Uniform Definability in Propositional Dependence Logic

Both propositional dependence logic and inquisitive logic are expressively complete. As a consequence, every formula with intuitionistic disjunction or intuitionistic implication can be translated equivalently into a formula in the language of propositional dependence logic without these two connectives. We show that although such a (non-compositional) translation exists, neither intuitionistic disjunction nor intuitionistic implication is uniformly definable in propositional dependence logic

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

New Optimal Binary Sequences with Period 4p4p via Interleaving Ding-Helleseth-Lam Sequences

Binary sequences with optimal autocorrelation play important roles in radar, communication, and cryptography. Finding new binary sequences with optimal autocorrelation has been an interesting research topic in sequence design. Ding-Helleseth-Lam sequences are such a class of binary sequences of period $p$, where $p$ is an odd prime with $p\equiv 1(\bmod~4)$. The objective of this letter is to present a construction of binary sequences of period $4p$ via interleaving four suitable Ding-Helleseth-Lam sequences. This construction generates new binary sequences with optimal autocorrelation which can not be produced by earlier ones