Corrections to Dasgupta’s “Hermitian Morita equivalences between maximal orders in central simple algebras” (Illinois J. Math. 53 (2009) 723–736) follow. $\operatorname{char} K=2$ was not considered in the paper, nor was the fact that $\delta^{-1}y_1^\beta y_1^{-1}\epsilon$ could be equal to $-1$ at a couple of points in the proof of Proposition 3 of the paper. Changes in the proof of Proposition 3, in its statement and the corresponding changes in the statements of Theorems 1, 2 and 3 follow.

We continue our study of the homological properties of the purely inseparable extensions $ \mathrm{H} \hookrightarrow\!\!\sqrt[\mathcal{P}^*]{\mathrm{H}}$ of integrally closed unstable Noetherian integral domains over the Steenrod algebra. It turns out that the projective dimension of $\mathrm {H}$ is a lower bound for the projective dimension of $\!\!\sqrt[\mathcal{P}^*]{\mathrm{H}}$. Furthermore, $\operatorname{depth} (\mathrm{H}) \geq\operatorname{depth}(\!\!\sqrt[\mathcal{P}^*]{\mathrm{H}})$, where $\operatorname{depth}$ denotes the depth. Moreover, both algebras have the same global dimension. We apply these results to extension $\mathbb F[V_{\bullet}]^G \hookrightarrow\mathbb F[V]^G$ of rings of invariants.

Let $\mathcal{F}$ be a compact Hausdorff foliation on a compact manifold. Let ${E_2^{>0,\bullet}}=\bigoplus\{E_2^{p,q} : p>0,q\geq0\}$ be the subalgebra of cohomology classes with positive transverse degree in the $E_2$ term of the spectral sequence of the foliation. We prove that the saturated transverse Lusternik-Schnirelmann category of $\mathcal {F}$ is bounded below by the length of the cup product in ${E_2^{>0,\bullet}}$. Other cohomological bounds are discussed.

In this paper, we investigate $L^p$-boundedness properties for the one-dimensional higher order Riesz transforms associated with Laguerre operators. We also prove that the $k$-th Riesz transform is a principal value singular integral operator (modulus a constant times of the function when $k$ is even). To establish our results, we exploit a new estimate connecting Riesz transforms in the Hermite and Laguerre settings in dimension one.

We give necessary conditions for real algebraic hypersurfaces (possibly with singularities) to contain nontrivial germs of complex hypersurfaces. Moreover, if a real hypersurface $S$ in $\mathbf{C}^2$ is defined by a real polynomial of a sufficiently general form and if $S$ contains a nontrivial analytic disk, then, using the above result, we show that $S$ must contain certain complex lines.

In this paper, we give explicit descriptions of all numerical semigroups, generated by three positive integer numbers, that are the set of solutions of a Diophantine inequality of the form $ax \operatorname{mod} b\leq x$.

We construct a sequence ${a_n}$ such that for any aperiodic measure-preserving system $(X, \Sigma, m, T)$ the ergodic averages \begin{equation*} A_Nf(x) = \frac{1}{N} \sum_{n=1}^N f\bigl(T^{a_n}x\bigr) \end{equation*} converge a.e. for all $f$ in $L \log\log(L)$ but fail to have a finite limit for an $f \in L^1$. In fact, we show that for each Orlicz space properly contained in $L^1$ there is a sequence along which the ergodic averages converge for functions in the Orlicz space, but diverge for all $f \in L^1$. Our method, introduced by A. Bellow and extended by K. Reinhold and M. Wierdl, is perturbation.

In the present paper, it is proved that any complete local domain of mixed characteristic has a weakly almost Cohen–Macaulay algebra $B$ in the sense that a system of parameters is a weakly almost regular sequence in $B$, which is a notion defined via a valuation. In fact, the central idea of this result originates from the main statement obtained by Heitmann to prove the Monomial Conjecture in dimension 3. A weakly almost Cohen–Macaulay algebra is constructed over the absolute integral closure of a complete local domain by applying the methods of Fontaine rings and Witt vectors. A connection of the main theorem with the Monomial Conjecture is also discussed.

We construct a flexible (non-embedded) suspension with a hexagonal equator in Euclidean 3-space. It is known that the volume bounded by such a suspension is well defined and constant during the flex. We study its properties related to the Strong Bellows Conjecture which reads as follows: if a, possibly singular, polyhedron $\mathcal P$ in Euclidean 3-space is obtained from another, possibly singular, polyhedron $\mathcal Q$ by a continuous flex, then $\mathcal P$ and $\mathcal Q$ have the same Dehn invariants. It is well known that if $\mathcal P$ and $\mathcal Q$ are embedded, with the same volume and the same Dehn invariant, then they are scissors congruent.

We prove that $D_*$, the set of all parameters $\lambda\in\mathbb{C}\setminus\{0\}$ for which the cubic polynomial $f_\lambda$ is parabolic and has no other parabolic or finite attracting periodic cycles, contains a deleted neighborhood $D_0$ of the origin 0. Our main result is that if $D_0$ is sufficiently small then the function $D_0\ni\lambda\mapsto\operatorname{HD}(J(f_\lambda))\in\mathbb{R}$ is real-analytic. This function ascribes to the polynomial $f_\lambda$ the Hausdorff dimension of its Julia set $J(f_\lambda)$. The theory of parabolic and hyperbolic graph directed Markov systems with infinite number of edges is used in the proofs.

We consider the integrals of the $r$-mean curvatures $S_r$ of a complete hypersurface $M$ in the space form $\mathcal{Q}_c^{n+1}$. Among other results, we prove that $\int_MS_r\,dM=\infty$ for a complete properly immersed hypersurfaces in a space form with $S_r\geq0$, $S_r\not\equiv0$ and $S_{r+1}\equiv0$ for some $r\le n-1$.

We consider a fixed free and proper action of a locally compact group $G$ on a space $T$, and actions $\alpha : G\to\operatorname{Aut} A$ on $C^*$-algebras for which there is an equivariant embedding of $(C_0(T),\mathrm{rt})$ in $(M(A),\alpha)$. A recent theorem of Rieffel implies that $\alpha$ is proper and saturated with respect to the subalgebra $C_c(T)AC_c(T)$ of $A$, so that his general theory of proper actions gives a Morita equivalence between $A\rtimes_{\alpha,r}G$ and a generalised fixed-point algebra $A^\alpha$. Here we investigate the functor $(A,\alpha)\mapsto A^\alpha$ and the naturality of Rieffel's Morita equivalence, focusing in particular on the relationship between the different functors associated to subgroups and quotients. We then use the results to study induced representations for crossed products by coactions of homogeneous spaces $G/H$ of $G$, which were previously shown by an Huef and Raeburn to be fixed-point algebras for the dual action of $H$ on the crossed product by $G$.

We introduce cone bilipschitz equivalences between metric spaces. These are maps, more general than quasi-isometries, that induce a bilipschitz homeomorphism between asymptotic cones. Nontrivial examples appear in the context of Lie groups, and we thus prove that the study of asymptotic cones of connected Lie groups can be reduced to that of solvable Lie groups of a special form. We also focus on asymptotic cones of nilpotent groups.

Let $G$ be a connected complex reductive linear algebraic group, and let $K \subset G$ be a maximal compact subgroup of it. Let $E_G$ be a holomorphic principal $G$-bundles over the complex projective line ${\mathbb C}{\mathbb P}^1$ and $E_K \subset E_G$ a $C^\infty$ reduction of structure group of $E_G$ to $K$. We consider all pairs $(E_G ,E_K)$ of this type such that the total space of $E_K$ is equipped with a $C^\infty$ lift of the standard action of $\operatorname{SU}(2)$ on ${\mathbb C}{\mathbb P}^1$ which satisfies the following two conditions: the actions of $K$ and $\operatorname{SU}(2)$ on $E_K$ commute, and for each element $g \in \operatorname{SU}(2)$, the induced action of $g$ on $E_G$ is holomorphic. We give a classification of the isomorphism classes of all such objects.

This paper is concerned with dynamics in general ultrametric spaces. We prove the Fixed Point theorem, the Stability theorem, the Common Point theorem, the Local Fixed Point theorem, the Attractor theorem and we derive conditions for a mapping to be surjective. We also discuss tightly continuous mappings and prove a theorem about the transfer of the property of principal completeness by a mapping between ultrametric spaces.

Multiple scalar integral representations for traces of operator derivatives are obtained and applied in the proof of existence of the higher order spectral shift functions.

In this paper, we introduce a new triangulated category for rational surface singularities which in the non-Gorenstein case acts as a substitute for the stable category of matrix factorizations. The category is formed as a stable quotient of the Frobenius category of special CM modules, and we classify the relatively projective-injective objects and thus describe the AR quiver of the quotient. Connections to the corresponding reconstruction algebras are also discussed.

Let $X$ be a (quasi-)Banach space. Let $d=(d_n)_{n\geq1}$ be an $X$-valued sequence of random variables adapted to a filtration $(\mathcal{F}_n)_{n\geq1}$ on a probability space $(\Omega,\mathcal{A},\mathbb{P})$, define $\mathcal{F}_{\infty}:=\sigma(\mathcal{F}_n : n\geq1)$ and let $e=(e_n)_{n\geq1}$ be a $\mathcal{F}_{\infty}$-conditionally independent sequence on $(\Omega,\mathcal{A},\mathbb{P})$ such that $\mathcal{L}(d_n | \mathcal{F}_{n-1}) = \mathcal{L}(e_n | \mathcal{F}_{\infty})$ for all $n\geq1$ ($\mathcal{F}_0=\{\Omega, \varnothing\}$). If there exists a $p\in(0,\infty)$ and a constant $D_p$ independent of $d$ and $e$ such that one has, for all $n\geq1$, $$(^{*})\qquad \mathbb{E}\Biggl \Vert \sum_{k=1}^{n}d_{k}\Biggr \Vert ^{p}\leq D_{p}^{p}\mathbb{E}\Biggl \Vert \sum_{k=1}^{n}e_{k}\Biggr \Vert ^{p},$$ then $X$ is said to satisfy the decoupling inequality for $p$. It has been proven that $X$ is a UMD space if and only if both $(^{*})$ and the reverse estimate hold for some (all) $p\in(1,\infty)$. However, in earlier work we proved that the space $L^1$, which is not a UMD space, satisfies the decoupling inequality for all $p\geq1$.

Here, we prove that if the decoupling inequality is satisfied in $X$ for some $p\in(0,\infty)$ then it is satisfied for all $p\in(0,\infty)$. We consider the behavior of the constant $D_p$ in $(^{*})$. We examine its relation to the norm of the Hilbert transform on $L^p(X)$ and show that if $X$ is a Hilbert space then there exists a universal constant $D$ such that $(^{*})$ holds with $D_p=D$, for all $p\in[1,\infty)$.

An important motivation to study decoupling inequalities is that they play a key role in the recently developed theory for stochastic integration in Banach spaces. We extend the available theory, proving a $p$th-moment Burkholder-Davis-Gundy inequality for the stochastic integral of an $X$-valued process, where $X$ is a UMD space and $p\in(0,\infty)$.

We study the weak Lefschetz property and the Hilbert function of level Artinian monomial almost complete intersections in three variables. Several such families are shown to have the weak Lefschetz property if the characteristic of the base field is zero or greater than the maximal degree of any minimal generator of the ideal. Two of the families have an interesting relation to tilings of hexagons by lozenges. This lends further evidence to a conjecture by Migliore, Miró-Roig, and the second author. Finally, using our results about the weak Lefschetz property, we show that the Hilbert function of each level Artinian monomial almost complete intersection in three variables is peaked strictly unimodal.

We classify the hypersurfaces of $\mathbb {S}^n\times \mathbb{R}$ and $\mathbb{H}^n\times \mathbb {R}$ with constant sectional curvature and dimension $n\geq3$.