2,452 research outputs found
On an extension of the Landau-Gonek formula
We prove an extension of the Landau-Gonek formula. As an application we
recover unconditionally some of the consequences of a pair correlation estimate
that previously was known under the Riemann hypothesis. As one corollary we
prove that at least two-thirds of the zeros of the zeta function are simple
under a zero density hypothesis, which is weaker than the Riemann hypothesis.
The results in this paper can be viewed as pair correlation estimates
independent of the Riemann hypothesis.Comment: 17 pages, comments are welcom
The distribution of -tuples of reduced residues
In 1940 Paul Erd\H{o}s made a conjecture about the distribution of reduced
residues. Here we study the distribution of -tuple of reduced residues.Comment: To appear in Mathematik
On binary and quadratic divisor problems
We study the shifted convolution sum of the divisor function and some other
arithmetic functions.Comment: 24 page
Pivotal Objects in Monoidal Categories and Their Hopf Monads
An object in a monoidal category is called pivotal if its
left dual and right dual objects are isomorphic. Given such an object and a
choice of dual , we construct the category , of objects
which intertwine with and in a compatible manner. We show that this
category lifts the monoidal structure of and the closed structure
of , when is closed. If has suitable
colimits we show that is monadic and thereby construct a
family of Hopf monads on arbitrary closed monoidal categories . We
also introduce the pivotal cover of a monoidal category and extend our work to
arbitrary pivotal diagrams.Comment: 34 pages, 6 figures, minor corrections and addition of Section 6.2
and Example 5.8. Comments are welcome
Distribution of squares modulo a composite number
In this paper we study the distribution of squares modulo a square-free
number . We also look at inverse questions for the large sieve in the
distribution aspect and we make improvements on existing results on the
distribution of -tuples of reduced residues.Comment: 22 pages, to appear in IMR
A passive Stokes flow rectifier for Newtonian fluids
Non-linear effects of the Navier-Stokes equations disappear under the Stokes
regime of Newtonian fluid flows disallowing the fluid flow rectification. Here
we show mathematically and experimentally that passive flow rectification of
Newtonian fluids is obtainable under the Stokes regime of both compressible and
incompressible flows by introducing nonlinearity into the otherwise linear
Stokes equations. Asymmetric flow resistances arise in shallow nozzle/diffuser
microchannels with deformable ceiling, in which the fluid flow is governed by a
non-linear coupled fluid-solid mechanics equation. Fluid flow rectification has
been demonstrated for low-Reynolds-number flows (Re ~ O(0.001)-O(1)) of common
Newtonian fluids such as air, water, and alcohol. This mechanism can pave the
way for regulating the low-Reynolds-number fluid flows with potential
applications in precise low-flow-rate micropumps, drug delivery systems, etc
First-Order Adaptive Sample Size Methods to Reduce Complexity of Empirical Risk Minimization
This paper studies empirical risk minimization (ERM) problems for large-scale
datasets and incorporates the idea of adaptive sample size methods to improve
the guaranteed convergence bounds for first-order stochastic and deterministic
methods. In contrast to traditional methods that attempt to solve the ERM
problem corresponding to the full dataset directly, adaptive sample size
schemes start with a small number of samples and solve the corresponding ERM
problem to its statistical accuracy. The sample size is then grown
geometrically -- e.g., scaling by a factor of two -- and use the solution of
the previous ERM as a warm start for the new ERM. Theoretical analyses show
that the use of adaptive sample size methods reduces the overall computational
cost of achieving the statistical accuracy of the whole dataset for a broad
range of deterministic and stochastic first-order methods. The gains are
specific to the choice of method. When particularized to, e.g., accelerated
gradient descent and stochastic variance reduce gradient, the computational
cost advantage is a logarithm of the number of training samples. Numerical
experiments on various datasets confirm theoretical claims and showcase the
gains of using the proposed adaptive sample size scheme
Adaptive Newton Method for Empirical Risk Minimization to Statistical Accuracy
We consider empirical risk minimization for large-scale datasets. We
introduce Ada Newton as an adaptive algorithm that uses Newton's method with
adaptive sample sizes. The main idea of Ada Newton is to increase the size of
the training set by a factor larger than one in a way that the minimization
variable for the current training set is in the local neighborhood of the
optimal argument of the next training set. This allows to exploit the quadratic
convergence property of Newton's method and reach the statistical accuracy of
each training set with only one iteration of Newton's method. We show
theoretically and empirically that Ada Newton can double the size of the
training set in each iteration to achieve the statistical accuracy of the full
training set with about two passes over the dataset
Experimental and theoretical investigation of a low-Reynolds-number flow through deformable shallow microchannels with ultra-low height-to-width aspect ratios
The emerging field of deformable microfluidics widely employed in the
Lab-on-a-Chip and MEMS communities offers an opportunity to study a relatively
under-examined physics. The main objective of this work is to provide a deeper
insight into the underlying coupled fluid-solid interactions of a
low-Reynolds-number fluid flow through a shallow deformable microchannel with
ultra-low height-to-width ratios. The fabricated deformable microchannels of
several microns in height and few millimeters in width, whose aspect ratio is
about two orders of magnitude smaller than that of the previous reports, allow
us to investigate the fluid flow characteristics spanning a variety of distinct
regimes from small wall deflections, where the deformable microchannel
resembles its corresponding rigid one, to wall deflections much larger than the
original height, where the height-independent characteristic behavior emerges.
The effects of the microchannel geometry, membrane properties, and pressure
difference across the channel are represented by a lumped variable called
flexibility parameter. Under the same pressure drop across different channels,
any difference in their geometries is reflected into the flexibility parameter
of the channels, which can potentially cause the devices to operate under
distinct regimes of the fluid-solid characteristics. For a fabricated
microchannel with given membrane properties and channel geometry, on the other
hand, a sufficiently large change in the applied pressure difference can alter
the flow-structure behavior from one characteristic regime to another. By
appropriately introducing the flexibility parameter and the dimensionless
volumetric flow rate, a master curve is found for the fluid flow through any
long and shallow deformable microchannel. A criterion is also suggested for
determining whether the coupled or decoupled fluid-solid mechanics should be
considered.Comment: 22 pages, 9 figures, submitted to Microfluidics and Nanofluidics on
March 26th, 201
Global Convergence of Online Limited Memory BFGS
Global convergence of an online (stochastic) limited memory version of the
Broyden-Fletcher- Goldfarb-Shanno (BFGS) quasi-Newton method for solving
optimization problems with stochastic objectives that arise in large scale
machine learning is established. Lower and upper bounds on the Hessian
eigenvalues of the sample functions are shown to suffice to guarantee that the
curvature approximation matrices have bounded determinants and traces, which,
in turn, permits establishing convergence to optimal arguments with probability
1. Numerical experiments on support vector machines with synthetic data
showcase reductions in convergence time relative to stochastic gradient descent
algorithms as well as reductions in storage and computation relative to other
online quasi-Newton methods. Experimental evaluation on a search engine
advertising problem corroborates that these advantages also manifest in
practical applications.Comment: 37 page
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