Multi-bump orbits homoclinic to resonance bands
Tasso
J.
Kaper;
Gregor
Kovacic
3835-3887
Abstract: We establish the existence of several classes of multi-bump orbits homoclinic to resonance bands for completely-integrable Hamiltonian systems subject to small-amplitude Hamiltonian or dissipative perturbations. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The homoclinic orbits, which include multi-bump \v{S}ilnikov orbits, connect equilibria and periodic orbits in the resonance band. The main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands.
A new uncountably categorical group
Andreas
Baudisch
3889-3940
Abstract: We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CM-trivial.
On the homology spectral sequence for topological Hochschild homology
Thomas
J.
Hunter
3941-3953
Abstract: Marcel Bökstedt has computed the homotopy type of the topological Hochschild homology of $\Bbb Z/p$ using his definition of topological Hochschild homology for a functor with smash product. Here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of \begin{math}S\end{math}-algebras as introduced by Elmendorf, Kriz, Mandell and May. We give algebraic arguments based on naturality properties of the topological Hochschild homology spectral sequence. In the process we demonstrate the utility of the unstable ``lower'' notation for the Dyer-Lashof algebra.
Sharp upper bound for the first non-zero Neumann eigenvalue for bounded domains in rank-1 symmetric spaces
A.
R.
Aithal;
G.
Santhanam
3955-3965
Abstract: In this paper, we prove that for a bounded domain $\Omega$ in a rank-$1$ symmetric space, the first non-zero Neumann eigenvalue $\mu _{1}(\Omega )\leq \mu _{1}(B(r_{1}))$ where $B(r_{1})$ denotes the geodesic ball of radius $r_{1}$ such that \begin{equation*}vol(\Omega )=vol(B(r_{1}))\end{equation*} and equality holds iff $\Omega =B(r_{1})$. This result generalises the works of Szego, Weinberger and Ashbaugh-Benguria for bounded domains in the spaces of constant curvature.
Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of classical type
Tom
Halverson;
Arun
Ram
3967-3995
Abstract: In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of the Iwahori-Hecke algebras of types $A_{n-1}, B_n$, and $D_n$. Our method is a generalization of a derivation of the Murnaghan-Nakayama formula for the irreducible characters of the symmetric group given by Curtis Greene. Greene's approach is to sum up the diagonal entries of the matrices of certain cycle permutations in Young's seminormal representations. The analogues of the Young seminormal representations for the Iwahori-Hecke algebras of types $A_{n-1},B_n$, and $D_n$ were given by Hoefsmit.
Invariants of piecewise-linear 3-manifolds
John
W.
Barrett;
Bruce
W.
Westbury
3997-4022
Abstract: This paper presents an algebraic framework for constructing invariants of closed oriented 3-manifolds by taking a state sum model on a triangulation. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. A significant feature is that the tensor category is not required to be braided. The main examples are constructed from the categories of representations of involutive Hopf algebras and of quantised enveloping algebras at a root of unity.
On Baire-1/4 functions
Vassiliki
Farmaki
4023-4041
Abstract: We give descriptions of the spaces $D(K)$ (i.e. the space of differences of bounded semicontinuous functions on $K$) and especially of $B_{1/4}(K)$ (defined by Haydon, Odell and Rosenthal) as well as for the norms which are defined on them. For example, it is proved that a bounded function on a metric space $K$ belongs to $B_{1/4}(K)$ if and only if the $\omega ^{ % \mathrm {th}}$-oscillation, $\mathrm {osc}_{\omega }f$, of $f$ is bounded and in this case $\| f\|_{1/4}=\|\, |f|+ \widetilde {\mathrm {osc}}_{\omega } f\|_{\infty }$. Also, we classify $B_{1/4}(K)$ into a decreasing family $(S_{\xi }(K))_{1\leq \xi <\omega _1}$ of Banach spaces whose intersection is equal to $D(K)$ and $S_1 (K)=B_{1/4}(K)$. These spaces are characterized by spreading models of order $\xi$ equivalent to the summing basis of $c_0$, and for every function $f$ in $S_{\xi }(K)$ it is valid that $\mathrm {osc}_{\omega ^{\xi }}f$ is bounded. Finally, using the notion of null-coefficient of order $\xi$ sequence, we characterize the Baire-1 functions not belonging to $S_{\xi }(K)$.
Transfinite multifractal dimension spectrums
Stanley
C.
Williams
4043-4081
Abstract: The first order theory of the decomposition of measures with respect to dimension which has been developed by Kahane, Katznelson, Cutler, and others is extended through transfinite recursion to a $\omega _1$-order theory. Necessary and sufficient conditions for a finite regular Borel measure on $% [0,d]^{\omega _1}$ to be a $\omega _1$-order multispectrum for a finite Borel measure on $\mathbb {R}^d$ is given.
Competitive exclusion and coexistence for competitive systems on ordered Banach spaces
S.
B.
Hsu;
H.
L.
Smith;
Paul
Waltman
4083-4094
Abstract: The dynamics of competitive maps and semiflows defined on the product of two cones in respective Banach spaces is studied. It is shown that exactly one of three outcomes is possible for two viable competitors. Either one or the other population becomes extinct while the surviving population approaches a steady state, or there exists a positive steady state representing the coexistence of both populations.
Extremal problems and symmetrization for plane ring domains
A.
Yu.
Solynin;
M.
Vuorinen
4095-4112
Abstract: We show that Teichmüller's classical lower bound for the capacity of a ring domain, obtained by circular symmetrization, can be replaced by an explicit one which is almost always better. The proof is based on a duplication formula for the solution of an associated extremal problem. Some inequalities are obtained for conformal invariants.
Single generator problem
Jun-ichi
Tanaka
4113-4129
Abstract: Using the Stone-\v{C}ech compactification $\beta \mathbf Z$ of integers, we introduce a free extension of an almost periodic flow. Together with some properties of outer functions, we see that, in a certain class of ergodic Hardy spaces $H^p(\mu )$, $1\le p\le \infty$, the corresponding subspaces $H_0^p(\mu )$ are all singly generated. This shows the existence of maximal weak-$^*$ Dirichlet algebras, different from $H^\infty$ of the disc, for which the single generator problem is settled.
Wiener's test for space-time random walks and its applications
Yasunari
Fukai;
Kôhei
Uchiyama
4131-4152
Abstract: This paper establishes a criterion for whether a $d$-dimensional random walk on the integer lattice $\mathbf {Z}^{d}$ visits a space-time subset infinitely often or not. It is a precise analogue of Wiener's test for regularity of a boundary point with respect to the classical Dirichlet problem. The test obtained is applied to strengthen the harder half of Kolmogorov's test for the random walk.
Hyponormality and spectra of Toeplitz operators
Douglas
R.
Farenick;
Woo
Young
Lee
4153-4174
Abstract: This paper concerns algebraic and spectral properties of Toeplitz operators $T_{\varphi }$, on the Hardy space $H^{2}({\mathbb {T}})$, under certain assumptions concerning the symbols $\varphi \in L^{\infty }({\mathbb {T}})$. Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at $T_{\varphi }$, for each quasicontinuous $\varphi$. Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.
Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts
D.-Q.
Zhang
4175-4184
Abstract: Let $(S, \Delta )$ be a log surface with at worst log canonical singularities and reduced boundary $\Delta$ such that $-(K_{S}+\Delta )$ is nef and big. We shall prove that $S^{o} = S - Sing S - \Delta$ either has finite fundamental group or is affine-ruled. Moreover, $\pi _{1}(S^{o})$ and the structure of $S$ are determined in some sense when $\Delta = 0$.
On polarized surfaces $(X,L)$ with $h^0(L)>0$, $\kappa (X)=2$, and $g(L)=q(X)$
Yoshiaki
Fukuma
4185-4197
Abstract: Let $X$ be a smooth projective surface over $\mathbb {C}$ and $L$ an ample Cartier divisor on $X$. If the Kodaira dimension $\kappa (X)\leq 1$ or $\operatorname {dim}H^{0}(L)>0$, the author proved $g(L)\geq q(X)$, where $q(X)=\operatorname {dim}H^{1}(\mathcal {O}_{X})$. If $\kappa (X)\leq 1$, then the author studied $(X,L)$ with $g(L)=q(X)$. In this paper, we study the polarized surface $(X,L)$ with $\kappa (X)=2$, $g(L)=q(X)$, and $\operatorname {dim}H^{0}(L)>0$.
K-theoretic classification for certain inductive limit $Z_2$ actions on real rank zero $C^*$-algebras
Hongbing
Su
4199-4230
Abstract: In this paper a K-theoretic classification is given of the C$^*$-algebra dynamical systems $(A, \alpha , Z_2)= \lim \limits_\to (A_n, {\alpha }_n, Z_2)$ where $A$ is of real rank zero, each $A_n$ is a finite direct sum of matrix algebras over finite connected graphs, and each $\alpha _n$ is induced by an action on each component of the spectrum of $A_n$. Corresponding to the trivial actions is the K-theoretic classification for real rank zero C$^*$-algebras that can be expressed as finite direct sums of matrix algebras over finite graphs obtained in Mem. Amer. Math. Soc. no. 547, vol. 114.
Separable Banach space theory needs strong set existence axioms
A.
James
Humphreys;
Stephen
G.
Simpson
4231-4255
Abstract: We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, $\Pi ^1_1$ comprehension, is needed to prove such basic facts as the existence of the weak-$*$ closure of any norm-closed subspace of $\ell _1=c_0^*$. This is in contrast to earlier work in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for $\Pi ^0_2$ sentences. En route to our main results, we prove the Krein-\v{S}mulian theorem in $\mathsf {ACA}_0$, and we give a new, elementary proof of a result of McGehee on weak-$*$ sequential closure ordinals.
Gross spaces
Saharon
Shelah;
Otmar
Spinas
4257-4277
Abstract: A Gross space is a vector space $E$ of infinite dimension over some field $F$, which is endowed with a symmetric bilinear form $\Phi :E^{2} \rightarrow F$ and has the property that every infinite dimensional subspace $U\subseteq E$ satisfies dim$U^{\perp }<$ dim$E$. Gross spaces over uncountable fields exist (in certain dimensions). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariant b equals $\omega _{1}$, a Gross space in dimension $\omega _{1}$ exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.