A join theorem for the computably enumerable degrees
Carl
G.
Jockusch Jr.;
Angsheng
Li;
Yue
Yang
2557-2568
Abstract: It is shown that for any computably enumerable (c.e.) degree $\mathbf{w}$, if $\mathbf{w\not=0}$, then there is a c.e. degree $\mathbf{a}$ such that $\mathbf{a}$ is low$_2$and $\mathbf{a\lor w}$ is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low$_2$ c.e. degrees are not elementarily equivalent as partial orderings.
Thomason's theorem for varieties over algebraically closed fields
Mark
E.
Walker
2569-2648
Abstract: We present a novel proof of Thomason's theorem relating Bott inverted algebraic $K$-theory with finite coefficients and étale cohomology for smooth varieties over algebraically closed ground fields. Our proof involves first introducing a new theory, which we term algebraic $K$-homology, and proving it satisfies étale descent (with finite coefficients) on the category of normal, Cohen-Macaulay varieties. Then, we prove algebraic $K$-homology and algebraic $K$-theory (each taken with finite coefficients) coincide on smooth varieties upon inverting the Bott element.
Subvarieties of general type on a general projective hypersurface
Gianluca
Pacienza
2649-2661
Abstract: We study subvarieties of a general projective degree $d$ hypersurface $X_d\subset \mathbf{P}^n$. Our main theorem, which improves previous results of L. Ein and C. Voisin, implies in particular the following sharp corollary: any subvariety of a general hypersurface $X_{d}\subset {\mathbf P}^n$, for $n\geq 6$ and $d\geq 2n-2$, is of general type.
Sums of squares in real rings
José
F.
Fernando;
Jesús
M.
Ruiz;
Claus
Scheiderer
2663-2684
Abstract: Let $A$ be an excellent ring. We show that if the real dimension of $A$ is at least three then $A$ has infinite Pythagoras number, and there exists a positive semidefinite element in $A$ which is not a sum of squares in $A$.
$L^{2}$-metrics, projective flatness and families of polarized abelian varieties
Wing-Keung
To;
Lin
Weng
2685-2707
Abstract: We compute the curvature of the $L^{2}$-metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kähler manifolds. As an application, we show that the $L^{2}$-metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kähler manifold, this leads to the poly-stability of the direct image with respect to any Kähler form on the parameter space.
Fundamental solutions for non-divergence form operators on stratified groups
Andrea
Bonfiglioli;
Ermanno
Lanconelli;
Francesco
Uguzzoni
2709-2737
Abstract: We construct the fundamental solutions $\Gamma$ and $\gamma$for the non-divergence form operators $\,{\textstyle\sum_{i,\,j}\,} a_{i,\,j}(x,t)\,X_iX_j\,-\,\partial_t\,$ and ${\,\textstyle\sum_{i,\,j}}\,a_{i,\,j}(x)\,X_iX_j$, where the $X_i$'s are Hörmander vector fields generating a stratified group $\mathbb{G}$ and $(a_{i,j})_{i,j}$ is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates of $\Gamma$ and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of $\Gamma$ allow us to construct $\gamma$ using both $t$-saturation and approximation arguments.
Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation
Víctor
Padrón
2739-2756
Abstract: In this paper we study the equation \begin{displaymath}u_t=\Delta(\phi(u) - \lambda f(u) + \lambda u_t) + f(u) \end{displaymath} in a bounded domain of $\mathbb{R} ^d$, $d\ge1$, with homogeneous boundary conditions of the Neumann type, as a model of aggregating population with a migration rate determined by $\phi$, and total birth and mortality rates characterized by $f$. We will show that the aggregating mechanism induced by $\phi(u)$ allows the survival of a species in danger of extinction. Numerical simulations suggest that the solutions stabilize asymptotically in time to a not necessarily homogeneous stationary solution. This is shown to be the case for a particular version of the function $\phi(u)$.
Dynamical approach to some problems in integral geometry
Boris
Paneah
2757-2780
Abstract: As is well known, the main problem in integral geometry is to reconstruct a function in a given domain $D$, where its integrals over a family of subdomains in $D$ are known. Such a problem is interesting not only as an object of pure analysis, but also in connection with various applications in practical disciplines. The most remarkable example of such a connection is the Radon problem and tomography. In this paper we solve one of these problems when $D$ is a bounded domain in ${\mathbb{R}}^2$ with a piecewise smooth boundary. Some intermediate results related to dynamical systems with two generators and to some functional-integral equations are new and interesting per se. As an application of the results obtained we briefly study a boundary problem for a general third order hyperbolic partial differential equation in a bounded domain $D\subset {\mathbb{R}}^2$ with data on the whole boundary $\partial D$.
The peak algebra and the descent algebras of types B and D
Marcelo
Aguiar;
Nantel
Bergeron;
Kathryn
Nyman
2781-2824
Abstract: We show the existence of a unital subalgebra $\mathfrak{P}_n$ of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that $\mathfrak{P}_n$ is the image of the descent algebra of type B under the map to the descent algebra of type A which forgets the signs, and also the image of the descent algebra of type D. The algebra $\mathfrak{P}_n$ contains a two-sided ideal $\overset{\circ}{\mathfrak{P}}_n$ which is defined in terms of interior peaks. This object was introduced in previous work by Nyman (2003); we find that it is the image of certain ideals of the descent algebras of types B and D. We derive an exact sequence of the form $0\to\overset{\circ}{\mathfrak{P}}_n \to\mathfrak{P}_n\to\mathfrak{P}_{n-2}\to 0$. We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions $n$ and $\lfloor\frac{n}{2}\rfloor+1)$ by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces $\mathfrak{P}_n$ and $\overset{\circ}{\mathfrak{P}}_n$ over $n\geq 0$ and explain the connection with previous work of Stembridge (1997); we also obtain new properties of his descents-to-peaks map and construct a type B analog.
Combinatorial properties of Thompson's group $F$
Sean
Cleary;
Jennifer
Taback
2825-2849
Abstract: We study some combinatorial consequences of Blake Fordham's theorems on the word metric of Thompson's group $F$ in the standard two generator presentation. We explore connections between the tree pair diagram representing an element $w$ of $F$, its normal form in the infinite presentation, its word length, and minimal length representatives of it. We estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates. In addition we explore some properties of the Cayley graph of $F$ with respect to the two generator finite presentation. Namely, we exhibit the form of ``dead end'' elements in this Cayley graph, and show that it has no ``deep pockets''. Finally, we discuss a simple method for constructing minimal length representatives for strictly positive or negative words.
Metrical diophantine approximation for continued fraction like maps of the interval
Andrew
Haas;
David
Molnar
2851-2870
Abstract: We study the metrical properties of a class of continued fraction-like mappings of the unit interval, each of which is defined as the fractional part of a Möbius transformation taking the endpoints of the interval to zero and infinity.
The ABC theorem for higher-dimensional function fields
Liang-Chung
Hsia;
Julie
Tzu-Yueh
Wang
2871-2887
Abstract: We generalize the ABC theorems to the function field of a variety over an algebraically closed field of arbitrary characteristic which is non-singular in codimension one. We also obtain an upper bound for the minimal order sequence of Wronskians over such function fields of positive characteristic.
A separable Brown-Douglas-Fillmore theorem and weak stability
Huaxin
Lin
2889-2925
Abstract: We give a separable Brown-Douglas-Fillmore theorem. Let $A$ be a separable amenable $C^*$-algebra which satisfies the approximate UCT, $B$ be a unital separable amenable purely infinite simple $C^*$-algebra and $h_1, \, h_2: A\to B$ be two monomorphisms. We show that $h_1$ and $h_2$ are approximately unitarily equivalent if and only if $[h_1]=[h_2]\,\,\,\,{\rm in}\,\,\, KL(A,B).$ We prove that, for any $\varepsilon>0$ and any finite subset $\mathcal{F}\subset A$, there exist $\delta>0$ and a finite subset $\mathcal{G}\subset A$ satisfying the following: for any amenable purely infinite simple $C^*$-algebra $B$ and for any contractive positive linear map $L: A\to B$ such that \begin{displaymath}\Vert L(ab)-L(a)L(b)\Vert<\delta\quad{and}\quad \Vert L(a)\Vert\ge (1/2)\Vert a\Vert \end{displaymath} for all $a\in \mathcal{G},$ there exists a homomorphism $h: A\to B$such that \begin{displaymath}\Vert h(a)-L(a)\Vert<\varepsilon\,\,\,\,\,{\rm for\,\,\,all}\,\,\, a\in \mathcal{F} \end{displaymath} provided, in addition, that $K_i(A)$ are finitely generated. We also show that every separable amenable simple $C^*$-algebra $A$ with finitely generated $K$-theory which is in the so-called bootstrap class is weakly stable with respect to the class of amenable purely infinite simple $C^*$-algebras. As an application, related to perturbations in the rotation $C^*$-algebras studied by U. Haagerup and M. Rørdam, we show that for any irrational number $\theta$ and any $\varepsilon>0$ there is $\delta>0$ such that in any unital amenable purely infinite simple $C^*$-algebra $B$ if \begin{displaymath}\Vert uv-e^{i\theta\pi}vu\Vert<\delta \end{displaymath} for a pair of unitaries, then there exists a pair of unitaries $u_1$ and $v_1$ in $B$ such that \begin{displaymath}u_1v_1=e^{i\theta\pi}v_1u_1,\,\,\,\,\,\Vert u_1-u\Vert<\varepsilon\quad\text{and} \quad\Vert v_1-v\Vert<\varepsilon. \end{displaymath}
Maps between non-commutative spaces
S.
Paul
Smith
2927-2944
Abstract: Let $J$ be a graded ideal in a not necessarily commutative graded $k$-algebra $A=A_0 \oplus A_1 \oplus \cdots$ in which $\dim_k A_i < \infty$ for all $i$. We show that the map $A \to A/J$ induces a closed immersion $i:\operatorname{Proj}_{nc} A/J \to \operatorname{Proj}_{nc}A$ between the non-commutative projective spaces with homogeneous coordinate rings $A$ and $A/J$. We also examine two other kinds of maps between non-commutative spaces. First, a homomorphism $\phi:A \to B$ between not necessarily commutative $\mathbb{N}$-graded rings induces an affine map $\operatorname{Proj}_{nc} B \supset U \to \operatorname{Proj}_{nc} A$from a non-empty open subspace $U \subset \operatorname{Proj}_{nc} B$. Second, if $A$ is a right noetherian connected graded algebra (not necessarily generated in degree one), and $A^{(n)}$ is a Veronese subalgebra of $A$, there is a map $\operatorname{Proj}_{nc} A \to \operatorname{Proj}_{nc} A^{(n)}$; we identify open subspaces on which this map is an isomorphism. Applying these general results when $A$ is (a quotient of) a weighted polynomial ring produces a non-commutative resolution of (a closed subscheme of) a weighted projective space.
Koszul homology and extremal properties of Gin and Lex
Aldo
Conca
2945-2961
Abstract: For every homogeneous ideal $I$ in a polynomial ring $R$ and for every $p\leq\dim R$ we consider the Koszul homology $H_i(p,R/I)$ with respect to a sequence of $p$ of generic linear forms. The Koszul-Betti number $\beta_{ijp}(R/I)$ is, by definition, the dimension of the degree $j$ part of $H_i(p,R/I)$. In characteristic $0$, we show that the Koszul-Betti numbers of any ideal $I$ are bounded above by those of the gin-revlex $\mathrm{Gin}(I)$ of $I$ and also by those of the Lex-segment $\mathrm{Lex}(I)$ of $I$. We show that $\beta_{ijp}(R/I)=\beta_{ijp}(R/\mathrm{Gin}(I))$ iff $I$ is componentwise linear and that and $\beta_{ijp}(R/I)=\beta_{ijp}(R/\mathrm{Lex}(I))$ iff $I$is Gotzmann. We also investigate the set $\mathrm{Gins}(I)$ of all the gin of $I$ and show that the Koszul-Betti numbers of any ideal in $\mathrm{Gins}(I)$are bounded below by those of the gin-revlex of $I$. On the other hand, we present examples showing that in general there is no $J$ is $\mathrm{Gins}(I)$such that the Koszul-Betti numbers of any ideal in $\mathrm{Gins}(I)$ are bounded above by those of $J$.
Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves
A.
Lins
Neto;
J.
C. Canille
Martins
2963-2988
Abstract: In this paper we consider the problem of uniformization of the leaves of a holomorphic foliation by curves in a complex manifold $M$. We consider the following problems: 1. When is the uniformization function $\lambda _{g}$, with respect to some metric $g$, continuous? It is known that the metric $\frac{g}{4\lambda _{g}}$ induces the Poincaré metric on the leaves. 2. When is the metric $\frac{g}{4\lambda _{g}}$ complete? We extend the concept of ultra-hyperbolic metric, introduced by Ahlfors in 1938, for singular foliations by curves, and we prove that if there exists a complete ultra-hyperbolic metric $g$, then $\lambda _{g}$ is continuous and $\frac{g}{4\lambda _{g}}$ is complete. In some local cases we construct such metrics, including the saddle-node (Theorem 1) and singularities given by vector fields with the first non-zero jet isolated (Theorem 2). We also give an example where for any metric $g$, $\frac{g}{4\,\lambda _{g}}$ is not complete (§3.2).