A rigid subspace of the real line whose square is a homogeneous subspace of the plane
L.
Brian
Lawrence
2535-2556
Abstract: Working in ZFC, we give an example as indicated in the title.
Cycles on curves over global fields of positive characteristic
Reza
Akhtar
2557-2569
Abstract: Let $k$ be a global field of positive characteristic, and let $\sigma: X \longrightarrow \operatorname{Spec} k$ be a smooth projective curve. We study the zero-dimensional cycle group $V(X) =\operatorname{Ker}(\sigma_*: SK_1(X) \rightarrow K_1(k))$ and the one-dimensional cycle group $W(X) =\operatorname{coker}(\sigma^*: K_2(k) \rightarrow H^0_{Zar}(X, \mathcal{K}_2))$, addressing the conjecture that $V(X)$ is torsion and $W(X)$ is finitely generated. The main idea is to use Abhyankar's Theorem on resolution of singularities to relate the study of these cycle groups to that of the $K$-groups of a certain smooth projective surface over a finite field.
On nonlinear wave equations with degenerate damping and source terms
Viorel
Barbu;
Irena
Lasiecka;
Mohammad
A.
Rammaha
2571-2611
Abstract: In this article we focus on the global well-posedness of the differential equation $u_{tt}- \Delta u + \vert u\vert^k\partial j(u_t) = \vert u\vert^{ p-1}u \, \text{ in } \Omega \times (0,T)$, where $\partial j$ is a sub-differential of a continuous convex function $j$. Under some conditions on $j$ and the parameters in the equations, we obtain several results on the existence of global solutions, uniqueness, nonexistence and propagation of regularity. Under nominal assumptions on the parameters we establish the existence of global generalized solutions. With further restrictions on the parameters we prove the existence and uniqueness of a global weak solution. In addition, we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent $p$ is greater than the critical value $k+m$, and the initial energy is negative. We also address the issue of propagation of regularity. Specifically, under some restriction on the parameters, we prove that solutions that correspond to any regular initial data such that $u_0\in H^2(\Omega)\cap H^1_0(\Omega)$, $u_1 \in H^1_0(\Omega)$ are indeed strong solutions.
The Bergman metric and the pluricomplex Green function
Zbigniew
Blocki
2613-2625
Abstract: We improve a lower bound for the Bergman distance in smooth pseudoconvex domains due to Diederich and Ohsawa. As the main tool we use the pluricomplex Green function and an $L^2$-estimate for the $\overline\partial$-operator of Donnelly and Fefferman.
Estimates of the derivatives for parabolic operators with unbounded coefficients
Marcello
Bertoldi;
Luca
Lorenzi
2627-2664
Abstract: We consider a class of second-order uniformly elliptic operators $\mathcal{A}$ with unbounded coefficients in $\mathbb{R}^N$. Using a Bernstein approach we provide several uniform estimates for the semigroup $T(t)$ generated by the realization of the operator $\mathcal{A}$ in the space of all bounded and continuous or Hölder continuous functions in $\mathbb{R}^N$. As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation $\lambda u-\mathcal{A}u=f$ ($\lambda>0$) and the nonhomogeneous Dirichlet Cauchy problem $D_tu=\mathcal{A}u+g$. Then, we prove two different kinds of pointwise estimates of $T(t)$ that can be used to prove a Liouville-type theorem. Finally, we provide sharp estimates of the semigroup $T(t)$ in weighted $L^p$-spaces related to the invariant measure associated with the semigroup.
On homeomorphism groups of Menger continua
Jan
J.
Dijkstra
2665-2679
Abstract: It is shown that the homeomorphism groups of the (generalized) Sierpinski carpet and the universal Menger continua are not zero-dimensional. These results were corollaries to a 1966 theorem of Brechner. New proofs were needed because we also show that Brechner's proof is inadequate. The method by which we obtain our results, the construction of closed imbeddings of complete Erdos space in the homeomorphism groups, is of independent interest.
Good measures on Cantor space
Ethan
Akin
2681-2722
Abstract: While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The clopen values set for a full, nonatomic measure $\mu$ is the countable dense subset $\{ \mu(U) : U$ is clopen$\}$ of the unit interval. It is a topological invariant for the measure. For the class of good measures it is a complete invariant. A full, nonatomic measure $\mu$ is good if whenever $U, V$ are clopen sets with $\mu(U) < \mu(V)$, there exists $W$ a clopen subset of $V$ such that $\mu(W) = \mu(U)$. These measures have interesting dynamical properties. They are exactly the measures which arise from uniquely ergodic minimal systems on Cantor space. For some of them there is a unique generic measure-preserving homeomorphism. That is, within the Polish group of such homeomorphisms there is a dense, $G_{\delta}$ conjugacy class.
Brauer groups of genus zero extensions of number fields
Jack
Sonn;
John
Swallow
2723-2738
Abstract: We determine the isomorphism class of the Brauer groups of certain nonrational genus zero extensions of number fields. In particular, for all genus zero extensions $E$ of the rational numbers $\mathbb{Q}$ that are split by $\mathbb{Q} (\sqrt{2})$, $\operatorname{Br}(E)\cong \operatorname{Br}(\mathbb{Q} (t))$.
Finite dimensional representations of invariant differential operators
Ian
M.
Musson;
Sonia
L.
Rueda
2739-2752
Abstract: Let $k$ be an algebraically closed field of characteristic $0$, $Y=k^{r}\times {(k^{\times})}^{s}$, and let $G$ be an algebraic torus acting diagonally on the ring of algebraic differential operators $\mathcal{D} (Y)$. We give necessary and sufficient conditions for $\mathcal{D}(Y)^G$ to have enough simple finite dimensional representations, in the sense that the intersection of the kernels of all the simple finite dimensional representations is zero. As an application we show that if $K\longrightarrow GL(V)$ is a representation of a reductive group $K$ and if zero is not a weight of a maximal torus of $K$ on $V$, then $\mathcal{D} (V)^K$ has enough finite dimensional representations. We also construct examples of FCR-algebras with any integer GK dimension $\geq 3$.
Homotopical localizations of module spectra
Carles
Casacuberta;
Javier
J.
Gutiérrez
2753-2770
Abstract: We prove that stable $f$-localizations (where $f$ is any map of spectra) preserve ring spectrum structures and module spectrum structures, under suitable hypotheses, and we use this fact to describe all possible localizations of the integral Eilenberg-MacLane spectrum $H{\mathbb{Z} }$. As a consequence of this study, we infer that localizations of stable GEMs are stable GEMs, and it also follows that there is a proper class of nonequivalent stable localizations.
Commutative ideal theory without finiteness conditions: Primal ideals
Laszlo
Fuchs;
William
Heinzer;
Bruce
Olberding
2771-2798
Abstract: Our goal is to establish an efficient decomposition of an ideal $A$ of a commutative ring $R$ as an intersection of primal ideals. We prove the existence of a canonical primal decomposition: $A = \bigcap_{P \in \mathcal{X}_A}A_{(P)}$, where the $A_{(P)}$ are isolated components of $A$ that are primal ideals having distinct and incomparable adjoint primes $P$. For this purpose we define the set $\operatorname{Ass}(A)$ of associated primes of the ideal $A$ to be those defined and studied by Krull. We determine conditions for the canonical primal decomposition to be irredundant, or residually maximal, or the unique representation of $A$ as an irredundant intersection of isolated components of $A$. Using our canonical primal decomposition, we obtain an affirmative answer to a question raised by Fuchs, and also prove for $P \in \operatorname{Spec}R$ that an ideal $A \subseteq P$ is an intersection of $P$-primal ideals if and only if the elements of $R \setminus P$ are prime to $A$. We prove that the following conditions are equivalent: (i) the ring $R$ is arithmetical, (ii) every primal ideal of $R$ is irreducible, (iii) each proper ideal of $R$ is an intersection of its irreducible isolated components. We classify the rings for which the canonical primal decomposition of each proper ideal is an irredundant decomposition of irreducible ideals as precisely the arithmetical rings with Noetherian maximal spectrum. In particular, the integral domains having these equivalent properties are the Prüfer domains possessing a certain property.
Poisson brackets associated to the conformal geometry of curves
G.
Marí Beffa
2799-2827
Abstract: In this paper we present an invariant moving frame, in the group theoretical sense, along curves in the Möbius sphere. This moving frame will describe the relationship between all conformal differential invariants for curves that appear in the literature. Using this frame we first show that the Kac-Moody Poisson bracket on $Lo(n+1,1)^\ast$ can be Poisson reduced to the space of conformal differential invariants of curves. The resulting bracket will be the conformal analogue of the Adler-Gel'fand-Dikii bracket. Secondly, a conformally invariant flow of curves induces naturally an evolution on the differential invariants of the flow. We give the conditions on the invariant flow ensuring that the induced evolution is Hamiltonian with respect to the reduced Poisson bracket. Because of a certain parallelism with the Euclidean case we study what we call Frenet and natural cases. We comment on the implications for completely integrable systems, and describe conformal analogues of the Hasimoto transformation.
Saddle surfaces in singular spaces
Dimitrios
E.
Kalikakis
2829-2841
Abstract: The notion of a saddle surface is well known in Euclidean space. In this work we extend the idea of a saddle surface to geodesically connected metric spaces. We prove that any solution of the Dirichlet problem for the Sobolev energy in a nonpositively curved space is a saddle surface. Further, we show that the space of saddle surfaces in a nonpositively curved space is a complete space in the Fréchet distance. We also prove a compactness theorem for saddle surfaces in spaces of curvature bounded from above; in spaces of constant curvature we obtain a stronger result based on an isoperimetric inequality for a saddle surface. These results generalize difficult theorems of S.Z. Shefel' on compactness of saddle surfaces in a Euclidean space.
Weighted estimates in $L^{2}$ for Laplace's equation on Lipschitz domains
Zhongwei
Shen
2843-2870
Abstract: Let $\Omega \subset \mathbb{R}^{d}$, $d\ge 3$, be a bounded Lipschitz domain. For Laplace's equation $\Delta u=0$ in $\Omega$, we study the Dirichlet and Neumann problems with boundary data in the weighted space $L^{2}(\partial \Omega ,\omega _{\alpha }d\sigma )$, where $\omega _{\alpha }(Q) =\vert Q-Q_{0}\vert^{\alpha }$, $Q_{0}$ is a fixed point on $\partial \Omega$, and $d\sigma$ denotes the surface measure on $\partial \Omega$. We prove that there exists $\varepsilon =\varepsilon (\Omega )\in (0,2]$ such that the Dirichlet problem is uniquely solvable if $1-d<\alpha <d-3+\varepsilon$, and the Neumann problem is uniquely solvable if $3-d-\varepsilon <\alpha <d-1$. If $\Omega$ is a $C^{1}$ domain, one may take $\varepsilon =2$. The regularity for the Dirichlet problem with data in the weighted Sobolev space $L^{2}_{1}(\partial \Omega ,\omega _{\alpha }d\sigma )$ is also considered. Finally we establish the weighted $L^{2}$ estimates with general $A_{p}$weights for the Dirichlet and regularity problems.
Dimension of families of determinantal schemes
Jan
O.
Kleppe;
Rosa
M.
Miró-Roig
2871-2907
Abstract: A scheme $X\subset \mathbb{P} ^{n+c}$ of codimension $c$ is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous $t \times (t+c-1)$ matrix and $X$ is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$ we denote by $W(\underline{b};\underline{a})\subset \operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$(resp. $W_s(\underline{b};\underline{a})$) the locus of good (resp. standard) determinantal schemes $X\subset \mathbb{P} ^{n+c}$ of codimension $c$ defined by the maximal minors of a $t\times (t+c-1)$ matrix $(f_{ij})^{i=1,...,t}_{j=0,...,t+c-2}$ where $f_{ij}\in k[x_0,x_1,...,x_{n+c}]$ is a homogeneous polynomial of degree $a_j-b_i$. In this paper we address the following three fundamental problems: To determine (1) the dimension of $W(\underline{b};\underline{a})$ (resp. $W_s(\underline{b};\underline{a})$) in terms of $a_j$ and $b_i$, (2) whether the closure of $W(\underline{b};\underline{a})$ is an irreducible component of $\operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$, and (3) when $\operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$ is generically smooth along $W(\underline{b};\underline{a})$. Concerning question (1) we give an upper bound for the dimension of $W(\underline{b};\underline{a})$ (resp. $W_s(\underline{b};\underline{a})$) which works for all integers $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$, and we conjecture that this bound is sharp. The conjecture is proved for $2\le c\le 5$, and for $c\ge 6$ under some restriction on $a_0,a_1,...,a_{t+c-2}$and $b_1,...,b_t$. For questions (2) and (3) we have an affirmative answer for $2\le c \le 4$ and $n\ge 2$, and for $c\ge 5$ under certain numerical assumptions.
Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities
Didier
Smets
2909-2938
Abstract: We study a time-independent nonlinear Schrödinger equation with an attractive inverse square potential and a nonautonomous nonlinearity whose power is the critical Sobolev exponent. The problem shares a strong resemblance with the prescribed scalar curvature problem on the standard sphere. Particular attention is paid to the blow-up possibilities, i.e. the critical points at infinity of the corresponding variational problem. Due to the strong singularity in the potential, some new phenomenon appear. A complete existence result is obtained in dimension 4 using a detailed analysis of the gradient flow lines.
Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module
Volodymyr
Mazorchuk;
Catharina
Stroppel
2939-2973
Abstract: We investigate certain singular categories of Harish-Chandra bimodules realized as the category of $\mathfrak{p}$-presentable modules in the principal block of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$. This category is equivalent to the module category of a properly stratified algebra. We describe the socles and endomorphism rings of standard objects in this category. Further, we consider translation and shuffling functors and their action on the standard modules. Finally, we study a graded version of this category; in particular, we give a graded version of the properly stratified structure, and use graded versions of translation functors to categorify a parabolic Hecke module.