Image of the Burau Representation at dd-th Roots of unity

We prove that the image of the Full braid group $B_{n+1}$ on $n+1$ strands under the Burau representation, evaluated at a primitive $d$-th root of unity is arithmetic provided $n\geq d$.Comment: To appear in Annals of Mathematics. arXiv admin note: text overlap with arXiv:1204.477

Hierarchy of the Selberg zeta functions

We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained.Comment: 14 page

Cache-Oblivious Persistence

Partial persistence is a general transformation that takes a data structure and allows queries to be executed on any past state of the structure. The cache-oblivious model is the leading model of a modern multi-level memory hierarchy.We present the first general transformation for making cache-oblivious model data structures partially persistent

Improved Implementation of Point Location in General Two-Dimensional Subdivisions

We present a major revamp of the point-location data structure for general two-dimensional subdivisions via randomized incremental construction, implemented in CGAL, the Computational Geometry Algorithms Library. We can now guarantee that the constructed directed acyclic graph G is of linear size and provides logarithmic query time. Via the construction of the Voronoi diagram for a given point set S of size n, this also enables nearest-neighbor queries in guaranteed O(log n) time. Another major innovation is the support of general unbounded subdivisions as well as subdivisions of two-dimensional parametric surfaces such as spheres, tori, cylinders. The implementation is exact, complete, and general, i.e., it can also handle non-linear subdivisions. Like the previous version, the data structure supports modifications of the subdivision, such as insertions and deletions of edges, after the initial preprocessing. A major challenge is to retain the expected O(n log n) preprocessing time while providing the above (deterministic) space and query-time guarantees. We describe an efficient preprocessing algorithm, which explicitly verifies the length L of the longest query path in O(n log n) time. However, instead of using L, our implementation is based on the depth D of G. Although we prove that the worst case ratio of D and L is Theta(n/log n), we conjecture, based on our experimental results, that this solution achieves expected O(n log n) preprocessing time.Comment: 21 page

Statistical properties of spectral fluctuations for a quantum system with infinitely many components

Extending the idea formulated in Makino {\it{et al}}[Phys.Rev.E {\bf{67}},066205], that is based on the Berry--Robnik approach [M.V. Berry and M. Robnik, J. Phys. A {\bf{17}}, 2413], we investigate the statistical properties of a two-point spectral correlation for a classically integrable quantum system. The eigenenergy sequence of this system is regarded as a superposition of infinitely many independent components in the semiclassical limit. We derive the level number variance (LNV) in the limit of infinitely many components and discuss its deviations from Poisson statistics. The slope of the limiting LNV is found to be larger than that of Poisson statistics when the individual components have a certain accumulation. This property agrees with the result from the semiclassical periodic-orbit theory that is applied to a system with degenerate torus actions[D. Biswas, M.Azam,and S.V.Lawande, Phys. Rev. A {\bf 43}, 5694].Comment: 6 figures, 10 page

Nodal domains of Maass forms I

This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the $L^2$-restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex $L^\infty$-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.Comment: To appear in GAF

Delocalization of slowly damped eigenmodes on Anosov manifolds

We look at the properties of high frequency eigenmodes for the damped wave equation on a compact manifold with an Anosov geodesic flow. We study eigenmodes with spectral parameters which are asymptotically close enough to the real axis. We prove that such modes cannot be completely localized on subsets satisfying a condition of negative topological pressure. As an application, one can deduce the existence of a "strip" of logarithmic size without eigenvalues below the real axis under this dynamical assumption on the set of undamped trajectories.Comment: 28 pages; compared with version 1, minor modifications, add two reference

Monodromy of Cyclic Coverings of the Projective Line

We show that the image of the pure braid group under the monodromy action on the homology of a cyclic covering of degree d of the projective line is an arithmetic group provided the number of branch points is sufficiently large compared to the degree.Comment: 47 pages (to appear in Inventiones Mathematicae

Expansion in SL_d(Z/qZ), q arbitrary

Let S be a fixed finite symmetric subset of SL_d(Z), and assume that it generates a Zariski-dense subgroup G. We show that the Cayley graphs of pi_q(G) with respect to the generating set pi_q(S) form a family of expanders, where pi_q is the projection map Z->Z/qZ

Level of kidney function as a risk factor for atherosclerotic cardiovascular outcomes in the community

AbstractObjectivesThe goal of this study was to determine whether the level of kidney function is an independent risk factor for atherosclerotic cardiovascular disease (ASCVD) outcomes in the Atherosclerosis Risk in Communities (ARIC) study, a prospective cohort study of subjects aged 45 to 64 years.BackgroundThe level of kidney function is now recognized as a risk factor for ASCVD outcomes in patients at high risk for ASCVD, but it remains unknown whether the level of kidney function is a risk factor for ASCVD outcomes in the community.MethodsCox proportional-hazards regression was used to evaluate the association of glomerular filtration rate (GFR) with ASCVD after adjustment for the major ASCVD risk factors in 15,350 subjects. We searched for nonlinear relationships between GFR and ASCVD.ResultsDuring a mean follow-up time of 6.2 years, 965 (6.3%) of subjects had ASCVD events. Subjects with GFR of 15 to 59 ml/min/1.73 m2(n = 444, hazard ratio 1.38 [1.02, 1.87]) and 60 to 89 ml/min/1.73 m2(n = 7,665, hazard ratio 1.16 [1.00, 1.34]) had an increased adjusted risk of ASCVD compared with subjects with GFR of 90 to 150 ml/min/1.73 m2. Each 10 ml/min/1.73 m2lower GFR was associated with an adjusted hazard ratio of 1.05 (1.02, 1.09), 1.07 (1.01, 1.12), and 1.06 (0.99, 1.13) for ASCVD, de novo ASCVD, and recurrent ASCVD, respectively. A nonlinear model did not fit the data better than a linear model.ConclusionsThe level of GFR is an independent risk factor for ASCVD and de novo ASCVD in the ARIC study