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 kk-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 $k$-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 $P$ in a monoidal category $\mathcal{C}$ is called pivotal if its left dual and right dual objects are isomorphic. Given such an object and a choice of dual $Q$, we construct the category $\mathcal{C}(P,Q)$, of objects which intertwine with $P$ and $Q$ in a compatible manner. We show that this category lifts the monoidal structure of $\mathcal{C}$ and the closed structure of $\mathcal{C}$, when $\mathcal{C}$ is closed. If $\mathcal{C}$ has suitable colimits we show that $\mathcal{C}(P,Q)$ is monadic and thereby construct a family of Hopf monads on arbitrary closed monoidal categories $\mathcal{C}$. 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 $q$. 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 $s$-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

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

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

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