Boundary slopes of punctured tori in 3-manifolds
C.
McA.
Gordon
1713-1790
Abstract: Let $M$ be an irreducible 3-manifold with a torus boundary component $T$, and suppose that $r,s$ are the boundary slopes on $T$ of essential punctured tori in $M$, with their boundaries on $T$. We show that the intersection number $\Delta(r,s)$ of $r$ and $s$ is at most $8$. Moreover, apart from exactly four explicit manifolds $M$, which contain pairs of essential punctured tori realizing $\Delta(r,s)=8,8,7$ and 6 respectively, we have $\Delta(r,s)\le 5$. It follows immediately that if $M$ is atoroidal, while the manifolds $M(r), M(s)$ obtained by $r$- and $s$-Dehn filling on $M$ are toroidal, then $\Delta(r,s)\le 8$, and $\Delta(r,s)\le 5$ unless $M$ is one of the four examples mentioned above. Let $\mathcal{H}_0$ be the class of 3-manifolds $M$ such that $M$ is irreducible, atoroidal, and not a Seifert fibre space. By considering spheres, disks and annuli in addition to tori, we prove the following. Suppose that $M\in \mathcal{H}_0$, where $\partial M$ has a torus component $T$, and $\partial M-T\ne \varnothing$. Let $r,s$ be slopes on $T$ such that $M(r), M(s)\notin \mathcal{H}_0$. Then $\Delta(r,s)\le 5$. The exterior of the Whitehead sister link shows that this bound is best possible.
Integration of singular braid invariants and graph cohomology
Michael
Hutchings
1791-1809
Abstract: We prove necessary and sufficient conditions for an arbitrary invariant of braids with $m$ double points to be the ``$m^{th}$ derivative'' of a braid invariant. We show that the ``primary obstruction to integration'' is the only obstruction. This gives a slight generalization of the existence theorem for Vassiliev invariants of braids. We give a direct proof by induction on $m$ which works for invariants with values in any abelian group. We find that to prove our theorem, we must show that every relation among four-term relations satisfies a certain geometric condition. To find the relations among relations we show that $H_1$ of a variant of Kontsevich's graph complex vanishes. We discuss related open questions for invariants of links and other things.
Integral representation of continuous comonotonically additive functionals
Lin
Zhou
1811-1822
Abstract: In this paper, I first prove an integral representation theorem: Every quasi-integral on a Stone lattice can be represented by a unique upper-continuous capacity. I then apply this representation theorem to study the topological structure of the space of all upper-continuous capacities on a compact space, and to prove the existence of an upper-continuous capacity on the product space of infinitely many compact Hausdorff spaces with a collection of consistent finite marginals.
Kronecker conjugacy of polynomials
Peter
Müller
1823-1850
Abstract: Let $f,g\in \mathbb{Z}[X]$ be non-constant polynomials with integral coefficients. In 1968 H. Davenport raised the question as to when the value sets $f(\mathbb{Z})$ and $g(\mathbb{Z})$ are the same modulo all but finitely many primes. The main progress until now is M. Fried's result that $f$ and $g$ then differ by a linear substitution, provided that $f$ is functionally indecomposable. We extend this result to polynomials $f$ of composition length $2$. Also, we study the analog when $\mathbb{Z}$ is replaced by the integers of a number field. The above number theoretic property translates to an equivalent property of subgroups of a finite group, known as Kronecker conjugacy, a weakening of conjugacy which has been studied by various authors under different assumptions and in other contexts. We also give a simplified and strengthened version of the Galois theoretic translation to finite groups.
On the hyperbolicity of small cancellation groups and one-relator groups
S.
V.
Ivanov;
P.
E.
Schupp
1851-1894
Abstract: In the article, a result relating to maps (= finite planar connected and simply connected 2-complexes) that satisfy a $C(p)\&T(q)$ condition (where $(p,q)$ is one of $(3,6)$, $(4,4)$, $(6,3)$ which correspond to regular tessellations of the plane by triangles, squares, hexagons, respectively) is proven. On the base of this result a criterion for the Gromov hyperbolicity of finitely presented small cancellation groups satisfying non-metric $C(p)\&T(q)$-conditions is obtained and a complete (and explicit) description of hyperbolic groups in some classes of one-relator groups is given: All one-relator hyperbolic groups with $> 0$ and $\le 3$ occurrences of a letter are indicated; it is shown that a finitely generated one-relator group $G$ whose reduced relator $R$ is of the form $R \equiv a T_{0} a T_{1} \dots a T_{n-1}$, where the words $T_{i}$ are distinct and have no occurrences of the letter $a^{\pm 1}$, is not hyperbolic if and only if one has in the free group that (1) $n=2$ and $T_{0} T_{1}^{-1}$ is a proper power; (2) $n = 3$ and for some $i$ it is true (with subscripts $\operatorname{mod} 3$) that $T_{i} T_{i+1}^{-1} T_{i} T_{i+2}^{-1} = 1$; (3) $n = 4$ and for some $i$ it is true (with subscripts $\operatorname{mod} 4$) that $T_{i} T_{i+1}^{-1} T_{i+2} T_{i+3}^{-1} = 1$.
A convexity theorem in the scattering theory for the Dirac operator
K.
L.
Vaninsky
1895-1911
Abstract: The Dirac operator enters into zero curvature representation for the cubic nonlinear Schrödinger equation. We introduce and study a conformal map from the upper half-plane of the spectral parameter of the Dirac operator into itself. The action variables turn out to be limiting boundary values of the imaginary part of this map. We describe the image of the momentum map (convexity theorem) in the simplest case of a potential from the Schwartz class. We apply this description to the invariant manifolds for the nonlinear Schrödinger equation.
Linkage and sums of ideals
Mark
R.
Johnson
1913-1930
Abstract: It is shown (under mild conditions) that the sum of transversal ideals in a regular local ring cannot lie in the linkage class of a complete intersection. For a sum of geometrically linked Cohen-Macaulay ideals, we compute the depths of the conormal module and the first Koszul homology. As applications, we construct general examples of ideals which are strongly Cohen-Macaulay, strongly nonobstructed but not in the linkage class of a complete intersection, and Gorenstein ideals which are strongly nonobstructed but not syzygetic.
The $L_2$-localization of $W(n)$
Robert
D.
Thompson
1931-1944
Abstract: In this paper we analyze the localization of $W(n)$, the fiber of the double suspension map $S^{2n-1}\to \Omega^{2}S^{2n+1}$, with respect to $E(2)$. If four cells at the bottom of $D_pM^{2np-1}$, the $p$th extended power spectrum of the Moore spectrum, are collapsed to a point, then one obtains a spectrum $C$. Let $QM^{2np-1}\to QC$ be the James-Hopf map followed by the collapse map. Then we show that the secondary suspension map $BW(n)\to QM^{2np-1}$ has a lifting to the fiber of $QM^{2np-1}\to QC$ and this lifting is shown to be a $v_2$-periodic equivalence, hence an $E(2)$-equivalence.
Extension and approximation of CR functions on tube manifolds
André
Boivin;
Roman
Dwilewicz
1945-1956
Abstract: A complete generalization of the classical Bochner theorem for infinite tubes is given.
Generalized Hestenes' Lemma and extension of functions
Massimo
Ferrarotti;
Leslie
C.
Wilson
1957-1975
Abstract: Suppose we have an $m$-jet field on $V\subset \mathbf{R}^{n}$ which is a Whitney field on the nonsingular part $M$ of $V$. We show that, under certain hypotheses about the relationship between geodesic and euclidean distance on $V$, if the field is flat enough at the singular part $S$, then it is a Whitney field on $V$ (the order of flatness required depends on the coefficients in the hypotheses). These hypotheses are satisfied when $V$ is subanalytic. In Section II, we show that a $C^{2}$ function $f$ on $M$ can be extended to one on $V$ if the differential $df$ goes to $0$ faster than the order of divergence of the principal curvatures of $M$ and if the first covariant derivative of $df$ is sufficiently flat. For the general case of $C^{m}$ functions with $m >2$, we give a similar result for $\operatorname{codim} M=1$ in Section III.
Hyperbolic complete minimal surfaces with arbitrary topology
F.
J.
López
1977-1990
Abstract: We show a method to construct orientable minimal surfaces in $\Bbb R^3$ with arbitrary topology. This procedure gives complete examples of two different kinds: surfaces whose Gauss map omits four points of the sphere and surfaces with a bounded coordinate function. We also apply these ideas to construct stable minimal surfaces with high topology which are incomplete or complete with boundary.
Prime and semiprime semigroup algebras of cancellative semigroups
M.
V.
Clase
1991-2007
Abstract: Necessary and sufficient conditions are given for a semigroup algebra of a cancellative semigroup to be prime and semiprime. These conditions were proved necessary by Okninski; our contribution is to show that they are also sufficient. The techniques used in the proof are a new variation on the $\Delta$-methods which were developed originally for group algebras.
Composition factors of indecomposable modules
Maria
Izabel
Ramalho Martins
2009-2031
Abstract: Let $\Lambda$ be a connected, basic finite dimensional algebra over an algebraically closed field. Our main aim is to prove that if $\Lambda$ is biserial, its ordinary quiver has no loop and every indecomposable $\Lambda$-module is uniquely determined by its composition factors, then each indecomposable $\Lambda$-module is multiplicity-free.
Equivalence of norms on operator space tensor products of $C^\ast$-algebras
Ajay
Kumar;
Allan
M.
Sinclair
2033-2048
Abstract: The Haagerup norm $\Vert \cdot \Vert _{h}$ on the tensor product $A\otimes B$ of two $C^*$-algebras $A$ and $B$ is shown to be Banach space equivalent to either the Banach space projective norm $\Vert \cdot \Vert _{\gamma }$ or the operator space projective norm $\Vert \cdot \Vert _{\wedge }$ if and only if either $A$ or $B$ is finite dimensional or $A$ and $B$ are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on $A\otimes B$ if and only if $A$ or $B$ is subhomogeneous.
Carleson conditions for asymptotic weights
Michael
Brian
Korey
2049-2069
Abstract: The doubling and $A_\infty$ conditions are characterized in terms of convolution with rapidly decreasing kernels. The Carleson-measure criterion for $A_\infty$ of Fefferman, Kenig, and Pipher is extended to the case when all bounds become optimally small in the asymptotic limit.
Local spectra and individual stability of uniformly bounded $C_0$-semigroups
Charles
J. K.
Batty;
Jan
van Neerven;
Frank
Räbiger
2071-2085
Abstract: We study the asymptotic behaviour of individual orbits $T(\cdot )x$ of a uniformly bounded $C_{0}$-semigroup $\{T(t)\}_{t\ge 0}$ with generator $A$ in terms of the singularities of the local resolvent $(\lambda -A)^{-1}x$ on the imaginary axis. Among other things we prove individual versions of the Arendt-Batty-Lyubich-Vu theorem and the Katznelson-Tzafriri theorem.
Tauberian theorems and stability of solutions of the Cauchy problem
Charles
J. K.
Batty;
Jan
van Neerven;
Frank
Räbiger
2087-2103
Abstract: Let $f : \mathbb{R}_{+} \to X$ be a bounded, strongly measurable function with values in a Banach space $X$, and let $iE$ be the singular set of the Laplace transform $\widetilde f$ in $i\mathbb{R}$. Suppose that $E$ is countable and $\alpha \left \| \int _{0}^{\infty }e^{-(\alpha + i\eta ) u} f(s+u) \, du \right \| \to 0$ uniformly for $s\ge 0$, as $\alpha \searrow 0$, for each $\eta$ in $E$. It is shown that \begin{displaymath}\left \| \int _{0}^{t} e^{-i\mu u} f(u) \, du - \widetilde f(i\mu ) \right \| \to 0\end{displaymath} as $t\to \infty$, for each $\mu$ in $\mathbb{R} \setminus E$; in particular, $\|f(t)\| \to 0$ if $f$ is uniformly continuous. This result is similar to a Tauberian theorem of Arendt and Batty. It is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on $BUC(\mathbb{R}_{+}, X)$, and it implies several results concerning stability of solutions of Cauchy problems.
Minimizing the Laplacian of a function squared with prescribed values on interior boundaries- Theory of polysplines
Ognyan
Iv.
Kounchev
2105-2128
Abstract: In this paper we consider the minimization of the integral of the Laplacian of a real-valued function squared (and more general functionals) with prescribed values on some interior boundaries $\Gamma$, with the integral taken over the domain D. We prove that the solution is a biharmonic function in $D$ except on the interior boundaries $\Gamma$, and satisfies some matching conditions on $\Gamma$. There is a close analogy with the one-dimensional cubic splines, which is the reason for calling the solution a polyspline of order 2, or biharmonic polyspline. Similarly, when the quadratic functional is the integral of $(\Delta ^{q}f)^{2}, \, q$ a positive integer, then the solution is a polyharmonic function of order $2q, \, \Delta ^{2q}f(x) = 0,$ for $x \in D\setminus \Gamma$, satisfying matching conditions on $\Gamma$, and is called a polyspline of order $2q$. Uniqueness and existence for polysplines of order $2q$, provided that the interior boundaries $\Gamma$ are sufficiently smooth surfaces and $\partial D \subseteq \Gamma$, is proved. Three examples of data sets $\Gamma$ possessing symmetry are considered, in which the computation of polysplines is reduced to computation of one-dimensional $L-$splines.
The average edge order of triangulations of 3-manifolds with boundary
Makoto
Tamura
2129-2140
Abstract: Feng Luo and Richard Stong introduced the average edge order $\mu _0(K)$ of a triangulation $K$ and showed in particular that for closed 3-manifolds $\mu _0(K)$ being less than 4.5 implies that $K$ is on $S^3$. In this paper, we establish similar results for 3-manifolds with non-empty boundary; in particular it is shown that $\mu _0(K)$ being less than 4 implies that $K$ is on the 3-ball.