Hausdorff measures, dimensions and mutual singularity
Manav
Das
4249-4268
Abstract: Let $(X,d)$ be a metric space. For a probability measure $\mu$ on a subset $E$of $X$ and a Vitali cover $V$ of $E$, we introduce the notion of a $b_{\mu}$-Vitali subcover $V_{\mu}$, and compare the Hausdorff measures of $E$with respect to these two collections. As an application, we consider graph directed self-similar measures $\mu$ and $\nu$ in $\mathbb{R}^{d}$ satisfying the open set condition. Using the notion of pointwise local dimension of $\mu$with respect to $\nu$, we show how the Hausdorff dimension of some general multifractal sets may be computed using an appropriate stochastic process. As another application, we show that Olsen's multifractal Hausdorff measures are mutually singular.
First countable, countably compact spaces and the continuum hypothesis
Todd
Eisworth;
Peter
Nyikos
4269-4299
Abstract: We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of $\omega_1$ with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah's iteration theorems appearing in Chapters V and VIII of Proper and improper forcing (1998), as well as Eisworth and Roitman's (1999) iteration theorem. We close the paper with a ZFC example (constructed using Shelah's club-guessing sequences) that shows similar results do not hold for closed pre-images of $\omega_2$.
Classification problems in continuum theory
Riccardo
Camerlo;
Udayan
B.
Darji;
Alberto
Marcone
4301-4328
Abstract: We study several natural classes and relations occurring in continuum theory from the viewpoint of descriptive set theory and infinite combinatorics. We provide useful characterizations for the relation of likeness among dendrites and show that it is a bqo with countably many equivalence classes. For dendrites with finitely many branch points the homeomorphism and quasi-homeomorphism classes coincide, and the minimal quasi-homeomorphism classes among dendrites with infinitely many branch points are identified. In contrast, we prove that the homeomorphism relation between dendrites is $S_\infty$-universal. It is shown that the classes of trees and graphs are both $\mathrm{D}_{2}({{\boldsymbol \Sigma_{3}^{0}}})$-complete, the class of dendrites is ${{\boldsymbol\Pi_{3}^{0}}}$-complete, and the class of all continua homeomorphic to a graph or dendrite with finitely many branch points is ${{\boldsymbol\Pi_{3}^{0}}}$-complete. We also show that if $G$ is a nondegenerate finitely triangulable continuum, then the class of $G$-like continua is ${\boldsymbol\Pi_{2}^{0}}$-complete.
Towers of 2-covers of hyperelliptic curves
Nils
Bruin;
E.
Victor
Flynn
4329-4347
Abstract: In this article, we give a way of constructing an unramified Galois-cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian $2$-group. The construction does not make use of the embedding of the curve in its Jacobian, and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplication-by-$2$ map of an embedding of the curve in its Jacobian. We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. In particular the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2-covers of hyperelliptic curves. As an application, we determine the rational points on the genus $2$ curve arising from the question of whether the sum of the first $n$ fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds.
Uniform properties of rigid subanalytic sets
Leonard
Lipshitz;
Zachary
Robinson
4349-4377
Abstract: In the context of rigid analytic spaces over a non-Archimedean valued field, a rigid subanalytic set is a Boolean combination of images of rigid analytic maps. We give an analytic quantifier elimination theorem for (complete) algebraically closed valued fields that is independent of the field; in particular, the analytic quantifier elimination is independent of the valued field's characteristic, residue field and value group, in close analogy to the algebraic case. This provides uniformity results about rigid subanalytic sets. We obtain uniform versions of smooth stratification for subanalytic sets and the \Lojasiewicz inequalities, as well as a unfiorm description of the closure of a rigid semianalytic set.
On the power series coefficients of certain quotients of Eisenstein series
Bruce
C.
Berndt;
Paul
R.
Bialek
4379-4412
Abstract: In their last joint paper, Hardy and Ramanujan examined the coefficients of modular forms with a simple pole in a fundamental region. In particular, they focused on the reciprocal of the Eisenstein series $E_6(\tau)$. In letters written to Hardy from nursing homes, Ramanujan stated without proof several more results of this sort. The purpose of this paper is to prove most of these claims.
Functional equations and their related operads
Vahagn
Minasian
4413-4443
Abstract: Using functional equations, we define functors that generalize standard examples from calculus of one variable. Examples of such functors are discussed, and their Taylor towers are computed. We also show that these functors factor through objects enriched over the homology of little $n$-cubes operads and discuss the relationship between functors defined via functional equations and operads. In addition, we compute the differentials of the forgetful functor from the category of $n$-Poisson algebras in terms of the homology of configuration spaces.
Riemannian nilmanifolds and the trace formula
Ruth
Gornet
4445-4479
Abstract: This paper examines the clean intersection hypothesis required for the expression of the wave invariants, computed from the asymptotic expansion of the classical wave trace by Duistermaat and Guillemin. The main result of this paper is the calculation of a necessary and sufficient condition for an arbitrary Riemannian two-step nilmanifold to satisfy the clean intersection hypothesis. This condition is stated in terms of metric Lie algebra data. We use the calculation to show that generic two-step nilmanifolds satisfy the clean intersection hypothesis. In contrast, we also show that the family of two-step nilmanifolds that fail the clean intersection hypothesis are dense in the family of two-step nilmanifolds. Finally, we give examples of nilmanifolds that fail the clean intersection hypothesis.
On the absolutely continuous spectrum of one-dimensional quasi-periodic Schrödinger operators in the adiabatic limit
Alexander
Fedotov;
Frédéric
Klopp
4481-4516
Abstract: In this paper we study the spectral properties of families of quasi-periodic Schrödinger operators on the real line in the adiabatic limit in the case when the adiabatic iso-energetic curves are extended along the position direction. We prove that, in energy intervals where this is the case, most of the spectrum is purely absolutely continuous in the adiabatic limit, and that the associated generalized eigenfunctions are Bloch-Floquet solutions. RÉSUMÉ. Cet article est consacré à l'étude du spectre de certaines familles d'équations de Schrödinger quasi-périodiques sur l'axe réel lorsque les variétés iso-énergetiques adiabatiques sont étendues dans la direction des positions. Nous démontrons que, dans un intervalle d'énergie où ceci est le cas, le spectre est dans sa majeure partie purement absolument continu et que les fonctions propres généralisées correspondantes sont des fonctions de Bloch-Floquet.
On the mod $p$ cohomology of $BPU(p)$
Ales
Vavpetic;
Antonio
Viruel
4517-4532
Abstract: We study the mod $p$ cohomology of the classifying space of the projective unitary group $PU(p)$. We first prove that conjectures due to J.F. Adams and Kono and Yagita (1993) about the structure of the mod $p$ cohomology of the classifying space of connected compact Lie groups hold in the case of $PU(p)$. Finally, we prove that the classifying space of the projective unitary group $PU(p)$ is determined by its mod $p$ cohomology as an unstable algebra over the Steenrod algebra for $p>3$, completing previous work by Dwyer, Miller and Wilkerson (1992) and Broto and Viruel (1998) for the cases $p=2,3$.
Free and semi-inert cell attachments
Peter
Bubenik
4533-4553
Abstract: Let $Y$ be the space obtained by attaching a finite-type wedge of cells to a simply-connected, finite-type CW-complex. We introduce the free and semi-inert conditions on the attaching map which broadly generalize the previously-studied inert condition. Under these conditions we determine $H_*(\Omega Y;R)$ as an $R$-module and as an $R$-algebra, respectively. Under a further condition we show that $H_*(\Omega Y;R)$ is generated by Hurewicz images. As an example we study an infinite family of spaces constructed using only semi-inert cell attachments.
Aleksandrov surfaces and hyperbolicity
Byung-Geun
Oh
4555-4577
Abstract: Aleksandrov surfaces are a generalization of two-dimensional Riemannian manifolds, and it is known that every open simply-connected Aleksandrov surface is conformally equivalent either to the unit disc (hyperbolic case) or to the plane (parabolic case). We prove a criterion for hyperbolicity of Aleksandrov surfaces which have nice tilings and where negative curvature dominates. We then apply this to generalize a result of Nevanlinna and give a partial answer for his conjecture about line complexes.
Canonical varieties with no canonical axiomatisation
Ian
Hodkinson;
Yde
Venema
4579-4605
Abstract: We give a simple example of a variety $\mathbf{V}$ of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety $\mathbf{RRA}$ of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many non-canonical sentences. Using probabilistic methods of Erdos, we construct an infinite sequence $G_0,G_1,\ldots$ of finite graphs with arbitrarily large chromatic number, such that each $G_n$ is a bounded morphic image of $G_{n+1}$ and has no odd cycles of length at most $n$. The inverse limit of the sequence is a graph with no odd cycles, and hence is 2-colourable. It follows that a modal algebra (respectively, a relation algebra) obtained from the $G_n$ satisfies arbitrarily many axioms from a certain axiomatisation of $\mathbf{V} (\mathbf{RRA})$, while its canonical extension satisfies only a bounded number of them. First-order compactness will now establish that $\mathbf{V} (\mathbf{RRA})$ has no canonical axiomatisation. A variant of this argument shows that all axiomatisations of these classes have infinitely many non-canonical sentences.
Nonautonomous Kato classes of measures and Feynman-Kac propagators
Archil
Gulisashvili
4607-4632
Abstract: The behavior of the Feynman-Kac propagator corresponding to a time-dependent measure on $R^n$ is studied. We prove the boundedness of the propagator in various function spaces on $R^n$, and obtain a uniqueness theorem for an exponentially bounded distributional solution to a nonautonomous heat equation.
Irregular hypergeometric systems associated with a singular monomial curve
María
Isabel
Hartillo-Hermoso
4633-4646
Abstract: In this paper we study irregular hypergeometric systems defined by one row. Specifically, we calculate slopes of such systems. In the case of reduced semigroups, we generalize the case studied by Castro and Takayama. In all the cases we find that there always exists a slope with respect to a hyperplane of this system. Only in the case of an irregular system defined by a $1\times 2$integer matrix we might need a change of coordinates to study slopes at infinity. In the other cases slopes are always at the origin, defined with respect to a hyperplane. We also compute all the $L$-characteristic varieties of the system, so we have a section of the Gröbner fan of the module defined by the hypergeometric system.
Unipotent flat bundles and Higgs bundles over compact Kähler manifolds
Silke
Lekaus
4647-4659
Abstract: We characterize those unipotent representations of the fundamental group $\pi_1(X,x)$ of a compact Kähler manifold $X$, which correspond to a Higgs bundle whose underlying Higgs field is equal to zero. The characterization is parallel to the one that R. Hain gave of those unipotent representations of $\pi_1(X,x)$ that can be realized as the monodromy of a flat connection on the holomorphically trivial vector bundle. We see that Hain's result and ours follow from a careful study of Simpson's correspondence between Higgs bundles and local systems.
Outer factorizations in one and several variables
Michael
A.
Dritschel;
Hugo
J.
Woerdeman
4661-4679
Abstract: A multivariate version of Rosenblum's Fejér-Riesz theorem on outer factorization of trigonometric polynomials with operator coefficients is considered. Due to a simplification of the proof of the single variable case, new necessary and sufficient conditions for the multivariable outer factorization problem are formulated and proved.