Let $\left( X,\,g \right)$ be a complete noncompact Kähler manifold, of dimension $n\,\ge \,2$, with positive Ricci curvature and of standard type (see the definition below). N. Mok proved that $X$ can be compactified, i.e., $X$ is biholomorphic to a quasi-projective variety. The aim of this paper is to prove that the ${{L}^{2}}$ holomorphic sections of the line bundle $K_{X}^{-q}$ and the volume form of the metric $g$ have no essential singularities near the divisor at infinity. As a consequence we obtain a comparison between the volume forms of the Kähler metric $g$ and of the Fubini-Study metric induced on $X$. In the case of ${{\dim}_{\mathbb{C}}}\,X\,=\,2$, we establish a relation between the number of components of the divisor $D$ and the dimension of the ${{H}^{i}}(\bar{X},\,\Omega \frac{1}{X}(\log \,D))$.

For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated.

The first main result of the paper is a criterion for a partially commutative group $\mathbb{G}$ to be a domain. It allows us to reduce the study of algebraic sets over $\mathbb{G}$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\mathbb{G}$ (of a coordinate group over $\mathbb{G}$) to the elementary theories of the direct factors of $\mathbb{G}$ (to the elementary theory of coordinate groups of irreducible algebraic sets).

Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group $\mathbb{H}$. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of $\mathbb{H}$ has quantifier elimination and that arbitrary first-order formulas lift from $\mathbb{H}$ to $\mathbb{H}\,*\,F$, where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.

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

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 $A$ most of whose formal fibers are geometrically regular.

We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.

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

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

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

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

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

A theorem of Tietze and Nakajima, from 1928, asserts that if a subset $X$ of ${{\mathbb{R}}^{n}}$ is closed, connected, and locally convex, then it is convex. We give an analogous “local to global convexity” theorem when the inclusion map of $X$ to ${{\mathbb{R}}^{n}}$ is replaced by a map from a topological space $X$ to ${{\mathbb{R}}^{n}}$ that satisfies certain local properties. Our motivation comes from the Condevaux–Dazord–Molino proof of the Atiyah–Guillemin–Sternberg convexity theorem in symplectic geometry.

We describe a noncommutative deformation theory for presheaves and sheaves of modules that generalizes the commutative deformation theory of these global algebraic structures and the noncommutative deformation theory of modules over algebras due to Laudal.

In the first part of the paper, we describe a noncommutative deformation functor for presheaves of modules on a small category and an obstruction theory for this functor in terms of global Hochschild cohomology. An important feature of this obstruction theory is that it can be computed in concrete terms in many interesting cases.

In the last part of the paper, we describe a noncommutative deformation functor for quasi-coherent sheaves of modules on a ringed space $\left( X,\,\mathcal{A} \right)$. We show that for any good $\mathcal{A}$-affine open cover $\cup$ of $X$, the forgetful functor $\text{QCoh}\mathcal{A}\,\to \,\text{PreSh}\left( \cup ,\,\mathcal{A} \right)$ induces an isomorphism of noncommutative deformation functors.

Applications. We consider noncommutative deformations of quasi-coherent $\mathcal{A}$-modules on $X$ when $\left( X,\,\mathcal{A} \right)\,=\,\left( X,\,{{\mathcal{O}}_{X}} \right)$ is a scheme or $\left( X,\,\mathcal{A} \right)\,=\,\left( X,\,\mathcal{D} \right)$ is a $\text{D}$-scheme in the sense of Beilinson and Bernstein. In these cases, we may use any open affine cover of $X$ closed under finite intersections to compute noncommutative deformations in concrete terms using presheaf methods. We compute the noncommutative deformations of the left ${{\mathcal{D}}_{X}}$-module ${{\mathcal{D}}_{X}}$ when $X$ is an elliptic curve as an example.

We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.

In this paper we consider an elliptic system with an inverse square potential and critical Sobolev exponent in a bounded domain of ${{\mathbb{R}}^{N}}$. By variational methods we study the existence results.

We decompose the restriction of ramified principal series representations of the $p$-adic group $\text{GL}\left( 3,\,\text{k} \right)$ to its maximal compact subgroup $K\,=\,\text{GL}\left( 3,\,\mathcal{R} \right)$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples.

A triangulation of a hyperbolic 3-manifold is $L$-thick if each tetrahedron having all vertices in the thick part of $M$ is $L$-bilipschitz diffeomorphic to the standard Euclidean tetrahedron. We show that there exists a fixed constant $L$ such that every complete hyperbolic 3-manifold has an $L$-thick geodesic triangulation. We use this to prove the existence of universal bounds on the principal curvatures of ${{\pi }_{1}}$-injective surfaces and strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds.

If an algebraic torus $T$ acts on a complex projective algebraic variety $X$, then the affine scheme $\text{Spec}\,H_{T}^{*}\left( X;\,\mathbb{C} \right)$ associated with the equivariant cohomology is often an arrangement of linear subspaces of the vector space $H_{2}^{T}\left( X;\,\mathbb{C} \right)$. In many situations the ordinary cohomology ring of $X$ can be described in terms of this arrangement.

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

This paper answers a question of Broomhead, Montaldi and Sidorov about the existence of gaskets of a particular type related to the Sierpiński sieve. These gaskets are given by iterated function systems that do not satisfy the open set condition. We use the methods of Ngai and Wang to compute the dimension of these gaskets.

We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at $q\,=\,1$, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in our previous article. We deduce Vafa–Intriligator type formulas for the Gromov–Witten invariants.

In this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann-Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions 2 and 3. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing $\text{S}$-curvature. This means that reversible non-Berwaldian Finsler spaces with vanishing $\text{S}$-curvature may exist at large. Hence the generalized volume comparison theorems due to $\text{Z}$. Shen are valid for a rather large class of Finsler spaces.

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.