Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots
Mario
Eudave Muñoz
463-501
Abstract: We consider composite links obtained by bandings of another link. It is shown that if a banding of a split link yields a composite knot then there is a decomposing sphere crossing the band in one arc, unless there is such a sphere disjoint from the band. We also prove that if a banding of the trivial knot yields a composite knot or link then there is a decomposing sphere crossing the band in one arc. The last theorem implies, via double branched covers, that the only way we can get a reducible manifold by surgery on a strongly invertible knot is when the knot is cabled and the surgery is via the slope of the cabling annulus.
Finite codimensional subalgebras of Stein algebras and semiglobally Stein algebras
Hà Huy Khoái;
Nguyen Văn Khuê
503-508
Abstract: The following theorem is proved: For each finite codimensional subalgebra $A$ of a Stein algebra $B$ there exists a natural number $n$ such that $B$ is algebraically isomorphic to $A \oplus {{\mathbf{C}}^n}$.
A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in $L\sp p(0,1)$
K. P.
Choi
509-529
Abstract: Let $1 < p < \infty$. Let $d = ({d_1},{d_2}, \ldots)$ be a real-valued martingale difference sequence, $\theta = ({\theta _1},{\theta _2}, \ldots)$ is a predictable sequence taking values in $ [0,1]$. We show that the best constant of the inequality, $\displaystyle {\left\Vert {\sum\limits_{k = 1}^n {{\theta _k}{d_k}} } \right\Ve... ...p}{\left\Vert {\sum\limits_{k = 1}^n {{d_k}} } \right\Vert _p}, \quad n \geq 1,$ satisfies $\displaystyle {c_p} = \frac{p}{2} + \frac{1}{2}\;\log \;\left({\frac{{1 + y}}{2}} \right) + \frac{{{\alpha _2}}}{p} + \cdots,$ where $\gamma = {e^{ - 2}}$ and ${\alpha _2} = {\left[ {\frac{1}{2}\;\log \;\frac{{1 + \gamma }}{2}} \right]^2}... ... \;\frac{{1 + \gamma }}{2} - 2{\left({\frac{\gamma }{{1 + \gamma }}} \right)^2}$. The best constant equals the unconditional basis constant of a monotone basis of $ {L^p}(0,1)$.
Invariant subspaces with finite codimension in Bergman spaces
Alexandru
Aleman
531-544
Abstract: For an arbitrary bounded domain in $ \mathbb{C}$ there are described those finite codimensional subspaces of the Bergman space that are invariant under multiplication by $z$.
Intersection theory of moduli space of stable $n$-pointed curves of genus zero
Sean
Keel
545-574
Abstract: We give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the Chow ring.
Lattice-ordered groups whose lattices determine their additions
Paul F.
Conrad;
Michael R.
Darnel
575-598
Abstract: In this paper it is shown that several large and important classes of lattice-ordered groups, including the free abelian lattice-ordered groups, have their group operations completely determined by the underlying lattices, or determined up to $l$-isomorphism.
Justification of multidimensional single phase semilinear geometric optics
Jean-Luc
Joly;
Jeffrey
Rauch
599-623
Abstract: For semilinear strictly hyperbolic systems $ Lu= f(x,u)$, we construct and justify high frequency nonlinear asymptotic expansions of the form $\displaystyle {u^\varepsilon }(x)\sim\sum\limits_{j\, \geq \,0} {{\varepsilon ^... ...phi \,(x)/\varepsilon}, \quad L{u^\varepsilon } - f(x,{u^\varepsilon })\sim 0 .$ The study of the principal term of such expansions is called nonlinear geometric optics in the applied literature. We show (i) formal expansions with periodic profiles ${U_j}$ can be computed to all orders, (ii) the equations for the profiles from (i) are solvable, and (iii) there are solutions of the exact equations which have the formal series as high frequency asymptotic expansion.
Superharmonic functions on foliations
S. R.
Adams
625-635
Abstract: We use techniques from geometric analysis to prove that any positive, leafwise superharmonic, measurable function on a Riemannian measurable foliation with transverse invariant measure, finite total volume and complete leaves is, in fact, constant on a.e. leaf.
Generalized Szeg\H o theorems and asymptotics of cumulants by graphical methods
Florin
Avram
637-649
Abstract: We obtain some general asymptotics results about a class of deterministic sums called "sums with dependent indices," which generalize a classical theorem of Szegö. The above type of sums is encountered when establishing convergence to the Gaussian distribution of sums of Wick products by the method of cumulants. Our asymptotic results reduce in this situation the proof of the central limit theorem to the study of the connectivity of a family of associated graphs.
Central limit theorems for sums of Wick products of stationary sequences
Florin
Avram;
Robert
Fox
651-663
Abstract: We show, by the method of cumulants, that checking whether the central limit theorem for sums of Wick powers of a stationary sequence holds can be reduced to the study of an associated graph problem (see Corollary 1). We obtain thus central limit theorems under various integrability conditions on the cumulant spectral functions (Theorems 2, 3).
Symmetry of knots and cyclic surgery
Shi Cheng
Wang;
Qing
Zhou
665-676
Abstract: If a nontorus knot $ K$ admits a symmetry which is not a strong inversion, then there exists no nontrivial cyclic surgery on $K$. No surgery on a symmetric knot can produce a fake lens space or a $3$-manifold $M$ with $ \vert{\pi _1}(M)\vert= 2$. This generalizes the result of Culler-Gordon-Luecke-Shalen-Bleiler-Scharlemann and supports the conjecture that no nontrivial surgery on a nontrivial knot yields a $ 3$-manifold $M$ with $\vert{\pi _1}(M)\vert < 5$.
Stable and uniformly stable unit balls in Banach spaces
Antonio
Suárez Granero
677-695
Abstract: Let $X$ be a Banach space with closed unit ball $ {B_X}$ and, for $x \in X$, $r \geq 0$, put $B(x;r)= \{ u \in X:\vert\vert u - x\vert\vert \leq r\}$ and $V(x,r)= {B_X} \cap B(x;r)$. We say that $ {B_X}$ (or in general a convex set) is stable if the midpoint map $ {\Phi _{1/2}}:{B_X} \times {B_X} \to {B_X}$, with $ {\Phi _{1/2}}(u,\upsilon)= \frac{1}{2}(u + \upsilon)$, is open. We say that $ {B_X}$ is uniformly stable (US) if there is a map $\alpha :(0,2] \to (0,2]$, called a modulus of uniform stability, such that, for each $x,y \in {B_X}$ and $r \in (0,2],V(\frac{1} {2}(x + y);\alpha (r)) \subseteq \frac{1} {2}(V(x;r) + V(y;r))$. Among other things, we see: (i) if $ \dim X \geq 3$, then $ X$ admits an equivalent norm such that ${B_X}$ is not stable; (ii) if $\dim X < \infty$, ${B_X}$ is stable iff ${B_x}$ is US; (iii) if $X$ is rotund, $X$ is uniformly rotund iff ${B_X}$ is US; (iv) if $X$ is $ 3.2.{\text{I.P}}$, $ {B_X}$ is US and $\alpha (r)= r/2$ is a modulus of US; (v) $ {B_X}$ is US iff ${B_{{X^{ \ast \ast }}}}$ is US and $ X$, ${X^{ \ast \ast}}$ have (almost) the same modulus of US; (vi) ${B_X}$ is stable (resp. US) iff ${B_{C(K,X)}}$ is stable (resp. US) for each compact $ K$ iff ${B_{A(K,X)}}$ is stable (resp. US) for each Choquet simplex $K$; (vii) ${B_X}$ is stable iff ${B_{{L_p}(\mu,X)}}$ is stable for each measure $ \mu$ and $1 \leq p < \infty $.
Analytic operator valued function space integrals as an ${\scr L}(L\sb p,L\sb {p'})$ theory
Kun Soo
Chang;
Kun Sik
Ryu
697-709
Abstract: The existence of an analytic operator-valued function space integral as an $ \mathcal{S}({L_p},{L_{p^{\prime}}})$ theory $ (1 \leq p \leq 2)$ has been established for certain functionals involving the Lebesgue measure. Recently, Johnson and Lapidus proved the existence of the integral as an operator on $ {L_2}$ for certain functionals involving any Borel measure. We establish the existence of the integral as an operator from $ {L_p}$ to $ {L_{p^{\prime}}}\;({1 < p < 2} )$ for certain functionals involving some Borel measures.
Isomorphism invariants for abelian groups
D. M.
Arnold;
C. I.
Vinsonhaler
711-724
Abstract: Let $A= ({A_1},\ldots,{A_n})$ be an $n$-tuple of subgroups of the additive group, $Q$, of rational numbers and let $G(A)$ be the kernel of the summation map ${A_1} \oplus \cdots \oplus {A_n} \to \sum \;{A_i}$ and $G[A]$ the cokernel of the diagonal embedding $\cap \,{A_1} \to {A_1} \oplus \cdots \oplus {A_n}$. A complete set of isomorphism invariants for all strongly indecomposable abelian groups of the form $ G(A)$, respectively, $ G[A]$, is given. These invariants are then extended to complete sets of isomorphism invariants for direct sums of such groups and for a class of mixed abelian groups properly containing the class of Warfield groups.
A general condition for lifting theorems
E. Arthur
Robinson
725-755
Abstract: We define a general condition, called stability on extensions $T$ of measure preserving transformations $ S$. Stability is defined in terms of relative unique ergodicity, and as a joining property. Ergodic compact group extensions are stable, and moreover stable extensions satisfy lifting theorems similar to those satisfied by group extensions. In general, stable extensions have relative entropy zero. In the class of continuous flow extensions over strictly ergodic homeomorphisms, stable extensions are generic.
The complete integral closure of $R[X]$
Thomas G.
Lucas
757-768
Abstract: For a reduced ring $ R$ that is completely integrally closed it is not always the case that the corresponding polynomial ring $R[X]$ is completely integrally closed. In this paper the question of when $R[X]$ is completely integrally closed is shown to be related to the question of when $R$ is completely integrally closed in $T(R[X])$ the total quotient ring of $ R[X]$. A characterization of the complete integral closure of $R[X]$ is given in the main theorem and this result is used to characterize the complete integral closure of the semigroup ring $ R[S]$ when $S$ is a torsion-free cancellative monoid.
Trace functions in the ring of fractions of polycyclic group rings
A. I.
Lichtman
769-781
Abstract: Let $KG$ be the group ring of a polycyclic-by-finite group $G$ over a field $K$ of characteristic zero, $R$ be the Goldie ring of fractions of $ KG$, $S$ be an arbitrary subring of ${R_{n \times n}}$. We prove that the intersection of the commutator subring $[S,S]$ with the center $Z(S)$ is nilpotent. This implies the existence of a nontrivial trace function in ${R_{n \times n}}$.
Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations
E.
DiBenedetto;
Y. C.
Kwong
783-811
Abstract: We establish an intrinsic Harnack estimate for nonnegative weak solutions of the singular equation $(1.1)$ below, for $m$ in the optimal range $((N - 2)_+/N,1)$. Intrinsic means that, due to the singularity, the space-time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself. Consequences are, sharp supestimates on the solutions and decay rates as $ t$ approaches the extinction time. Analogous results are shown for $p$-laplacian type equations.
A phenomenon of reciprocity in the universal Steenrod algebra
Luciano
Lomonaco
813-821
Abstract: In this paper we compute the cohomology algebra of certain subalgebras $ {L_r}$ and certain quotients ${K_s}$ of the $\bmod\, 2$ universal Steenrod algebra $ Q$, the algebra of cohomology operations for $ {H_\infty }$-ring spectra (see $[$M$]$). We prove that $\displaystyle \operatorname{Ext}_{{L_r}}({F_2},{F_2}) \cong {K_{ - k + 1}}, \qquad \operatorname{Ext}_{{K_s}}({F_2},{F_2}) \cong {L_{ - s + 1}}$ with $r$, $s$ integers and $r \leq 1$, $s \geq 0$. We also observe that some of the algebras ${L_r}$, ${K_s}$ are well known objects in stable homotopy theory and in fact our computation generalizes the fact that ${H^{\ast} }({A_L}) \cong \Lambda ^{{\text{opp}}}$ and ${H^{\ast} }({\Lambda ^{{\text{opp}}}}) \cong {A_L}$ (see, for instance, $ [$P$]$). Here ${A_L}$ is the Steenrod algebra for simplicial restricted Lie algebras and $\Lambda$ is the ${E_1}$-term of the Adams spectral sequence discovered in $[$B-S$]$.
The ``Defektsatz'' for central simple algebras
Joachim
Gräter
823-843
Abstract: Let $Q$ be a central simple algebra finite-dimensional over its center $F$ and let $V$ be a valuation ring of $F$. Then $V$ has an extension to $Q$, i.e., there exists a Dubrovin valuation ring $ B$ of $Q$ satisfying $ V= F \cap B$. Generally, the number of extensions of $V$ to $Q$ is not finite and therefore the so-called intersection property of Dubrovin valuation rings ${B_1}, \ldots,{B_n}$ is introduced. This property is defined in terms of the prime ideals and the valuation overrings of the intersection ${B_1} \cap \cdots \; \cap {B_n}$. It is shown that there exists a uniquely determined natural number $n$ depending only on $V$ and having the following property: If ${B_1}, \ldots,{B_k}$ are extensions of $V$ having the intersection property then $ k \leq n$ and $ k= n$ holds if and only if $ {B_1} \cap \cdots \cap {B_k}$ is integral over $V$. Let $n$ be the extension number of $V$ to $Q$. There exist extensions ${B_1}, \cdots,{B_n}$ of $V$ having the intersection property and if $ {R_1}, \ldots,{R_n}$ are also extensions of $V$ having the intersection property then $ {B_1} \cap \cdots \cap {B_n}$ and $ {R_1} \cap \cdots \cap {R_n}$ are conjugate. The main result regarding the extension number is the Defektsatz: $[Q:F]= {f_B}(Q/F){e_B}(Q/F){n^2}{p^d}$, where $ {f_B}(Q/F)$ is the residue degree, $ {e_B}(Q/F)$ the ramification index, $n$ the extension number, $p = \operatorname{char}(V/J(V))$, and $d$ a natural number.
A deformation of tori with constant mean curvature in ${\bf R}\sp 3$ to those in other space forms
Masaaki
Umehara;
Kotaro
Yamada
845-857
Abstract: It is shown that tori with constant mean curvature in ${\mathbb{R}^3}$ constructed by Wente $ [7]$ can be deformed to tori with constant mean curvature in the hyperbolic $ 3$-space or the $ 3$-sphere.
Functors on the category of finite sets
Randall
Dougherty
859-886
Abstract: Given a covariant or contravariant functor from the category of finite sets to itself, one can define a function from natural numbers to natural numbers by seeing how the functor maps cardinalities. In this paper we answer the question: what numerical functions arise in this way from functors? The sufficiency of the conditions we give is shown by simple constructions of functors. In order to show the necessity, we analyze the way in which functions in the domain category act on members of objects in the range category, and define combinatorial objects describing this action; the permutation groups in the domain category act on these combinatorial objects, and the possible sizes of orbits under this action restrict the values of the numerical function. Most of the arguments are purely combinatorial, but one case is reduced to a statement about permutation groups which is proved by group-theoretic methods.
Affine $3$-spheres with constant affine curvature
Martin A.
Magid;
Patrick J.
Ryan
887-901
Abstract: We classify the affine hyperspheres in ${R^4}$ which have constant curvature in the affine metric $h$ and whose Pick invariant is nonzero. In particular, the metric $h$ must be flat.
On compactly supported spline wavelets and a duality principle
Charles K.
Chui;
Jian-zhong
Wang
903-915
Abstract: Let $\cdots \subset{V_{ - 1}} \subset{V_0} \subset{V_1} \subset \cdots$ be a multiresolution analysis of $ {L^2}$ generated by the $ m$th order $B$-spline ${N_m}(x)$. In this paper, we exhibit a compactly supported basic wavelet $ {\psi _m}(x)$ that generates the corresponding orthogonal complementary wavelet subspaces $\cdots,{W_{ - 1}},{W_0},{W_1}, \ldots$. Consequently, the two finite sequences that describe the two-scale relations of ${N_m}(x)$ and $ {\psi _m}(x)$ in terms of $ {N_m}(2x - j),j \in \mathbb{Z}$, yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases $ \{ {\tilde N_m}(x - j)\}$ and $ \{ {\tilde \psi _m}(x - j)\}$, relative to $ \{ {N_m}(x - j)\}$ and $\{ {\psi _m}(x - j)\}$, respectively.
The kernel-trace approach to right congruences on an inverse semigroup
Mario
Petrich;
Stuart
Rankin
917-932
Abstract: A kernel-trace description of right congruences on an inverse semigroup is developed. It is shown that the trace mapping is a complete $\cap$homomorphism but not a $ \vee$-homomorphism. However, the trace classes are intervals in the complete lattice of right congruences. In contrast, each kernel class has a maximum element, namely the principal right congruence on the kernel, but in general there is no minimum element in a kernel class. The kernel mapping preserves neither intersections nor joins. The set of axioms presented in [7] for right kernel systems is reviewed. A new set of axioms is obtained as a consequence of the fact that a right congruence is the intersection of the principal right congruences on the idempotent classes. Finally, it is shown that even though a congruence on a regular semigroup is the intersection of the principal congruences on the idempotent classes, the situation is not the same for right congruences on a regular semigroup. Right congruences on a regular, even orthodox, semigroup are not, in general, determined by their idempotent classes.