We give a two dimensional extension of the three distance theorem. Let $\theta $ be in ${{\mathbf{R}}^{2}}$ and let $q$ be in $\mathbf{N}$. There exists a triangulation of ${{\mathbf{R}}^{2}}$ invariant by ${{\mathbf{Z}}^{2}}$-translations, whose set of vertices is ${{\mathbf{Z}}^{2}}\,+\,\{0,\,\theta ,\,\ldots ,\,q\theta \}$, and whose number of different triangles, up to translations, is bounded above by a constant which does not depend on $\theta $ and $q$.

Let $C$ be the class of convex sequences of real numbers. The quadratic irrational numbers can be partitioned into two types as follows. If $\alpha$ is of the first type and $\left( {{c}_{k}} \right)\,\in C$, then $\sum{{{(-1)}^{\left\lfloor k\alpha \right\rfloor }}}{{c}_{k}}$ converges if and only if ${{c}_{k}}\log k\to 0$. If $\alpha$ is of the second type and $\left( {{c}_{k}} \right)\,\in C$, then $\sum{{{(-1)}^{\left\lfloor k\alpha \right\rfloor }}}{{c}_{k}}$ converges if and only if $\sum{{{c}_{k}}/k}$ converges. An example of a quadratic irrational of the first type is $\sqrt{2}$, and an example of the second type is $\sqrt{3}$. The analysis of this problem relies heavily on the representation of $\alpha$ as a simple continued fraction and on properties of the sequences of partial sums $S\left( n \right)\,=\,{{\sum\nolimits_{k=1}^{n}{\left( -1 \right)}}^{\left\lfloor k\alpha \right\rfloor }}$ and double partial sums $T\left( n \right)\,=\,\sum\nolimits_{k=1}^{n}{\,S\left( k \right)}$.

We consider maximal regularity in the ${{H}^{p}}$ sense for the Cauchy problem ${{u}^{\prime }}(t)+Au(t)=f(t)(t\,\in \mathbb{R})$, where $A$ is a closed operator on a Banach space $X$ and $f$ is an $X$-valued function defined on $\mathbb{R}$. We prove that if $X$ is an AUMD Banach space, then $A$ satisfies ${{H}^{p}}$-maximal regularity if and only if $A$ is Rademacher sectorial of type $<\frac{\pi }{2}.$ Moreover we find an operator $A$ with ${{H}^{p}}$-maximal regularity that does not have the classical ${{L}^{p}}$-maximal regularity. We prove a related Mikhlin type theorem for operator valued Fourier multipliers on Hardy spaces ${{H}^{p}}(\mathbb{R};\,X)$, in the case when $X$ is an AUMD Banach space.

Let $G$ be a locally compact group, and let ${{A}_{\text{cb}}}(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that ${{C}^{*}}(G)$ is residually finite-dimensional, we show that ${{A}_{\text{cb}}}(G)$ is operator amenable. In particular, ${{A}_{\text{cb}}}({{\mathbb{F}}_{2}})$ is operator amenable even though ${{\mathbb{F}}_{2}}$, the free group in two generators, is not an amenable group. Moreover, we show that if $G$ is a discrete group such that ${{A}_{\text{cb}}}(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$ if and only if it has an approximate identity bounded in the $\text{cb}$-multiplier norm.

Two formulas for the multiplicity of the fiber cone $F\left( I \right)\,=\,\oplus _{n=0}^{\infty }{{I}^{n}}/\text{m}{{I}^{n}}$ of an $\text{m}$-primary ideal of a $d$-dimensional Cohen–Macaulay local ring $\left( R,m \right)$ are derived in terms of the mixed multiplicity ${{e}_{d-1}}\left( \text{m }|I \right)$, the multiplicity $e\left( I \right)$, and superficial elements. As a consequence, the Cohen–Macaulay property of $F\left( I \right)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of $I$ and lengths of certain ideals. We also characterize the Cohen–Macaulay and Gorenstein properties of fiber cones of $\text{m}$-primary ideals with a $d$-generated minimal reduction $J$ satisfying $\ell \left( {{I}^{2}}/JI \right)\,=\,1$ or $l\left( Im \right)Jm) = 1$.

We study the geometry of the set of closed extensions of index 0 of an elliptic differential cone operator and its model operator in connection with the spectra of the extensions, and we give a necessary and sufficient condition for the existence of rays of minimal growth for such operators.

For a commutative local ring $R$, consider (noncommutative) $R$-algebras $\Lambda$ of the form $\Lambda \,=\,\text{En}{{\text{d}}_{R}}\left( M \right)$ where $M$ is a reflexive $R$-module with nonzero free direct summand. Such algebras $\Lambda$ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of Spec $R$. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal $\mathbb{C}$-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra $\Lambda$ with finite global dimension and which is maximal Cohen-Macaulay over $R$ (a “noncommutative crepant resolution of singularities”). We produce algebras $\Lambda \,=\,\text{En}{{\text{d}}_{R}}\left( M \right) $ having finite global dimension in two contexts: when $R$ is a reduced one-dimensional complete local ring, or when $R$ is a Cohen-Macaulay local ring of finite Cohen–Macaulay type. If in the latter case $R$ is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh.

In this paper we study the Chen–Ruan cohomology ring of weighted projective spaces. Given a weighted projective space ${{\mathbf{P}}^{n}}_{{{q}_{0}},\ldots ,{{q}_{n}}}$, we determine all of its twisted sectors and the corresponding degree shifting numbers. The main result of this paper is that the obstruction bundle over any 3-multisector is a direct sum of line bundles which we use to compute the orbifold cup product. Finally we compute the Chen–Ruan cohomology ring of weighted projective space $\mathbf{P}_{1,\,2,\,2,\,3,3,\,{{3}^{\centerdot }}}^{5}$.

Let $K$ be a real quadratic field, and $p$ a rational prime which is inert in $K$. Let $\alpha $ be a modular unit on ${{\Gamma }_{0}}(N)$. In an earlier joint article with Henri Darmon, we presented the definition of an element $u\left( \alpha ,\,\text{ }\!\!\tau\!\!\text{ } \right)\,\in \,K_{P}^{\times }$ attached to $\alpha $ and each $\tau \,\in \,K$. We conjectured that the $p$-adic number $u(\alpha ,\,\tau )$ lies in a specific ring class extension of $K$ depending on $\tau $, and proposed a “Shimura reciprocity law” describing the permutation action of Galois on the set of $u(\alpha ,\,\tau )$. This article provides computational evidence for these conjectures. We present an efficient algorithm for computing $u(\alpha ,\,\tau )$, and implement this algorithm with the modular unit $\alpha (z)\,=\,\Delta {{(z)}^{2}}\,\Delta (4z)\,/\,\Delta {{(2z)}^{3}}$. Using $p\,=\,3,\,5,\,7\,and\,11$, and all real quadratic fields $K$ with discriminant $D\,<\,500$ such that 2 splits in $K$ and $K$ contains no unit of negative norm, we obtain results supporting our conjectures. One of the theoretical results in this paper is that a certain measure used to define $u(\alpha ,\,\tau )$ is shown to be $\mathbf{Z}$-valued rather than only ${{\mathbf{Z}}_{P}}\,\cap \,\mathbf{Q}-$valued; this is an improvement over our previous result and allows for a precise definition of $u(\alpha ,\,\tau )$, instead of only up to a root of unity.

We define the higher order Riesz transforms and the Littlewood-Paley $g$-function associated to the differential operator ${{L}_{\lambda }}f(\theta )\,=\,-{f}''(\theta )-2\lambda \cot \theta {f}'(\theta )+{{\lambda }^{2}}f(\theta )$. We prove that these operators are Calderón–Zygmund operators in the homogeneous type space $((0,\,\pi ),\,{{(\sin t)}^{2\lambda }}dt)$. Consequently, ${{L}^{p}}$ weighted, ${{H}^{1}}\,-\,{{L}^{1}}$ and ${{L}^{\infty }}\,-\,BMO$ inequalities are obtained.

Cubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in ${{\mathbb{R}}^{d}}$ in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris–Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.

Let $\phi (n)$ be the Euler phi-function, define ${{\phi }_{0}}\left( n \right)\,=\,n$ and ${{\phi }_{k+1}}\left( n \right)\,=\,\phi \left( {{\phi }_{k}}\left( n \right) \right)$ for all $k\ge 0$. We will determine an asymptotic formula for the set of integers $n$ less than $x$ for which ${{\phi }_{k}}\left( n \right)$ is $y$-smooth, conditionally on a weak form of the Elliott–Halberstam conjecture.

We study the cardinal invariants of analytic $P$-ideals, concentrating on the ideal $Z$ of asymptotic density zero. Among other results we prove $\min \,\{\mathfrak{b},\,\operatorname{cov}(\mathcal{N})\,\}\,\le \,\operatorname{cov}*\,(Z)\,\le \,\max \{\mathfrak{b},\,\text{non(}\mathcal{N}\text{)}\}$.

Let $G$ be a locally compact group and $\pi $ a representation of $G$ by weakly* continuous isometries acting in a dual Banach space $E$. Given a probability measure $\mu $ on $G$, we study the Choquet–Deny equation $\pi (\mu )x\,=\,x,\,x\,\in \,E$. We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a “Poisson formula” on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law $\mu $. The relation between the space of solutions of the Choquet–Deny equation in $E$ and the space of bounded harmonic functions can be understood in terms of a construction resembling the ${{W}^{*}}$-crossed product and coinciding precisely with the crossed product in the special case of the Choquet–Deny equation in the space $E\,=\,B({{L}^{2}}(G))$ of bounded linear operators on ${{L}^{2}}(G)$. Other general properties of the Choquet–Deny equation in a Banach space are also discussed.

Let $A$ be a separable amenable purely infinite simple ${{C}^{*}}$-algebra which satisfies the Universal Coefficient Theorem. We prove that $A$ is weakly semiprojective if and only if ${{K}_{i}}(A\text{)}$ is a countable direct sum of finitely generated groups $\left( i\,=\,0,\,1 \right)$. Therefore, if $A$ is such a ${{C}^{*}}$-algebra, for any $\varepsilon \,>\,0$ and any finite subset $\mathcal{F}\,\subset \,A$ there exist $\delta \,>\,0$ and a finite subset $G\,\subset \,A$ satisfying the following: for any contractive positive linear map $L\,:\,A\,\to \,B$ (for any ${{C}^{*}}$-algebra $B$) with $||L\left( ab \right)\,-\,L\left( a \right)L\left( b \right)||\,<\,\delta$ for $a,\,b\,\in \,\mathcal{G}$ there exists a homomorphism $h:\,A\,\to \,B$ such that $||\,h\left( a \right)\,-\,L\left( a \right)||\,<\,\varepsilon$ for $a\,\in \,\mathcal{F}$.

We consider the $p$-Yang-Mills functional $\left( p\,\ge \,2 \right)$ defined as $Y{{M}_{p}}(\nabla ):=\frac{1}{p}{{\int }_{M}}{{\left\| {{R}^{\nabla }} \right\|}^{p}}$. We call critical points of $Y{{M}_{p}}(\cdot )$ the p-Yang–Mills connections, and the associated curvature ${{R}^{\nabla }}$ the $p$-Yang-Mills fields. In this paper, we prove gap properties and instability theorems for $p$-Yang-Mills fields over submanifolds in ${{\mathbb{R}}^{n+k}}$ and ${{\mathbb{S}}^{n+k}}$.

We determine which isogeny classes of supersingular abelian surfaces over a finite field $k$ of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus 2. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the number of curves, up to $k$-isomorphism, leading to the same zeta function.

In this paper we prove the unitarity of duals of tempered representations supported on minimal parabolic subgroups for split classical $p$-adic groups. We also construct a family of unitary spherical representations for real and complex classical groups.

Let $(Y,\,T)$ be a minimal suspension flow built over a dynamical system $(X,\,S)$ and with (strictly positive, continuous) ceiling function $f:\,X\,\to \,\mathbb{R}$. We show that the eigenvalues of $(Y,\,T)$ are contained in the range of a trace on the ${{K}_{0}}$-group of $(X,\,S)$. Moreover, a trace gives an order isomorphism of a subgroup of ${{K}_{0}}\left( C(X)\,{{\rtimes }_{S}}\,\mathbb{Z} \right)$ with the group of eigenvalues of $(Y,\,T)$. Using this result, we relate the values of $t$ for which the time-$t$ map on the minimal suspension flow is minimal with the $K$-theory of the base of this suspension.

Let $X$ be a locally finite, connected graph without vertices of degree 1. Non-backtracking random walk moves at each step with equal probability to one of the “forward” neighbours of the actual state, i.e., it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of $X$. Thus we obtain for infinite $X$ that the $n$-step non-backtracking transition probabilities tend to zero, and we can also compute their limit when $X$ is finite. This provides a short proof of an old result concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when $X$ is non-regular, but small cycles are dense in$X$, we show that the graph $X$ is non-amenable if and only if the non-backtracking $n$-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen.