Showing papers on "Section (fiber bundle) published in 2002"

Empirical Process Techniques for Dependent Data

Abstract: Let (X k) k≥1 be a sequence of random variables with common distribution function F(x) = P(X 1 ≤ x). Define the empirical distribution function $$ {{F}_{n}}(x) = \frac{1}{n}\# \{ 1 \leqslant i \leqslant n:{{X}_{1}} \leqslant x\} , $$ and the empirical process by \( \sqrt {n} ({{F}_{n}}(x) - F(x)) \) In this chapter we provide a survey of classical as well as modern techniques in the study of empirical processes of dependent data. We begin with a sketch of the early roots of the field in the theory of uniform distribution mod 1, of sequences defined by X k = {n k ω}, ω ∈ [0, 1], dating back to Weyl’s celebrated 1916 paper. In the second section we provide the essential tools of empirical process theory, and we prove Donsker’s classical empirical process invariance principle for i.i.d. processes. The third section provides an introduction to the subject of weakly dependent random variables. We introduce a variety of mixing concepts, provide necessary technical tools like correlation and moment inequalities, and prove central limit theorems for partial sums. The empirical process of weakly dependent data is investigated in the fourth section, where we put special emphasis on almost sure approximation techniques. The fifth section is devoted to the empirical distribution of U-statistics, defined as $$ Un(x) = {{\left( {\begin{array}{*{20}{c}} n \\ 2 \\ \end{array} } \right)}^{{ - 1}}}\# \{ 1 \leqslant i < j \leqslant n:h({{X}_{i}},{{X}_{j}}) \leqslant x\} $$ for some symmetric kernel h. We give some applications, e.g., to dimension estimation in the analysis of time series, and prove weak convergence of the corresponding empirical process. Empirical processes of long-range dependent data are the topic of the sixth section. We give an introduction to the area of long-range dependent processes, provide important technical tools for the study of their partial sums and investigate the limit behavior of the empirical process. It turns out that the limit process is of a completely different type as in the case of independent or weakly dependent data, and that this has important consequences for various functionals of the empirical process. The final section is devoted to pair correlations, i.e., U-statistics empirical processes over short intervals associated with the kernel h(x, y) = |x − y|

Generalized Bundle Methods

Abstract: We study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties. This setting covers several algorithms from the literature that have been so far regarded as distinct. Under a different hypothesis on the stabilizing term and/or the function to be minimized, we prove finite termination, asymptotic convergence, and finite convergence to an optimal point, with or without limits on the number of serious steps and/or requiring the proximal parameter to go to infinity. The convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm, i.e., the management of the bundle of subgradients ($\beta$-strategy) and of the proximal parameter (t-strategy). We extensively exploit a dual view of bundle methods, which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space, where nonlinear subproblems are approximately solved at each step with an inner linearization approach. This allows us to precisely characterize the changes in the subproblems during the serious steps, since the dual problem is not tied to the local concept of $\varepsilon$-subdifferential. For some of the proofs, a generalization of inf-compactness, called *-compactness, is required; this concept is related to that of asymptotically well-behaved functions.

A momentum construction for circle-invariant Kähler metrics

Abstract: Examples of Kahler metrics of constant scalar curvature are relatively scarce. Over the past two decades, several workers in geometry and physics have used symmetry reduction to construct complete Kahler metrics of constant scalar curvature by ODE methods. One fruitful idea-the Calabi ansatz-is to begin with an Hermitian line bundle p: (L, h) → (M, g M ) over a Kahler manifold, and to search for Kahler forms ω = p*ωM + dd c f(t) in some disk subbundle, where t is the logarithm of the norm function and f is a function of one variable. Our main technical result (Theorem A) is the calculation of the scalar curvature for an arbitrary Kahler metric g arising from the Calabi ansatz. This suggests geometric hypotheses (which we call a-constancy) to impose upon the base metric g M and Hermitian structure h in order that the scalar curvature of g be specified by solving an ODE. We show that σ-constancy is necessary and sufficient for the Calabi ansatz to work in the following sense. Under the assumption of σ-constancy, the disk bundle admits a one-parameter family of complete Kahler metrics of constant scalar curvature that restrict to g M on the zero section (Theorems B and D); an analogous result holds for the punctured disk bundle (Theorem C). A simple criterion determines when such a metric is Einstein. Conversely, in the absence of σ-constancy the Calabi ansatz yields at most one metric of constant scalar curvature, in either the disk bundle or the punctured disk bundle (Theorem E). Many of the metrics constructed here seem to be new, including a complete, negative Einstein-Kahler metric on the disk subbundle of a stable vector bundle over a Riemann surface of genus at least two, and a complete, scalar-flat Kahler metric on C 2 .

Support Vector Machines

Abstract: This chapter gives a short introduction to support vector machines, the basic learning method used, extended, and analyzed for text classification throughout this work. Support vector machines [Cortes and Vapnik, 1995][Vapnik, 1998] were developed by Vapnik et al. based on the Structural Risk Minimization principle [Vapnik, 1982] from statistical learning theory. The idea of structural risk minimization is to find a hypothesis h from a hypothesis space H for which one can guarantee the lowest probability of error Err(h) for a given training sample S $$({\vec x_1},{y_1}), \ldots ,({\vec x_n},{y_n}) {\vec x_i} \in {\Re ^N},{y_i} \in \{ - 1, + 1\}$$ (3.1) of n examples. The following upper bound connects the true error of a hypothesis h with the error Err train (h) of h on the training set and the complexity of h [Vapnik, 1998] (see also Section 1.1).

Deformations of special Lagrangian submanifolds

Abstract: In this thesis we study the deformations of special Lagrangian submanifolds X ⊆ M sitting inside a Calabi-Yau manifold (M, g, J,Ω). Let N be the normal bundle of X, and identify N ∼= T ∗X via the complex structure J and induced metric on X. Then using the exponential map one can identify small 1-forms ξ on X with submanifolds Xξ ⊆ M close to X. In the case that X is compact, McLean [50, Theorem 3-6], showed that the small 1-forms ξ parameterising special Lagrangian submanifolds Xξ ⊆ M form a smooth manifold M ⊆ C∞(T ∗X) of dimension b(X), the first Betti number of X. We give a full proof of this result, including the necessary details which were absent from [50]. In fact our result Theorem 3.21 is an extension of the original McLean theorem, in that we show that the special Lagrangian deformations M persist under (certain types of) perturbations of the ambient Calabi-Yau structure. We then go on to consider the situation when X ⊆ C is non-compact, but asymptotic to a cone C ⊆ C at a specified rate α < 1 of decay. Provided that α is not too negative, it turns out that for almost all α there is again a smooth manifold Mα ⊆ C∞(T ∗X) parameterising the special Lagrangian submanifolds Xξ ⊆ C which are near to X and decay towards C at rate α. The main result here is Theorem 6.45, which also gives the dimensions of the smooth manifold Mα. It turns out that for small rates of decay, dimMα depends only on the topology of X, whereas for higher rates dimMα will also depend on analytic data got from the link Σ := S2n−1 ∩ C of the cone C. Along the way to proving Theorem 6.45 we develop a theory of analysis for asymptotically conical Riemannian manifolds, expanding on the existing theory of Lockhart and McOwen [46] and Lockhart [45] for damped Sobolev spaces. In particular, in Section 6.1.1 we give the relevant details for damped Holder spaces. We finish in Section 6.3 by applying our theory to some specific examples, and prove the existence of special Lagrangian submanifolds in Xξ ⊆ C which were previously unknown.

Relative Hitchin--Kobayashi correspondences for principal pairs

Abstract: A principal pair consists of a holomorphic principal $G$-bundle together with a holomorphic section of an associated Kaehler fibration. Such objects support natural gauge theoretic equations coming from a moment map condition, and also admit a notion of stability based on Geometric Invariant Theory. The Hitchin--Kobayashi correspondence for principal pairs identifies stability as the condition for the existence of solutions to the equations. In this paper we generalize these features in a way which allows the full gauge group of the principal bundle to be replaced by certain proper subgroups. Such a generalization is needed in order to use principal pairs as a general framework for describing augmented holomorphic bundles. We illustrate our results with applications to well known examples.

Smooth geometric evolutions of hypersurfaces

Abstract: We consider the gradient flow associated to the following functionals:¶¶\( {\cal F}_m(\varphi) = \int_M 1 + |\bigtriangledown^m u|^2\,d\mu \).¶The functionals are defined on hypersurfaces immersed in \( {\mathbb R}^{n+1} \) via a map \( \varphi : M \to {\mathbb R}^{n+1} \), where M is smooth closed and connected n-dimensional manifold without boundary. Here \( \mu \) and \( \bigtriangledown \) are respectively the canonical measure and the Levi—Civita connection of the Riemannian manifold (M,g), where the metric g is obtained by pulling back on M the usual metric of \( {\mathbb R}^{n+1} \) with the map \( \mu \). The symbol \( \bigtriangledown^m \) denotes the mth iterated covariant derivative and \( u \) is a unit normal local vector field to the hypersurface.¶Our main result is that if the order of derivation \( m \in {\mathbb N} \) is strictly larger than the integer part of n/2 then singularities in finite time cannot occur during the evolution.¶These geometric functionals are related to similar ones proposed by Ennio De Giorgi, who conjectured for them an analogous regularity result. In the final section we discuss the original conjecture of De Giorgi and some related problems.

Lagrangians for the Gopakumar-Vafa conjecture

Abstract: This article explains how to construct immersed Lagrangian submanifolds in C^2 that are asymptotic at large distance from the origin to a given braid in the 3-sphere. The self-intersections of the Lagrangians are related to the crossings of the braid. These Lagrangians are then used to construct immersed Lagrangians in the vector bundle O(-1) oplus O(-1) over the Riemann sphere which are asymptotic at large distance from the zero section to braids.

First magnetostratigraphic study of the Pondaung formation: Implications for the age of the middle eocene anthropoids of Myanmar

Abstract: We report the results of a magnetostratigraphic investigation to improve the stratigraphical and chronological resolution of the Pondaung Formation of central Myanmar. A total of 98 samples were collected from 45 sites through a 319‐m‐thick section at the fossiliferous locality of Yashe Kyitchaung or the Primate Resort (yielding primate species Bahinia pondaungensis, Amphipithecus mogaungensis, and Myanmarpithecus yarshensis) near the Bahin village. Thermal and alternating field demagnetization allowed separation of two remanence components. The high‐temperature component is interpreted as the characteristic magnetization. Rock magnetic experiments show that the remanence magnetization is mainly carried by magnetite. This investigation documents normal polarity remanent magnetization, with a mean direction \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usep...

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

Abstract: We consider the numerical treatment of second kind integral equations on the real line of the form $$\phi (s) = \psi (s) + \int_{ - \infty }^{ + \infty } {K(s - t)z(t)} \phi (t)dt, s \in \mathbb{R},$$ (abbreviated? =? +K z ?) in which? ?L 1(?),z ?L ? (?), and? ?BC(?), the space of bounded continuous functions on ?, are assumed known and? ?BC(?) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [?A, A]) via bounds on (I ? K z )?1 as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on ? is then analysed: in the case whenz is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases wherez is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that ifz (related to the boundary impedance in the application) takes values in an appropriate compact subsetQ of the complex plane, then the difference between?(s) and its finite section approximation computed numerically using the iterative scheme proposed is ≤C 1[khlog(1/kh)+(1??)?1/2(kA)?1/2] in the interval [??A, ?A] (?<1), forkh sufficiently small, wherek is the wavenumber andh the grid spacing. Moreover this numerical approximation can be computed in ≤C 2 N logN operations, whereN = 2A/h is the number of degrees of freedom. The values of the constantsC 1 andC 2 depend only on the setQ and not on the wavenumberk or the support ofz.

Invariants of Finite Group Schemes

Abstract: Let $G$ be a finite group scheme operating on an algebraic variety $X$ , both defined over an algebraically closed field $k$ . The paper first investigates the properties of the quotient morphism $X\longrightarrow X/G$ over the open subset of $X$ consisting of points whose stabilizers have maximal index in $G$ . Given a $G$ -linearized coherent sheaf on $X$ , it describes similarly an open subset of $X$ over which the invariants in the sheaf behave nicely in some way. The points in $X$ with linearly reductive stabilizers are characterized in representation theoretic terms. It is shown that the set of such points is nonempty if and only if the field of rational functions $k(X)$ is an injective $G$ -module. Applications of these results to the invariants of a restricted Lie algebra ${\frak g}$ operating on the function ring $k[X]$ by derivations are considered in the final section. Furthermore, conditions are found ensuring that the ring $k[X]^{\frak g}$ is generated over the subring of $p$ th powers in $k[X]$ , where $p={\rm char}\,k>0$ , by a given system of invariant functions and is a locally complete intersection.

Fractal study of magnetic domain patterns

Abstract: Fractal geometry is introduced into the analysis of ``two-phase'' magnetic domain patterns. The line-measuring dimension ${D}_{\mathrm{line}}$ is selected to quantitatively describe the ``line structure'' patterns of the multi-branched domains (MBD's) formed in garnet bubble films, and a meaningful ${D}_{\mathrm{line}}$ can be related to the numbers of vertical Bloch lines in their walls, i.e., to the hardness of the MBD's. For quantitatively describing the ``plane-filling'' domain patterns of magnetic materials with uniaxial anisotropy, such as corrugation and spike, even ``flower,'' domains, the box-counting dimension ${D}_{\mathrm{box}}$ is selected. For describing the series of domains of Co and Dy-NdFeB single crystals due to branching process, ${D}_{\mathrm{line}}$ and ${D}_{\mathrm{box}}$ are used in section. Our results show that two phase domain patterns possess fractal natures, and can be described by fractal dimensions.

Experimental investigation of zoisite–clinozoisite phase equilibria in the system CaO–Fe 2 O 3 –Al 2 O 3 –SiO 2 –H 2 O

Abstract: The system Ca2Al3Si3O11(O/OH)–Ca2Al2FeSi3O11(O/OH), with emphasis on the Al-rich portion, was investigated by synthesis experiments at 0.5 and 2.0 GPa, 500–800 °C, using the technique of producing overgrowths on natural seed crystals. Electron microprobe analyses of overgrowths up to >100 µm wide have located the phase transition from clinozoisite to zoisite as a function of P–T–Xps and a miscibility gap in the clinozoisite solid solution. The experiments confirm a narrow, steep zoisite–clinozoisite two-phase loop in T–Xps section. Maximum and minimum iron contents in coexisting zoisite and clinozoisite are given by $${\rm X}_{{\rm ps}}^{{\rm zo}} {\rm (max) = 1}{\rm .9*10}^{ - 4} T{\rm + 3}{\rm .1*10}^{ - 2} P - {\rm 5}{\rm .36*10}^{ - 2} $$ and $${\rm X}_{{\rm ps}}^{{\rm czo}} {\rm (min)} = {\rm (4}{\rm .6} * {\rm 10}^{ - {\rm 4}} - 4 * {\rm 10}^{ - {\rm 5}} P{\rm )}T + {\rm 3}{\rm .82} * {\rm 10}^{ - {\rm 2}} P - {\rm 8}{\rm .76} * {\rm 10}^{ - {\rm 2}} $$ (P in GPa, T in °C). The iron-free end member reaction clinozoisite = zoisite has equilibrium temperatures of 185±50 °C at 0.5 GPa and 0±50 °C at 2.0 GPa, with ΔH r 0=2.8±1.3 kJ/mol and ΔS r 0=4.5±1.4 J/mol×K. At 0.5 GPa, two clinozoisite modifications exist, which have compositions of clinozoisite I ~0.15 to 0.25 Xps and clinozoisite II >0.55 Xps. The upper thermal stability of clinozoisite I at 0.5 GPa lies slightly above 600 °C, whereas Fe-rich clinozoisite II is stable at 650 °C. The schematic phase relations between epidote minerals, grossular-andradite solid solutions and other phases in the system CaO–Al2O3–Fe2O3–SiO2–H2O are shown.

Theta Functions and Szeg\"o Kernels

Abstract: We study relations between two fundamental constructions associated to vector bundles on a smooth complex projective curve: the theta function (a section of a line bundle on the moduli space of vector bundles) and the Szeg\"o kernel (a section of a vector bundle on the square of the curve). Two types of relations are demonstrated. First, we establish a higher--rank version of the prime form, describing the pullback of determinant line bundles by difference maps, and show the theta function pulls back to the determinant of the Szeg\"o kernel. Next, we prove that the expansion of the Szeg\"o kernel at the diagonal gives the logarithmic derivative of the theta function over the moduli space of bundles for a fixed, or moving, curve. In particular, we recover the identification of the space of connections on the theta line bundle with moduli space of flat vector bundles, when the curve is fixed. When the curve varies, we identify this space of connections with the moduli space of {\em extended connections}, which we introduce.

Numerical Solution of Improper Integrals with Valid Implementation

Abstract: In this paper, two theorems are explained which are used in order to find the improper integral I = \({\int_a^\infty}\)f(x)dx numerically. It has been proved in [4], one can use the Trapezoidal and Simpson rules to find the definite integral Im = \({\int_a^\infty}\)f(x)dx numerically using the CESTAC (Control et Estimation Stochastique des Arrondis de Calculs ) method which is based on the stochastic arithmetic, [5-8,12]. These theorems are developed on the improper integrals. Then, the CESTAC method and stochastic arithmetic are used to validate the results and implement the numerical examples. By using this method, one can find the optimal integer number m ≥ 1 such that I ~ Im. In the last section two examples are solved. The programs have been provided with Fortran 90.

Partially Integrable Almost CR Manifolds of CR Dimension and Codimension Two

Abstract: We extend the results of [11] on embedded CR manifolds of CR dimension and codimension two to abstract partially integrable almost CR manifolds. We prove that points on such manifolds fall into three different classes, two of which (the hyperbolic and the elliptic points) always make up open sets. We prove that manifolds consisting entirely of hyperbolic (respectively elliptic) points admit canonical Cartan connections. More precisely, these structures are shown to be exactly the normal parabolic geometries of types $(PSU(2, 1) \times PSU(2, 1), B \times B)$, respectively $(PSL(3, \mathbb{C}), B)$, where $B$ indicates a Borel subgroup. We then show how general tools for parabolic geometries can be used to obtain geometric interpretations of the torsion part of the harmonic components of the curvature of the Cartan connection in the elliptic case.

Rare and forbidden decays

Abstract: In these lectures I first cover radiative and semileptonic B decays, including the QCD corrections for the quark subprocesses. The exclusive modes and the evaluation of the hadronic matrix elements, i.e. the relevant hadronic form factors, are the second step. Small effects due to the long-distance, spectator contributions, etc. are discussed next. The second section we started with non-leptonic decays, typically $B \to \pi\pi, K\pi, \rho\pi, ...$ We describe in more detail our prediction for decays dominated by the $b\to s \eta_c$ transition. Reports on the most recent experimental results are given at the end of each subsection. In the second part of the lectures I discuss decays forbidden by the Lorentz and gauge invariance, and due to the violation of the angular moment conservation, generally called the Standard Model-forbiden decays. However, the non-commutative QED and/or non-commutative Standard Model (NCSM), developed in a series of works in the last few years allow some of those decay modes. These are, in the gauge sector, $Z\to \gamma\gamma, gg$, and in the hadronic sector, flavour changing decays of the type $K\to \pi \gamma$, $B\to K\gamma$, etc. We shall see, for example, that the flavour changing decay $D^+_s\to \pi^+ \gamma$ dominates over other modes, because the processes occur via charged currents, i.e. on the quark level it arises from the point-like photon $\times$ current $\times$ current interactions. In the last section we present the transition rate of ``transverse plasmon'' decay into a neutrino--antineutrino pair via noncommutative QED, i.e. $\gamma_{;\rm pl};\to u\bar u$. Such decays gives extra contribution to the mechanism for the energy loss in stars.

Connes’ Trace Formula and Dirac Realization of Maxwell and Yang-Mills Action

Abstract: The essential ingredients of the trace formula [37,38] of A. Connes are operator algebraic constructs over the W*-algebra B(\( \mathcal{H}\)) of all bounded linear operators over some infinite dimensional separable Hilbert space \( \mathcal{H}\) which are known as Dixmier traces. The constructions will be explained in this section. For a general operator-algebraic background the reader is referred e.g. to [64,187,209,117,118].

Global geometrised Rankin-Selberg method for GL(n)

Abstract: Following G. Laumon [12], to a nonramified $\ell$-adic local system $E$ of rank $n$ on a curve $X$ one associates a complex of $\ell$-adic sheaves $\sb n\mathscr {K}\sb E$ on the moduli stack of rank $n$ vector bundles on $X$ with a section, which is cuspidal and satisfies the Hecke property for $E$. This is a geometric counterpart of the well-known construction due to J. Shalika [19] and I. Piatetski-Shapiro [18]. We express the cohomology of the tensor product $\sb n\mathscr {K}\sb {E\sb 1}\otimes \sb n\mathscr {K}\sb {E\sb 2}$ in terms of cohomology of the symmetric powers of $X$. This may be considered as a geometric interpretation of the local part of the classical Rankin-Selberg method for ${\rm GL}(n)$ in the framework of the geometric Langlands program.

Very ample linear systems on blowings-up at general points of smooth projective varieties

Abstract: Let X be a smooth projective variety, let L be a very ample invertible sheaf on X and assume N+1 = dim(H 0 (X, L)), the dimension of the space of global sections of L. Let P 1 ,…, Pt be general points on X and consider the blowing-up π: Y → X of X at those points. Let E i = π -1 (P i ) be the exceptional divisors of this blowing-up. Consider the invertible sheaf M:= π * (L) O Y (-E 1 - … - E t ) on Y. In case t 3 (most invertible sheaves on X satisfy that property on the Clifford index), then M is very ample if t < N - 5. Examples show that the condition on the Clifford index cannot be omitted.

Measurement of a small vertical emittance with a laser wire beam profile monitor

Abstract: We describe in this paper a measurement of vertical emittance in the Accelerator Test Facility (ATF) damping ring at KEK with a laser wire beam profile monitor. This monitor is based on the Compton scattering process of electrons with a laser light target which is produced by injecting a cw laser beam into a Fabry-Perot optical cavity. We installed the monitor at a straight section of the damping ring and measured the vertical emittance with three different ring conditions. In all cases, the ATF ring was operated at 1.28 GeV in a single bunch mode. When the ring was tuned for ultralow emittance, the vertical emittance of ${\ensuremath{\epsilon}}_{y}=(1.18\ifmmode\pm\else\textpm\fi{}0.08)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}\text{ }\mathrm{mrad}$ was achieved. This shows that the ATF damping ring has realized its target value also vertically.

An Equation of Goormaghtigh and Diophantine Approximations

Abstract: An equation of Goormaghtigh (1917) states $$\frac{{{x^m} - 1}} {{x - 1}} = \frac{{{y^n} - 1}} {{y - 1}}\quad {\text{in}}\,{\text{integers}}\quad {\text{x}} > {\text{1}},{\text{y}} > {\text{1}},{\text{m}} > {\text{2}},{\text{n}} > {\text{2}}.$$ (1) We give an account of recent results on (1) and we refer to [26] for a survey. In fact the present article can be viewed as updating Section 3 of [26]. Further, we shall consider an extension of (1) with m = 3 and derive a new result from a recent theorem of Bilu, Hanrot and Voutier [4] on primitive divisors of Lucas and Lehmer sequences. We shall also discuss some general results on diophantine approximations by applying them to (1). All the constants appearing in this article are effectively computable. This means that they can be determined explicitly in terms of various parameters involved. By C = C(ϰ), we understand that C is a number depending only on Κ.

Local geometrized Rankin-Selberg method for GL_n

Abstract: Following Laumon [10], to a nonramified l-adic local system E of rank n on a curve X one associates a complex of l-adic sheaves $_n{\cal K}_E$ on the moduli stack of rank n vector bundles on X with a section, which is cuspidal and satisfies Hecke property for E. This is a geometric counterpart of the well-known construction due to Shalika [17] and Piatetski-Shapiro [16]. We express the cohomology of the tensor product $_n{\cal K}_{E_1}\otimes {_n{\cal K}_{E_2}}$ in terms of cohomology of the symmetric powers of X. This may be considered as a geometric interpretation of the local part of the classical Rankin-Selberg method for GL(n) in the framework of the geometric Langlands program.

The asymptotic growth of equivariant sections of positive and big line bundles

Abstract: If a finite group acts holomorphically on a pair (X,L), where X is a complex projective manifold and L a line bundle on it, for every k the space of holomorphic global section of the k-th power of L splits equivariantly according to the irreducible representations of G. We consider the asymptotic growth of these summands, under various positivity conditions on L. The methods apply also to the context of almost complex quantization.

Methods for Determining Soil Moisture with Ground Penetrating Radar (GPR)

Abstract: The soil water content has a dominant influence on the dielectric permittivity E of porous media because of the high permittivity of water compared with the matrix. The well known Maxwell’s relation which is valid for most natural soils combines the permittivity and the phase velocity v of electromagnetic waves where co is the velocity of light in vacuum, $$\varepsilon = \left( {\frac{{c_0 }} { u }} \right)^2 \cdot $$ (1) Thus the water content can be quantified by determining this velocity. There exists a wide variety of possibilities to determine the phase velocity of the ground penetrating radar waves, whereby two of them will be introduced in the following section.

Riemannian metrics on locally projectively flat manifolds

Abstract: The expression [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] transforms as a symmetric (0, 2) tensor under projective coordinate changes of a domain in R n so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold, the section u can be regarded as a metric potential analogous to the local potential in Kahlergeometry. Let M be a compact locally projectively flat manifold. We prove that if u is a negative section of the dual of the tautological bundle such that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] is a Riemannian metric, then M is projectively equivalent to a quotient of a bounded convex domain in R n . The same is true for such manifolds M with boundary if u = 0 on the boundary. This theorem is an analog of a result of Schoen and Yau in locally conformally flat geometry. The proof involves affine differential geometry techniques developed by Cheng and Yau.

The ring of $ α $-quotients

Abstract: This paper introduces, for each regular, uncountable cardinal \( \alpha \), the ring of quotients of the commutative ring A obtained by the direct limit ¶¶$ Q_\alpha A = \def\limind{\mathop{\oalign{\hfil$\rm lim$\hfil\cr$\longrightarrow$\cr}}} {\limind_} {\hom_A} {(I,A)} $¶¶ where I ranges over the filter of base of ideals which can be generated by fewer than \( \alpha \) elements of A. This is the ring of $ \alpha $ -quotients. It is shown that \( Q_\alpha A \) is the least \( \alpha \)-selfinjected ring of quotients of A; that is to say, having the injective property relative to maps out of ideals generated by fewer than a elements. For semiprime rings, the ring of \( \alpha \)-quotients of A has the $ \alpha $ -splitting property: if D and D' are two subsets of size \( $<$ \alpha \), such that dd' = 0, for each \( d \in D \) and \( d' \in D' \), and \( D \bigcup D' \) generates a dense ideal, then there is an idempotent e such that de = d, for all \( d \in D \) and de = 0, for all \( d \in D' \). The paper examines the least ring of quotients \( Q{^S}{_\alpha}A \) with the \( \alpha \)-splitting property. The application of greatest interest here is to archimedean f-rings. A considerable amount of attention is paid to the maximal \( \ell \)-ideal spaces of the rings \( Q_\alpha A \) and \( Q{^S}{_\alpha} A \). It is shown that the minimum \( \alpha \)-cloz cover of mA, the space of maximal \( \ell \)-ideals of A, is none other than \( \textrm{m}Q{^S}{_\alpha}A \). Applied to C(X), the ring of continuous real valued functions on a compact Hausdorff space X, it turns out that the minimum \( \alpha \)-cloz cover of X is, in fact, \( \textrm{m}Q{_\alpha}C(X) \), as long as X is zero-dimensional. Moreover, it is shown that A has the \( \alpha \)-splitting property if and only if mA is \( \alpha \)-cloz. The final section is devoted to the cardinal \( w_1 \), and to the question of whether the minimum quasi F-cover of mA is \( \textrm{m}Q_{w1}A \). This is shown to be so, provided that A is complemented or mA is zero-dimensional.

On prime ends and plane continua

Abstract: Let $f$ be a conformal map of the unit disk ${\bb D}$ onto the domain $G \subset \hat{\bb C} = {\bb C} \cup \{\infty\}$ . We shall always use the spherical metric in $\hat{\bb C}$ . Caratheodory [ 3 ] introduced the concept of a prime end of $G$ in order to describe the boundary behaviour of $f$ in geometric terms; see for example [ 6 , Chapter 9] or [ 12 , Section 2.4]. There is a bijective map $\hat{f}$ of ${\bb T} = \partial {\bb D}$ onto the set of prime ends of $G$ .

Vacuum Polarization Effects in the Global Monopole Spacetime in the Presence of Wu-Yang Magnetic Monopole

Abstract: In this paper we consider the presence of the Wu-Yang magnetic monopole in the global monopole spacetime and their influence on the vacuum polarization effects around these two monopoles placed together. According to Wu-Yang [Nucl. Phys. {\bf B107}, 365 (1976)] the solution of the Klein-Gordon equation in such an external field will not be an ordinary function but, instead, {\it section}. Because of the peculiar radial symmetry of the global monopole spacetime, it is possible to cover its space section by two overlapping regions, needed to define the singularity free vector potential, and to study the quantum effects due to a charged scalar field in this system. In order to develop this analysis we construct the explicit Euclidean scalar Green {\it section} associated with a charged massless field in a global monopole spacetime in the presence of the Abelian Wu-Yang magnetic monopole. Having this Green section it is possible to study the vacuum polarization effects. We explicitly calculate the renormalized vacuum expectation value $ _{Ren.}$, associated with a charged scalar field operator and the respective energy-momentum tensor, $ _{Ren.}$, which are expressed in terms of the parameter which codify the presence of the global and magnetic monopoles.

Geometric Phase and Modulo Relations for Probability Amplitudes as Functions on Complex Parameter Spaces

Abstract: We investigate general differential relations connecting the respective behavior s of the phase and modulo of probability amplitudes of the form $\amp{\psi_f}{\psi}$, where $\ket{\psi_f}$ is a fixed state in Hilbert space and $\ket{\psi}$ is a section of a holomorphic line bundle over some complex parameter space. Amplitude functions on such bundles, while not strictly holomorphic, nevertheless satisfy generalized Cauchy-Riemann conditions involving the U(1) Berry-Simon connection on the parameter space. These conditions entail invertible relations between the gradients of the phase and modulo, therefore allowing for the reconstruction of the phase from the modulo (or vice-versa) and other conditions on the behavior of either polar component of the amplitude. As a special case, we consider amplitude functions valued on the space of pure states, the ray space ${\cal R} = {\mathbb C}P^n$, where transition probabilities have a geometric interpretation in terms of geodesic distances as measured with the Fubini-Study metric. In conjunction with the generalized Cauchy-Riemann conditions, this geodesic interpretation leads to additional relations, in particular a novel connection between the modulus of the amplitude and the phase gradient, somewhat reminiscent of the WKB formula. Finally, a connection with geometric phases is established.