Extensions of a result of Elekes and R\'onyai

Many problems in combinatorial geometry can be formulated in terms of curves or surfaces containing many points of a cartesian product. In 2000, Elekes and R\'onyai proved that if the graph of a polynomial contains $cn^2$ points of an $n\times n\times n$ cartesian product in $\mathbb{R}^3$, then the polynomial has the form $f(x,y)=g(k(x)+l(y))$ or $f(x,y)=g(k(x)l(y))$. They used this to prove a conjecture of Purdy which states that given two lines in $\mathbb{R}^2$ and $n$ points on each line, if the number of distinct distances between pairs of points, one on each line, is at most $cn$, then the lines are parallel or orthogonal. We extend the Elekes-R\'onyai Theorem to a less symmetric cartesian product. We also extend the Elekes-R\'onyai Theorem to one dimension higher on an $n\times n\times n\times n$ cartesian product and an asymmetric cartesian product. We give a proof of a variation of Purdy's conjecture with fewer points on one of the lines. We finish with a lower bound for our main result in one dimension higher with asymmetric cartesian product, showing that it is near-optimal.Comment: 23 page

Simultaneous Arithmetic Progressions on Algebraic Curves

A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (x_i, y_\sigma(i)), where x_i and y_i are arithmetic progressions and \sigma is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over Q. We show that 4319 is such a bound for curves over R. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the 'crossing inequality' gives a lower bound. Together these bound the length of an s.a.p. on the curve. We then use a similar method to extend the result to arbitrary real algebraic curves. Instead of considering s.a.p.'s we consider k^2/3 points in a grid. The number of crossings is bounded by Bezout's Theorem. We then give another proof using a result of Jarnik bounding the number of grid points on a convex curve. This result applies as any real algebraic curve can be broken up into convex and concave parts, the number of which depend on the degree. Lastly, these results are extended to complex algebraic curves.Comment: 11 pages, 6 figures, order of email addresses corrected 12 pages, closing remarks, a reference and an acknowledgment adde

Activity Recognition based on a Magnitude-Orientation Stream Network

The temporal component of videos provides an important clue for activity recognition, as a number of activities can be reliably recognized based on the motion information. In view of that, this work proposes a novel temporal stream for two-stream convolutional networks based on images computed from the optical flow magnitude and orientation, named Magnitude-Orientation Stream (MOS), to learn the motion in a better and richer manner. Our method applies simple nonlinear transformations on the vertical and horizontal components of the optical flow to generate input images for the temporal stream. Experimental results, carried on two well-known datasets (HMDB51 and UCF101), demonstrate that using our proposed temporal stream as input to existing neural network architectures can improve their performance for activity recognition. Results demonstrate that our temporal stream provides complementary information able to improve the classical two-stream methods, indicating the suitability of our approach to be used as a temporal video representation.Comment: 8 pages, SIBGRAPI 201

TE Wave Measurement and Modeling

In the TE wave method, microwaves are coupled into the beam-pipe and the effect of the electron cloud on these microwaves is measured. An electron cloud (EC) density can then be calculated from this measurement. There are two analysis methods currently in use. The first treats the microwaves as being transmitted from one point to another in the accelerator. The second more recent method, treats the beam-pipe as a resonant cavity. This paper will summarize the reasons for adopting the resonant TE wave analysis as well as give examples from CESRTA and DA{\Phi}NE of resonant beam-pipe. The results of bead-pull bench measurements will show some possible standing wave patterns, including a cutoff mode (evanescent) where the field decreases exponentially with distance from the drive point. We will outline other recent developments in the TE wave method including VORPAL simulations of microwave resonances, as well as the simulation of transmission in the presence of both an electron cloud and magnetic fields.Comment: Presented at ECLOUD'12: Joint INFN-CERN-EuCARD-AccNet Workshop on Electron-Cloud Effects, La Biodola, Isola d'Elba, Italy, 5-9 June 2012; CERN-2013-002, pp. 193-20

Are all Credit Default Swap databases equal?

The presence of different prices in different databases for the same securities can impair the comparability of research efforts and seriously damage the management decisions based upon such research. In this study we compare the six major sources of corporate Credit Default Swap prices: GFI, Fenics, Reuters EOD, CMA, Markit and JP Morgan, using the most liquid single name 5-year CDS of the components of the leading market indexes, iTraxx (European firms) and CDX (US firms) for the period from 2004 to 2010. We find systematic differences between the data sets implying that deviations from the common trend among prices in the different databases are not purely random but are explained by idiosyncratic factors as well as liquidity, global risk and other trading factors. The lower is the amount of transaction prices available the higher is the deviation among databases. Our results suggest that the CMA database quotes lead the price discovery process in comparison with the quotes provided by other databases. Several robustness tests confirm these results.Credit Default Swap prices, Databases, Liquidity

Lexicographic choice functions without archimedeanicity

We investigate the connection between choice functions and lexicographic probabilities, by means of the convexity axiom considered by Seidenfeld, Schervisch and Kadane (2010) but without imposing any Archimedean condition. We show that lexicographic probabilities are related to a particular type of sets of desirable gambles, and investigate the properties of the coherent choice function this induces via maximality. Finally, we show that the convexity axiom is necessary but not sufficient for a coherent choice function to be the infimum of a class of lexicographic ones

Experimental demonstration of a dual-frequency laser free from anti-phase noise

A reduction of more than 20 dB of the intensity noise at the anti-phase relaxation oscillation frequency is experimentally demonstrated in a two-polarization dual-frequency solid-state laser without any optical or electronic feedback loop. Such a behavior is inherently obtained by aligning the two orthogonally polarized oscillating modes with the crystallographic axes of a (100)-cut neodymium-doped yttrium aluminum garnet active medium. The anti-phase noise level is shown to increase as soon as one departs from this peculiar configuration, evidencing the predominant role of the nonlinear coupling constant. This experimental demonstration opens new perspectives on the design and realization of extremely low noise dual-frequency solid-state lasers

Loss of brain inter-frequency hubs in Alzheimer's disease

Alzheimer's disease (AD) causes alterations of brain network structure and function. The latter consists of connectivity changes between oscillatory processes at different frequency channels. We proposed a multi-layer network approach to analyze multiple-frequency brain networks inferred from magnetoencephalographic recordings during resting-states in AD subjects and age-matched controls. Main results showed that brain networks tend to facilitate information propagation across different frequencies, as measured by the multi-participation coefficient (MPC). However, regional connectivity in AD subjects was abnormally distributed across frequency bands as compared to controls, causing significant decreases of MPC. This effect was mainly localized in association areas and in the cingulate cortex, which acted, in the healthy group, as a true inter-frequency hub. MPC values significantly correlated with memory impairment of AD subjects, as measured by the total recall score. Most predictive regions belonged to components of the default-mode network that are typically affected by atrophy, metabolism disruption and amyloid-beta deposition. We evaluated the diagnostic power of the MPC and we showed that it led to increased classification accuracy (78.39%) and sensitivity (91.11%). These findings shed new light on the brain functional alterations underlying AD and provide analytical tools for identifying multi-frequency neural mechanisms of brain diseases.Comment: 27 pages, 6 figures, 3 tables, 3 supplementary figure