Orthogonal Polynomials, Asymptotics and Heun Equations

The Painlev\'{e} equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable ``deformation factors", usually dependent on a ``time variable'' $t$. From ladder operators one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlev\'e and related functions appear as the residues of these rational functions. We will be interested in the situation when $n$, the order of the Hankel matrix and also the degree of the polynomials $P_n(x)$ orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by $P_n(x)$ are particular cases of Heun equations when $n$ is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equa\-tions. Heun equations are of considerable importance in mathematical physics and in the special cases they degenerate to the hypergeometric and confluent hypergeometric equations. In this paper we look at three type of weights: the Jacobi type, which are are supported $(0,1]$ the Laguerre type and the weights deformed by the indicator function of $(a,b)$ $\chi_{(a,b)}$ and the step function $\theta(x)$

Understanding and controlling the chemical evolution and polysulfide-blocking ability of lithium–sulfur battery membranes cast from polymers of intrinsic microporosity

By understanding how PIM-1 membrane reactivity affects its ion-transport selectivity, it’s possible to intervene to stabilize it in a Li–S battery.</p

Shaker/Kv1 potassium channel SHK-1 protects against pathogen infection and oxidative stress in C. elegans

The Shaker/Kv1 subfamily of voltage-gated potassium (K+) channels is essential for modulating membrane excitability. Their loss results in prolonged depolarization and excessive calcium influx. These channels have also been implicated in a variety of other cellular processes, but the underlying mechanisms remain poorly understood. Through comprehensive screening of K+ channel mutants in C. elegans, we discovered that shk-1 mutants are highly susceptible to bacterial pathogen infection and oxidative stress. This vulnerability is associated with reduced glycogen levels and substantial mitochondrial dysfunction, including decreased ATP production and dysregulated mitochondrial membrane potential under stress conditions. SHK-1 is predominantly expressed and functions in body wall muscle to maintain glycogen storage and mitochondrial homeostasis. RNA-sequencing data reveal that shk-1 mutants have decreased expression of a set of cation-transporting ATPases (CATP), which are crucial for maintaining electrochemical gradients. Intriguingly, overexpressing catp-3, but not other catp genes, restores the depolarization of mitochondrial membrane potential under stress and enhances stress tolerance in shk-1 mutants. This finding suggests that increased catp-3 levels may help restore electrochemical gradients disrupted by shk-1 deficiency, thereby rescuing the phenotypes observed in shk-1 mutants. Overall, our findings highlight a critical role for SHK-1 in maintaining stress tolerance by regulating glycogen storage, mitochondrial homeostasis, and gene expression. They also provide insights into how Shaker/Kv1 channels participate in a broad range of cellular processes