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

Happiness is from the soul: The nature and origins of our happiness concept

What is happiness? Is happiness about feeling good or about being good? Across five studies, we explored the nature and origins of our happiness concept developmentally and crosslinguistically. We found that surprisingly, children as young as age 4 viewed morally bad people as less happy than morally good people, even if the characters all have positive subjective states (Study 1). Moral character did not affect attributions of physical traits (Study 2), and was more powerfully weighted than subjective states in attributions of happiness (Study 3). Moreover, moral character but not intelligence influenced children and adults’ happiness attributions (Study 4). Finally, Chinese people responded similarly when attributing happiness with two words, despite one (“Gao Xing”) being substantially more descriptive than the other (“Kuai Le”) (Study 5). Therefore, we found that moral judgment plays a relatively unique role in happiness attributions, which is surprisingly early emerging and largely independent of linguistic and cultural influences, and thus likely reflects a fundamental cognitive feature of the mind

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