183 research outputs found
On multiplicative functions which are small on average
Let be a completely multiplicative function that assumes values inside
the unit disc. We show that if \sum_{n2, for
some , then either is small on average or pretends to be
for some .Comment: 51 pages. Slightly strengthened Theorem 1.2 and simplified its
statement. Removed Remark 1.3. Other minor changes and corrections. To appear
in Geom. Funct. Ana
The pair correlation of zeros of DirichletL-functions and primes in arithmetic progressions
Ecology and Evolution of Drosophila ambochila, A Rare Picture-Winged Species Endemic to the Wai'anae Range of O'ahu, Hawaiian Islands
The rare O'ahu picture-winged fly Drosophila ambochila Hardy
& Kaneshiro is endemic to two windward ravines in the Wai'anae Mountains
that harbor its host plant. Drosophila ambochila is an ecological specialist that
breeds on Pisonia stems and trunks in an intermediate stage of decay. By providing
field-collected females with suitable substrate material, we have been
able to observe the oviposition behavior of this species in the laboratory and
obtain F 1 larvae. In nature, females oviposit each batch of mature eggs ("'4050)
in a single cluster, by repeatedly inserting their long ovipositor into the
same crack or beetle hole in the decaying Pisonia bark. Ovipositor, ovary, and
egg morphology are characteristic of bark-breeding Hawaiian Drosophila, but
SEM studies revealed a distinctive chorionic ultrastructure for the eggs of this
species. Larval salivary chromosome analyses indicated that the O'ahu D. ambochila
is most closely related to D. alsophila from the island of Hawai'i and
have helped to resolve the phylogenetic relationships among six of the nine species
belonging to the vesciseta subgroup of the glabriapex species group
The size of Selmer groups for the congruent number problem, II
The oldest problem in the theory of elliptic curves is to determine which positive integers D can be the common difference of a three term arithmetic progres-sion of squares of rational numbers. Such integers D are known as congruent numbers. Equivalently one may ask which elliptic curve
Bandlimited approximations to the truncated Gaussian and applications
In this paper we extend the theory of optimal approximations of functions in the -metric by entire functions of prescribed
exponential type (bandlimited functions). We solve this problem for the
truncated and the odd Gaussians using explicit integral representations and
fine properties of truncated theta functions obtained via the maximum principle
for the heat operator. As applications, we recover most of the previously known
examples in the literature and further extend the class of truncated and odd
functions for which this extremal problem can be solved, by integration on the
free parameter and the use of tempered distribution arguments. This is the
counterpart of the work \cite{CLV}, where the case of even functions is
treated.Comment: to appear in Const. Appro
Improved model identification for non-linear systems using a random subsampling and multifold modelling (RSMM) approach
In non-linear system identification, the available observed data are conventionally partitioned into two parts: the training data that are used for model identification and the test data that are used for model performance testing. This sort of 'hold-out' or 'split-sample' data partitioning method is convenient and the associated model identification procedure is in general easy to implement. The resultant model obtained from such a once-partitioned single training dataset, however, may occasionally lack robustness and generalisation to represent future unseen data, because the performance of the identified model may be highly dependent on how the data partition is made. To overcome the drawback of the hold-out data partitioning method, this study presents a new random subsampling and multifold modelling (RSMM) approach to produce less biased or preferably unbiased models. The basic idea and the associated procedure are as follows. First, generate K training datasets (and also K validation datasets), using a K-fold random subsampling method. Secondly, detect significant model terms and identify a common model structure that fits all the K datasets using a new proposed common model selection approach, called the multiple orthogonal search algorithm. Finally, estimate and refine the model parameters for the identified common-structured model using a multifold parameter estimation method. The proposed method can produce robust models with better generalisation performance
Some extremal functions in Fourier analysis, III
We obtain the best approximation in , by entire functions of
exponential type, for a class of even functions that includes
, where , and , where . We also give periodic versions of these results where the
approximating functions are trigonometric polynomials of bounded degree.Comment: 26 pages. Submitte
Close to Uniform Prime Number Generation With Fewer Random Bits
In this paper, we analyze several variants of a simple method for generating
prime numbers with fewer random bits. To generate a prime less than ,
the basic idea is to fix a constant , pick a
uniformly random coprime to , and choose of the form ,
where only is updated if the primality test fails. We prove that variants
of this approach provide prime generation algorithms requiring few random bits
and whose output distribution is close to uniform, under less and less
expensive assumptions: first a relatively strong conjecture by H.L. Montgomery,
made precise by Friedlander and Granville; then the Extended Riemann
Hypothesis; and finally fully unconditionally using the
Barban-Davenport-Halberstam theorem. We argue that this approach has a number
of desirable properties compared to previous algorithms.Comment: Full version of ICALP 2014 paper. Alternate version of IACR ePrint
Report 2011/48
On some problems involving Hardy's function
Some problems involving the classical Hardy function are discussed. In particular we discuss the odd moments of
, the distribution of its positive and negative values and the primitive
of . Some analogous problems for the mean square of are
also discussed.Comment: 15 page
Short proofs of some extremal results III
We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short
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