We define and study the secondary Chern–Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with non-isolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.

It has been shown that a holomorphic function $f$ in the unit ball ${{\mathbb{B}}_{n}}$ of ${{\mathbb{C}}_{n}}$ belongs to the weighted Bergman space $A_{\alpha }^{p},\,p\,>\,n\,+\,1\,+\alpha $, if and only if the function $\left| f(z)\,-\,f(w) \right|/\left| 1\,-\,\left\langle z,\,w \right\rangle \right|$ is in ${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\beta }}\,\times \,d{{v}_{\beta }})$, where $\beta \,=\,(p\,+\,\alpha \,-\,n\,-\,1)/2$ and $d{{v}_{\beta }}(z)\,=\,{{(1\,-\,{{\left| z \right|}^{2}})}^{\beta }}\,dv(z)$. In this paper we consider the range $0\,<\,p\,<\,n\,+\,1\,+\,\alpha $ and show that in this case, $f\,\in \,A_{\alpha }^{p}\,(\text{i})$ (i) if and only if the function $\left| f(z)\,-\,f(w) \right|/\left| 1\,-\,\left\langle z,\,w \right\rangle \right|$ is in ${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\alpha }}\,\times \,d{{v}_{\alpha }})$, (ii) if and only if the function $\left| f(z)\,-\,f(w) \right|/\left| z\,-\,w \right|$ is in ${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\alpha }}\,\times \,d{{v}_{\alpha }})$. We think the revealed difference in the weights for the double integrals between the cases $0\,<\,p\,<\,n\,+\,1\,+\,\alpha $ and $p\,>\,n\,+\,1\,+\,\alpha $ is particularly interesting.

Frey and Jarden asked if any abelian variety over a number field $K$ has the infinite Mordell–Weil rank over the maximal abelian extension ${{K}^{\text{ab}}}$. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve $C$ over $K$ such that $\sharp C\left( {{K}^{\text{ab}}} \right)\,=\,\infty $ and for any abelian variety of $\text{G}{{\text{L}}_{2}}$-type with trivial character.

In this paper, we consider the Cauchy problem

$$\left\{ \begin{align} & {{u}_{t}}=\Delta ({{u}^{m}}),\,\,\,\,\,x\in {{\mathbb{R}}^{N}},t>0,N\ge 3, \\ & u(x,0)={{u}_{0}}(x),\,\,\,\,\,x\in {{\mathbb{R}}^{N}}. \\ \end{align} \right.$$

We will prove that

(i) for ${{m}_{c}}\,<\,m,\,{{m}_{0}}\,<\,1,\,\left| u(x,\,t,m)-u(x,\,t,{{m}_{0}}) \right|\,\to \,0$ as $m\,\to \,{{m}_{0}}$ uniformly on every compact subset of ${{\mathbb{R}}^{N}}\,\times \,{{\mathbb{R}}^{+}}$, where ${{m}_{c}}\,=\,\frac{{{(N-2)}_{+}}}{N}$;

(ii) there is a ${{C}^{*}}$ that explicitly depends on $m$ such that

$${{\left\| u(\cdot ,\cdot ,m)-u(\cdot ,\cdot ,1) \right\|}_{{{L}^{2}}({{\mathbb{R}}^{N}}\times {{\mathbb{R}}^{+}})}}\le {{C}^{*}}\left| m-1 \right|.$$

We obtain nontrivial estimates of quadratic character sums of division polynomials ${{\text{ }\!\!\psi\!\!\text{ }}_{n}}\left( P \right)$, $n\,=\,1,\,2,\,\ldots $ , evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least ${{q}^{{}^{1}/{}_{2+\varepsilon }}}$ for some fixed $\varepsilon \,>\,0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author.

We show that if $R\,=\,{{\oplus }_{n\in \mathbb{N}0}}\,{{R}_{n}}$ is a Noetherian homogeneous ring with local base ring $({{R}_{0}},\,{{m}_{0}})$, irrelevant ideal ${{R}_{+}}$, and $M$ a finitely generated graded $R$-module, then $H_{{{m}_{0}}R}^{j}\,(H_{R+}^{t}\,(M))$ is Artinian for $j\,=\,0,\,1$ where $t\,=\,\inf ${$i\in {{\mathbb{N}}_{0}}:H_{R+}^{i}(M)$ is not finitely generated}. Also, we prove that if $\text{cd(}{{R}_{+}},M)\,=\,2$, then for each $i\,\in \,{{\mathbb{N}}_{0}},\,H_{{{m}_{0}}R}^{i}\,(H_{R+}^{2}\,(M))$ is Artinian if and only if $H_{{{m}_{0}}R}^{i+2}(H_{R+}^{1}(M))$ is Artinian, where $ \text{cd(}{{R}_{+}},\,M)$ is the cohomological dimension of $M$ with respect to ${{R}_{+}}$. This improves some results of R. Sazeedeh.

Let $R$ be a commutative ring with identity, and let $M$ be a unitary module over $R$. We call $M$$\text{H}$-smaller ($\text{HS}$ for short) if and only if $M$ is infinite and $\left| M/N \right|\,<\,\,\left| M \right|$ for every nonzero submodule $N$ of $M$. After a brief introduction, we show that there exist nontrivial examples of HS modules of arbitrarily large cardinality over Noetherian and non-Noetherian domains. We then prove the following result: suppose $M$ is faithful over $R$, $R$ is a domain (we will show that we can restrict to this case without loss of generality), and $K$ is the quotient field of $R$. If $M$ is $\text{HS}$ over $R$, then $R$ is $\text{HS}$ as a module over itself, $R\,\subseteq \,M\,\subseteq \,K$, and there exists a generating set $S$ for $M$ over $R$ with $\left| S \right|\,<\,\left| R \right|$. We use this result to generalize a problem posed by Kaplansky and conclude the paper by answering an open question on Jónsson modules.

We prove metric rigidity for complete manifolds supporting solutions of certain second order differential systems, thus extending classical works on a characterization of space-forms. Along the way, we also discover new characterizations of space-forms. We next generalize results concerning metric rigidity via equations involving vector fields.

Let $\left| K \right|$ be the metric polyhedron of a simplicial complex $K$. In this paper, we characterize a simplicial subdivision ${{K}^{\prime }}$ of $K$ preserving the metric topology for $\left| K \right|$ as the one such that the set ${{K}^{\prime }}(0)$ of vertices of ${{K}^{\prime }}$ is discrete in $\left| K \right|$. We also prove that two such subdivisions of $K$ have such a common subdivision.

We determine the order of the automorphism group Aut$(W)$ for each member $W$ of an important family of finite $p$-groups that may be constructed as iterated regular wreath products of cyclic groups. We use a method based on representation theory.

We show that the Schrödinger equation is a lift of Newton's third law of motion $\nabla _{{\dot{\mu }}}^{\mathcal{W}}\,\dot{\mu }\,=\,-{{\nabla }^{\mathcal{W}}}\,F\left( \mu \right)$ on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potential $\mu \,\to \,F\left( \mu \right)$ is the sum of the total classical potential energy $\left\langle V,\,\mu \right\rangle $ of the extended system and its Fisher information $\frac{{{\hbar }^{2}}}{8}\,{{\int{\left| \nabla \,\text{1n}\,\mu \right|}}^{2}}\,d\mu $. The precise relation is established via a well-known (Madelung) transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space. All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures.

Under the assumption that $\mu $ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a Hörmander-type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case.

Let ${{M}^{m}}$ be an $m$-dimensional, closed and smooth manifold, equipped with a smooth involution $T\,:\,{{M}^{m}}\,\to \,{{M}^{m}}$ whose fixed point set has the form ${{F}^{n}}\,\bigcup \,{{F}^{j}}$, where ${{F}^{n}}$ and ${{F}^{j}}$ are submanifolds with dimensions $n$ and $j$, ${{F}^{j}}$ is indecomposable and $n\,>\,j$. Write $n\,-\,j\,={{2}^{p}}q$, where $q\,\ge \,1$ is odd and $p\,\ge \,0$, and set $m(n\,-\,j)\,=\,2n\,+\,p\,-q\,+\,1$ if $p\,\le \,q\,+\,1$ and $m(n\,-\,j)\,=\,2n\,+\,{{2}^{p-q}}$ if $p\,\ge \,q$. In this paper we show that $m\,\le \,m(n-j)+2j+1$. Further, we show that this bound is almost best possible, by exhibiting examples $({{M}^{m(n-j)+2j}},\,T)$ where the fixed point set of $T$ has the form ${{F}^{n}}\,\bigcup \,{{F}^{j}}$ described above, for every $2\,\le \,j\,<\,n$ and $j$ not of the form ${{2}^{t}}\,-\,1$ (for $j\,=\,0$ and 2, it has been previously shown that $m(n\,-\,j)\,+\,2j$ is the best possible bound). The existence of these bounds is guaranteed by the famous 5/2-theorem of J. Boardman, which establishes that under the above hypotheses $m\,\le \,\frac{5}{2}\,n$.

In this paper, we give a new proof of the Onofri-type inequality

$$\int_{S}{{{e}^{2u}}\,d{{s}^{2}}\,\le \,4\pi (\beta \,+\,1)\,\text{exp}}\left\{ \frac{1}{4\pi (\beta \,+\,1)}{{\int_{S}{\left| \nabla u \right|}}^{2}}\,d{{s}^{2}}\,+\,\frac{1}{2\pi (\beta \,+\,1)}\,\int_{S}{u\,d{{s}^{2}}} \right\}$$

on the sphere $S$ with Gaussian curvature 1 and with conical singularities divisor $\mathcal{A}\,=\,\beta \,\cdot \,{{p}_{1}}\,+\,\beta \,\cdot \,{{p}_{2}}$ for $\beta \in \,(-1,\,0)$; here ${{p}_{1}}$ and ${{p}_{2}}$ are antipodal.

In this paper we study left invariant Einstein–Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein–Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein–Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

The purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions, and equivariant forms.

In this paper we give the form of every multiplicative Hausdorff prime matrix, thus answering a long-standing open question.

The notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of Galvin's theorem is used in the proof. An alternative proof of the dichotomy result for sequences in Banach spaces is also sketched, using the Galvin–Prikry theorem.

${{L}_{p}}$ stability and exponential stability are two important concepts for nonlinear dynamic systems. In this paper, we prove that a nonlinear exponentially bounded Lipschitzian semigroup is exponentially stable if and only if the semigroup is ${{L}_{p}}$ stable for some $p\,>\,0$. Based on the equivalence, we derive two sufficient conditions for exponential stability of the nonlinear semigroup. The results obtained extend and improve some existing ones.

We consider the linearization of the three-dimensional water waves equation with surface tension about a flat interface. Using oscillatory integral methods, we prove that solutions of this equation demonstrate dispersive decay at the somewhat surprising rate of ${{t}^{-5/6}}$. This rate is due to competition between surface tension and gravitation at $O(1)$ wave numbers and is connected to the fact that, in the presence of surface tension, there is a so-called “slowest wave”. Additionally, we combine our dispersive estimates with ${{L}^{2}}$ type energy bounds to prove a family of Strichartz estimates.