A Lie theoretic categorification of the coloured Jones polynomial

We use the machinery of categorified Jones-Wenzl projectors to construct a categorification of a type A Reshetikhin-Turaev invariant of oriented framed tangles where each strand is labelled by an arbitrary finite-dimensional representation. As a special case, we obtain a categorification of the coloured Jones polynomial of links.     

The metabolites and biotransformation pathways in vivo after oral administration of ocotillol,RT5, and PF11

Ocotillol, pseudo-ginsenoside RT5 (RT₅), and pseudo-ginsenoside F₁₁ (PF₁₁) are ocotillol-type saponins that have the same aglycone structure but with different numbers of glucose at the C-6 position. In this study, the metabolites of ocotillol, RT₅, and PF₁₁ in rat plasma, stomach, intestine, urine, and feces after oral administration were investigated by ultra-performance liquid chromatography coupled with time-of-flight mass spectrometry. The results showed that RT₅ was easily biotransformed into metabolites in vivo, whereas PF₁₁ and RT₅ were difficult to be biotransformed. Hydrogenation, dehydrogenation, dehydration, deglycosylation, deoxygenation, hydration, phosphorylation, deoxidation, glucuronidation, and reactions combining amino acid were speculated to be involved in the biotransformation of ocotillol, RT₅, and PF₁₁. Based on the structural analysis of metabolites, it was deduced that hydrogenation, dehydration, deoxidation, and reactions combining amino acid occurred on the aglycone structure, whereas deglycosylation, hydration, and phosphorylation occurred on the glycosyl chain. Further, metabolites in plasma, urine, feces, and tissues were different: First, glucuronidation products were found in urine, stomach, intestine, and feces, but not in plasma. Second, the ocotillol prototype was not identified in urine samples. Third, the RT₅ prototype was found in stomach, intestine, feces, and urine, but not in plasma.     

Minimal kernels of Dirac operators along maps

Let M be a closed spin manifold and let N be a closed manifold. For maps and Riemannian metrics g on M and h on N, we consider the Dirac operator of the twisted Dirac bundle . To this Dirac operator one can associate an index in . If M is 2‐dimensional, one gets a lower bound for the dimension of the kernel of out of this index. We investigate the question whether this lower bound is obtained for generic tupels .     

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

We provide a bound on a distance between finitely supported elements and general elements of the unit sphere of ${?}^2\left({\mathbb{N}}^1\right)$. We use this bound to estimate the Wasserstein-2 distance between random variables represented by linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure related to Nourdin–Peccati’s Malliavin–Stein method. The main application is towards the computation of quantitative rates of convergence to elements of the second Wiener chaos. In particular, we explicit these rates for non-central asymptotic of sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.     

Existence of homoclinic solutions for nonlinear second-order coupled systems

This work presents sufficient conditions for the existence of homoclinic solutions for second order coupled discontinuous systems of differential equations on the real line without the usual growth condition in the literature.The arguments apply the fixed point theory, Green's functions technique, $L^1$-Carathéodory functions, lower and upper solutions and Schauder's fixed point theorem.     

Er3+单掺、Er3+/Yb3+共掺杂Ca12Al14O32F2的 制备及上转换发光性质

利用高温固相法成功制备了Er~(3+)单掺、Er~(3+)/Yb~(3+)共掺杂Ca_(12)Al_(14)O_(32)F_2上转换发光样品。在980 nm激光激发下,Er~(3+)单掺和Er~(3+)/Yb~(3+)共掺杂样品均呈现出较强的绿光(528,549 nm)和较弱的红光(655 nm)发射,分别归因于Er~(3+)离子的~2H_(11/2),~4S_(3/2)→~4I_(15/2)和~4F_(9/2)→~4I_(15/2)能级跃迁。随着Er离子浓度的增加,单掺杂样品上转换发光强度先增大后减小,最佳掺杂浓度为0.8%。共掺杂Yb~(3+)后,Er~(3+)的发光强度明显增大。还原气氛下合成的样品上转换发光强度增大约两倍,可能和笼中阴离子基团变化有关。发光强度和激发光功率的关系表明所得上转换发射为双光子吸收过程,借助Er~(3+)-Yb~(3+)体系能级结构详细讨论了上转换发射的跃迁机制。