Let \left( T,\,M \right)$\left( T,\,M \right)$ be a complete local (Noetherian) ring such that \dim\,T\,\ge \,2$\dim\,T\,\ge \,2$ and \left| T \right|\,=\,\left| T/M \right|$\left| T \right|\,=\,\left| T/M \right|$ and let {{\left\{ {{p}_{i}} \right\}}_{i\in \Im }}${{\left\{ {{p}_{i}} \right\}}_{i\in \Im }}$ be a collection of elements of T$T$ indexed by a set \mathcal{J}$\mathcal{J}$ so that \left| \mathcal{J} \right|\,<\,\left| T \right|$\left| \mathcal{J} \right|\,<\,\left| T \right|$. For each i\,\in \,\mathcal{J}$i\,\in \,\mathcal{J}$, let {{C}_{i}}:=\left\{ {{Q}_{i1}},...,{{Q}_{i{{n}_{i}}}} \right\}${{C}_{i}}:=\left\{ {{Q}_{i1}},...,{{Q}_{i{{n}_{i}}}} \right\}$ be a set of nonmaximal prime ideals containing {{p}_{i}}${{p}_{i}}$ such that the {{Q}_{ij}}${{Q}_{ij}}$ are incomparable and {{p}_{i}}\in {{Q}_{jk}}${{p}_{i}}\in {{Q}_{jk}}$ if and only if i\,=\,j$i\,=\,j$. We provide necessary and sufficient conditions so that T$T$ is the \mathbf{m}$\mathbf{m}$-adic completion of a local unique factorization domain \left( A,\,\mathbf{m} \right)$\left( A,\,\mathbf{m} \right)$, and for each i\,\in \,\mathcal{J}$i\,\in \,\mathcal{J}$, there exists a unit {{t}_{i}}${{t}_{i}}$ of T$T$ so that {{p}_{i}}{{t}_{i}}\in A${{p}_{i}}{{t}_{i}}\in A$ and {{C}_{i}}${{C}_{i}}$ is the set of prime ideals Q$Q$ of T$T$ that are maximal with respect to the condition that Q\cap A={{p}_{i}}{{t}_{i}}A$Q\cap A={{p}_{i}}{{t}_{i}}A$.

We then use this result to construct a (nonexcellent) unique factorization domain containing many ideals for which tight closure and completion do not commute. As another application, we construct a unique factorization domain A$A$ most of whose formal fibers are geometrically regular.

For a positive finite measure d\mu \left( \mathbf{u} \right)$d\mu \left( \mathbf{u} \right)$ on {{\mathbb{R}}^{d}}${{\mathbb{R}}^{d}}$ normalized to satisfy \int{_{{{\mathbb{R}}^{d}}}d\mu \left( \mathbf{u} \right)}=1$\int{_{{{\mathbb{R}}^{d}}}d\mu \left( \mathbf{u} \right)}=1$ , the dilated average of f\left( \mathbf{x} \right)$f\left( \mathbf{x} \right)$ is given by

{{A}_{t}}\,f\left( \mathbf{x} \right)\,=\,\int{_{{{\mathbb{R}}^{d}}}\,f\left( \mathbf{x}\,-\,t\mathbf{u}\, \right)}d\mu \left( \mathbf{u} \right) $${{A}_{t}}\,f\left( \mathbf{x} \right)\,=\,\int{_{{{\mathbb{R}}^{d}}}\,f\left( \mathbf{x}\,-\,t\mathbf{u}\, \right)}d\mu \left( \mathbf{u} \right)$$

It will be shown that under some mild assumptions on d\mu \left( \mathbf{u} \right)$d\mu \left( \mathbf{u} \right)$ one has the equivalence

||{A_t}f - f|{|_B} \approx \inf \left\{ {\left( {||f - g|{|_B} + {t^2}||P\left( D \right)g|{|_B}} \right):P\left( D \right)g \in B} \right\}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\rm{for}}{\mkern 1mu} {\mkern 1mu} {\rm{t}}{\mkern 1mu} {\rm{ > }}\,{\rm{0,}} $$||{A_t}f - f|{|_B} \approx \inf \left\{ {\left( {||f - g|{|_B} + {t^2}||P\left( D \right)g|{|_B}} \right):P\left( D \right)g \in B} \right\}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\rm{for}}{\mkern 1mu} {\mkern 1mu} {\rm{t}}{\mkern 1mu} {\rm{ > }}\,{\rm{0,}}$$

where \varphi \left( t \right)\approx \psi \left( t \right)$\varphi \left( t \right)\approx \psi \left( t \right)$ means {{c}^{-1}}\le \varphi \left( t \right)/\psi \left( t \right)\le c${{c}^{-1}}\le \varphi \left( t \right)/\psi \left( t \right)\le c$ , B$B$ is a Banach space of functions for which translations are continuous isometries and P\left( D \right)$P\left( D \right)$ is an elliptic differential operator induced by \mu $\mu$. Many applications are given, notable among which is the averaging operator with d\mu \left( \mathbf{u} \right)\,=\,\frac{1}{m\left( S \right)}{{\chi }_{S}}\left( \mathbf{u} \right)d\mathbf{u},$d\mu \left( \mathbf{u} \right)\,=\,\frac{1}{m\left( S \right)}{{\chi }_{S}}\left( \mathbf{u} \right)d\mathbf{u},$ where S$S$ is a bounded convex set in {{\mathbb{R}}^{d}}${{\mathbb{R}}^{d}}$ with an interior point, m\left( S \right)$m\left( S \right)$ is the Lebesgue measure of S$S$, and {{\chi }_{S}}\left( \mathbf{u} \right)${{\chi }_{S}}\left( \mathbf{u} \right)$ is the characteristic function of S$S$. The rate of approximation by averages on the boundary of a convex set under more restrictive conditions is also shown to be equivalent to an appropriate K$K$-functional.

Let {{M}_{n}}${{M}_{n}}$ be the algebra of all n\,\times \,n$n\,\times \,n$ matrices over \mathbb{C}$\mathbb{C}$. We say that A,B\in {{M}_{n}}$A,B\in {{M}_{n}}$ quasi-commute if there exists a nonzero \xi \,\in \,\mathbb{C}$\xi \,\in \,\mathbb{C}$ such that AB\,=\,\xi BA$AB\,=\,\xi BA$. In the paper we classify bijective not necessarily linear maps \Phi :{{M}_{n}}\to {{M}_{n}}$\Phi :{{M}_{n}}\to {{M}_{n}}$ which preserve quasi-commutativity in both directions.

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet–Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

This paper studies the relationship between vector-valued \text{BMO}$\text{BMO}$ functions and the Carleson measures defined by their gradients. Let dA$dA$ and dm$dm$ denote Lebesgue measures on the unit disc D$D$ and the unit circle \mathbb{T}$\mathbb{T}$, respectively. For 1\,<\,q\,<\,\infty $1\,<\,q\,<\,\infty$ and a Banach space B$B$, we prove that there exists a positive constant c$c$ such that

\underset{{{z}_{0}}\in D}{\mathop{\sup }}\,{{\int }_{D}}{{\left( 1-\left| z \right| \right)}^{q-1}}{{\left\| \nabla f\left( z \right) \right\|}^{q}}{{P}_{{{Z}_{0}}}}\left( z \right)dA\left( z \right)\le {{c}^{q}}\underset{{{z}_{0}}\in D}{\mathop{\sup }}\,{{\int }_{\mathbb{T}}}{{\left\| f\left( z \right)-f\left( {{z}_{0}} \right) \right\|}^{q}}{{P}_{{{z}_{0}}}}\left( z \right)dm\left( z \right) $$\underset{{{z}_{0}}\in D}{\mathop{\sup }}\,{{\int }_{D}}{{\left( 1-\left| z \right| \right)}^{q-1}}{{\left\| \nabla f\left( z \right) \right\|}^{q}}{{P}_{{{Z}_{0}}}}\left( z \right)dA\left( z \right)\le {{c}^{q}}\underset{{{z}_{0}}\in D}{\mathop{\sup }}\,{{\int }_{\mathbb{T}}}{{\left\| f\left( z \right)-f\left( {{z}_{0}} \right) \right\|}^{q}}{{P}_{{{z}_{0}}}}\left( z \right)dm\left( z \right)$$

holds for all trigonometric polynomials f$f$ with coefficients in B$B$ if and only if B$B$ admits an equivalent norm which is q$q$-uniformly convex, where

{{P}_{{{z}_{0}}}}\left( z \right)=\frac{1-|{{z}_{0}}{{|}^{2}}}{|1-{{{\bar{z}}}_{0}}z{{|}^{2}}}. $${{P}_{{{z}_{0}}}}\left( z \right)=\frac{1-|{{z}_{0}}{{|}^{2}}}{|1-{{{\bar{z}}}_{0}}z{{|}^{2}}}.$$

The validity of the converse inequality is equivalent to the existence of an equivalent q$q$-uniformly smooth norm.

We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, {{L}^{1}}\left( G \right)${{L}^{1}}\left( G \right)$, and the Fourier algebra, A\left( G \right)$A\left( G \right)$, of a locally compact group G$G$.

A set of necessary and sufficient conditions for the Brascamp–Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank 1. This complements the result of Barthe concerning the case where the linear maps all have rank 1. Pushing our analysis further, we describe the case where the maps have either rank 1 or rank 2.

A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp–Lieb inequality to hold. We present an algorithm which generates such a list.

Let \mathcal{T}$\mathcal{T}$ be the {{C}^{*}}${{C}^{*}}$-algebra generated by the Toeplitz operators \left\{ {{T}_{\varphi }}:\varphi \in {{L}^{\infty }}\left( S,d\sigma \right) \right\}$\left\{ {{T}_{\varphi }}:\varphi \in {{L}^{\infty }}\left( S,d\sigma \right) \right\}$ on the Hardy space {{H}^{2}}\left( S \right)${{H}^{2}}\left( S \right)$ of the unit sphere in {{C}^{n}}${{C}^{n}}$. It is well known that \mathcal{T}$\mathcal{T}$ is contained in the essential commutant of \left\{ {{T}_{\varphi }}:\varphi \in \text{VMO}\cap {{L}^{\infty }}\left( S,d\sigma \right) \right\}$\left\{ {{T}_{\varphi }}:\varphi \in \text{VMO}\cap {{L}^{\infty }}\left( S,d\sigma \right) \right\}$. We show that the essential commutant of \left\{ {{T}_{\varphi }}:\varphi \in \text{VMO}\cap {{L}^{\infty }}\left( S,d\sigma \right) \right\}$\left\{ {{T}_{\varphi }}:\varphi \in \text{VMO}\cap {{L}^{\infty }}\left( S,d\sigma \right) \right\}$ is strictly larger than \mathcal{T}$\mathcal{T}$.

Let {{G}_{n}}=\text{S}{{\text{p}}_{n}}\left( F \right)${{G}_{n}}=\text{S}{{\text{p}}_{n}}\left( F \right)$ be the rank n$n$ symplectic group with entries in a nondyadic p$p$-adic field F$F$. We further let {{\tilde{G}}_{n}}${{\tilde{G}}_{n}}$ be the metaplectic extension of {{G}_{n}}\,\text{by}\,{{\mathbb{C}}^{1}}=\left\{ z\in {{\mathbb{C}}^{\times }}|\,|z|=1 \right\}${{G}_{n}}\,\text{by}\,{{\mathbb{C}}^{1}}=\left\{ z\in {{\mathbb{C}}^{\times }}|\,|z|=1 \right\}$ defined using the Leray cocycle. In this paper, we aim to demonstrate the complete list of reducibility points of the genuine principal series of {{\tilde{G}}_{2}}${{\tilde{G}}_{2}}$. In most cases, we will use some techniques developed by Tadić that analyze the Jacquet modules with respect to all of the parabolics containing a fixed Borel. The exceptional cases involve representations induced from unitary characters \chi $\chi$ with {{\chi }^{2}}=1${{\chi }^{2}}=1$. Because such representations \pi $\pi$ are unitary, to show the irreducibility of \pi $\pi$, it suffices to show that {{\dim}_{\mathbb{C}}}\,\text{Ho}{{\text{m}}_{{\tilde{G}}}}\left( \pi ,\,\pi \right)\,=\,1${{\dim}_{\mathbb{C}}}\,\text{Ho}{{\text{m}}_{{\tilde{G}}}}\left( \pi ,\,\pi \right)\,=\,1$ . We will accomplish this by examining the poles of certain intertwining operators associated to simple roots. Then some results of Shahidi and Ban decompose arbitrary intertwining operators into a composition of operators corresponding to the simple roots of {{\tilde{G}}_{2}}.${{\tilde{G}}_{2}}.$ We will then be able to show that all such operators have poles at principal series representations induced from quadratic characters and therefore such operators do not extend to operators in \text{Ho}{{\text{m}}_{{{{\tilde{G}}}_{2}}}}\left( \pi ,\,\pi \right)$\text{Ho}{{\text{m}}_{{{{\tilde{G}}}_{2}}}}\left( \pi ,\,\pi \right)$ for the \pi $\pi$ in question.