The u-invariant of a field F, u = u(F), is defined to be the maximum of the dimensions of anisotropic quadratic forms over F. If F is a non-formally real field with a finite number q of square classes then it is known that u ≦ q. The purpose of this note is to give some necessary and sufficient conditions for equality in terms of the structure of the Witt ring of F.

The fractional Galois ideal is a conjectural improvement on the higher Stickelberger ideals defined at negative integers, and is expected to provide non-trivial annihilators for higher $K$-groups of rings of integers of number fields. In this article, we extend the definition of the fractional Galois ideal to arbitrary (possibly infinite and non-abelian) Galois extensions of number fields under the assumption of Stark's conjectures and prove naturality properties under canonical changes of extension. We discuss applications of this to the construction of ideals in non-commutative Iwasawa algebras.

Let F be a totally real field, and let $E/F$ be a CM quadratic extension. We construct a p-adic L-function attached to Hida families for the group $\mathrm {GL}_{2/F}\times \mathrm {Res}_{E/F}\mathrm {GL}_{1}$ . It is characterized by an exact interpolation property for critical Rankin–Selberg L-values, at classical points corresponding to representations $\pi \boxtimes \chi $ with the weights of $\chi $ smaller than the weights of $\pi $ .

Our p-adic L-function agrees with previous results of Hida when $E/F$ splits above p or $F=\mathbf {Q}$ , and it is new otherwise. Exploring a method that should bear further fruits, we build it as a ratio of families of global and local Waldspurger zeta integrals, the latter constructed using the local Langlands correspondence in families.

In an appendix of possibly independent recreational interest, we give a reality-TV-inspired proof of an identity concerning double factorials.

The purpose of this note is to show that the values of an irreducible (Brauer) character are the characteristic values of a matrix with non-negative rational integers. The construction of these integral matrices is done by a description of a representation of the Grothendieck ring of the category of modules over the group algebra. In particular a result of Solomon on characters and a result of Burnside on vanishing of a non-linear character on some conjugate class are generalized.

Dans [11] (page 32 formule (5)), Tricomi donne la formule suivante:

0.1

Dans [8] (page 704, formule 36) Meixner donne une formule analogue.

La formule (0.1) peut donc être considérée comme la formule de Meixner-Tricomi, la formule de Meixner citée étant antérieure, semblet-il, à celle de Tricomi.

L'objet de ce travail est de généraliser le résultat (0.1) qui sera considéré comme la formule de Meixner-Tricomi.

Pour cela, on va considérer l'équation différentielle ordinaire suivante:

0.2

Combings of compact, oriented, 3-dimensional manifolds $M$ are homotopy classes of nowhere vanishing vector fields. The Euler class of the normal bundle is an invariant of the combing, and it only depends on the underlying $\text{Spi}{{\text{n}}^{\text{c}}}$-structure. A combing is called torsion if this Euler class is a torsion element of ${{H}^{2}}(M;\,\mathbb{Z})$. Gompf introduced a $\mathbb{Q}$-valued invariant ${{\theta }_{G}}$ of torsion combings on closed 3-manifolds, and he showed that ${{\theta }_{G}}$ distinguishes all torsion combings with the same $\text{Spi}{{\text{n}}^{\text{c}}}$-structure. We give an alternative definition for ${{\theta }_{G}}$ and we express its variation as a linking number. We define a similar invariant ${{p}_{1}}$ of combings for manifolds bounded by ${{S}^{2}}$. We relate ${{p}_{1}}$ to the $\Theta$-invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formula $\Theta \,=\,\frac{1}{4}{{p}_{1}}\,+\,6\text{ }\!\!\lambda\!\!\text{ }\left( {\hat{M}} \right)$, where $\text{ }\!\!\lambda\!\!\text{ }$ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.

For the $q$-series $\sum\nolimits_{n=0}^{\infty }{{{a}^{n}}{{q}^{b{{n}^{2}}+cn}}/}\,{{(q)}_{n}}$ we construct a companion $q$-series such that the asymptotic expansions of their logarithms as $q\,\to \,{{1}^{-}}$ differ only in the dominant few terms. The asymptotic expansion of their quotient then has a simple closed form; this gives rise to a new $q$–hypergeometric identity. We give an asymptotic expansion of a general class of $q$-series containing some of Ramanujan's mock theta functions and Selberg's identities.

In [2], J. Gross provides an infinite collection of topologically distinct irreducible homology 3-spheres. In this paper, we construct for any finitely generated abelian group A, an infinite collection {Mi} of topologically distinct irreducible closed 3-manifolds such that H1(Mi) = A for each i.

The proof consists of first constructing a closed irreducible 3-manifold MA with H(MA) = A, and then providing a method for producing more such manifolds with the same first homology group.

All maps and spaces in this paper are assumed to be in the piecewise linear category, and all subspaces are assumed to be piecewise linear subspaces.

A 3-manifold M is irreducible if each 2-sphere in M bounds a 3-cell in M. A compact 2-manifold (or surface) F in a compact 3-manifold M is properly embedded in M if F ∩ bdM = bdF.

We make conjectures on the moments of the central values of the family of all elliptic curves and on the moments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of $L$-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family.

Furthermore, as arithmetical applications, we make a conjecture on the distribution of ${{a}_{p}}$'s amongst all rank 2 elliptic curves and show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec).

The $ER\left( 2 \right)$-cohomology of $B\mathbb{Z}/\left( {{2}^{q}} \right)$ and $\mathbb{C}{{\mathbb{P}}^{n}}$ are computed along with the Atiyah–Hirzebruch spectral sequence for $ER{{\left( 2 \right)}^{*}}\left( \mathbb{C}{{\mathbb{P}}^{\infty }} \right)$. This, along with other papers in this series, gives us the $ER\left( 2 \right)$-cohomology of all Eilenberg–MacLane spaces.

It was shown in (1) that a complete set of bounded observables is metrically complete. However, an extra axiom was needed to prove this result (1, footnote, p. 436). In this note we prove the above-mentioned result without the extra axiom. We also show that there is an abundance of pure states if M is closed in the weak topology and give a necessary and sufficient condition for the latter to be the case.

We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.

The following problem was the starting point for this investigation: Can every desarguesian affine Hjelmslev plane be embedded into a desarguesian projective Hjelmslev plane [8]? An affine Hjelmslev plane is called desarguesian if it can be coordinatized by a right chain ring R with a maximal ideal J(R) consisting of two-sided zero divisors. A projective Hjemslev plane is called desarguesian if the coordinate ring is in addition a left chain ring, i.e. a chain ring. This leads to the algebraic version of the above problem, namely the embedding of right chain rings into suitable chain rings. We can prove the following result.

We address the problem of computing the fundamental group of a symplectic ${{S}^{1}}$-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian ${{S}^{1}}$-action. Several examples are presented to illustrate our main results.

In [4], H. Kamowitz considered the condition, to be satisfied by a bounded operator N on a Hilbert space , that

for all operators X on . Kamowitz discovered that such an N must be normal and its spectrum must lie on a line or a circle; that is, N must be of the form αJ + β, where α and β are complex numbers and J is either Hermitian or unitary. G. Weiss [5] showed that the Hilbert-Schmidt norm behaves differently: N need only be normal in order that

for all finite-rank operators X, and in fact this condition is equivalent to normality. Actually, the result in [5] removes the restriction that X be finite-rank, that is, if N is normal and X is any bounded operator, then

Let $P\in \mathbb{Z}\left[ n \right]$ with $P(0)\,=\,0\,\text{and}\,\varepsilon \,>\,0$. We show, using Fourier analytic techniques, that if $N\ge \exp \exp \left( C{{\varepsilon }^{-1}}\log {{\varepsilon }^{-1}} \right)\,\text{and}\,A\,\subseteq \,\left\{ 1,\,.\,.\,.\,,\,N \right\}$ then there must exist $n\in \mathbb{N}$ such that

$$\frac{\left| A\cap \left( A+P\left( n \right) \right) \right|}{N}>{{\left( \frac{\left| A \right|}{N} \right)}^{2}}-\,\varepsilon $$

In addition to this we show, using the same Fourier analytic methods, that if $A\subseteq \mathbb{N}$, then the set of $\varepsilon $-optimal return times

$$R\left( A,P,\varepsilon \right)=\left\{ n\in \mathbb{N}:\delta \left( A\cap \left. \left( A+P\left( n \right) \right) \right)> \right.\delta {{\left( A \right)}^{2}}-\varepsilon \right\}$$

is syndetic for every $\varepsilon >0$. Moreover, we show that $R\left( A,\,P,\,\varepsilon \right)$ is dense in every sufficiently long interval, in particular we show that there exists an $L=L\left( \varepsilon ,P,A \right)$ such that

$$\left| R\left( A,P,\varepsilon \right)\cap I \right|\ge c\left( \varepsilon ,P \right)\left| I \right|$$

for all intervals $I$ of natural numbers with $\left| I \right|\,\ge \,L\,\text{and}\,c\left( \varepsilon ,\,P \right)\,=\,\exp \exp \,\left( -C\,{{\varepsilon }^{-1}}\,\log {{\varepsilon }^{-1}} \right).$

The permanent of an n × n matrix A = (aij) is defined as

where Sn is the symmetric group of order n. For a survey article on permanents the reader is referred to [2]. An unresolved conjecture due to van der Waerden states that if A is an n × n doubly stochastic matrix; then per (A) ≧ n!/nn, with equality if and only if A = Jn = (1/n).

In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure $\mu $ that naturally induces an embedding of the anisotropic fractional Sobolev class $\dot{\Lambda }_{\alpha ,K}^{1,1}$ into the $\mu $-based-Lebesgue-space $L_{\mu }^{n/\beta }\,\text{with}\,0<\beta \le n$. Also, we investigate the anisotropic fractional $\alpha $-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as $\alpha \to {{0}^{+}}$, will be provided.

J. Bellaïche and M. Dimitrov showed that the $p$-adic eigencurve is smooth but not étale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the overconvergent cuspidal Eisenstein points, being the base change lift for $\text{GL}(2)_{/F}$ of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.

We study harmonic analysis on the symmetric space $\text{GL}_n \times \text{GL}_n \backslash \text{GL}_{2n}$ . We prove several standard results, e.g. Shalika germ expansion of orbital integrals, representability of the Fourier transform of orbital integrals and representability of spherical characters. These properties are not expected to hold for symmetric spaces in general.