A compact Riemannian manifold (M, g) of dimension 3 or higher admits a metric of constant (positive or negative) sectional curvature if the following conditions hold: the diameter is bounded from above, the part of the Ricci curvature which lies below some fixed negative number is bounded in LP norm for p > n/2, and the metric is almost spherical or almost hyperbolic in the LP sense. The idea of the proof is to obtain stronger (i.e. L∞) pinching by deforming the initial metric using the Ricci flow, thus reducing the problem to the theorems of Gromov in the case rg < 0 and of Grove, Karcher and Ruh in the case rg > 0. The reduced curvature tensor changes along the flow according to the heat equation, which implies a weak nonlinear parabolic inequality for its norm. The iteration method of De Giorgi, Nash and Moser is applied to obtain the estimate for the maximum norm of the reduced curvature tensor. The crucial step in the iteration consists of controlling the Sobolev constant of the appropriate imbedding (which also changes along the flow, but behaves well) by the isoperimetric constant, which, in turn, can be bounded in terms independent of the particular manifold.

In [13] we characterized the completely regular Hausdorff spaces as the class of spaces whose topology is generated by an Alexandrofï base. A space may have more than one Alexandrofï base and each such base determines a Hausdorff compactification .

Let $\mathcal{F}$ be a coherent rank 2 sheaf on a scheme $Y\,\subset \,{{\mathbb{P}}^{n}}$ of dimension at least two and let $X\,\subset \,Y$ be the zero set of a section $\sigma \,\in \,{{H}^{0}}\left( \mathcal{F} \right)$. In this paper, we study the relationship between the functor that deforms the pair $\left( \mathcal{F},\,\sigma \right)$ and the two functors that deform $\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two forgetful maps between the functors, we prove that the scheme structure of e.g., the moduli scheme ${{\text{M}}_{\text{Y}}}\left( P \right)$ of stable sheaves on a threefold $Y$ at $\left( \mathcal{F} \right)$, and the scheme structure at ($X$) of the Hilbert scheme of curves on $Y$ become closely related. Using this relationship, we get criteria for the dimension and smoothness of ${{\text{M}}_{\text{Y}}}\left( P \right)$ at $\left( \mathcal{F} \right)$, without assuming $\text{Ex}{{\text{t}}^{2}}\left( \mathcal{F},\,\mathcal{F} \right)\,=\,0$. For reflexive sheaves on $Y\,=\,{{\mathbb{P}}^{3}}$ whose deficiency module $M\,=\,H_{*}^{1}\left( \mathcal{F} \right)$ satisfies $_{0}\text{Ex}{{\text{t}}^{2}}\left( M,\,M \right)\,=\,0$ (e.g., of diameter at most 2), we get necessary and sufficient conditions of unobstructedness that coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of $H_{*}^{0}\left( \mathcal{F} \right)$. Moreover, we show that every irreducible component of ${{\text{M}}_{\mathbb{P}}}^{3}\left( P \right)$ containing a reflexive sheaf of diameter one is reduced (generically smooth) and we compute its dimension. We also determine a good lower bound for the dimension of any component of ${{\text{M}}_{\mathbb{P}}}^{3}\left( P \right)$ that contains a reflexive stable sheaf with “small” deficiency module $M$.

An explicit formula forthe mean value of $\vert L(1,\chi )\vert ^2$ is known, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Lately, the authors obtained that the mean value of $\vert L(1,\chi )\vert ^2$ is asymptotic to $\pi ^2/6$, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\ \ \pmod {2d}$ which are trivial on a subgroup H of odd order d of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$, provided that $d\ll \frac {\log p}{\log \log p}$. Bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Here, for a given integer $d_0>1$, we consider the same questions for the nonprimitive odd Dirichlet characters $\chi '$ modulo $d_0p$ induced by the odd primitive characters $\chi $ modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,\chi ')\vert ^2$ is asymptotic to $\frac {\pi ^2}{6}\prod _{q\mid d_0}\left (1-\frac {1}{q^2}\right )$, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order $d\ll \frac {\log p}{\log \log p}$. As a consequence, we improve the previous bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$. Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on d is essentially sharp.

In this paper, we study the stability of the ring solution of the N-body problem in the entire sphere $\mathbb {S}^2$ by using the logarithmic potential proposed in Boatto et al. (2016, Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 472, 20160020) and Dritschel (2019, Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 377, 20180349), derived through a definition of central force and Hodge decomposition theorem for 1-forms in manifolds. First, we characterize the ring solution and study its spectral stability, obtaining regions (spherical caps) where the ring solution is spectrally stable for $2\leq N\leq 6$, while, for $N\geq 7$, the ring is spectrally unstable. The nonlinear stability is studied by reducing the system to the homographic regular polygonal solutions, obtaining a 2-d.o.f. Hamiltonian system, and therefore some classic results on stability for 2-d.o.f. Hamiltonian systems are applied to prove that the ring solution is unstable at any parallel where it is placed. Additionally, this system can be reduced to 1-d.o.f. by using the angular momentum integral, which enables us to describe the phase portraits and use them to find periodic ring solutions to the full system. Some of those solutions are numerically approximated.

In a previous article, we studied the distribution of “low–lying” zeros of the family of quadratic Dirichlet $L$–functions assuming the Generalized Riemann Hypothesis for all Dirichlet $L$–functions. Even with this very strong assumption, we were limited to using weight functions whose Fourier transforms are supported in the interval (−2, 2). However, it is widely believed that this restriction may be removed, and this leads to what has become known as the One-Level Density Conjecture for the zeros of this family of quadratic $L$-functions. In this note, we make use of Weil's explicit formula as modified by Besenfelder to prove an analogue of this conjecture.

This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.

0. Introduction. In this short note we give an explicit computation of the harmonic measure of a half space x > 0 in the 3-dimensional Heisenberg group in terms of a degenerate hypergeometric function. A probabilistic argument reduces the whole problem to a Hermite-type equation on a half line, that we can solve in terms of the function G(l/4, 1/2; x2).

A preliminary attempt to compute this kernel was done in [1] p. 107 and, cited by Huber [4]. Unfortunately a small mistake was made in [1] and the problem was still open until now. The first author is very grateful to Prof. Huber for having pointed out the weak argument of [1]. Since that time, other harmonic measures and even Green functions have been explicitly computed (see [2]).

We give a new proof of the Hansen–Mullen irreducibility conjecture. The proof relies on an application of a (seemingly new) sufficient condition for the existence of elements of degree $n$ in the support of functions on finite fields. This connection to irreducible polynomials is made via the least period of the discrete Fourier transform $\left( \text{DFT} \right)$ of functions with values in finite fields. We exploit this relation and prove, in an elementary fashion, that a relevant function related to the $\text{DFT}$ of characteristic elementary symmetric functions (that produce the coefficients of characteristic polynomials) satisfies a simple requirement on the least period. This bears a sharp contrast to previous techniques employed in the literature to tackle the existence of irreducible polynomials with prescribed coefficients.

1. Our aim in this paper is to determine a necessary and sufficient condition for N∅rlund summability of Fourier series and to include a wider class of classical results. A Fourier series, of a Lebesgue-integrable function, is said to be summable at a point by N∅rlund method (N, pn), as defined by Hardy [1], if pn → Σpn → ∞, and the point is in a certain subset of the Lebesgue set. The following main results are known.

If 0 → A → C → B → 0 is an exact sequence of abelian groups, if ƒ is a 2-cocyle for this extension, if α ∈ End A, and if β ∈ End B, then a necessary and sufficient condition that α extend to an endomorphism γ of C which induces β is that (M) αƒ and ƒβ be cohomologous ; see Montgomery (2). We shall extend this result to the case where 1 → A → G → B → 1 is an exact sequence of groups and A is abelian.

Many years ago, I defined (8) three types of exceptional unode on an algebraic surface, which I called U*8, U*9, U*10, corresponding, on a non-singular model of the surface, to sets of six, seven, and eight rational curves, each of grade — 2, with the intersection patterns represented by the Coxeter-Dynkin graphs now usually known as E6, E7, E8:

where each dot represents a curve, and linked dots intersecting curves. In each case we shall denote the curves in the horizontal sequence by S1, s2, … from left to right, and the extra curve meeting s3 by s*.

Let ℳ(N, N) be the space of all N × N real matrices and let 𝒢(N) be the set of all linear subspaces of ℝN. The maps ker and coker from ℳ(N, N) onto 𝒢(N) induce two quotient topologies, the right and left respectively. A quasibundle over a space X is defined as a continuous map from X into 𝒢(N)\ it is a right quasibundle if 𝒢(N) = ℳ(N,N)/ ker and a left quasibundle if 𝒢(N) = ℳ(N,N)/ coker. The following is established. Theorem: Let ξ be a left quasibundle over a closed subset of some Euclidean space. Then the following statements are equivalent: (i) ξ has enough sections pointwise. (ii) Sections zero at infinity over closed subsets may be extended globally, (iii) A vector subbundle over a closed subset extends to a vector subbundle over a neighborhood, (iv) ξ is a fibrewise sum of local vector subbundles. (v) There exist finitely many global sections spanning ξ. (vi) ξ is an image quasibundle. (vii) ξ results from a Swan construction. These results are used to prove a version of the Hirsch-Smale immersion theorem for locally compact subsets of Euclidean space.

We study families of metrics on automorphic vector bundles associated with representations of the modular group. These metrics are defined using an Eisenstein series construction. We show that in certain cases, the residue of these Eisenstein metrics at their rightmost pole is a harmonic metric for the underlying representation of the modular group. The last section of the paper considers the case of a family of representations that are indecomposable but not irreducible. The analysis of the corresponding Eisenstein metrics, and the location of their rightmost pole, is an open question whose resolution depends on the asymptotics of matrix-valued Kloosterman sums.

The upper light cone L in R3 is a homogeneous space of the 3-dimensional Lorentz group G. It may be identified with the space of horocycles in the upper hyperboloide H which is the symmetric space associated to G. There exists a duality between H and L (see [5] p. 144) and a general procedure leads to a generalized Radon transform:

and the dual Radon transform

These operations commute with the natural action of the group G.

Let PGL(2, ƒ) denote the group of all Moebius transformations

over a field F. The object of this paper is to prove the following theorem.

Theorem 1. G is an S2-group and the centre of G is trivial if and only if G is isomorphic to a group PGL(2, ƒ), char ƒ ≠ 2.

This theorem was proved for finite groups in (1). The present paper extends the result to infinite groups and also improves the method of proof used in that paper. Many of the theorems given there were proved for infinite groups and are used here with appropriate references.

Wick polynomials and Wick products are studied in the context of noncommutative probability theory. It is shown that free, Boolean, and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf-algebraic approach to cumulants and Wick products in classical probability theory.

The existence of nonzero fixed points of positive contractions in L1 spaces has received considerable attention in recent years. In 1966, Dean and Sucheston [1] and independently, Neveu [5] showed that a positive contraction has a strictly positive invariant function if and only if for any measurable subset A with positive measure, where 1 is the constant function of value one.

Let F be a linear, but not necessarily closed, subspace of L2[X, dm], where (X,,m) is a σ-finite measure space with the Borel subsets of the locally compact space X. If u and v are measureable functions, then v is called a normalized contraction of u if and Assume that F is stable under normalized contractions, that is, if u ∈ F and v is a normalized contraction of u, then v ∈ F.