Quasi-planar graphs have a linear number of edges

A graph is calledquasi-planar if it can be drawn in the plane so that no three of its edges are pairwise crossing. It is shown that the maximum number of edges of a quasi-planar graph withn vertices isO(n).Work on this paper by Pankaj K. Agarwal, Boris Aronov and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work on this paper by Pankaj K. Agarwal has also been supported by NSF Grant CCR-93-01259, by an Army Research Office MURI grant DAAH04-96-1-0013, by an NYI award, and by matching funds from Xerox Corporation. Work on this paper by Boris Aronov has also been supported by NSF Grant CCR-92-11541 and by a Sloan Research Fellowship. Work on this paper by János Pach, Richard Pollack, and Micha Sharir has been supported by NSF Grants CCR-91-22103 and CCR-94-24398. Work by János Pach was also supported by Grant OTKA-4269 and by a CUNY Research Award. Work by Richard Pollack was also supported by NSF Grants CCR-94-02640 and DMS-94-00293. Work by Micha Sharir was also supported by NSF Grant CCR-93-11127, by a Max-Planck Research Award, and by grants from the Israel Science Fund administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Part of the work on this paper was done during the participation of the first four authors in the Special Semester on Computational and Combinatorial Geometry organized by the Mathematical Research Institute of Tel Aviv University, Spring 1995.     

How much are increasing sets positively correlated?

Two upward directed sets of sequences of zeroes and ones are positively correlated. We provide a lower bound on the correlation, in function of how much the two sets simultaneously depend on the same coordinates.Work partially supported by an NSF grant.     

Uniform non-amenability of free Burnside groups

The aim of this paper is to show that free Burnside groups of sufficiently large odd exponent are non-amenable in a certain strong sense. More precisely their left regular representations are isolated from the trivial representation uniformly on finite generating sets. It follows that free Burnside groups are of uniform exponential growth. This answers a question of de la Harpe. This work has been partially supported by the NSF grant DMS-0605093 and by the RFBR Grant # 05-01-00892. Received: 25 April 2006     

Geometric rigidity of transcendental meromorphic functions

The classes of dynamically and geometrically tame functions meromorphic outside a small set are introduced. The Julia sets of geometrically tame functions are proven to be either geometrical circle (in ) or to have Hausdorff dimension strictly larger than 1. Vast classes of dynamically and geometrically tame functions are identified. The research of both authors was supported in part by the Polish KBN Grant No 2 PO3A 034 25, the Warsaw University of Technology Grant no 504G 11200043000 and by the NSF/PAN grant INT-0306004. The research of the second author was supported in part by the NSF Grant DMS 0400481.     

A note on a theorem of Yamamuro

Conditions are presented which are necessary for the existence of a regular fixed point of aC ¹ map.Work supported in part by NSF Grant No. MCS 77-15509.Work supported in part by ONR Grant No. N0014-75-C-0495 and NSF Grant No. Eng. 76-81058.     

Space tilings and substitutions

We generalize the study of symbolic dynamical systems of finite type and ² action, and the associated use of symbolic substitution dynamical systems, to dynamical systems with ² action. The new systems are associated with tilings of the plane. We generalize the classical technique of the matrix of a substitution to include the geometrical information needed to study tilings, and we utilize rotation invariance to eliminate discrete spectrum. As an example we prove that the pinwheel tilings have no discrete spectrum.Research supported in part by NSF Grant No. DMS-9304269 and Texas ARP Grant 003658-113     

Countably determined sets and a conjecture of C.W. Henson

Partially supported by NSF grants DMS-9208071 and DMS-9100383     

Phase transitions in Mandelbrot's percolation process in three dimensions

Summary We study the phase structure and transitions in three-dimensional Mandelbrot percolation—a process which generates random fractal sets. We establish the existence of three distinct phase transitions, and we show that two of these transitions, corresponding to percolation across the initial set by paths and sheets, are discontinuous.Work partially supported by NSF Grant: DMS 8806552Work partially supported by NSF Grant: DMS 8902152     

Mathematical problems of stochastic quantum mechanics: Harmonic analysis on phase space and quantum geometry

In this paper we review the mathematical methods and problems that are specific to the programme of stochastic quantum mechanics and quantum spacetime. The physical origin of these problems is explained, and then the mathematical models are developed. Three notions emerge as central to the programme: positive operator-valued (POV) measures on a Hilbert space, reproducing kernel Hilbert spaces, and fibre bundle formulations of quantum geometries. A close connection between the first two notions is shown to exist, which provides a natural setting for introducing a fibration on the associated overcomplete family of vectors. The introduction of group covariance leads to an extended version of harmonic analysis on phase space. It also yields a theory of induced group representations, which extends the results of Mackey on imprimitivity systems for locally compact groups to the more general case of systems of covariance. Quantum geometries emerge as fibre bundles whose base spaces are manifolds of mean stochastic locations for quantum test particles (i.e., spacetime excitons) that display a phase space structure, and whose fibres and structure groups contain, respectively, the aforementioned overcomplete families of vectors and unitary group representations of phase space systems of covariance.Work supported in part by the Natural Science and Engineering Research Council of Canada (NSERC) grants.     

A Robin formula for the Fekete-Leja transfinite diameter

We generalize the classical Robin formula to higher dimensions. Work supported in part by NSF grant DMS-0300784.     

The non-orientable genus of some metacyclic groups

We describe non-orientable, octagonal embeddings for certain 4-valent, bipartite Cayley graphs of finite metacyclic groups, and give a class of examples for which this embedding realizes the non-orientable genus of the group. This yields a construction of Cayley graphs for which is arbitrarily large, where and are the orientable genus and the non-orientable genus of the Cayley graph.Work supported in part by the Research Council of Slovenia, Yugoslavia and NSF Contract DMS-8717441.Supported by NSF Contract DMS-8601760.     

Asymptotics of R-, MD- and LAD-estimators in linear regression models with long range dependent errors

Summary This paper establishes the uniform closeness of a weighted residual empirical process to its natural estimate in the linear regression setting when the errors are Gaussian, or a function of Gaussian random variables, that are strictly stationary and long range dependent. This result is used to yield the asymptotic uniform linearity of a class of rank statistics in linear regression models with long range dependent errors. The latter result, in turn, yields the asymptotic distribution of the Jaeckel (1972) rank estimators. The paper also studies the least absolute deviation and a class of certain minimum distance estimators of regression parameters and the kernel type density estimators of the marginal error density when the errors are long range dependent.Research of this author was partly supported by the NSF grant: DMS-9102041     

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted) universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. Finally, we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups. The first author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021. The second author was supported in part by NSF Grants DMS00-70433 and DMS05-00106.     

Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle

In this work, we study surfaces over convex regions in ² which are evolving by the mean curvature flow. Here, we specify the angle of contact of the surface to the boundary cylinder. We prove that solutions converge to ones moving only by translation.Partially supported by the NSF grant no: DMS-9100383Partially supported by the NSF grant no: DMS 9108269.A01     

Percolation on random Johnson–Mehl tessellations and related models

We make use of the recent proof that the critical probability for percolation on random Voronoi tessellations is 1/2 to prove the corresponding result for random Johnson–Mehl tessellations, as well as for two-dimensional slices of higher-dimensional Voronoi tessellations. Surprisingly, the proof is a little simpler for these more complicated models. B. Bollobás’s research was supported in part by NSF grants CCR-0225610 and DMS-0505550 and ARO grant W911NF-06-1-0076. O. Riordan’s research was supported by a Royal Society Research Fellowship.     

<Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Bold">1,</Emphasis><Emphasis Type="Italic">α</Emphasis></Superscript> regularity for infinity harmonic functions in two dimensions

We propose a new method for showing C ^{1, α} regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE. LCE is supported in part by NSF Grant DMS-0500452. OS was supported in part by the Miller Institute for Basic Research in Science, Berkeley.     

Regular Resolution Lower Bounds For The Weak Pigeonhole Principle

We prove that any regular resolution proof for the weak pigeon hole principle, with n holes and any number of pigeons, is of length , (for some global constant > 0).* Research supported by NSF grant CCR-9820831, US-Israel BSF grant 98-00349, and an NSERC grant. Research supported by US-Israel BSF grant 98-00349, and NSF grant CCR-9987077.     

Stochastic partial differential equations in Hölder spaces

Summary We prove the existence and regularity of solutions to stochastic partial differential equations of parabolic Itô type in Hölder spaces under the usual sublinear growth and local Lipschitz conditions. Some examples are given to which our main theorems apply.The work of the first author was supported in part by the NSF grant DMS-91-01360     

Onset and structure of interfaces in a Kawasaki+Glauber interacting particle system

Summary We investigate the spatial structure of typical configurations of a reaction-diffusion spin system (Kawasaki+Glauber model), following the noise induced escape from an unstable spatially homogeneous state. After the escape, the system will be locally in a stationary phase, but will display a globally nontrivial spatial behavior, characterized by large clusters of the (two) different phases. The system can be spatially rescaled according to the typical linear dimension of the clusters and, on this space scale, regions of the opposite phases are separated by smooth (hyper) surfaces, called interfaces. The location of the interfaces is determined by means of the zero-level set of the trajectories of a Gaussian random field. This paper is devoted primarily to the characterization of the structure which appears on a finer scale (the hydrodynamical one) at the interface. A better understanding of the dynamics of the escape (especially in its last and nonlinear stage) leads to substantial improvements of the results in [7, 12].This research has been partly supported by NSF grant DMR 92-13424 and by a CNR fellowship     

On chordal and bilateral SLE in multiply connected domains

We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner’s equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to first recover the motion of the moduli of the domain and then, second, the curve in the interior. For random curves in the interior we show that the induced random motion on the boundary is not Markov if the domain is multiply connected, but that the random motion on the boundary together with the random motion of the moduli forms a Markov process. In the chordal case, we show that this Markov process satisfies Brownian scaling and discuss how this limits the possible conformally invariant random non-self-crossing curves. We show that the possible candidates are labeled by two functions, one homogeneous of degree zero, the other homogeneous of degree minus one, which describes the interaction of the random curve with the boundary. We show that the random curve has the locality property for appropriate choices of the interaction term. The research of the first author was supported by NSA grant H98230-04-1-0039. The research of the second author was supported by a grant from the Max-Planck-Gesellschaft.