29 research outputs found
Asymptotic formula for the moments of Minkowski question mark function in the interval [0,1]
In this paper we prove the asymptotic formula for the moments of Minkowski
question mark function, which describes the distribution of rationals in the
Farey tree. The main idea is to demonstrate that certain a variation of a
Laplace method is applicable in this problem, hence the task reduces to a
number of technical calculations.Comment: 11 pages, 1 figure (final version). Lithuanian Math. J. (to appear
Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function
This paper continues investigations on the integral transforms of the
Minkowski question mark function. In this work we finally establish the
long-sought formula for the moments, which does not explicitly involve regular
continued fractions, though it has a hidden nice interpretation in terms of
semi-regular continued fractions. The proof is self-contained and does not rely
on previous results by the author.Comment: 8 page
On Ramanujan's Q-function
This study provides a detailed analysis of a function which Knuth discovered to play a central role in the analysis of hashing with linear probing. The function, named after Knuth Q(n), is related to several of Ramanujan's investigations. It surfaces in the analysis of a variety of algorithms ans discrete probability problems including hashing, the birthday paradox, random mapping statistics, the "rho" method for integer factorization, union-find algorithms, optimum caching, and the study of memory conflicts. A process related to the complex asymptotic methods of singularity analysis and saddle point integrals permits to precisely quantify the behaviour of the Q(n) function. in this way, tight bounds are derived. They answer a question of Knuth (the art of Computer Programming, vol. 1, 1968), itself a rephrasing of earlier questions of Ramanujan in 1911-1913
Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals
We investigate the existence of the meromorphic extension of the spectral
zeta function of the Laplacian on self-similar fractals using the classical
results of Kigami and Lapidus (based on the renewal theory) and new results of
Hambly and Kajino based on the heat kernel estimates and other probabilistic
techniques. We also formulate conjectures which hold true in the examples that
have been analyzed in the existing literature
On the Lebesgue measure of Li-Yorke pairs for interval maps
We investigate the prevalence of Li-Yorke pairs for and
multimodal maps with non-flat critical points. We show that every
measurable scrambled set has zero Lebesgue measure and that all strongly
wandering sets have zero Lebesgue measure, as does the set of pairs of
asymptotic (but not asymptotically periodic) points.
If is topologically mixing and has no Cantor attractor, then typical
(w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally
admits an absolutely continuous invariant probability measure (acip), then
typical pairs have a dense orbit for . These results make use of
so-called nice neighborhoods of the critical set of general multimodal maps,
and hence uniformly expanding Markov induced maps, the existence of either is
proved in this paper as well.
For the setting where has a Cantor attractor, we present a trichotomy
explaining when the set of Li-Yorke pairs and distal pairs have positive
two-dimensional Lebesgue measure.Comment: 41 pages, 3 figure
Point sets on the sphere with small spherical cap discrepancy
In this paper we study the geometric discrepancy of explicit constructions of
uniformly distributed points on the two-dimensional unit sphere. We show that
the spherical cap discrepancy of random point sets, of spherical digital nets
and of spherical Fibonacci lattices converges with order . Such point
sets are therefore useful for numerical integration and other computational
simulations. The proof uses an area-preserving Lambert map. A detailed analysis
of the level curves and sets of the pre-images of spherical caps under this map
is given
Quasi-Monte Carlo rules for numerical integration over the unit sphere
We study numerical integration on the unit sphere using equal weight quadrature rules, where the weights are such
that constant functions are integrated exactly.
The quadrature points are constructed by lifting a -net given in the
unit square to the sphere by means of an area
preserving map. A similar approach has previously been suggested by Cui and
Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2].
We prove three results. The first one is that the construction is (almost)
optimal with respect to discrepancies based on spherical rectangles. Further we
prove that the point set is asymptotically uniformly distributed on
. And finally, we prove an upper bound on the spherical cap
-discrepancy of order (where denotes the
number of points). This slightly improves upon the bound on the spherical cap
-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm.
Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the
-nets lifted to the sphere have spherical cap
-discrepancy converging with the optimal order of
Virtual Ontogeny of Cortical Growth Preceding Mental Illness
Background: Morphology of the human cerebral cortex differs across psychiatric disorders, with neurobiology and developmental origins mostly undetermined. Deviations in the tangential growth of the cerebral cortex during pre/perinatal periods may be reflected in individual variations in cortical surface area later in life. Methods: Interregional profiles of group differences in surface area between cases and controls were generated using T1-weighted magnetic resonance imaging from 27,359 individuals including those with attention-deficit/hyperactivity disorder, autism spectrum disorder, bipolar disorder, major depressive disorder, schizophrenia, and high general psychopathology (through the Child Behavior Checklist). Similarity of interregional profiles of group differences in surface area and prenatal cell-specific gene expression was assessed. Results: Across the 11 cortical regions, group differences in cortical area for attention-deficit/hyperactivity disorder, schizophrenia, and Child Behavior Checklist were dominant in multimodal association cortices. The same interregional profiles were also associated with interregional profiles of (prenatal) gene expression specific to proliferative cells, namely radial glia and intermediate progenitor cells (greater expression, larger difference), as well as differentiated cells, namely excitatory neurons and endothelial and mural cells (greater expression, smaller difference). Finally, these cell types were implicated in known pre/perinatal risk factors for psychosis. Genes coexpressed with radial glia were enriched with genes implicated in congenital abnormalities, birth weight, hypoxia, and starvation. Genes coexpressed with endothelial and mural genes were enriched with genes associated with maternal hypertension and preterm birth. Conclusions: Our findings support a neurodevelopmental model of vulnerability to mental illness whereby prenatal risk factors acting through cell-specific processes lead to deviations from typical brain development during pregnancy
Secant and cosecant sums and Bernoulli-Nörlund polynomials
We give explicit formulæ for sums of even powers of secant and cosecant values in terms of Bernoulli numbers and central factorial numbers. © 2007 NISC Pty Ltd.Articl
On a constant arising in the analysis of bit comparisons in quickselect
A (real) constant that appears as the factor of the leading term of the average number of bit comparisons required by quickselect, and was originally given in terms of complex numbers, is expressed using real numbers alone. A further representation is derived which is converging very quickly. Methods include residue calculus and the Euler-MacLaurin summation formula. © 2008 NISC Pty Ltd.Articl