A model in which GCH holds at successors but fails at limits
James
Cummings
1-39
Abstract: Starting with GCH and a $ {\mathcal{P}_3}\kappa$-hypermeasurable cardinal, a model is produced in which $ {2^\lambda } = {\lambda ^ + }$ if $\lambda$ is a successor cardinal and ${2^\lambda } = {\lambda ^{ + + }}$ if $ \lambda$ is a limit cardinal. The proof uses a Reverse Easton extension followed by a modified Radin forcing.
Continuation theorems for periodic perturbations of autonomous systems
Anna
Capietto;
Jean
Mawhin;
Fabio
Zanolin
41-72
Abstract: It is first shown in this paper that, whenever it exists, the coincidence degree of the left-hand member of an autonomous differential equation $ {\text{g}}$. This result provides efficient continuation theorems specially for $\omega$-periodic perturbations of autonomous systems. Extensions to differential equations in flow-invariant ENR's are also given.
Two characteristic numbers for smooth plane curves of any degree
Paolo
Aluffi
73-96
Abstract: We use a sequence of blow-ups over the projective space parametrizing plane curves of degree $d$ to obtain some enumerative results concerning smooth plane curves of arbitrary degree. For $d = 4$, this gives a first modern verification of results of H. G. Zeuthen.
Compactifications of locally compact groups and closed subgroups
A. T.
Lau;
P.
Milnes;
J. S.
Pym
97-115
Abstract: Let $G$ be a locally compact group with closed normal subgroup $N$ such that $G/N$ is compact. In this paper, we construct various semigroup compactifications of $G$ from compactifications of $N$ of the same type. This enables us to obtain specific information about the structure of the compactifi cation of $G$ from the structure of the compactification of $ N$. Our results seem to be interesting and new even when $G$ is the additive group of real numbers and $N$ is the integers. Applications and other examples are given.
On the distribution of extremal points of general Chebyshev polynomials
András
Kroó;
Franz
Peherstorfer
117-130
Abstract: For a linear subspace $ {\mathcal{U}_n} = {\operatorname{span}}[{\varphi _1}, \ldots ,{\varphi _n}]$ in $ C[a,b]$ we introduce general Chebyshev polynomials as solutions of the minimization problem $ {\operatorname{min}_{{a_i}}}{\left\Vert {{\varphi _n} - \sum\nolimits_{i = 1}^{n - 1} {{a_i}{\varphi _i}} } \right\Vert _C}$. For such a Chebyshev polynomial we study the distribution of its extremal points (maximum and minimum points) in terms of structural and approximative properties of $ {\mathcal{U}_n}$.
Ces\`aro summability of double Walsh-Fourier series
F.
Móricz;
F.
Schipp;
W. R.
Wade
131-140
Abstract: We introduce quasi-local operators (these include operators of Calderón-Zygmund type), a hybrid Hardy space $ {{\mathbf{H}}^\sharp }$ of functions of two variables, and we obtain sufficient conditions for a quasi-local maximal operator to be of weak type $(\sharp ,1)$. As an application, we show that Cesàro means of the double Walsh-Fourier series of a function $f$ converge a.e. when $f$ belongs to $ {{\mathbf{H}}^\sharp }$. We also obtain the dyadic analogue of a summability result of Marcienkiewicz and Zygmund valid for all $f \in {L^1}$ provided summability takes place in some positive cone.
Extending cellular cohomology to $C\sp *$-algebras
Ruy
Exel;
Terry A.
Loring
141-160
Abstract: A filtration on the $ K$-theory of $ {C^*}$-algebras is introduced. The relative quotients define groups ${H_n}(A),n \geq 0$, for any $ {C^*}$-algebra $ A$, which we call the spherical homology of $A$. This extends cellular cohomology in the sense that $\displaystyle {H_n}(C(X)) \otimes {\mathbf{Q}} \cong {H^n}(X;{\mathbf{Q}})$ for $ X$ a finite CW-complex. While no extension of cellular cohomology which is derived from a filtration on $K$-theory can be additive, Morita-invariant, and continuous, ${H_n}$ is shown to be infinitely additive, Morita invariant for unital ${C^*}$-algebras, and continuous in limited cases.
Maximal entropy odd orbit types
William
Geller;
Juán
Tolosa
161-171
Abstract: A periodic orbit of a continuous map of an interval induces in a natural way a cyclic permutation, called its type. We consider a family of orbit types of period $n$ congruent to $1$ ( $ \operatorname{mod} 4$) introduced recently by Misiurewicz and Nitecki. We prove that the Misiurewicz-Nitecki orbit types and their natural generalizations to the remaining odd periods $n$ have maximal entropy among all orbit types of period $n$, and even among all $n$-permutations.
A decomposition theorem for the spectral sequence of Lie foliations
Jesús A.
Alvarez López
173-184
Abstract: For a Lie $\mathfrak{g}$-foliation $ \mathcal{F}$ on a closed manifold $M$, there is an "infinitesimal action of $\mathfrak{g}$ on $M$ up to homotopy along the leaves", in general it is not an action but defines an action of the corresponding connected simply connected Lie group $\mathfrak{S}$ on the term ${E_1}$ of the spectral sequence associated to $\mathcal{F}$. Even though ${E_1}$ in general is infinite-dimensional and non-Hausdorff (with the topology induced by the $ {\mathcal{C}^\infty }$-topology), it is proved that this action can be averaged when $ \mathfrak{S}$ is compact, obtaining a tensor decomposition theorem of $ {E_2}$. It implies duality in the whole term ${E_2}$ for Riemannian foliations on closed oriented manifolds with compact semisimple structural Lie algebra.
A damped hyperbolic equation on thin domains
Jack K.
Hale;
Geneviève
Raugel
185-219
Abstract: For a damped hyperbolic equation in a thin domain in ${{\mathbf{R}}^3}$ over a bounded smooth domain in $ {{\mathbf{R}}^2}$, it is proved that the global attractors are upper semicontinuous. It is shown also that a global attractor exists in the case of the critical Sobolev exponent.
On the linear independence of certain cohomology classes in the classifying space for subfoliations
Demetrio
Domínguez
221-232
Abstract: The purpose of this paper is to establish the linear independence of certain cohomology classes in the Haefliger classifying space $ B{\Gamma _{({q_1},{q_2})}}$ for sub-foliations of codimension $({q_{1,}}{q_2})$. The classes considered are of secondary type, not belonging to the subalgebra of $ H(B{\Gamma _{({q_1},{q_2})}},R)$ generated by the union of the universal characteristic classes for foliations of codimension $ {q_1}$ and ${q_2}$ respectively, and are elements of the kernel of the canonical homomorphism $ H(B{\Gamma _{({q_1},{q_2})}},R) \to H(B{\Gamma _{{q_1}}} \times B{\Gamma _d},R)$ with $ d = {q_2} - {q_1} > 0$.
Poincar\'e-Lefschetz duality for the homology Conley index
Christopher
McCord
233-252
Abstract: The Conley index for continuous dynamical systems is defined for (one-sided) semiflows. For (two-sided) flows, there are two indices defined: one for the forward flow; and one for the reverse flow. In general, the two indices give different information about the flow; but for flows on orientable manifolds, there is a duality isomorphism between the homology Conley indices of the forward and reverse flows. This duality preserves the algebraic structure of many of the constructions of the Conley index theory: sums and products; continuation; attractor-repeller sequences and connection matrices.
The modular representation theory of $q$-Schur algebras
Jie
Du
253-271
Abstract: We developed some basic theory of characteristic zero modular representations of $q$-Schur algebras. We described a basis of the $ q$-Schur algebra in terms of the relative norm which was first introduced by P. Hoefsmit and L. Scott, and studied the product of two such basis elements. We also defined the defect group of a primitive idempotent in a $q$-Schur algebra and showed that such a defect group is just the vertex of the corresponding indecomposable $ {\mathcal{H}_F}$-module.
The Green correspondence for the representations of Hecke algebras of type $A\sb {r-1}$
Jie
Du
273-287
Abstract: We first prove the conjecture mentioned by Leonard K. Jones in his thesis. By applying this conjecture, we obtain that the vertex of an indecomposable ${\mathcal{H}_F}$-module is an $l$-parabolic subgroup. Finally, we establish the Green correspondence for the representations of Hecke algebras of type $ {A_{r - 1}}$.
The determination of minimal projections and extensions in $L\sp 1$
B. L.
Chalmers;
F. T.
Metcalf
289-305
Abstract: Equations are derived which are shown to be necessary and sufficient for finite rank projections in ${L^1}$ to be minimal. More generally, these equations are also necessary and sufficient to determine operators of minimal norm which extend a fixed linear action on a given finite-dimensional subspace of ${L^1}$ and thus may be viewed as an extension of the Hahn-Banach theorem to higher dimensions in the $ {L^1}$ setting. These equations are solved in terms of an ${L^1}$ best approximation problem and the required orthogonality conditions. Moreover, this solution has a simple geometric interpretation. Questions of uniqueness are considered and a number of examples are given to illustrate the usefulness of these equations in determining minimal projections and extensions, including the minimal $ {L^1}$ projection onto the quadratics.
A relationship between the Jones and Kauffman polynomials
Christopher
King
307-323
Abstract: A simple relationship is presented between the Kauffman polynomial of a framed link $L$ and the Jones polynomial of a derived link $ \tilde L$. The link is $ \tilde L$ obtained by splitting each component of $L$ into two parallel strands, using the framing to determine the linking number of the strands. The relation is checked in several nontrivial examples, and a proof of the general result is given.
A modification of Shelah's oracle-c.c. with applications
Winfried
Just
325-356
Abstract: A method of constructing iterated forcing notions that has a scope of applications similar to Shelah's oracle-c.c. is presented. This method yields a consistency result on homomorphisms of quotient algebras of the Boolean algebra $ \mathcal{P}(\omega )$. Also, it is shown to be relatively consistent with ZFC that the Boolean algebra of Lebesgue measurable subsets of the unit interval has no projective lifting.
Moderate deviations and associated Laplace approximations for sums of independent random vectors
A.
de Acosta
357-375
Abstract: Let $\{ {X_j}\}$ be an i.i.d. sequence of Banach space valued r.v.'s and let ${S_n} = \sum\nolimits_{j = 1}^n {{X_j}}$. For certain positive sequences $ {b_n} \to \infty$, we determine the exact asymptotic behavior of $E{\operatorname{exp}}\{ (b_n^2/n)\Phi ({S_n}/{b_n})\}$, where $\Phi$ is a smooth function. We also prove a large deviation principle for $\{ \mathcal{L}({S_n}/{b_n})\}$.
Young measures and an application of compensated compactness to one-dimensional nonlinear elastodynamics
Peixiong
Lin
377-413
Abstract: We study the existence problem for the equations of $1$-dimensional nonlinear elastodynamics. We obtain the convergence of ${L^p}(p < \infty )$ bounded approximating sequences generated by the method of vanishing viscosity and the Lax-Friedrichs scheme. The analysis uses Young measures, Lax entropies, and the method of compensated compactness.
Generalized potentials and obstacle scattering
Richard L.
Ford
415-431
Abstract: Potential scattering theory is a very well-developed and understood subject. Scattering for Schràdinger operators represented formally by $- \Delta + V$, where $ V$ is a generalized function such as a $\delta$-function, is less understood and requires form perturbation techniques. A general scattering theory for a large class of such singular perturbations of the Laplacian is developed. The theory has application to obstacle scattering. One considers an alternative mathematical model of an obstacle in ${{\mathbf{R}}^n}$. Instead of representing the obstacle by deleting the region inhabited by the obstacle from $ {{\mathbf{R}}^n}$, the surface of the obstacle is treated as impenetrable. The impenetrable surface is understood to be the limiting case of a sequence of highly spiked potentials whose support converges to the surface of the obstacle and whose peaks grow without bound. The limiting case is identified as a $\delta$-function acting on the surface of the obstacle. Hamiltonians for the limiting case are constructed and the conditions governing the existence and completeness of the associated wave operators are determined through application of the general theory.