Showing papers on "Section (fiber bundle) published in 1973"

P-ADIC Properties of Modular Schemes and Modular Forms

Abstract: This expose represents an attempt to understand some of the recent work of Atkin, Swinnerton-Dyer, and Serre on the congruence properties of the q-expansion coefficients of modular forms from the point of view of the theory of moduli of elliptic curves, as developed abstractly by Igusa and recently reconsidered by Deligne. In this optic, a modular form of weight k and level n becomes a section of a certain line bundle \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } ^{ \otimes k} \) on the modular variety Mn which “classifies” elliptic curves with level n structure (the level n structure is introduced for purely technical reasons). The modular variety Mn is a smooth curve over ℤ[l/n], whose “physical appearance” is the same whether we view it over ℂ (where it becomes ϕ(n) copies of the quotient of the upper half plane by the principal congruence subgroup Г(n) of SL(2,ℤ)) or over the algebraic closure of ℤ/pℤ, (by “reduction modulo p”) for primes p not dividing n. This very fact rules out the possibility of obtaining p-adic properties of modular forms simply by studying the geometry of Mn ⊗ℤ/pℤ and its line bundles \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } ^{ \otimes k} \); we can only obtain the reductions modulo p of identical relations which hold over ℂ.

Stable vector bundles on algebraic surfaces. II

Abstract: This paper is a continuation of “Stable vector bundles on algebraic surfaces” [10]. For simplicity we deal with non-singular projective varieties over the field of complex numbers. Let W be a variety whose fundamental group is solvable, let H be an ample line bundle on W, and let f: V → W be an unramified covering. Then we show in section 1 that if E is an f*H-stable vector bundle on V then f*E is a direct sum of H-stable vector bundles. In particular f*L is a direct sum of simple vector bundles if L is a line bundle on V.

Properties of Optimal Extended-Valued Stopping Rules for $S_n/n^1$

Abstract: Let $X_1, X_2, \cdots$ be i.i.d. nondegenerate mean zero random variables with common distribution function $F$ such that $EX_1^+ \log^+ X_1 0$. Let $b^{-1}(x) = 1/(\int^\infty_x (y/x)\log (y/x) dF(y))$. We find that $P(X_n \geqq b(n) \mathrm{i.o.}) = 1 \Rightarrow P(\tau 1, P(\tau 0$. Moreover, if $g(n) earrow$ and $\sum^\infty_{n=1} g(2^n)^{-1} < \infty$ then $P(|X_n| \geqq b(n)g(n)\mathrm{i.o.}) = 1 \Rightarrow P(\tau < \infty) = 1$. Examples satisfying these latter conditions are given. An outgrowth of this work is that for any i.i.d. sequence $\{X_n\}$ of mean zero random variables and $c < \frac{1}{4}, P(S_n \geqq cE|S_n|\mathrm{i.o.}) = 1$. The importance of this result stems from the fact that we may also have $S_n(\log n)^\varepsilon/n \rightarrow -\infty$ in probability (see [15]). In order to be completely rigorous, a section was included which provides a useful characterization of the general form of the class of optimal rules for sequential decision problems.

Generic Vector Bundle Maps

Abstract: Publisher Summary This chapter presents the theorem relating the singularities of a generic vector bundle map from one vector bundle to another with the characteristic classes of the two vector bundles. The proof that is presented is because of Mather; it is more elegant and conceptually simpler than the original proof. The stress is on the main ideas that could be used to study homological properties of generic singularities of other structures as well. The chapter discusses the various techniques to prove a stronger result with no genericity assumptions for analytic maps including the present theorem as a special case. It presents the definition and discusses the main properties of vector bundle maps.

On sequential distinguishability

Abstract: Let $X_1, X_2,\cdots$ be a sequence of independent and identically distributed random variables governed by an unknown member of a countable family $\mathscr{P} = \{P_\theta: \theta \in \Omega\}$ of probability measures. The family $\mathscr{P}$ is said to be sequentially distinguishable if for any $\varepsilon (0 < \varepsilon < 1)$ there exist a stopping time $t$ and a terminal decision function $\delta(X_1,\cdots, X_t)$ such that $P_\theta\{t < \infty\} = 1 \forall\theta\in\Omega$ and $\sup_{\theta\in\Omega} P_\theta(\delta(X_1,\cdots, X_t) eq \theta) \leqq \varepsilon$. Robbins [12] defined a general stopping time (see Section 2) as an approach to this problem. This paper is a study of this stopping time with applications to some exponential distributions.

Some Monotonicity Properties of Bessel Functions

Abstract: It is proved that the sequence \[\left\{ {\int_{C_ u k}^{C_ u ,k + 1} {t^{\gamma - 1} \left| {\mathcal{C}_ u (t)} \right|dt} } \right\}_{k = \kappa }^\infty \] is decreasing for all $ u $, for $ - \infty < \gamma < \frac{3}{2}$, and for suitable $\varkappa $, where $C_ u (t)$is an arbitrary Bessel function of order $ u $ and $c_{ u k} $ its kth positive zero. This subsumes and unifies results obtained by G. Szego and R. G. Cooke, extending and sharpening some. For one of his results Szego used a Sturm comparison theorem which is shown here to permit the requisite generalization and to incorporate and extend other results originally proved by quite different methods. Auxiliary results are derived. Various remarks are collected in the final section.

Surfaces with a parallel isoperimetric section

Abstract: This announcement is a continuation of Chen [1] (also, Yau [3]). We shall present additional theorems relating surfaces in a space form with a parallel normal section. Let M be a surface in an m-dimensional Riemannian manifold R with the induced normal connection D. For a unit normal section £ on M (that is, a unit normal vector field of M in R\\ let Aç be the second fundamental tensor with respect to £; if we have DC = 0 identically, then £ is called a parallel section; if the trace of A% is constant (respectively, zero), then £ is called an isoperimetric section (respectively, minimal section) on M ; if the determinant of A4 is nowhere zero, then £ is called a nondegenerate section; if Aç vanishes identically, then £ is called a geodesic section; and if ^ is not proportional to the identity transformation everywhere, then Ç is called a umbilical-free section.

(Topological) Semivector Spaces: Convexity and Fixed Point Theory

Abstract: Without speaking too roughly, (topological) semivector spaces are to (topological) semigroups as (topological) vector spaces are to (topological) groups. In Section 1 we introduce the notion of a (topological) semivector space. In Section 2 we study convexity in semivector spaces, in particular “Pointwise convexity,” i.e., convexity of singletons. The main results of the section are astructuretheorem (2.7), acancellationtheorem (2.11), and anembeddingtheorem (2.14). In Section 3 we identify and briefly study a hierarchy of local convexity properties in topological semivector spaces. Lastly, in Section 4 we presentfixedpointtheorems for compact convex subsets enjoying one or another local convexity property in a pointwise convex topological semivector space, indicating associatedminmaxtheorems.

Elementary Properties and Examples of Analytic Functions

Abstract: In this section the definition and basic properties of a power series will be given. The power series will then be used to give examples of analytic functions. Before doing this it is necessary to give some elementary facts on infinite series in ℂ whose statements for infinite series in ℝ should be well known to the reader. If a n is in ℂ for every n ≥ 0 then the series \(\sum\limits_{n = 0}^\infty {{a_n}} \) converges to z iff for every e > 0 there is an integer N such that \(|\sum\limits_{n = 0}^m {{a_n} - | < \in } \) whenever m ≥ N. The series ∑ a n converges absolutely if ∑ |a n | converges.