Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles

In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. $(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters $\alpha=\beta$). By exploiting the even parity of the weight, a doubling of the interval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter $\alpha=-1/2,$ and $\alpha=1/2$ and the (shifted) JUE with weights $x^{1/2}(1-x)^{\beta}$ and $x^{-1/2}(1-x)^{\beta}$ The $\sigma$ function, namely, the derivative of the log of the smallest eigenvalue distributions of the finite-$n$ LUE or the JUE, satisfies the Jimbo-Miwa-Okamoto $\sigma$ form of $P_{V}$ and $P_{VI}$, although in the shift Jacobi case, with the weight $x^{\alpha}(1-x)^{\beta},$ the $\beta$ parameter does not show up in the equation. We also obtain the asymptotic expansions for the smallest eigenvalue distributions of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, and obtained the constants in the asymptotic expansion of the gap probablities, expressed in term of the Barnes $G-$ function valuated at special point.Comment: 38 page

Painlev\'e III′' and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight

In this paper, we study the Hankel determinant generated by a singularly perturbed Gaussian weight $$w(x,t)=\mathrm{e}^{-x^{2}-\frac{t}{x^{2}}},\;\;x\in(-\infty, \infty),\;\;t>0.$$ By using the ladder operator approach associated with the orthogonal polynomials, we show that the logarithmic derivative of the Hankel determinant satisfies both a non-linear second order difference equation and a non-linear second order differential equation. The Hankel determinant also admits an integral representation involving a Painlev\'e III$'$. Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. $n\rightarrow\infty$ and $t\rightarrow 0$ such that $s=(2n+1)t$ is fixed. The asymptotic expansions of the scaled Hankel determinant for large $s$ and small $s$ are established, from which Dyson's constant appears.Comment: 22 page

SDRL: Interpretable and Data-efficient Deep Reinforcement Learning Leveraging Symbolic Planning

Deep reinforcement learning (DRL) has gained great success by learning directly from high-dimensional sensory inputs, yet is notorious for the lack of interpretability. Interpretability of the subtasks is critical in hierarchical decision-making as it increases the transparency of black-box-style DRL approach and helps the RL practitioners to understand the high-level behavior of the system better. In this paper, we introduce symbolic planning into DRL and propose a framework of Symbolic Deep Reinforcement Learning (SDRL) that can handle both high-dimensional sensory inputs and symbolic planning. The task-level interpretability is enabled by relating symbolic actions to options.This framework features a planner -- controller -- meta-controller architecture, which takes charge of subtask scheduling, data-driven subtask learning, and subtask evaluation, respectively. The three components cross-fertilize each other and eventually converge to an optimal symbolic plan along with the learned subtasks, bringing together the advantages of long-term planning capability with symbolic knowledge and end-to-end reinforcement learning directly from a high-dimensional sensory input. Experimental results validate the interpretability of subtasks, along with improved data efficiency compared with state-of-the-art approaches

THE CATALYTIC UREASE SUBUNIT UREC IS CRITICAL FOR BIFIDOBACTERIUM LONGUM UREA UTILIZATION

In the first study, we investigated the utilization of a human milk nitrogen source, urea, by Bifidobacterium. Urea accounts for ~15% in human milk, which is an abundant non-protein nitrogen (NPN). Some bifidobacteria are found to harbor urease gene clusters that potentially enable their hydrolysis of the human milk urea. However, the underlying mechanisms are still unclear. To incisively link the urease gene cluster with bifidobacterial urea utilization, chemical mutagenesis (i.e. ethyl methanesulfonate) was performed on the urease-positive Bifidobacterium longum subsp. suis UMA399. Mutants were selected on differential media and genetic lesions were identified using whole genome sequencing. A mutant that did not exhibit urease activity, or utilize urea as a primary nitrogen source, was selected for further characterization. We found that a single-point mutation was located on the urease catalytic subunit ureC gene to prompt a substitution at residue 343 from glutamic acid to lysine (E343K). Recombinantly expressed and purified mutant UreC exhibits the loss of urease function. The mutation was complemented by expressing the wild-type UreC in the mutated strain. The restoration of urease activity and urea utilization approached levels exhibited by the wild-type strain. Thus, UreC is essential for the bifidobacterial urea utilization phenotype. In the ongoing research, we are exploring the ability of Bifidobacterium to utilize cysteine, a sulfur-containing proteinogenic amino acid. Previous studies have shown most Bifidobacterium cannot grow without cysteine (cysteine auxotrophic). It will be interesting to clarify why bifidobacteria cannot synthesize cysteine and how they assimilate cysteine from the gut environment as a necessity for propagation. Thus, we first evaluated bifidobacterial strains on their ability to grow on different sole nitrogen sources as well as sulfur sources. We found that only B. boum LMG10736 was able to grow in methionine as a sole nitrogen source, the rest of the strains are all cysteine auxotroph. However, B. boum LMG10736 was not able to utilize sulfate and sulfide for its growth. We therefore proposed that the methionine degradation pathway may be silenced under the transcriptional or translational regulations