Severi degrees on toric surfaces

Ardila and Block used tropical results of Brugalle and Mikhalkin to count nodal curves on a certain family of toric surfaces. Building on a linearity result of the first author, we revisit their work in the context of the Goettsche-Yau-Zaslow formula for counting nodal curves on arbitrary smooth surfaces, addressing several questions they raised by proving stronger versions of their main theorems. In the process, we give new combinatorial formulas for the coefficients arising in the Goettsche-Yau-Zaslow formulas, and give correction terms arising from rational double points in the relevant family of toric surfaces.Comment: 35 pages, 1 figure, 1 tabl

Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata

In this paper, we develop Leray-Serre-type spectral sequences to compute the intersection homology of the regular neighborhood and deleted regular neighborhood of the bottom stratum of a stratified PL-pseudomanifold. The E^2 terms of the spectral sequences are given by the homology of the bottom stratum with a local coefficient system whose stalks consist of the intersection homology modules of the link of this stratum (or the cone on this link). In the course of this program, we establish the properties of stratified fibrations over unfiltered base spaces and of their mapping cylinders. We also prove a folk theorem concerning the stratum-preserving homotopy invariance of intersection homology.Comment: To appear in Topology and Its Applications; see also http://www.math.yale.edu/~friedman

Linear spectral statistics of eigenvectors of anisotropic sample covariance matrices

Consider sample covariance matrices of the form $Q:=\Sigma^{1/2} X X^* \Sigma^{1/2}$, where $X=(x_{ij})$ is an $n\times N$ random matrix whose entries are independent random variables with mean zero and variance $N^{-1}$, and $\Sigma$ is a deterministic positive-definite matrix. We study the limiting behavior of the eigenvectors of $Q$ through the so-called eigenvector empirical spectral distribution (VESD) $F_{\mathbf u}$, which is an alternate form of empirical spectral distribution with weights given by $|\mathbf u^\top \xi_k|^2$, where $\mathbf u$ is any deterministic unit vector and $\xi_k$ are the eigenvectors of $Q$. We prove a functional central limit theorem for the linear spectral statistics of $F_{\mathbf u}$, indexed by functions with H{\"o}lder continuous derivatives. We show that the linear spectral statistics converge to universal Gaussian processes both on global scales of order 1, and on local scales that are much smaller than 1 and much larger than the typical eigenvalues spacing $N^{-1}$. Moreover, we give explicit expressions for the means and covariance functions of the Gaussian processes, where the exact dependence on $\Sigma$ and $\mathbf u$ allows for more flexibility in the applications of VESD in statistical estimations of sample covariance matrices.Comment: 60 pages, 2 figure

Realization spaces of arrangements of convex bodies

We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial complexity of the bodies and the topological complexity of their realization space. First, we show that every combinatorial type is realizable and its realization space is contractible under mild assumptions. Second, we prove a universality theorem that says the restriction of the realization space to arrangements polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set

Optimal Watermark Embedding and Detection Strategies Under Limited Detection Resources

An information-theoretic approach is proposed to watermark embedding and detection under limited detector resources. First, we consider the attack-free scenario under which asymptotically optimal decision regions in the Neyman-Pearson sense are proposed, along with the optimal embedding rule. Later, we explore the case of zero-mean i.i.d. Gaussian covertext distribution with unknown variance under the attack-free scenario. For this case, we propose a lower bound on the exponential decay rate of the false-negative probability and prove that the optimal embedding and detecting strategy is superior to the customary linear, additive embedding strategy in the exponential sense. Finally, these results are extended to the case of memoryless attacks and general worst case attacks. Optimal decision regions and embedding rules are offered, and the worst attack channel is identified.Comment: 36 pages, 5 figures. Revised version. Submitted to IEEE Transactions on Information Theor

Some Remarks on the Notion of Bohr Chaos and Invariant Measures

The notion of Bohr chaos was introduced in [3, 4]. We answer a question raised in [3] of whether a non uniquely ergodic minimal system of positive topological entropy can be Bohr chaotic. We also prove that all systems with the specification property are Bohr chaotic, and by this line of thought give an independent proof (and stengthening) of theorem 1 of [3] for the case of invertible systems. In addition, we present an obstruction for Bohr chaos: a system with fewer than a continuum of ergodic invariant probability measures cannot be Bohr chaotic