Towards an ‘average’ version of the Birch and Swinnerton-Dyer conjecture

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The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s,E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a family. Mestre (1986) [Mes] showed the first zero occurs by , where NE is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller . We significantly improve on Mestre's result by averaging over a one-parameter family of elliptic curves E over Q(T). We assume GRH, Tate's conjecture if E is not a rational surface, and either the ABC or the Square-Free Sieve Conjecture if the discriminant has an irreducible polynomial factor of degree at least 4. We find non-trivial upper and lower bounds for the average number of normalized zeros in intervals on the order of (which is the expected scale). Our results may be interpreted as providing further evidence in support of the Birch and Swinnerton-Dyer conjecture, as well as the Katz-Sarnak density conjecture from random matrix theory (as the number of zeros predicted by random matrix theory lies between our upper and lower bounds). These methods may be applied to additional families of L-functions.

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Meta densities and the shape of their sample clouds

This paper compares the shape of the level sets for two multivariate densities. The densities are positive and continuous, and have the same dependence structure. The density f is heavy-tailed. It decreases at the same rate-up to a positive constant-along all rays. The level sets {f>c} for c↓0, have a limit shape, a bounded convex set. We transform each of the coordinates to obtain a new density g with Gaussian marginals. We shall also consider densities g with Laplace, or symmetric Weibull marginal densities. It will be shown that the level sets of the new light-tailed density g also have a limit shape, a bounded star-shaped set. The boundary of this set may be written down explicitly as the solution of a simple equation depending on two positive parameters. The limit shape is of interest in the study of extremes and in risk theory, since it determines how the extreme observations in different directions relate. Although the densities f and g have the same copula-by construction-the shapes of the level sets are not related. Knowledge of the limit shape of the level sets for one density gives no information about the limit shape for the other density.     

Irrationalité de Valeurs Associées a l'Exponentielle de Carlitz

Lete and be the Carlitz-module analogues of their usual counterparts. We have proved in [4]-that these elements of are algebraically independent over whenq3. We study here the remaining caseq=2 and prove among other things that 1,e, are linearly independent over .

     

Geometric Analysis and Symbol Calculus: Fourier Transform Magnetic Resonance Imaging and Wavelets

Due to its unequalled advantages, the magnetic resonance imaging (MRI) modality has truly revolutionized the diagnosis and evaluation of pathology. Because many morphological anatomic details that may not be visualized by other high tech imaging methods can now be readily shown by diagnostic MRI, it has already become the standard modality by which all other clinical imaging techniques are measured. The unique quantum physical basis of the MRI modality combined with the imaging capabilities of current computer technology has made this imaging modality a target of interdisciplinary interest for clinicians, physicists, biologists, engineers, and mathematicians. Due to the fact that MRI scanners perform corticomorphic processing, this modality is by far more complex than all the other high tech clinical imaging techniques. The purpose of this paper is to outline a phase coherent wavelet approach to Fourier transform MRI. It is based on distributional harmonic analysis on the Heisenberg nilpotent Lie group G and the associated symplectically invariant symbol calculus of pseudodifferential operators. The contour and contrast resolution of MRI scans which is controlled by symplectic filter bank processing gives the noninvasive MRI modality superiority over X-ray computed tomography (CT) in soft tissue differentiation.     

Collision problems of random walks in two-dimensional time

The collision problems of two-parameter random walks are studied. That is, some criteria have been established in terms of the characteristic functions of two or more mutually independent random walks in order to determine if they meet infinitly often in certain restricted time sets.     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

Conditions for the convergence in distribution of maxima of stationary normal processes

The asymptotic distribution of the maximum M_n=max_1?t?nξ_t in a stationary normal sequence ξ₁,ξ,… depends on the correlation r_t between ξ₀ and ξ_t. It is well known that if r_t log t → 0 as t → ∞ or if Σr²_t<∞, then the limiting distribution is the same as for a sequence of independent normal variables. Here it is shown that this also follows from a weaker condition, which only puts a restriction on the number of t-values for which rt log t islarge. The condition gives some insight into what is essential for this asymptotic behaviour of maxima. Similar results are obtained for a stationary normal process in continuous time.     

Page usage in a quadtree index

This paper provides a characterization of the storage needs of a quadtree when used as an index to access large volumes of 2-dimensional data. It is shown that the page occupancy for data in random order approaches 33%. A precise mathematical analysis that involves a modicum of hypergeometric functions and dilogarithms, together with some computer algebra is presented.A brief survey of the analysis of storage usage in tree structures is included. The 33% ratio for quadtrees is to be compared to the figures for binary search trees (50%), tries (69%), and quadtries (46%).The research of this author was done while visiting INRIA, Rocquencourt, France under support from the Ministry of Education of Japanese Government.Work of this author was supported in part by the Basic Research Action of the E.C. under contract No. 3075 (Project ALCOM).     

Sample path Large Deviations and optimal importance sampling for stochastic volatility models

Sample path Large Deviation Principles (LDP) of the Freidlin–Wentzell type are derived for a class of diffusions, which govern the price dynamics in common stochastic volatility models from Mathematical Finance. LDP are obtained by relaxing the non-degeneracy requirement on the diffusion matrix in the standard theory of Freidlin and Wentzell. As an application, a sample path LDP is proved for the price process in the Heston stochastic volatility model.     

Effect of an aliphatic spacer group on the adsorption mechanism on the colloidal silver surface of L‐proline phosphonodipeptides

A comparative study of molecular structures of five L ‐proline (L ‐Pro) phosphonodipeptides: L ‐Pro‐NH‐C(Me,Me)‐PO₃H₂ (P1), L ‐Pro‐NH‐C(Me,iPr)‐PO₃H₂ (P2), L ‐Pro‐L ‐NH‐CH(iBu)‐PO₃H₂ (P3), L ‐Pro‐L ‐NH‐CH(PA)‐PO₃H₂ (P4) and L ‐Pro‐L ‐NH‐CH(BA)‐PO₃H₂ (P5) has been carried out using Raman and absorption infrared techniques of molecular spectroscopy. The interpretation of the obtained spectra has been supported by density functional theory calculations (DFT) at the B3LYP; 6–31 + + G** level using Gaussian 2003 software. The surface‐enhanced Raman scattering (SERS) on Ag‐sol in aqueous solutions of these phosphonopeptides has also been investigated. The surface geometry of these molecules on a silver colloidal surface has been determined by observing the position and relative intensity changes of the Pro ring, amide, phosphonate and so‐called spacer (−R) groups vibrations of the enhanced bands in their SERS spectra. Results show that P4 and P5 adsorb onto the silver as anionic molecules mainly via the amide bond (∼1630, ∼1533, ∼1248, ∼800 and ∼565 cm⁻¹), Pro ring (∼956, ∼907 and ∼876 cm⁻¹) and carboxylate group (∼1395 and ∼909 cm⁻¹). Coadsorption of the imine nitrogen atom and PO group with the silver surface, possibly by formation of a weaker interaction with the metal, is also suggested by the enhancement of the bands at 1158 and 1248 cm⁻¹. P1, P2 and P3 show two orientations of their main chain on the silver surface resulting from different interactions of the  C CH₃,  NH and  CONH fragments with this surface. Bonding to the Ag surface occurs mainly through the imino atom (1166 cm⁻¹) for P2, while for P1 and P3 it occurs via the methyl group(s) (1194–1208 cm⁻¹). The amide group functionality (CONH) is practically not involved in the adsorption process for P1 and P2, whereas the C_s P bonds do assist in the adsorption. Copyright © 2008 John Wiley & Sons, Ltd.