Homology of pseudodifferential operators on manifolds with fibered cusps
Robert
Lauter;
Sergiu
Moroianu
3009-3046
Abstract: The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the $0$-dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case.
Moderate deviation principles for trajectories of sums of independent Banach space valued random variables
Yijun
Hu;
Tzong-Yow
Lee
3047-3064
Abstract: Let $\{X_n\}$ be a sequence of i.i.d. random vectors with values in a separable Banach space. Moderate deviation principles for trajectories of sums of $\{X_n\}$ are proved, which generalize related results of Borovkov and Mogulskii (1980) and Deshayes and Picard (1979). As an application, functional laws of the iterated logarithm are given. The paper also contains concluding remarks, with examples, on extending results for partial sums to corresponding ones for trajectory setting.
Weierstrass functions with random phases
Yanick
Heurteaux
3065-3077
Abstract: Consider the function \begin{displaymath}f_\theta(x)=\sum_{n=0}^{+\infty}b^{-n\alpha}g(b^nx+\theta_n),\end{displaymath} where $b>1$, $0<\alpha<1$, and $g$ is a non-constant 1-periodic Lipschitz function. The phases $\theta_n$ are chosen independently with respect to the uniform probability measure on $[0,1]$. We prove that with probability one, we can choose a sequence of scales $\delta_k\searrow 0$ such that for every interval $I$ of length $\vert I\vert=\delta_k$, the oscillation of $f_\theta$ satisfies $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^\alpha$. Moreover, the inequality $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^{\alpha+\varepsilon}$ is almost surely true at every scale. When $b$ is a transcendental number, these results can be improved: the minoration $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^\alpha$ is true for every choice of the phases $\theta_n$ and at every scale.
Explicit Lower bounds for residues at $s=1$ of Dedekind zeta functions and relative class numbers of CM-fields
Stéphane
Louboutin
3079-3098
Abstract: Let $S$ be a given set of positive rational primes. Assume that the value of the Dedekind zeta function $\zeta_K$ of a number field $K$ is less than or equal to zero at some real point $\beta$ in the range ${1\over 2} <\beta <1$. We give explicit lower bounds on the residue at $s=1$ of this Dedekind zeta function which depend on $\beta$, the absolute value $d_K$of the discriminant of $K$ and the behavior in $K$ of the rational primes $p\in S$. Now, let $k$ be a real abelian number field and let $\beta$ be any real zero of the zeta function of $k$. We give an upper bound on the residue at $s=1$ of $\zeta_k$which depends on $\beta$, $d_k$ and the behavior in $k$ of the rational primes $p\in S$. By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields $K$ which depend on the behavior in $K$ of the rational primes $p\in S$. We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.
Primitive free cubics with specified norm and trace
Sophie
Huczynska;
Stephen
D.
Cohen
3099-3116
Abstract: The existence of a primitive free (normal) cubic $x^3-ax^2+cx-b$ over a finite field $F$ with arbitrary specified values of $a$ ($\neq 0$) and $b$ (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.
D-log and formal flow for analytic isomorphisms of n-space
David
Wright;
Wenhua
Zhao
3117-3141
Abstract: Given a formal map $F=(F_1,\ldots,F_n)$ of the form $z+\text{higher-order}$ terms, we give tree expansion formulas and associated algorithms for the D-Log of $F$ and the formal flow $F_t$. The coefficients that appear in these formulas can be viewed as certain generalizations of the Bernoulli numbers and the Bernoulli polynomials. Moreover, the coefficient polynomials in the formal flow formula coincide with the strict order polynomials in combinatorics for the partially ordered sets induced by trees. Applications of these formulas to the Jacobian Conjecture are discussed.
A generalization of tight closure and multiplier ideals
Nobuo
Hara;
Ken-ichi
Yoshida
3143-3174
Abstract: We introduce a new variant of tight closure associated to any fixed ideal $\mathfrak{a}$, which we call $\mathfrak{a}$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $\tau (\mathfrak{a})$ of all $\mathfrak{a}$-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal $\tau (\mathfrak{a})$ and the multiplier ideal associated to $\mathfrak{a}$ (or, the adjoint of $\mathfrak{a}$ in Lipman's sense) in normal $\mathbb{Q}$-Gorenstein rings reduced from characteristic zero to characteristic $p \gg 0$. Also, in fixed prime characteristic, we establish some properties of $\tau (\mathfrak{a})$ similar to those of multiplier ideals (e.g., a Briançon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal $\tau (\mathfrak{a})$ and the F-rationality of Rees algebras.
Seshadri constants on Jacobian of curves
Jian
Kong
3175-3180
Abstract: We compute the Seshadri constants on the Jacobian of hyperelliptic curves, as well as of curves with genus three and four. For higher genus curves we conclude that if the Seshadri constants of their Jacobian are less than 2, then the curves must be hyperelliptic.
On the Clifford algebra of a binary form
Rajesh
S.
Kulkarni
3181-3208
Abstract: The Clifford algebra $C_f$ of a binary form $f$ of degree $d$is the $k$-algebra $k\{x, y\}/I$, where $I$ is the ideal generated by $\{(\alpha x + \beta y)^d - f(\alpha, \beta) \mid \alpha, \beta \in k\}$. $C_f$ has a natural homomorphic image $A_f$ that is a rank $d^2$ Azumaya algebra over its center. We prove that the center is isomorphic to the coordinate ring of the complement of an explicit $\Theta$-divisor in $\ensuremath{{Pic}_{C/k}^{d + g - 1}}$, where $C$ is the curve $(w^d - f(u, v))$ and $g$is the genus of $C$.
Projective normality of abelian varieties
Jaya
N.
Iyer
3209-3216
Abstract: We show that ample line bundles $L$ on a $g$-dimensional simple abelian variety $A$, satisfying $h^0(A,L)>2^g\cdot g!$, give projective normal embeddings, for all $g\geq 1$.
Clustering of critical points in Lefschetz fibrations and the symplectic Szpiro inequality
V.
Braungardt;
D.
Kotschick
3217-3226
Abstract: We prove upper bounds for the number of critical points in semi- stable symplectic Lefschetz fibrations. We also obtain a new lower bound for the number of nonseparating vanishing cycles in Lefschetz pencils and reprove the known lower bounds for the commutator lengths of Dehn twists.
On measures of maximal and full dimension for polynomial automorphisms of $\mathbb{C}^2$
Christian
Wolf
3227-3239
Abstract: For a hyperbolic polynomial automorphism of $\mathbb{C} ^2$, we show the existence of a measure of maximal dimension and identify the conditions under which a measure of full dimension exists.
Hausdorff dimension and asymptotic cycles
Mark
Pollicott
3241-3252
Abstract: We carry out a multifractal analysis for the asymptotic cycles for compact Riemann surfaces of genus $g \geq 2$. This describes the set of unit tangent vectors for which the associated orbit has a given asymptotic cycle in homology.
Constructions preserving Hilbert space uniform embeddability of discrete groups
Marius
Dadarlat;
Erik
Guentner
3253-3275
Abstract: Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences for the Novikov conjecture. Exactness, introduced and studied extensively by Kirchberg and Wassermann, is a functional analytic property of locally compact groups. Recently it has become apparent that, as properties of countable discrete groups, uniform embeddability and exactness are closely related. We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups. In particular, we prove that it is closed under direct and free products (with and without amalgam), inductive limits and certain extensions.
Vitali covering theorem in Hilbert space
Jaroslav
Tiser
3277-3289
Abstract: It is shown that the statement of the Vitali Covering Theorem does not hold for a certain class of measures in a Hilbert space. This class contains all infinite-dimensional Gaussian measures.
On idempotents in reduced enveloping algebras
George
B.
Seligman
3291-3300
Abstract: Explicit constructions are given for idempotents that generate all projective indecomposable modules for certain finite-dimensional quotients of the universal enveloping algebra of the Lie algebra $s\ell (2)$ in odd prime characteristic. The program is put in a general context, although constructions are only carried through in the case of $s\ell (2)$.
Stability of infinite-dimensional sampled-data systems
Hartmut
Logemann;
Richard
Rebarber;
Stuart
Townley
3301-3328
Abstract: Suppose that a static-state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-and-hold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space $X$ and the control space $U$ are Hilbert spaces, the system is of the form $\dot x(t) = Ax(t) + Bu(t)$, where $A$ is the generator of a strongly continuous semigroup on $X$, and the continuous time feedback is $u(t) = Fx(t)$. The answer to the above question is known to be ``yes'' if $X$ and $U$ are finite-dimensional spaces. In the infinite-dimensional case, if $F$ is not compact, then it is easy to find counterexamples. Therefore, we restrict attention to compact feedback. We show that the answer to the above question is ``yes'', if $B$ is a bounded operator from $U$ into $X$. Moreover, if $B$ is unbounded, we show that the answer ``yes'' remains correct, provided that the semigroup generated by $A$ is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.
Approximations for Gabor and wavelet frames
Deguang
Han
3329-3342
Abstract: Let $\psi$ be a frame vector under the action of a collection of unitary operators $\mathcal U$. Motivated by the recent work of Frank, Paulsen and Tiballi and some application aspects of Gabor and wavelet frames, we consider the existence and uniqueness of the best approximation by normalized tight frame vectors. We prove that for any frame induced by a projective unitary representation for a countable discrete group, the best normalized tight frame (NTF) approximation exists and is unique. Therefore it applies to Gabor frames (including Gabor frames for subspaces) and frames induced by translation groups. Similar results hold for semi-orthogonal wavelet frames.
Mean curvature flow, orbits, moment maps
Tommaso
Pacini
3343-3357
Abstract: Given a compact Riemannian manifold together with a group of isometries, we discuss MCF of the orbits and some applications: e.g., finding minimal orbits. We then specialize to Lagrangian orbits in Kaehler manifolds. In particular, in the Kaehler-Einstein case we find a relation between MCF and moment maps which, for example, proves that the minimal Lagrangian orbits are isolated.
Singular integrals on symmetric spaces, II
Alexandru
D.
Ionescu
3359-3378
Abstract: We extend some of our earlier results on boundedness of singular integrals on symmetric spaces of real rank one to arbitrary noncompact symmetric spaces. Our main theorem is a transference principle for operators defined by $\mathbb{K}$-bi-invariant kernels with certain large scale cancellation properties. As an application we prove $L^p$ boundedness of operators defined by Fourier multipliers that satisfy singular differential inequalities of the Hörmander-Michlin type.
West's problem on equivariant hyperspaces and Banach-Mazur compacta
Sergey
Antonyan
3379-3404
Abstract: Let $G$ be a compact Lie group, $X$ a metric $G$-space, and $\exp X$ the hyperspace of all nonempty compact subsets of $X$ endowed with the Hausdorff metric topology and with the induced action of $G$. We prove that the following three assertions are equivalent: (a) $X$ is locally continuum-connected (resp., connected and locally continuum-connected); (b) $\exp X$ is a $G$-ANR (resp., a $G$-AR); (c) $(\exp X)/G$ is an ANR (resp., an AR). This is applied to show that $(\exp G)/G$ is an ANR (resp., an AR) for each compact (resp., connected) Lie group $G$. If $G$ is a finite group, then $(\exp X)/G$ is a Hilbert cube whenever $X$ is a nondegenerate Peano continuum. Let $L(n)$ be the hyperspace of all centrally symmetric, compact, convex bodies $A\subset \mathbb{R}^n$, $n\ge 2$, for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing $A$, and let $L_0(n)$ be the complement of the unique $O(n)$-fixed point in $L(n)$. We prove that: (1) for each closed subgroup $H\subset O(n)$, $L_0(n)/H$ is a Hilbert cube manifold; (2) for each closed subgroup $K\subset O(n)$ acting non-transitively on $S^{n-1}$, the $K$-orbit space $L(n)/K$ and the $K$-fixed point set $L(n)[K]$ are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta $L(n)/O(n)$ and prove that $L_0(n)$ and $(\exp S^{n-1})\setminus\{S^{n-1}\}$ have the same $O(n)$-homotopy type.
Local solvability and hypoellipticity for semilinear anisotropic partial differential equations
Giuseppe
de Donno;
Alessandro
Oliaro
3405-3432
Abstract: We propose a unified approach, based on methods from microlocal analysis, for characterizing the local solvability and hypoellipticity in $C^\infty$ and Gevrey $G^\sigma$ classes of $2$-variable semilinear anisotropic partial differential operators with multiple characteristics. The conditions imposed on the lower-order terms of the linear part of the operator are optimal.