Geometric variational problems of statistical mechanics and of combinatorics

We present the geometric solutions of the various extremal problems of statistical mechanics and combinatorics. Together with the Wulff construction, which predicts the shape of the crystals, we discuss the construction which exhibits the shape of a typical Young diagram and of a typical skyscraper.Comment: 10 page

Deep Convolutional Ranking for Multilabel Image Annotation

Multilabel image annotation is one of the most important challenges in computer vision with many real-world applications. While existing work usually use conventional visual features for multilabel annotation, features based on Deep Neural Networks have shown potential to significantly boost performance. In this work, we propose to leverage the advantage of such features and analyze key components that lead to better performances. Specifically, we show that a significant performance gain could be obtained by combining convolutional architectures with approximate top-$k$ ranking objectives, as thye naturally fit the multilabel tagging problem. Our experiments on the NUS-WIDE dataset outperforms the conventional visual features by about 10%, obtaining the best reported performance in the literature

Subdifferentials of Performance Functions and Calculus of Coderivatives of Set-Valued Mappings

The paper contains calculus rules for coderivatives of compositions, sums and intersections of set-valued mappings. The types of coderivatives considered correspond to Dini-Hadamard and limiting Dini-Hadamard subdifferentials in Gˆateaux differentiable spaces, Fréchet and limiting Fréchet subdifferentials in Asplund spaces and approximate subdifferentials in arbitrary Banach spaces. The key element of the unified approach to obtaining various calculus rules for various types of derivatives presented in the paper are simple formulas for subdifferentials of marginal, or performance functions

Critical region for droplet formation in the two-dimensional Ising model

We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size $L^2$, inverse temperature \beta>\betac and overall magnetization conditioned to take the value \mstar L^2-2\mstar v_L, where \betac^{-1} is the critical temperature, \mstar=\mstar(\beta) is the spontaneous magnetization and $v_L$ is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when $v_L^{3/2} L^{-2}$ tends to a definite limit. Specifically, we identify a dimensionless parameter $\Delta$, proportional to this limit, a non-trivial critical value \Deltac and a function $\lambda_\Delta$ such that the following holds: For \Delta<\Deltac, there are no droplets beyond $\log L$ scale, while for \Delta>\Deltac, there is a single, Wulff-shaped droplet containing a fraction \lambda_\Delta\ge\lamc=2/3 of the magnetization deficit and there are no other droplets beyond the scale of $\log L$. Moreover, $\lambda_\Delta$ and $\Delta$ are related via a universal equation that apparently is independent of the details of the system.Comment: 48 pages, 2 figures, version to appear in Commun. Math. Phy

Pion form factor in QCD sum rules, local duality approach, and O(A_2) fractional analytic perturbation theory

Using the results on the electromagnetic pion Form Factor (FF) obtained in the $O(\alpha_s)$ QCD sum rules with non-local condensates \cite{BPS09} we determine the effective continuum threshold for the local duality approach. Then we apply it to construct the $O(\alpha_s^2)$ estimation of the pion FF in the framework of the fractional analytic perturbation theory.Comment: 4 pages, 2 figures, invited talk at the 3rd Joint International Hadron Structure'09 Conference, Tatranska Strba (Slovak Republic), Aug. 30--Sept. 3, 200