Sign rank versus VC dimension

This work studies the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is {three}. For $d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. For $d >2$, similar but slightly less accurate statements hold. {The lower bounds improve over previous ones by Ben-David et al., and the upper bounds are novel.} The lower bounds are obtained by probabilistic constructions, using a theorem of Warren in real algebraic topology. The upper bounds are obtained using a result of Welzl about spanning trees with low stabbing number, and using the moment curve. The upper bound technique is also used to: (i) provide estimates on the number of classes of a given VC dimension, and the number of maximum classes of a given VC dimension -- answering a question of Frankl from '89, and (ii) design an efficient algorithm that provides an $O(N/\log(N))$ multiplicative approximation for the sign rank. We also observe a general connection between sign rank and spectral gaps which is based on Forster's argument. Consider the $N \times N$ adjacency matrix of a $\Delta$ regular graph with a second eigenvalue of absolute value $\lambda$ and $\Delta \leq N/2$. We show that the sign rank of the signed version of this matrix is at least $\Delta/\lambda$. We use this connection to prove the existence of a maximum class $C\subseteq\{\pm 1\}^N$ with VC dimension $2$ and sign rank $\tilde{\Theta}(N^{1/2})$. This answers a question of Ben-David et al.~regarding the sign rank of large VC classes. We also describe limitations of this approach, in the spirit of the Alon-Boppana theorem. We further describe connections to communication complexity, geometry, learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC dimension". Additional results in this version: (i) Estimates on the number of maximum VC classes (answering a question of Frankl from '89). (ii) Estimates on the sign rank of large VC classes (answering a question of Ben-David et al. from '03). (iii) A discussion on the computational complexity of computing the sign-ran

Exoplanet HD 209458b : Evaporation strengthened

Following re-analysis of Hubble Space Telescope observations of primary transits of the extrasolar planet HD209458b at Lyman-alpha, Ben-Jaffel (2007, BJ007) claims that no sign of evaporation is observed. Here we show that, in fact, this new analysis is consistent with the one of Vidal-Madjar et al. (2003, VM003) and supports the detection of evaporation. The apparent disagreement is mainly due to the disparate wavelength ranges that are used to derive the transit absorption depth. VM003 derives a (15+/-4)% absorption depth during transit over the core of the stellar Lyman-alpha line (from -130 km/s to +100 km/s), and this result agrees with the (8.9+/-2.1)% absorption depth reported by BJ007 from a slightly expanded dataset but over a larger wavelength range (+/-200 km/s). These measurements agree also with the (5+/-2)% absorption reported by Vidal-Madjar et al. (2004) over the whole Lyman-alpha line from independent, lower-resolution data. We show that stellar Lyman-alpha variability is unlikely to significantly affect those detections. The HI atoms must necessarily have velocities above the escape velocities and/or be outside the Roche lobe, given the lobe shape and orientation. Absorption by HI in HD209458b's atmosphere has thus been detected with different datasets, and now with independent analyses. All these results strengthen the concept of evaporating hot-Jupiters, as well as the modelization of this phenomenon.Comment: To be published in ApJ

Art as a laboratory – Guy Ben-Ary’s work

Szykowna Sylwia, Art as a laboratory – Guy Ben-Ary’s work. “Images” vol. XXV, no. 34. Poznań 2019. Adam Mickiewicz University Press. Pp. 115–124. ISSN 1731-450X. DOI 10.14746/i.2019.34.07. The present paper deals with the work of an Israeli artist, Guy Ben-Ary. His work is a prime example of artistic practice in the field of bio art. Bio art provokes critical thinking about the place and role of people in today’s world. The main purpose of the article is to describe changes in contemporary artistic practices within the framework of art as a laboratory, the aim of which is to study reality.  Szykowna Sylwia, Art as a laboratory – Guy Ben-Ary’s work. “Images” vol. XXV, no. 34. Poznań 2019. Adam Mickiewicz University Press. Pp. 115–124. ISSN 1731-450X. DOI 10.14746/i.2019.34.07. The present paper deals with the work of an Israeli artist, Guy Ben-Ary. His work is a prime example of artistic practice in the field of bio art. Bio art provokes critical thinking about the place and role of people in today’s world. The main purpose of the article is to describe changes in contemporary artistic practices within the framework of art as a laboratory, the aim of which is to study reality

The Hodge theory of Soergel bimodules

We prove Soergel's conjecture on the characters of indecomposable Soergel bimodules. We deduce that Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems. Using results of Soergel one may deduce an algebraic proof of the Kazhdan-Lusztig conjecture.Comment: 44 pages. v2: many minor changes, final versio

The 6-vertex model and deformations of the Weyl character formula

We use statistical mechanics -- variants of the six-vertex model in the plane studied by means of the Yang-Baxter equation -- to give new deformations of Weyl's character formula for classical groups of Cartan type B, C, and D, and a character formula of Proctor for type BC. In each case, the corresponding Boltzmann weights are associated to the free fermion point of the six-vertex model. These deformations add to the earlier known examples in types A and C by Tokuyama and Hamel-King, respectively. A special case for classical types recovers deformations of the Weyl denominator formula due to Okada.Comment: v2: renamed the last family of models and showed their connection to character formulae for groups of type BC; addressed some issues in the proof of Lemma 6.2; updated abstrac