Quantum groups, differential calculi and the eigenvalues of the Laplacian
J.
Kustermans;
G.
J.
Murphy;
L.
Tuset
4681-4717
Abstract: We study $*$-differential calculi over compact quantum groups in the sense of S.L. Woronowicz. Our principal results are the construction of a Hodge operator commuting with the Laplacian, the derivation of a corresponding Hodge decomposition of the calculus of forms, and, for Woronowicz' first calculus, the calculation of the eigenvalues of the Laplacian.
$k$-hyponormality of finite rank perturbations of unilateral weighted shifts
Raúl
E.
Curto;
Woo
Young
Lee
4719-4737
Abstract: In this paper we explore finite rank perturbations of unilateral weighted shifts $W_{\alpha }$. First, we prove that the subnormality of $W_{\alpha }$ is never stable under nonzero finite rank perturbations unless the perturbation occurs at the zeroth weight. Second, we establish that 2-hyponormality implies positive quadratic hyponormality, in the sense that the Maclaurin coefficients of $D_{n}(s):=\text{det}\,P_{n}\,[(W_{\alpha }+sW_{\alpha }^{2})^{*},\, W_{\alpha }+s W_{\alpha }^{2}]\,P_{n}$are nonnegative, for every $n\ge 0$, where $P_{n}$ denotes the orthogonal projection onto the basis vectors $\{e_{0},\cdots ,e_{n}\}$. Finally, for $\alpha$ strictly increasing and $W_{\alpha }$ 2-hyponormal, we show that for a small finite-rank perturbation $\alpha ^{\prime }$ of $\alpha$, the shift $W_{\alpha ^{\prime }}$ remains quadratically hyponormal.
Canonical forms of Borel functions on the Milliken space
Olaf
Klein;
Otmar
Spinas
4739-4769
Abstract: The goal of this paper is to canonize Borel measurable mappings $\Delta\colon\Omega^\omega\to\mathbb{R}$, where $\Omega^\omega$ is the Milliken space, i.e., the space of all increasing infinite sequences of pairwise disjoint nonempty finite sets of $\omega$. This main result is a common generalization of a theorem of Taylor and a theorem of Prömel and Voigt.
A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior
Xinfu
Chen;
Shangbin
Cui;
Avner
Friedman
4771-4804
Abstract: In this paper we study a free boundary problem modeling the growth of radially symmetric tumors with two populations of cells: proliferating cells and quiescent cells. The densities of these cells satisfy a system of nonlinear first order hyperbolic equations in the tumor, and the tumor's surface is a free boundary $r=R(t)$. The nutrient concentration satisfies a diffusion equation, and $R(t)$ satisfies an integro-differential equation. It is known that this problem has a unique stationary solution with $R(t)\equiv R_s$. We prove that (i) if $\lim _{T\to \infty} \int^{T+1}_T \vert\dot R(t)\vert\,dt=0$, then $\lim_{t\to \infty}R(t)=R_s$, and (ii) the stationary solution is linearly asymptotically stable.
A simple algorithm for principalization of monomial ideals
Russell
A.
Goward Jr.
4805-4812
Abstract: In this paper, we give a simple constructive proof of principalization of monomial ideals and the global analog. This also gives an algorithm for principalization.
Cofinality of the nonstationary ideal
Pierre
Matet;
Andrzej
Roslanowski;
Saharon
Shelah
4813-4837
Abstract: We show that the reduced cofinality of the nonstationary ideal ${\mathcal N S}_\kappa$ on a regular uncountable cardinal $\kappa$ may be less than its cofinality, where the reduced cofinality of ${\mathcal N S}_\kappa$ is the least cardinality of any family ${\mathcal F}$ of nonstationary subsets of $\kappa$ such that every nonstationary subset of $\kappa$ can be covered by less than $\kappa$ many members of ${\mathcal F}$. For this we investigate connections of the various cofinalities of ${\mathcal N S}_\kappa$ with other cardinal characteristics of ${}^{\textstyle\kappa}\kappa$ and we also give a property of forcing notions (called manageability) which is preserved in ${<}\kappa$-support iterations and which implies that the forcing notion preserves non-meagerness of subsets of ${}^{\textstyle\kappa}\kappa$ (and does not collapse cardinals nor changes cofinalities).
Complete analytic equivalence relations
Alain
Louveau;
Christian
Rosendal
4839-4866
Abstract: We prove that various concrete analytic equivalence relations arising in model theory or analysis are complete, i.e. maximum in the Borel reducibility ordering. The proofs use some general results concerning the wider class of analytic quasi-orders.
Affine pseudo-planes and cancellation problem
Kayo
Masuda;
Masayoshi
Miyanishi
4867-4883
Abstract: We define affine pseudo-planes as one class of $\mathbb{Q}$-homology planes. It is shown that there exists an infinite-dimensional family of non-isomorphic affine pseudo-planes which become isomorphic to each other by taking products with the affine line $\mathbb{A} ^1$. Moreover, we show that there exists an infinite-dimensional family of the universal coverings of affine pseudo-planes with a cyclic group acting as the Galois group, which have the equivariant non-cancellation property. Our family contains the surfaces without the cancellation property, due to Danielewski-Fieseler and tom Dieck.
Extended Hardy-Littlewood inequalities and some applications
Hichem
Hajaiej
4885-4896
Abstract: We establish conditions under which the extended Hardy-Little- wood inequality \begin{displaymath}\int \limits_{\mathbb{R} ^N} H\big{(}\vert x\vert,\, u_1(x), ... ...ig{(}\vert x\vert,\, u_1^*(x), \ldots , u_m^*(x)\big{)}\, dx, \end{displaymath} where each $u_i$ is non-negative and $u_i^*$ denotes its Schwarz symmetrization, holds. We also determine appropriate monotonicity assumptions on $H$ such that equality occurs in the above inequality if and only if each $u_i$ is Schwarz symmetric. We end this paper with some applications of our results in the calculus of variations and partial differential equations.
A Gieseker type degeneration of moduli stacks of vector bundles on curves
Ivan
Kausz
4897-4955
Abstract: We construct a new degeneration of the moduli stack of vector bundles over a smooth curve when the curve degenerates to a singular curve which is irreducible with one double point. We prove that the total space of the degeneration is smooth and its special fibre is a divisor with normal crossings. Furthermore, we give a precise description of how the normalization of the special fibre of the degeneration is related to the moduli space of vector bundles over the desingularized curve.
Regulating flows, topology of foliations and rigidity
Sérgio
R.
Fenley
4957-5000
Abstract: A flow transverse to a foliation is regulating if, in the universal cover, an arbitrary orbit of the flow intersects every leaf of the lifted foliation. This implies that the foliation is $\mathbf{R}$-covered, that is, its leaf space in the universal cover is homeomorphic to the reals. We analyse the converse of this implication to study the topology of the leaf space of certain foliations. We prove that if a pseudo-Anosov flow is transverse to an $\mathbf{R}$-covered foliation and the flow is not an $\mathbf{R}$-covered Anosov flow, then the flow is regulating for the foliation. Using this we show that several interesting classes of foliations are not $\mathbf{R}$-covered. Finally we show a rigidity result: if an $\mathbf{R}$-covered Anosov flow is transverse to a foliation but is not regulating, then the foliation blows down to one topologically conjugate to the stable or unstable foliations of the transverse flow.
Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces
Siva
R.
Athreya;
Richard
F.
Bass;
Edwin
A.
Perkins
5001-5029
Abstract: We introduce a new method for proving the estimate \begin{displaymath}\left\Vert\frac{\partial^2 u}{\partial x_i \partial x_j} \right\Vert_{C^\alpha}\leq c\Vert f\Vert _{C^\alpha},\end{displaymath} where $u$ solves the equation $\Delta u-\lambda u=f$. The method can be applied to the Laplacian on $\mathbb{R}^\infty$. It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.
An algebraic approach to multiresolution analysis
Richard
Foote
5031-5050
Abstract: The notion of a weak multiresolution analysis is defined over an arbitrary field in terms of cyclic modules for a certain affine group ring. In this setting the basic properties of weak multiresolution analyses are established, including characterizations of their submodules and quotient modules, the existence and uniqueness of reduced scaling equations, and the existence of wavelet bases. These results yield some standard facts on classical multiresolution analyses over the reals as special cases, but provide a different perspective by not relying on orthogonality or topology. Connections with other areas of algebra and possible further directions are mentioned.
Filtrations in semisimple rings
D.
S.
Passman
5051-5066
Abstract: In this paper, we describe the maximal bounded $\mathbb{Z}$-filtrations of Artinian semisimple rings. These turn out to be the filtrations associated to finite $\mathbb{Z}$-gradings. We also consider simple Artinian rings with involution, in characteristic $\neq 2$, and we determine those bounded $\mathbb{Z}$-filtrations that are maximal subject to being stable under the action of the involution. Finally, we briefly discuss the analogous questions for filtrations with respect to other Archimedean ordered groups.
Minimal invariant tori in the resonant regions for nearly integrable Hamiltonian systems
Chong-Qing
Cheng
5067-5095
Abstract: Consider a real analytical Hamiltonian system of KAM type $H(p,q)$ $=N(p)+P(p,q)$ that has $n$ degrees of freedom ($n>2$) and is positive definite in $p$. Let $\Omega =\{\omega\in \mathbb R^n \vert\langle \bar k,\omega\rangle =0, \bar k\in\mathbb Z^n\}$. In this paper we show that for most rotation vectors in $\Omega$, in the sense of ($n-1$)-dimensional Lebesgue measure, there is at least one ($n-1$)-dimensional invariant torus. These tori are the support of corresponding minimal measures. The Lebesgue measure estimate on this set is uniformly valid for any perturbation.
The cyclic and simplicial cohomology of $l^1(\mathbf{N})$
Frédéric
Gourdeau;
B.
E.
Johnson;
Michael
C.
White
5097-5113
Abstract: Let $\mathcal{A}=l^1(\mathbf Z_+)$ be the unital semigroup algebra of $\mathbf N$. We show that the cyclic cohomology groups $\mathcal{H}^n(\mathcal{A},\mathcal{A}')$ vanish for $n\ge 2$. The results obtained are extended to unital algebras $l^1(S)$ for some other semigroups of $\mathbf{R}$.