Boolean functions: noise stability, non-interactive correlation distillation, and mutual information

Let $T_{\epsilon}$ be the noise operator acting on Boolean functions $f:\{0, 1\}^n\to \{0, 1\}$, where $\epsilon\in[0, 1/2]$ is the noise parameter. Given $\alpha>1$ and fixed mean $\mathbb{E} f$, which Boolean function $f$ has the largest $\alpha$-th moment $\mathbb{E}(T_\epsilon f)^\alpha$? This question has close connections with noise stability of Boolean functions, the problem of non-interactive correlation distillation, and Courtade-Kumar's conjecture on the most informative Boolean function. In this paper, we characterize maximizers in some extremal settings, such as low noise ($\epsilon=\epsilon(n)$ is close to 0), high noise ($\epsilon=\epsilon(n)$ is close to 1/2), as well as when $\alpha=\alpha(n)$ is large. Analogous results are also established in more general contexts, such as Boolean functions defined on discrete torus $(\mathbb{Z}/p\mathbb{Z})^n$ and the problem of noise stability in a tree model.Comment: Corrections of some inaccuracie

Boolean Functions: Noise Stability, Non-interactive Correlation Distillation, and Mutual Information

© 1963-2012 IEEE. Let T be the noise operator acting on Boolean functions f:{0,1nto 0, 1 , where in [0, 1/2] is the noise parameter. Given α >1 and fixed mean E f , which Boolean function f has the largest α -th moment E(Tf)α ? This question has close connections with noise stability of Boolean functions, the problem of non-interactive correlation distillation, and Courtade-Kumar's conjecture on the most informative Boolean function. In this paper, we characterize maximizers in some extremal settings, such as low noise (=(n) close to 0), high noise (=(n) close to 1/2), as well as when α =α (n) is large. Analogous results are also established in more general contexts, such as Boolean functions defined on discrete torus (Z/p Z)n and the problem of noise stability in a tree model

Complicate dynamical properties of a discrete slow-fast predator-prey model with ratio-dependent functional response

Abstract Using a semidiscretization method, we derive in this paper a discrete slow-fast predator-prey system with ratio-dependent functional response. First of all, a detailed study for the local stability of fixed points of the system is obtained by invoking an important lemma. In addition, by utilizing the center manifold theorem and the bifurcation theory some sufficient conditions are obtained for the transcritical bifurcation and Neimark-Sacker bifurcation of this system to occur. Finally, with the use of Matlab software, numerical simulations are carried out to illustrate the corresponding theoretical results and reveal some new dynamics of the system. Our results clearly demonstrate that the system is very sensitive to its fast time scale parameter variable

Research progress of corneal lymphangiogenesis in ocular diseases

Corneal lymphangiogenesis plays a crucial role in ocular diseases. Normally, the cornea lacks blood vessels and lymph vessels, which are essential for maintaining transparency and function of cornea. However, certain diseases or injuries will prompt angiogenesis and lymphangiogenesis in cornea, thus disrupting the structure and function of cornea. Although various drugs targeting corneal angiogenesis have been applied in clinical practice, there is still a gap in medications targeting corneal lymphangiogenesis. Therefore, this review will introduce the factors related to corneal lymphangiogenesis, introduce related ocular diseases, and analyze the current treatment status, which will provide more options and possibilities for the treatment of lymphangiogenesis in ocular diseases and provide guidance for future research and drug development