Generalized quotients in Coxeter groups
Anders
Björner;
Michelle L.
Wachs
1-37
Abstract: For ($W$, $S$) a Coxeter group, we study sets of the form $\displaystyle W/V = \{ w \in W\vert l(wv) = l(w) + l(v)\;{\text{for all}}\;v \in V\} ,$ , where $V \subseteq W$. Such sets $W/V$, here called generalized quotients, are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when $ V \subseteq S$ (i.e., for minimal coset representatives modulo a parabolic subgroup). We show that Bruhat intervals in $ W/V$, for general $V \subseteq W$, are lexicographically shellable. The Möbius function on $W/V$ under Bruhat order takes values in $\{ - 1,\,0,\, + 1\}$. For finite groups $ W$, generalized quotients are the same thing as lower intervals in the weak order. This is, however, in general not true. Connections with the weak order are explored and it is shown that $W/V$ is always a complete meet-semilattice and a convex order ideal as a subset of $W$ under weak order. Descent classes $ {D_I} = \{ w \in W\vert l(ws) < l(w) \Leftrightarrow s \in I,\;{\text{for all}}\;s \in S\}$, $ I \subseteq S$, are also analyzed using generalized quotients. It is shown that each descent class, as a poset under Bruhat order or weak order, is isomorphic to a generalized quotient under the corresponding ordering. The latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics. For instance, the set of standard Young tableaux of a fixed shape or the set of linear extensions of a rooted forest, suitably interpreted, form generalized quotients. We prove a factorization result for the quotients that come from rooted forests, which shows that algebraically these quotients behave as a system of minimal "coset" representatives of a subset which is in general not a subgroup. We also study the rank generating function for certain quotients in the symmetric group.
On the local boundedness of singular integral operators
Mark
Leckband
39-56
Abstract: The class of singular integral operators whose kernels satisfy the usual smoothness conditions is studied. Let such an operator be denoted by $K$. We establish necessary conditions that imply $K$ has local (weighted) ${L^p}$ norm inequalities. The underlying principle is as follows. If ${\chi _Q}$ is the characteristic function of a fixed cube $Q$ of ${R^n}$, or all of ${R^n}$, then $ K{\chi _Q}$ and (the adjoint of $K$) $ {K^{\ast}}{\chi _Q}$ determine the boundedness properties of $K$ for functions supported in a proper fraction of $Q$.
Weighted norm inequalities for potential operators
Martin
Schechter
57-68
Abstract: We give sufficient conditions for inequalities of the form $\displaystyle {\left( {\int {{{\left( {\int {G(x - y)f(y)\,d\mu (y)} } \right)}... ...1/q}}\, \leqslant C{\left( {\int {\vert f(y){\vert^p}d\nu (y)} } \right)^{1/p}}$ to hold for measurable functions $f$. We determine the dependence of the constant $C$ on the measures $\mu$, $\nu$, $\omega$ and give some applications.
Conformal distortion of boundary sets
D. H.
Hamilton
69-81
Abstract: Conformal maps $ f$ of the disk into itself have the property that $\dim {f^{ - 1}}(F) \leqslant \dim F$ for any set $F$ on the unit circle.
Subordination families and extreme points
Yusuf
Abu-Muhanna;
D. J.
Hallenbeck
83-89
Abstract: Let $s(F)$ denote the set of functions subordinate to a univalent function $F$ in $\Delta$ the unit disk. Let ${B_0}$ denote the set of functions $ \phi (z)$ analytic in $ \Delta$ satisfying $\vert\phi (z)\vert < 1$ and $\phi (0) = 0$. We prove that if $f = F \circ \phi$ is an extreme point of $ s(F)$, then $\phi$ is an extreme point of $ {B_0}$. Let $D = F(s)$ and $\lambda (w,\,\partial D)$ denote the distance between $w$ and $ \partial D$ (boundary of $ D$). We also prove that if $ \phi$ is an extreme point of ${B_0}$ and $ \vert\phi ({e^{it}})\vert < 1$ for almost all $t$, then $\int_0^{2\pi } {\log \lambda (F(\phi ({e^{it}}){e^{i\theta }}),\,\partial D)\,dt = - \infty }$ for almost all $\theta$.
Generalized Chebyshev polynomials associated with affine Weyl groups
Michael E.
Hoffman;
William Douglas
Withers
91-104
Abstract: We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in one-to-one correspondence with affine Weyl groups. For each such figure in $ n$-dimensional Euclidean space, we construct a sequence of polynomials ${P_k}:{{\mathbf{R}}^n} \to {{\mathbf{R}}^n}$ so that the mapping ${P_k}$ is conjugate to stretching the figure by a factor $k$ and folding it back onto itself. If $n = 1$ and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials $ {P_k}$ are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials.
Local uncertainty inequalities for locally compact groups
John F.
Price;
Alladi
Sitaram
105-114
Abstract: Let $G$ be a locally compact unimodular group equipped with Haar measure $m$, $\hat G$ its unitary dual and $\mu$ the Plancherel measure (or something closely akin to it) on $\hat G$. When $G$ is a euclidean motion group, a non-compact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given $\theta \in \left[ {0,\tfrac{1} {2}} \right.)$ there exists a constant $ {K_\theta }$ such that for all $f$ in a certain class of functions on $G$ and all measurable $E \subseteq \hat G$, $\displaystyle {\left( {\int_E {\operatorname{Tr} (\pi {{(f)}^{\ast}}\pi (f))\,d... ...\leqslant {K_\theta }\mu {(E)^\theta }\vert\vert{\phi _\theta }f\vert{\vert _2}$ where ${\phi _\theta }$ is a certain weight function on $ G$ (for which an explicit formula is given). When $G = {{\mathbf{R}}^k}$ the inequality has been established with ${\phi _\theta }(x) = \vert x{\vert^{k\theta }}$.
The Brauer group of graded continuous trace $C\sp *$-algebras
Ellen Maycock
Parker
115-132
Abstract: Let $X$ be a locally compact Hausdorff space. The graded Morita equivalence classes of separable, $ {{\mathbf{Z}}_2}$-graded, continuous trace $ {C^{\ast}}$-algebras which have spectrum $X$ form a group, ${\operatorname{GBr} ^\infty }(X)$, the infinite-dimensional graded Brauer group of $X$. Techniques from algebraic topology are used to prove that ${\operatorname{GBr} ^\infty }(X)$ is isomorphic via an isomorphism $w$ to the direct sum $\check{H}^1(X; \underline{\mathbf{Z}}_2) \oplus \check{H}^3 (X; \underline{\mathbf{Z}})$. The group ${\operatorname{GBr} ^\infty }(X)$ includes as a subgroup the ungraded continuous trace ${C^{\ast}}$-algebras, and the Dixmier-Douady invariant of such an ungraded $ {C^{\ast}}$-algebra is its image in $\check{H}^3 (X; \underline{\mathbf{Z}})$ under $w$.
Proper knot theory in open $3$-manifolds
Peter
Churchyard;
David
Spring
133-142
Abstract: This paper introduces a theory of proper knots, i.e., smooth proper embeddings of $ {{\mathbf{R}}^1}$ into open $3$-manifolds. Proper knot theory is distinguished by the fact that proper isotopies of knots are not ambient in general. A uniqueness theorem for proper knots is proved in case the target manifold is the interior of a one-dimensional handlebody.
The closing lemma for generalized recurrence in the plane
Maria Lúcia Alvarenga
Peixoto
143-158
Abstract: We prove a version of the Closing Lemma for ${C^r}$ vector fields on the plane, $r \geqslant 1$, and for a kind of recurrence obtained using the concept of prolongational limit sets. We call it generalized recurrence. Given a nonperiodic point $p$ in the generalized recurrent set we perturb the vector field in order to get a new vector field arbitrarily close to it, with a closed orbit through $ p$.
Stable manifolds in the method of averaging
Stephen
Schecter
159-176
Abstract: Consider the differential equation $\dot z = \varepsilon f(z,\,t,\,\varepsilon )$, where $f$ is $T$periodic in $t$ and $ \varepsilon > 0$ is a small parameter, and the averaged equation $\dot z = \overline f (z): = (1/T)\,\int_0^T {\,f(z,\,t,\,0)\,dt}$. Suppose the averaged equation has a hyperbolic equilibrium at $z = 0$ with stable manifold $\overline W$. Let ${\beta _\varepsilon }(t)$ denote the hyperbolic $ T$-periodic solution of $\dot z = \varepsilon f(z,\,t,\,\varepsilon )$ near $z \equiv 0$. We prove a result about smooth convergence of the stable manifold of ${\beta _\varepsilon }(t)$ to $\overline W \times {\mathbf{R}}$ as $\varepsilon \to 0$. The proof uses ideas of Vanderbauwhede and van Gils about contractions on a scale of Banach spaces.
Uniformly fat sets
John L.
Lewis
177-196
Abstract: In this paper we study closed sets $E$ which are "locally uniformly fat" with respect to a certain nonlinear Riesz capacity. We show that $ E$ is actually "locally uniformly fat" with respect to a weaker Riesz capacity. Two applications of this result are given. The first application is concerned with proving Sobolev-type inequalities in domains whose complements are uniformly fat. The second application is concerned with the Fekete points of $E$.
Euler spaces of analytic functions
James
Rovnyak
197-208
Abstract: A formula due to Euler and Legendre is used to construct finite-difference counterparts to the Dirichlet space. The spaces have integral representations and characterizations in terms of area integrals. Their reproducing kernels are logarithms of the reproducing kernels of the Newton spaces, which are counterparts to the Hardy class. A Hilbert space with reproducing kernel $\displaystyle \log [(1/\overline w z)\,\log \;1/(1 - \overline w z)]$ is also shown to exist and to be related to Bernoulli numbers and combinatorial theory.
On the complete ${\rm GL}(n,{\bf C})$-decomposition of the stable cohomology of ${\rm gl}\sb n(A)$
Phil
Hanlon
209-225
Abstract: Let $A$ be a graded, associative ${\mathbf{C}}$-algebra. For each $n$ let $g{l_n}(A)$ denote the Lie algebra of $n \times n$ matrices with entries from $ A$. In this paper we extend the Loday-Quillen theorem to nontrivial isotypic components of $ GL(n,\,{\mathbf{C}})$ acting on the Lie algebra cohomology of $g{l_n}(A)$. For $\alpha$ and $\beta$ partitions of some nonnegative integer $ m$ let $ {[\alpha ,\,\beta ]_n} \in {{\mathbf{Z}}^n}$ denote the maximal $ GL(n,\,{\mathbf{C}})$-weight given by $\displaystyle {[\alpha ,\,\beta ]_n} = \sum\limits_i {{\alpha _i}{e_i}} - \sum\limits_j {{\beta _j}{e_{n + 1 - j}}.}$ We show that the $ {[\alpha ,\,\beta ]_n}$-isotypic component of the Lie algebra cohomology of $ g{l_n}(A)$ stabilizes when $n \to \infty $ and is equal to $\displaystyle HR{C^{\ast}}(A) \otimes ({\tilde H^{\ast}}{(A;\,{\mathbf{C}})^{ \otimes m}} \otimes {S^\alpha } \otimes {S^\beta }){s_m}$ where ${\tilde H^{\ast}}(A;\,{\mathbf{C}})$ is the reduced Hochschild cohomology of $A$ with trivial coefficients, where $ HR{C^{\ast}}(A)$ is the graded exterior algebra generated by the cyclic cohomology of $A$, where $ {S^\alpha }$ and ${S^\beta }$ are the irreducible $ {S_m}$-modules indexed by $ \alpha$ and $ \beta$ and where the action of ${S_m}$ on $\tilde H{(A;\,{\mathbf{C}})^{ \otimes m}}$ is the exterior action.
Pseudo-orbit shadowing in the family of tent maps
Ethan M.
Coven;
Ittai
Kan;
James A.
Yorke
227-241
Abstract: We study the family of tent maps--continuous, unimodal, piecewise linear maps of the interval with slopes $\pm s$, $\sqrt 2 \leqslant s \leqslant 2$. We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters $s$, although they fail to have the shadowing property for an uncountable, dense set of parameters. We also show that for any tent map, every pseudo-orbit can be approximated by an actual orbit of a tent map with a perhaps slightly larger slope.
Smoothness up to the boundary for solutions of the nonlinear and nonelliptic Dirichlet problem
C. J.
Xu;
C.
Zuily
243-257
Abstract: For the Dirichlet problem associated with a general real second order p.d.e. $F(x,\,u,\,\nabla u,\,{\nabla ^2}u) = 0$ in a smooth open set $\Omega$ of $ {{\mathbf{R}}^n}$, we prove smoothness up to the boundary of the solution $ u$ for which the linearized characteristic form is nonnegative and satisfies Hörmander's brackets condition, the boundary of $ \Omega$ being noncharacteristic.
Some inequalities for singular convolution operators in $L\sp p$-spaces
Andreas
Seeger
259-272
Abstract: Suppose that a bounded function $m$ satisfies a localized multiplier condition ${\sup _{t > 0}}\vert\vert\phi m({t^P} \cdot )\vert{\vert _{{M_p}}} < \infty$, for some bump function $\phi$. We show that under mild smoothness assumptions $m$ is a Fourier multiplier in ${L^p}$. The approach uses the sharp maximal operator and Littlewood-Paley-theory. The method gives new results for lacunary maximal functions and for multipliers in Triebel-Lizorkin-spaces.
A Riemannian geometric invariant and its applications to a problem of Borel and Serre
Bang-Yen
Chen;
Tadashi
Nagano
273-297
Abstract: A new geometric invariant will be introduced, studied and determined on compact symmetric spaces.
Variational principles for Hill's spherical vortex and nearly spherical vortices
Yieh Hei
Wan
299-312
Abstract: In this paper, vortex rings are regarded as axisymmetric motions without swirl of an incompressible inviscid fluid in space, with vorticity confined to their finite cores. The main results of this paper are (H) Hill's spherical vortex is a "nondegenerate" local maximum of the energy function subject to a fixed impulse, among vortex rings. (N) Norbury's nearly spherical vortex is a "nondegenerate" local maximum of the energy function subject to a fixed impulse, and a fixed circulation. Estimates are established to overcome the discontinuity of vorticity distributions, and the singular behavior of Stoke's stream functions near the axis of symmetry. The spectral analysis involves the use of Legendre's functions.
Iwasawa's $\lambda\sp -$-invariant and a supplementary factor in an algebraic class number formula
Kuniaki
Horie
313-328
Abstract: Let $l$ be a prime number and $k$ an imaginary abelian field. Sinnott [12] has shown that the relative class number of $k$ is expressed by the so-called index of the Stickelberger ideal of $k$, with a "supplementary factor" $ {c^ - }$ in $ \mathbb{N}/2 = \{ n/2\vert n \in \mathbb{N}\}$, and that if $k$ varies through the layers of the basic $ {\mathbb{Z}_l}$-extension over an imaginary abelian field, then ${c^ - }$ becomes eventually constant. On the other hand, ${c^ - }$ can take any value in $\mathbb{N}/2$ as $k$ ranges over the imaginary abelian fields (cf. [10]). In this paper, we shall study relations between the supplementary factor $ {c^ - }$ and Iwasawa's ${\lambda ^ - }$-invariant for the basic $ {\mathbb{Z}_l}$-extension over $k$, our discussion being based upon some formulas of Kida [8, 9], those of Sinnott [12], and fundamental results concerning a finite abelian $ l$-group acted on by a cyclic group. As a consequence, we shall see that the ${\lambda ^ - }$-invariant goes to infinity whenever $k$ ranges over a sequence of imaginary abelian fields such that the $l$-part of ${c^ - }$ goes to infinity.
On the zero set of a holomorphic one-form on a compact complex manifold
Michael J.
Spurr
329-339
Abstract: On any compact complex surface $M$, divisors of nonnegative self-intersection which are contained in the zero set (or in the integral set) of a holomorphic $1$-form are shown to induce a fibration of $ M$ onto a Riemann surface. This result is extended to higher dimensions for $ M$ projective. Applications to zero sets of holomorphic $1$-forms on surfaces are given.
The Morse index theorem where the ends are submanifolds
Diane
Kalish
341-348
Abstract: In this paper the Morse Index Theorem is proven in the case where submanifolds $P$ and $Q$ are at the endpoints of a geodesic, $ \gamma$. At $ \gamma$, the index of the Hessian of the energy function defined on paths joining $P$ and $Q$ is computed using $P$-focal points, and a calculation at the endpoint of $\gamma$, involving the second fundamental form of $ Q$.
The blow-up surface for nonlinear wave equations with small spatial velocity
Avner
Friedman;
Luc
Oswald
349-367
Abstract: Consider the Cauchy problem for ${u_{tt}} - {\varepsilon ^2}\Delta u = f(u)$ in space dimension $ \leqslant 3$ where $ f(u)$ is superlinear and nonnegative. The solution blows up on a surface $t = {\phi _\varepsilon }(x)$. Denote by $t = \phi (x)$ the blow-up surface corresponding to $v'' = f(v)$. It is proved that $\vert{\phi _\varepsilon }(x) - \phi (x)\vert \leqslant C{\varepsilon ^2}$, $\vert\nabla ({\phi _\varepsilon }(x) - \phi (x))\vert \leqslant C{\varepsilon ^2}$ in a neighborhood of any point ${x_0}$ where $ \phi ({x_0}) < \infty$.
Traveling wave solutions of a gradient system: solutions with a prescribed winding number. I
David
Terman
369-389
Abstract: Consideration is given to a system of equations of the form ${u_t} = {u_{xx}} + \nabla F(u)$, $u \in {{\mathbf{R}}^2}$. In a previous paper [6], conditions of $F$ were given which guarantee that the system possesses infinitely many traveling wave solutions. The solutions are now characterized by how many times they wind around in phase space. A winding number for solutions is defined. It is demonstrated that for each positive integer $K$, there exists at least two traveling wave solutions, each with winding number $K$ or $K + 1$.
Traveling wave solutions of a gradient system: solutions with a prescribed winding number. II
David
Terman
391-412
Abstract: This paper completes the analysis begun in [2] concerning the existence of traveling wave solutions of a system of the form $ {u_t} = {u_{xx}} + \nabla F(u)$, $u \in {{\mathbf{R}}^2}$. In [2] a notion of winding number for solutions was defined, and the proof that there exists a traveling wave solution with a prescribed winding number was reduced to a purely algebraic problem. In this paper the algebraic problem is solved.
Spectrum reducing extension for one operator on a Banach space
C. J.
Read
413-429
Abstract: In this paper we show that, given an operator $T$ on a Banach space $X$, there is an extension $Y$ of $X$ such that $T$ extends in a natural way to an operator ${T^ \sim }$ on $Y$, and the spectrum of $ {T^ \sim }$ is the approximate point spectrum of $T$. This answers a question posed by Bollobás, and contributes to a theory investigated by Shilov, Arens, Bollobás, etc. The unusual transfinite construction is similar to that which we used earlier to find an inverse producing extension for a commutative unital Banach algebra which eliminates the residual spectrum of one element. We also give a counterexample, consisting of a Banach algebra $L$ containing elements ${g_1}$ and ${g_2}$ such that in no extension $L'$ of $L$ are the residual spectra of ${g_1}$ and ${g_{_2}}$ eliminated simultaneously.
Errata to: ``Where does the $L\sp p$-norm of a weighted polynomial live?''
H. N.
Mhaskar;
E. B.
Saff
431