# Math RSS Feeds

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## math updates on arXiv.org

The Colored Hofstadter Butterfly for the Honeycomb Lattice. (arXiv:1403.1270v1 [math-ph])

We rely on a recent method for determining edge spectra and we use it to compute the Chern numbers for Hofstadter models on the honeycomb lattice having rational magnetic flux per unit cell. Based on the bulk-edge correspondence, the Chern number $\sigma_H$ is given as the winding number of an eigenvector of a $2 \times 2$ transfer matrix, as a function of the quasi-momentum $k \in (0,2 \pi)$. This method is computationally efficient (of order $O(n^4)$ in the resolution of the desired image). It also shows that for the honeycomb lattice the solution for $\sigma_H$ for flux $p/q$ in the $r$-th gap conforms with the Diophantine equation $r=\sigma_H\cdot p+ s\cdot q$, which determines $\sigma_H \mod q$. A window such as $\sigma_H \in(-q/2,q/2)$, or possibly shifted, provides a natural further condition for $\sigma_H$, which however turns out not to be met. Based on extensive numerical calculations, we conjecture that the solution conforms with the relaxed condition $\sigma_H\in(-q,q)$.

Total Variation Regularisation in Measurement and Image space for PET reconstruction. (arXiv:1403.1272v1 [math.NA])

The aim of this paper is to test and analyze a novel technique for image reconstruction in positron emission tomography, which is based on (total variation) regularization on both the image space and the projection space. We formulate our variational problem considering both total variation penalty terms on the image and on an idealized sinogram to be reconstructed from a given Poisson distributed noisy sinogram. We prove existence, uniqueness and stability results for the proposed model and provide some analytical insight into the structures favoured by joint regularization.

For the numerical solution of the corresponding discretized problem we employ the split Bregman algorithm and extensively test the approach in comparison to standard total variation regularization on the image. The numerical results show that an additional penalty on the sinogram performs better on reconstructing images with thin structures.

Toroidal grid minors and stretch in embedded graphs. (arXiv:1403.1273v1 [math.CO])

We investigate the {\em toroidal expanse} of an embedded graph $G$, that is, the size of the largest toroidal grid contained in $G$ as a minor. In the course of this work we introduce a new embedding density parameter, the {\em stretch} of an embedded graph $G$, and use it to bound the toroidal expanse from above and from below within a constant factor depending only on the genus and the maximum degree. We also show that these parameters are tightly related to the planar {\em crossing number} of $G$. As a consequence of our bounds, we derive an efficient constant factor approximation algorithm for the toroidal expanse and for the crossing number of a surface-embedded graph with bounded maximum degree.

Almost optimal sparsification of random geometric graphs. (arXiv:1403.1274v1 [math.PR])

Authors: Nicolas Broutin, Luc Devroye, Gabor Lugosi

A random geometric irrigation graph $\Gamma_n(r_n,\xi)$ has $n$ vertices identified by $n$ independent uniformly distributed points $X_1,\ldots,X_n$ in the unit square $[0,1]^2$. Each point $X_i$ selects $\xi_i$ neighbors at random, without replacement, among those points $X_j$ ($j\neq i$) for which $\|X_i-X_j\| < r_n$, and the selected vertices are connected to $X_i$ by an edge. The number $\xi_i$ of the neighbors is an integer-valued random variable, chosen independently with identical distribution for each $X_i$ such that $\xi_i$ satisfies $1\le \xi_i \le \kappa$ for a constant $\kappa>1$. We prove that when $r_n = \gamma_n \sqrt{\log n/n}$ for $\gamma_n \to \infty$ with $\gamma_n =o(n^{1/6}/\log^{5/6}n)$, then the random geometric irrigation graph experiences explosive percolation in the sense that when $\mathbf \xi_i=1$, then the largest connected component has size $o(n)$ but if $\mathbf \xi_i >1$, then the size of the largest connected component is with high probability $n-o(n)$. This offers a natural non-centralized sparsification of a random geometric graph that is mostly connected.

Timing Side Channels in Shared Queues. (arXiv:1403.1276v1 [cs.IT])

Authors: Xun Gong, Negar Kiyavash

When two job processes meet at a single server queue, the queueing delays of one process are often affected by the other process. This causes a timing side channel leaking the job arrival pattern of one job sender to the other. In this work, we study the timing side channel arising in a job scheduler shared by a regular user and a malicious attacker. Utilizing the Shannon mutual information as a measure of information leakage between the user and attacker, we analyze privacy behaviors of common work-conserving schedulers. We find that the attacker can always learn perfectly the user's job arrival times in a longest-queue-first (LQF) scheduler, which also occurs to a first-come-first-serve (FCFS) and round-robin (RR) scheduler when the user's job arrival rate is very low. The complete information leakage in the low-rate traffic region is proven to be reduced by half in a work-conserving version of TDMA (WC-TDMA) scheduler, which turns out to be privacy-optimal in the class of deterministic-working-conserving (det-WC) schedulers, according to a universal lower bound on information leakage we derive for all det-WC schedulers.

Dynamic sampling schemes for optimal noise learning under multiple nonsmooth constraints. (arXiv:1403.1278v1 [math.OC])

We consider the bilevel optimisation approach proposed by De Los Reyes, Sch\"onlieb (2013) for learning the optimal parameters in a Total Variation (TV) denoising model featuring for multiple noise distributions. In applications, the use of databases (dictionaries) allows an accurate estimation of the parameters, but reflects in high computational costs due to the size of the databases and to the nonsmooth nature of the PDE constraints. To overcome this computational barrier we propose an optimisation algorithm that by sampling dynamically from the set of constraints and using a quasi-Newton method, solves the problem accurately and in an efficient way.

Plancherel-Rotach Asymptotics of Second-Order Difference Equations with Linear Coefficients. (arXiv:1403.1281v1 [math.CA])

Authors: Xiang-Sheng Wang

In this paper, we provide a complete Plancherel-Rotach asymptotic analysis of polynomials that satisfy a second-order difference equation with linear coefficients. According to the signs of the parameters, we classify the difference equations into six cases and derive explicit asymptotic formulas of the polynomials in the outer and oscillatory regions, respectively. It is remarkable that the zero distributions of the polynomials may locate on the imaginary line or even on a sideways Y-shape curve in some cases.

The Picard integral formulation of weighted essentially non-oscillatory schemes. (arXiv:1403.1282v1 [math.NA])

Numerical weighted essentially non-oscillatory (WENO) schemes have historically been formulated as a method of lines (MOL) or Taylor (Lax-Wendroff) method. In the MOL viewpoint, the partial differential equation (PDE) is treated as a large system of ordinary differential equations, to which an appropriate time-integrator is applied. In contrast, Lax-Wendroff discretizations immediately convert Taylor series in time to discrete spatial derivatives. We propose a Picard integral formulation that introduces new possibilities for combining space and time derivatives. In particular, we present a new class of methods by applying the WENO reconstruction procedure to the so-called "time-averaged" fluxes in place of the semi-discrete flux that would arise from a typical MOL formulation. Given that the classical WENO reconstruction coupled with forward Euler time stepping conserves mass, the Picard integral formulation is automatically conservative under any temporal discretization, and therefore lends itself to a multitude of options for further investigation, including Taylor, Runge-Kutta and linear-multistep discretizations. At present, our focus is on the discretization of the time-averaged fluxes with Taylor methods. We describe this new vantage by combining it with classical finite difference WENO schemes and apply it to hyperbolic conservation laws in one- and two-dimensions. Its effectiveness is demonstrated in a series of test cases, and the results are in agreement with current state of the art methods.

## Mathematics News -- ScienceDaily

Are you smarter than a 5-year-old? Preschoolers can do algebra
Millions of high school and college algebra students are united in a shared agony over solving for x and y, and for those to whom the answers don't come easily, it gets worse: Most preschoolers and kindergarteners can do some algebra before even entering a math class. A new study finds that most preschoolers and kindergarteners, or children between 4 and 6, can do basic algebra naturally.
Classroom focus on social, emotional skills can lead to academic gains, study shows
Classroom programs designed to improve elementary school students' social and emotional skills can also increase reading and math achievement, even if academic improvement is not a direct goal of the skills building, according to a study. The benefit holds true for students across a range of socio-economic backgrounds.
To teach scientific reproducibility, start young
In the wake of retraction scandals and studies showing reproducibility rates as low as 10 percent for peer-reviewed articles, the scientific community has focused attention on ways to improve transparency and duplication. A team of math and statistics professors has proposed a way to address one root of that problem: teach and emphasize reproducibility to aspiring scientists, using software that makes the concept feel logical rather than cumbersome.
The nature of color: New formula to calculate hue improves accuracy of color analysis
Color is crucial in ecological studies, playing an important role in studies of flower and fruit development, responses to heat/drought stress, and plant–pollinator communication. But, measuring color variation is difficult, and available formulas sometimes give misleading results. An improved formula to calculate hue (one of three variables characterizing color) has now been developed.
Math anxiety factors into understanding genetically modified food messages
People who feel intimidated by math may be less able to understand messages about genetically modified foods and other health-related information, according to researchers.
Probing the edge of chaos: How do variable physical characteristics behave at the point preceding onset of chaos?
The edge of chaos -- right before chaos sets in -- is a unique place. It is found in many dynamical systems that cross the boundary between a well-behaved dynamics and a chaotic one. Now, physicists have shown that the distribution -- or frequency of occurrence -- of the variables constituting the physical characteristics of such systems at the edge of chaos has a very different shape than previously reported distributions. This could help us better understand natural phenomena with a chaotic nature.
Optimizing custody is child's play for physicists
Ensuring that parents in recomposed families see their children regularly is a complex network problem, according to a new study. The lead researcher set out to resolve one of his real-life problems: finding a suitable weekend for both partners in his recomposed family to see all their children at the same time. He then joined forces with a mathematician and a complex systems expert. The answer they came up with is that such an agreement is not possible, in general.
Forest model predicts canopy competition: Airborne lasers help researchers understand tree growth
Scientists use measurements from airborne lasers to gauge changes in the height of trees in the forest. Tree height tells them things like how much carbon is being stored. But what accounts for height changes over time -- vertical growth or overtopping by a taller tree? A new statistical model helps researchers figure out what's really happening on the ground.