A Proof of the G\"ottsche-Yau-Zaslow Formula

Let S be a complex smooth projective surface and L be a line bundle on S. G\"ottsche conjectured that for every integer r, the number of r-nodal curves in |L| is a universal polynomial of four topological numbers when L is sufficiently ample. We prove G\"ottsche's conjecture using the algebraic cobordism group of line bundles on surfaces and degeneration of Hilbert schemes of points. In addition, we prove the the G\"ottsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasi-modular forms and two unknown series.Comment: 29 page

Adversarial Discriminative Domain Adaptation

Adversarial learning methods are a promising approach to training robust deep networks, and can generate complex samples across diverse domains. They also can improve recognition despite the presence of domain shift or dataset bias: several adversarial approaches to unsupervised domain adaptation have recently been introduced, which reduce the difference between the training and test domain distributions and thus improve generalization performance. Prior generative approaches show compelling visualizations, but are not optimal on discriminative tasks and can be limited to smaller shifts. Prior discriminative approaches could handle larger domain shifts, but imposed tied weights on the model and did not exploit a GAN-based loss. We first outline a novel generalized framework for adversarial adaptation, which subsumes recent state-of-the-art approaches as special cases, and we use this generalized view to better relate the prior approaches. We propose a previously unexplored instance of our general framework which combines discriminative modeling, untied weight sharing, and a GAN loss, which we call Adversarial Discriminative Domain Adaptation (ADDA). We show that ADDA is more effective yet considerably simpler than competing domain-adversarial methods, and demonstrate the promise of our approach by exceeding state-of-the-art unsupervised adaptation results on standard cross-domain digit classification tasks and a new more difficult cross-modality object classification task

Hidden regret in insurance markets: adverse and advantageous selection

We examine insurance markets with two types of customers: those who regret suboptimal decisions and those who don.t. In this setting, we characterize the equilibria under hidden information about the type of customers and hidden action. We show that both pooling and separating equilibria can exist. Furthermore, there exist separating equilibria that predict a positive correlation between the amount of insurance coverage and risk type, as in the standard economic models of adverse selection, but there also exist separating equilibria that predict a negative correlation between the amount of insurance coverage and risk type, i.e. advantageous selection. Since optimal choice of regretful customers depends on foregone alternatives, any equilibrium includes a contract which is o¤ered but not purchased

Exact Solution of an One Dimensional Deterministic Sandpile Model

Using the transfer matrix method, we give the exact solution of a deterministic sandpile model for arbitrary $N$, where $N$ is the size of a single toppling. The one- and two-point functions are given in term of the eigenvalues of an $N \times N$ transfer matrix. All the n-point functions can be found in the same way. Application of this method to a more general class of models is discussed. We also present a quantitative description of the limit cycle (attractor) as a multifractal.Comment: need RevTeX; to appear in Physical Review E January 6, (1995