We provide a local classification of \iis $f\colon\,L^p\times_\rho M^n\to\Q_c^{\,p+n+k}$ in codimensions $k=1, 2$ of warped products of Riemannian manifolds into space forms, under the assumptions that $n\geq k+1$ and that $N^{p+n}=L^p\times_\rho M^n$ has no points with the same constant sectional curvature $c$ as the ambient space form.

Let $D$ be a bounded domain in $\mathbb C^n$ with $C^2$ boundary. Let $H^p(D)$ be the Hardy space and $A^{p,\alpha}(D)$ be the space of holomorphic functions which are $L^p$-integrable with respect to the weighted measure $dV_\alpha(z)=\delta_D(z)^{\alpha-1}dV(z)$. We obtain some estimates on the mean growth of $H^p$ functions in $D$. Using these estimates, we can embed the $H^p(D)$ space into $A^{q,\beta}(D)$ for $0<p<q<\infty,\, \beta>0$ satisfying $n/p=(n+\beta)/q$. We also show that the condition of $C^2$-smoothness of the boundary of $D$ is an essential condition by giving a counter-example of a convex domain with $C^{1,\lambda}$ smooth boundary for $0<\lambda<1$ which does not satisfy the embedding result.

We prove that if $T = T^{eq}$ is a stable theory (without the finite cover property), then the theory $T_{P}$ of "beautiful pairs" eliminates imaginaries if and only if no infinite group is definable in a model of $T$.

Let $G$ be a discrete countable Abelian group. We construct an infinite measure preserving rank one action $T=(T_g)$ of $G$ such that (i) the transformation $T_g$ has infinite ergodic index but $T_g\times T_{2g}$ is not ergodic for any element $g$ of infinite order, (ii) $T_{g_1}\times\cdots\times T_{g_n}$ is conservative for every finite sequence $g_1,\dots, g_n\in G$. In the case $G=\mathbb{Z}$ this answers a question of C. Silva. Moreover, we show that

(i) every weakly stationary nonsingular Chacon transformation with 2-cuts is power weakly mixing and

(ii) every weakly stationary nonsingular Chacon$^*$ transformation with 2-cuts has infinite ergodic index but is not power weakly mixing.

We relate Orlicz-Hardy inequalities on a bounded Euclidean domain to certain fatness conditions on the complement. In the case of certain log-scale distortions of $L^n$, this relationship is necessary and sufficient, thus extending results of Ancona, Lewis, and Wannebo.

We study the interplay between the properties of the germ of a singular variety $N\subset \mathbb R^n$ given in the title and the algebra of vector fields tangent to $N$. The Poincare lemma property means that any closed differential $(p+1)$-form vanishing at any point of $N$ is a differential of a $p$-form which also vanishes at any point of $N$. In particular, we show that the classical quasi-homogeneity is not a necessary condition for the Poincare lemma property; it can be replaced by quasi-homogeneity with respect to a smooth submanifold of $\mathbb R^n$ or a chain of smooth submanifolds. We prove that $N$ is quasi-homogeneous if and only if there exists a vector field $V, V(0)=0,$ which is tangent to $N$ and has positive eigenvalues. We also generalize this theorem to quasi-homogeneity with respect to a smooth submanifold of $\mathbb R^n$.

We prove that for $p\in (0,\infty)$ an analytic univalent function in the unit disk belongs to the Hardy space $H^p$ if and only if it belongs to the Dirichlet type space $\mathcal {D}_{p-1}^p$.

J. M. Rosenblatt and G. A. Willis introduced the notion of configurations for finitely generated groups $G$. They characterised amenability of $G$ in terms of the configuration equations. In this paper we investigate which group properties can be characterised by configurations. It is proved that if $G_1$ and $G_2$ are two finitely generated groups having the same configuration sets and $G_1$ satisfies a semigroup law, then $G_2$ satisfies the same semigroup law. Furthermore, if $G_1$ is abelian then $G_1$ and $G_2$ are isomorphic.

Let $\Omega\subset\mathbb C^2$ be a smoothly bounded domain. We prove that if $\partial \Omega$ contains a (small) smooth curve of points of infinity type, then the automorphism group $\Aut(\Omega)$ is compact. This result implies the Greene-Krantz conjecture for a special class of domains. The proof makes no use of scaling techniques.

We prove several new inequalities for trigonometric sums in two variables. One of our results states that the double-inequality

\begin{align} -\frac{2}{3}(\sqrt{2}-1) &\leq \sum_{k=1}^{n}\frac{\cos((k-1/2)x)\sin((k-1/2)y)}{k-1/2}\leq 2 \end{align}

holds for all integers $n\geq 1$ and real numbers $x,y \in [0,\pi]$. Both bounds are best possible.

We prove that the best weak-type $(p,p)$ constant, $1\leq p\leq 2$, for orthogonal harmonic functions $u$ and $v$ with $v$ differentially subordinate to $u$ is

\[ K_p ={\left(\frac{1}{\pi}\int_{-\infty}^\infty \frac{{\left|\frac{2}{\pi} \log{|t|}\right|}^p}{t^2 + 1} dt\right)}^{-1}.\]

Let $p$ be a prime number, $K$ be a finite extension of the $p$-adic rational numbers containing a primitive $p^n$th root of unity, $R$ be the valuation ring of $K$ and $G$ be the cyclic group of order $p^n$. We define triangular Hopf orders over $R$ in $KG$, and show that there exist triangular Hopf orders with $n(n+1)/2$ parameters by showing that the linear duals of "sufficiently $p$-adic" formal group Hopf orders are triangular.

The notion of an algebra that is locally embeddable into finite dimensional algebras (LEF) and the notion of an LEF group was introduced by Gordon and Vershik in [GoVe]. M. Ziman proved in [Zi] that the group algebra of a group $G$ is an LEF algebra if and only if $G$ is an LEF group. He conjectured that an algebra generated as a vector space by a multiplicative subgroup $G$ of its invertible elements is an LEF algebra if and only if $G$ is an LEF group. In this paper we give a characterization of the invertible elements of an LEF algebra and use it to construct a counterexample to this conjecture.

Let $R$ be a commutative noetherian local ring. Denote by $\mod R$ the category of finitely generated $R$-modules, and by ${\mathcal G} (R)$ the full subcategory of $\mod R$ consisting of all $R$-modules of G-dimension zero. Suppose that $R$ is henselian and non-Gorenstein, and that there is a non-free $R$-module in ${\mathcal G} (R)$. Then it is known that ${\mathcal G} (R)$ is not contravariantly finite in $\mod R$ if $R$ has depth at most one. In this paper, we prove that the same statement holds if $R$ has depth two.

We prove that the capacity associated to the signed vector-valued Riesz kernel $\frac{x}{|x|^{1+\alpha}}$ in $\Rn$, $0<\alpha<n$, $\alpha\notin\Z$, vanishes on compact sets with finite $\alpha$-Hausdorff measure that satisfy an additional density condition.

We study the structure of the set of semidualizing complexes over a local ring. In particular, we prove that for a pair of semidualizing complexes $X_1$ and $X_2$ such that $G_{X_{2}}\dim X_{1}<\infty $ we have $X_2\simeq X_1\otimes^{L}_R\func{\mathbf{R}Hom}_R(X_{1},X_{2})$. Specializing to the case of semidualizing modules over artinian rings we obtain a number of quantitative results for rings possessing a configuration of semidualizing modules of special form. For rings with ${\mathfrak m}^3=0$ this condition reduces to the existence of a nontrivial semidualizing module and we prove a number of structural results in this case.

We prove a relative Fatou theorem for $\alpha$-harmonic functions on bounded Lipschitz domains $D$ that vanish outside $D$. We also discuss the case when the normalizing function corresponds to the Hausdorff measure on the boundary of $D$.

We show that the Fredholm spectrum of an evolution semigroup $\{E^t\}_{t\geq 0}$ is equal to its spectrum, and prove that the ranges of the operator $E^t-I$ and the generator ${\bf G}$ of the evolution semigroup are closed simultaneously. The evolution semigroup is acting on spaces of functions with values in a Banach space, and is induced by an evolution family that could be the propagator for a well-posed linear differential equation $u'(t)=A(t)u(t)$ with, generally, unbounded operators $A(t)$; in this case ${\bf G}$ is the closure of the operator $G$ given by $(Gu)(t)=-u'(t)+A(t)u(t)$.

In this paper we use renorming techniques to settle several questions on extremal structures of Banach spaces. We construct a unitary vector in a dual space which is not a weak$^\ast$-unitary. We construct an exposed point in the unit ball of a Banach space $X$ that remains exposed in the unit ball of $X^{(4)}$ but is not extreme in the unit ball of $X^{(6)}$.

We will show that the Camassa-Holm equation possesses periodic traveling wave solutions with spikes, i.e., peaks where the first derivative is unbounded. Moreover, we will show that such a solution can be chosen to be $\rho$-periodic for arbitrarily small $\rho>0$.

This family of solutions (parametrized by $\rho$) has the important property that, for $q \in [1,3)$, $\|u_0'\|_{L^q(\mathbb{T})}$ is uniformly bounded above and below, where $u_0$ is the initial data. Using this property with $q=2$ we are able to prove that the corresponding Cauchy problem is not locally well-posed in the Sobolev space $H^1(\mathbb{T})$. Similarly, we will show ill-posedness in the corresponding $L^q$ Sobolev space, $W^{1,q}(\mathbb{T})$, for any $q\in[1,3)$.

We estimate the probabilities that the Menezes-Okamoto-Vanstone reduction of the discrete logarithm problem on an elliptic curve $\E$ to the discrete logarithm problem in a certain finite field succeeds for various groups on points on $\E$. Our bounds imply that in all interesting cases these probabilities are exponentially small. This extends results of Balasubramanian and Koblitz who have treated the instance in which the order of the group of points on $\E$ is prime.

Let $H^2_m$ be the Hilbert function space on the unit ball in $\C{m}$ defined by the kernel $k(z,w) = (1-\langle z,w \rangle)^{-1}$. For any weak zero set of the multiplier algebra of $H^2_m$, we study a natural extremal function, $E$. We investigate the properties of $E$ and show, for example, that $E$ tends to $0$ at almost every boundary point. We also give several explicit examples of the extremal function and compare the behaviour of $E$ to the behaviour of $\delta^*$ and $g$, the corresponding extremal function for $H^\infty$ and the pluricomplex Green function, respectively.

The Hilbert matrix acts on Bergman spaces by multiplication on Taylor coefficients. We find an upper bound for the norm of the induced operator.