A modification of the Rubinstein-Thompson criterion for a 3-manifold to be the 3-sphere is proposed. Special cell decompositions, called $Q$-triangulations and irreducible $Q$-triangulations, for closed compact orientable 3-manifolds are introduced. It is shown that if a closed compact orientable 3-manifold $M^3$ is given by a triangulation (or by a $Q$-triangulation) then one can effectively decompose $M^3$ into a connected sum of finitely many 3-manifolds some of which are given by irreducible $Q$-triangulations and others are 2-sphere bundles over a circle. Furthermore, it is shown that the problem whether a 3-manifold given by an irreducible $Q$-triangulation is homeomorphic to the 3-sphere is in $\textup{\textbf{NP}}$, and the problem whether a $Q$-triangulation of a 3-manifold is irreducible is in $\textup{\textbf{coNP}}$.

Letting $ T_p$ denote the class of separable $p$-Banach spaces (for $0<p<1$) with trivial dual, we show that $ T_p$ does not have any projective spaces, \ie, there is no space $X$ in $ T_p$ such that every space in $ T_p$ is a quotient of $X$. In lieu of a projective space we construct the $L_p(w)$ spaces, which are structurally similar to the space $L_p$. We then define a particularly well behaved type of $L_p(w)$ space, namely the uniform $L_p(w)$ spaces, and we show that every space in $ T_p$ is a quotient of some uniform $L_p(w)$ space. We then define a badly behaved type of $L_p(w)$ space, namely the unbalanced biuniform L$_p$(w) spaces. If $ L_p(w)$ is unbalanced biuniform and $C$ denotes the one dimensional subspace of constant functions, then $ L_p(w)/C$ is a rigid space. We then show that each space in $ T_p$ is a quotient of one of these rigid spaces. This last result is used in an essential way to prove the nonexistence of a projective space in $ T_p$.

$L_p(w)$ spaces ($0<p<1$) were developed by J. W. Roberts to serve as a special class of trivial-dual spaces which admit compact operators and to provide counterexamples to various interesting problems. Roberts showed that any separable, trivial-dual $p$-Banach space is a quotient of some \emph{uniform $L_p(w)$ space}. Uniform $L_p(w)$ spaces are indexed by a sequence of finite dimensional spaces $\langle X_n \rangle$ in $L_p$ and a sequence of constants $\langle c_n \rangle$ such that $1 \leq c_0 \leq c_1 \leq c_2 \leq \cdots$. If $\langle c_n \rangle$ is bounded, the resulting space is isomorphic to $L_p$. Hence these spaces can be thought of as generalized $L_p$ spaces. We prove that if $c_n \uparrow \infty$, the corresponding $L_p(w)$ space admits compact operators and is thus not isomorphic to $L_p$. Further, we show that there is no non-zero continuous linear operator from $L_p$ into any $L_p(w)$, where $ c_n \uparrow \infty$. Using and sharpening a result of Roberts, we also demonstrate that for any separable, trivial-dual $p$-Banach space $S$ there exists a uniform $L_p(w)$ space $X_S$ with $\mathcal{L} (S,X_S)=\{0\}$.

Given a ring $C$ and a totally (resp. partially) ordered set of "monomials" $\mathfrak{M}$, Hahn (resp. Higman) defined the set of power series $C [[ \mathfrak{M} ]]$ with well-ordered (resp. Noetherian or well-quasi-ordered) support in $\mathfrak{M}$. This set $C [[ \mathfrak{M} ]]$ can usually be given a lot of additional structure: if $C$ is a field and $\mathfrak{M}$ a totally ordered group, then Hahn proved that $C [[ \mathfrak{M} ]]$ is a field. More recently, we have constructed fields of ``transseries'' of the form $C [[ \mathfrak{M} ]]$ on which we defined natural derivations and compositions. In this paper we develop an operator theory for generalized power series of the above form. We first study linear and multilinear operators. We next isolate a big class of so-called Noetherian operators $\Phi : C [[ \mathfrak{M} ]] \rightarrow C [[ \mathfrak{N}]]$, which include (when defined) summation, multiplication, differentiation, composition, etc. Our main result is the proof of an implicit function theorem for Noetherian operators. This theorem may be used to explicitly solve very general types of functional equations in generalized power series.

An action of a topological group $G$ on a topological space $X$ is orbit nonproper if, for some $x\in X$, the map $g\mapsto gx:G\to X$ is nonproper. We describe the collection of connected, simply connected Lie groups admitting a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold.

We study the Brill-Noether theory of special divisors of a general branched covering of the complex projective line with one total ramification point, while the other ramification points are ordinary.

We prove that for any given integer $n\geq 2$ and $q\in [1, n)$ there exists a constant $\epsilon= \epsilon(n,q)>0$ such that any $n$-dimensional complete Riemannian manifold with nonnegative Ricci curvature, in which the Sobolev inequality

\[ \left(\int_M|f|^{\frac {nq}{n-q}}\,dv\right)^{\frac{n-q}{nq}}\leq (K(n,q)+\epsilon)\left(\int_M|\nabla f|^q \,dv\right)^{\sfrac{1}{q}}, \,\,\forall f\in C_0^{\infty}(M) \]

holds with $K(n,q)$ the optimal constant of this inequality in the $n$-dimensional Euclidean space $R^n$, is diffeomorphic to~$R^n$.

The moduli space of five lines in $\mathbf{P}^2$ can be described by a quintic Del Pezzo surface in $\mathbf{P}^5$. Given five fixed lines in $\mathbf{P}^3$ and a fixed plane, we define a map from $\mathbf{P}^3$ to the quintic Del Pezzo surface by projecting the lines to the fixed plane, and taking the point on the Del Pezzo surface defined by the image lines as the image of the point of projection. We show that the fibers of this map are twisted cubic curves. Conversely, we show that the moduli space of curves in $\mathbf{P}^3$ with the five fixed lines as secants can be seen as isomorphic to the quintic Del Pezzo surface.

In this paper we study the boundary orbit accumulation points of smoothly bounded domains in $\mathbf{C}^2$ with non-compact automorphism group. We prove that a boundary orbit accumulation point cannot be exponentially flat. This confirms a version of a conjecture of Greene and Krantz. The proof uses a new result on the a priori convergence of convex scaling methods, which in particular implies the equivalence of two different scaling methods on convex domains.

Let $\{ T_t\}_{t>0}$ be the semigroup of linear operators generated by a Schrödinger operator $-A=\Delta -V$ on $\mathbb{R}^d$, where $V$ is a nonnegative nonzero potential satisfying a reverse Hölder inequality, and let $\int_0^\infty \lambda \, dE_A(\lambda )$ be the spectral resolution of $A$. We say that a function $f$ is an element of $H_A^1$ if the maximal function $\mathcal{M}f(x)=\sup_{t>0} |T_tf(x)|$ belongs to $L^1$. We prove that if a function $F$ satisfies a Mihlin condition with exponent $\alpha >d/2$ then the operator $F(A)=\int_0^\infty F(\lambda )\, dE_A(\lambda )$ is bounded on $H_A^1$.

The extension problem for positive definite generalized Toeplitz Kernels defined on the positive semi-axis is discussed, and criteria for the uniqueness of the extension are given. These results are applied to the generalized Nehari problem.

A Finsler manifold is a Riemannian manifold without the quadratic restriction. In this paper we introduce the energy functional, the Euler-Lagrange operator, and the stress-energy tensor for a smooth map $\phi$ from a Finsler manifold to a Riemannian manifold. We show that $\phi$ is an extremal of the energy functional if and only if $\phi$ satisfies the corresponding Euler-Lagrange equation. We also characterize weak Landsberg manifolds in terms of harmonicity and horizontal conservativity. Using the representation of a tension field in terms of geodesic coefficients, we construct new examples of harmonic maps from Berwald manifolds which are neither Riemannian nor Minkowskian.

We show that, in any topological space, boolean combinations of open sets have a canonical representation as a finite union of locally closed sets. As an application, if $\mathfrak M$ is a first-order topological structure, then sets definable in $\mathfrak M$ that are boolean combinations of open sets are boolean combinations of open definable sets.

We consider some semilinear (= semiaffine) and semialgebraic loci of o-minimal sets in euclidean spaces. Semilinear loci have good properties. Some of these properties hold for semialgebraic loci when we restrict to a smaller class of analytically o-minimal sets.

We investigate the $x$- and $y$-regularity of a bigraded $K$-algebra $R$ as introduced in \cite{ARCRNE}. These notions are used to study asymptotic properties of certain finitely generated bigraded modules. As an application we get for any equigenerated graded ideal $I$ upper bounds for the number $j_0$ for which $\operatorname{reg}(I^j)$ is a linear function for $j \geq j_0$. Finally, we give upper bounds for the $x$- and $y$-regularity of generalized Veronese algebras.

We investigate connections between pseudodifferential operators and linear ordinary differential equations via their respective links to automorphic forms. We also introduce Hecke operators on the space of pseudodifferential operators as well as on the space of certain meromorphic functions associated to ordinary differential equations and prove that the actions of those Hecke operators are compatible with the correspondence between pseudodifferential operators and linear ordinary differential equations.

In this paper we study the geometry of higher duals of a Banach space using techniques from the theory of $M$-ideals. We show that any Banach space that is an $M$-ideal in its bidual is an $M$-ideal in all duals of even order. As a consequence of this result, we show that continuous linear functionals on such spaces have unique norm preserving extensions to all duals of even order.

Some properties of and relationships between important classes of copulas are discussed. In particular, the class of associative copulas is shown to be compact, and the classes of strict and non-strict copulas are shown to be dense subsets of this class.

Let $x_0\in b\Omega $ in a smooth domain $\Omega \subset \C^n$, which is not assumed to be pseudoconvex. We define a finite type condition $R( L,x_0) $ for a vector field $L\in T^{1,0}( b\Omega)$, which equals the well-known type $c(L,x_0) $ in certain important cases. We prove that if $R(L,x_0)=m$, then a subelliptic estimate of order $1/m$ holds at $x_0$ for $(p,n-1)$ forms.