L80, 47L55, 47L40Acknowledgments. We would like to thank the referee for several constructive suggestions on the initial draft and for bringing to our attention the work in [8,9]. The first author was partially supported by a research grant from ECU and the second author by an NSERC research grant and start up funds from the University of Guelph. We thank David Pitts for enlightening conversations and Alex Kumjian for helpful comments on the literature.     

Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations

Summary We introduce a class of n×n structured matrices which includes three well-known classes of generalized companion matrices: tridiagonal plus rank-one matrices (comrade matrices), diagonal plus rank-one matrices and arrowhead matrices. Relying on the structure properties of , we show that if A then A=RQ , where A=QR is the QR decomposition of A. This allows one to implement the QR iteration for computing the eigenvalues and the eigenvectors of any A with O(n) arithmetic operations per iteration and with O(n) memory storage. This iteration, applied to generalized companion matrices, provides new O(n²) flops algorithms for computing polynomial zeros and for solving the associated (rational) secular equations. Numerical experiments confirm the effectiveness and the robustness of our approach.The results of this paper were presented at the Workshop on Nonlinear Approximations in Numerical Analysis, June 22 – 25, 2003, Moscow, Russia, at the Workshop on Operator Theory and Applications (IWOTA), June 24 – 27, 2003, Cagliari, Italy, at the Workshop on Numerical Linear Algebra at Universidad Carlos III in Leganes, June 16 – 17, 2003, Leganes, Spain, at the SIAM Conference on Applied Linear Algebra, July 15 – 19, 2003, Williamsburg, VA and in the Technical Report [8]. This work was partially supported by MIUR, grant number 2002014121, and by GNCS-INDAM. This work was supported by NSF Grant CCR 9732206 and PSC CUNY Awards 66406-0033 and 65393-0034.     

Interpolation and sampling hypersurfaces for the Bargmann-Fock space in higher dimensions

We study those smooth complex hypersurfaces W in having the property that all holomorphic functions of finite weighted L^p norm on W extend to entire functions with finite weighted L^p norm. Such hypersurfaces are called interpolation hypersurfaces. We also examine the dual problem of finding all sampling hypersurfaces, i.e., smooth hypersurfaces W in such that any entire function with finite weighted L^p norm is stably determined by its restriction to W. We provide sufficient geometric conditions on the hypersurface to be an interpolation or sampling hypersurface. The geometric conditions that imply the extension property and the restriction property are given in terms of some directional densities. The first author is supported by projects MTM 2005-08984-Co2-O2 and 2001SGR00611 The third author is partially supported by NSF grant DMS0400909     

Global Solutions to the Two Dimensional Quasi-Geostrophic Equation with Critical or Super-Critical Dissipation

The two dimensional quasi-geostrophic (2D QG) equation with critical and super-critical dissipation is studied in Sobolev space H^s(ℝ²). For critical case (α=), existence of global (large) solutions in H^s is proved for s≥ when is small. This generalizes and improves the results of Constantin, D. Cordoba and Wu [4] for s = 1, 2 and the result of A. Cordoba and D. Cordoba [8] for s=. For s≥1, these solutions are also unique. The improvement for pushing s down from 1 to is somewhat surprising and unexpected. For super-critical case (α ∈ (0,)), existence and uniqueness of global (large) solution in H^s is proved when the product is small for suitable s≥2−2α, p ∈ [1,∞] and β ∈ (0,1].     

Sectional curvatures of Kähler moduli

If X is a compact Kähler manifold of dimension n, we let denote the cone of Kähler classes, and the level set given by classes D with Dⁿ=1. This space is naturally a Riemannian manifold and is isometric to the manifold of Kähler forms with ⁿ some fixed volume form, equipped with the Hodge metric, as studied previously by Huybrechts. We study these spaces further, in particular their geodesics and sectional curvatures. Conjecturally, at least for Calabi–Yau manifolds and probably rather more generally, these sectional curvatures should be bounded between and zero. We find simple formulae for the sectional curvatures, and prove both the bounds hold for various classes of varieties, developing along the way a mirror to the Weil–Petersson theory of complex moduli. In the case of threefolds with h^1,1=3, we produce an explicit formula for this curvature in terms of the invariants of the cubic form. This enables us to check the bounds by computer for a wide range of examples. Finally, we explore the implications of the non-positivity of these curvatures.     

Simplicial complexes lying equivariantly over the affine building of GL(<Emphasis Type="Italic">N</Emphasis>)

Let G=GL(N,K), K a non-archimedean local field and X be the semisimple affine building of G over K. We construct a projective tower of G-sets with X(0)=X. They are obtained by using a minor modification in Bruhat and Tits original construction (an idea due to P. Schneider). Using the structure of X as an abstract building, we construct a projective tower of simplicial G-complexes such that, for each r, X_(r) is canonically a geometrical realization of X_r. In the case N=2, X_r has a natural two-sheeted covering _r and we show that the supercuspidal part of the cohomology space is characterized by a nice property.Mathematics Subject Classification (2000): 14R25, 20E42, 20G25, 55U10, 57S25     

Traps for reflected Brownian motion

Consider an open set , d ≥ 2, and a closed ball . Let denote the expectation of the hitting time of B for reflected Brownian motion in D starting from x ∈ D. We say that D is a trap domain if . A domain D is not a trap domain if and only if the reflecting Brownian motion in D is uniformly ergodic. We fully characterize the simply connected planar trap domains using a geometric condition and give a number of (less complete) results for d > 2. Research partially supported by NSF grant DMS-0303310. Research partially supported by NSF grant DMS-0303310. Research partially supported by NSF grant DMS-0201435.     

Un phénomène de concentration de genre

We show that the main part of the genus of the preimages of a projective line by a generic holomorphic endomorphism of goes to the support of the Green measure.

---

Résumé Nous montrons que le genre des préimages dune droite projective par un endomorphisme holomorphe générique de se concentre sur le support de la mesure de Green.A phenomenon of concentration of genus

---

---

Mathematics Subject Classification (2000): 32H50, 32U40, 32Q45     

Bootstrap percolation on the hypercube

In the bootstrap percolation on the n-dimensional hypercube, in the initial position each of the 2ⁿ sites is occupied with probability p and empty with probability 1−p, independently of the state of the other sites. Every occupied site remains occupied for ever, while an empty site becomes occupied if at least two of its neighbours are occupied. If at the end of the process every site is occupied, we say that the (initial) position spans the hypercube. We shall show that there are constants c₁,c₂>0 such that for the probability of spanning tends to 1 as n→∞, while for the probability tends to 0. Furthermore, we shall show that for each n the transition has a sharp threshold function. J. Balogh: work was done while at The University of Memphis, USA Research supported in part by NSF grant DMS0302804 Research supported in part by NSF grant ITR 0225610 and DARPA grant F33615-01-C-1900     

Integral structures in automorphic line bundles on the p-adic upper half plane

Given an automorphic line bundle of weight k on the Drinfeld upper half plane X over a local field K, we construct a GL₂(K)-equivariant integral lattice in as a coherent sheaf on the formal model underlying Here is ramified of degree 2. This generalizes a construction of Teitelbaum from the case of even weight k to arbitrary integer weight k. We compute and obtain applications to the de Rham cohomology H_dR¹( X, Sym_K^k(St)) with coefficients in the k-th symmetric power of the standard representation of SL₂(K) (where k0) of projective curves X uniformized by X: namely, we prove the degeneration of a certain reduced Hodge spectral sequence computing H_dR¹( X, Sym_K^k(St)), we re-prove the Hodge decomposition of H_dR¹( X, Sym_K^k(St)) and show that the monodromy operator on H_dR¹( X, Sym_K^k(St)) respects integral de Rham structures and is induced by a universal monodromy operator defined on , i.e. before passing to the -quotient.Mathematics Subject Classification (2000): 11F33, 11F12, 11G09, 11G18I wish to thank Peter Schneider and Jeremy Teitelbaum for generously providing me with some helpful private notes on their own work, and for their interest. I am also grateful to Matthias Strauch for useful discussions on odd weight modular forms. I thank Christophe Breuil for his interest and his insisting on lattices for the entire G-action. Finally I thank the referee for his suggestions concerning the presentation of several technical constructions.     

Packing directed circuits

We prove a conjecture of Younger, that for every integern0 there exists an integert0 such that for every digraphG, eitherG hasn vertex-disjoint directed circuits, orG can be made acyclic by deleting at mostt vertices.Research partially supported by DONET ECHM contract CHRXCT930090.Research partially supported by DIMACS, by NSF grant DMS-9401981 and by ONR grant N00014-92-J-1965, and partially performed under a consulting agreement with Bellcore.Research partially supported by DIMACS, by Université de Paris VI, by NSF grant DMS-9303761 and by ONR grant N00014-93-1-0325, and partially performed under a consulting agreement with Bellcore.