Degeneracy Implies Non-abelian Statistics

A non-abelian anyon can only occur in the presence of ground state degeneracy in the plane. It is conceivable that for some strange anyon with quantum dimension $>1$ that the resulting representations of all $n$-strand braid groups $B_n$ are overall phases, even though the ground state manifolds for $n$ such anyons in the plane are in general Hilbert spaces of dimensions $>1$. We observe that degeneracy is all that is needed: for an anyon with quantum dimension $>1$ the non-abelian statistics cannot all be overall phases on the degeneracy ground state manifold. Therefore, degeneracy implies non-abelian statistics, which justifies defining a non-abelian anyon as one with quantum dimension $>1$. Since non-abelian statistics presumes degeneracy, degeneracy is more fundamental than non-abelian statistics.Comment: State the main result as a theorem and add several clarification

The N-eigenvalue Problem and Two Applications

We consider the classification problem for compact Lie groups $G\subset U(n)$ which are generated by a single conjugacy class with a fixed number $N$ of distinct eigenvalues. We give an explicit classification when N=3, and apply this to extract information about Galois representations and braid group representations.Comment: 30 pages. version 3: many typos fixed, section 6 substantially reorganized. To appear in Int. Math. Res. No

Nanoporous silica-based protocells at multiple scales for designs of life and nanomedicine.

Various protocell models have been constructed de novo with the bottom-up approach. Here we describe a silica-based protocell composed of a nanoporous amorphous silica core encapsulated within a lipid bilayer built by self-assembly that provides for independent definition of cell interior and the surface membrane. In this review, we will first describe the essential features of this architecture and then summarize the current development of silica-based protocells at both micro- and nanoscale with diverse functionalities. As the structure of the silica is relatively static, silica-core protocells do not have the ability to change shape, but their interior structure provides a highly crowded and, in some cases, authentic scaffold upon which biomolecular components and systems could be reconstituted. In basic research, the larger protocells based on precise silica replicas of cells could be developed into geometrically realistic bioreactor platforms to enable cellular functions like coupled biochemical reactions, while in translational research smaller protocells based on mesoporous silica nanoparticles are being developed for targeted nanomedicine. Ultimately we see two different motivations for protocell research and development: (1) to emulate life in order to understand it; and (2) to use biomimicry to engineer desired cellular interactions

Rank-finiteness for modular categories

We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular category $\mathcal{C}$ with $N=ord(T)$, the order of the modular $T$-matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension $D^2$ in the Dedekind domain $\mathbb{Z}[e^{\frac{2\pi i}{N}}]$ is identical to that of $N$.Comment: 25 pages (last version). Version 2: removed weakly integral rank 6 and integral rank 7 section, improved rank 5 classification up to monoidal equivalence. Version 3: removed rank 5 classification (note title change)--this will be published separately. Significantly improved expositio

Multiply Robust Causal Inference with Double Negative Control Adjustment for Categorical Unmeasured Confounding

Unmeasured confounding is a threat to causal inference in observational studies. In recent years, use of negative controls to mitigate unmeasured confounding has gained increasing recognition and popularity. Negative controls have a longstanding tradition in laboratory sciences and epidemiology to rule out non-causal explanations, although they have been used primarily for bias detection. Recently, Miao et al. (2018) have described sufficient conditions under which a pair of negative control exposure and outcome variables can be used to nonparametrically identify the average treatment effect (ATE) from observational data subject to uncontrolled confounding. In this paper, we establish nonparametric identification of the ATE under weaker conditions in the case of categorical unmeasured confounding and negative control variables. We also provide a general semiparametric framework for obtaining inferences about the ATE while leveraging information about a possibly large number of measured covariates. In particular, we derive the semiparametric efficiency bound in the nonparametric model, and we propose multiply robust and locally efficient estimators when nonparametric estimation may not be feasible. We assess the finite sample performance of our methods in extensive simulation studies. Finally, we illustrate our methods with an application to the postlicensure surveillance of vaccine safety among children

Sociability is decreased following deletion of the _trpc4_ gene

Shyness and social anxiety are predominant features of some psychiatric disorders including autism, schizophrenia, anxiety and depression. Understanding the cellular and molecular determinants of sociability may reveal therapeutic approaches to treat individuals with these disorders and improve their quality of life. Previous experiments from our laboratory have identified selective mRNA and protein expression of a nonselective cation channel known as the canonical transient receptor potential channel 4 (TRPC4s) in brain regions implicated in emotional regulation and anxiety. TRPC4 is highly expressed in the corticolimbic regions of the mammalian brain. We hypothesized that robust corticolimbic expression of TRPC4 may regulate the brain’s response to emotion and anxiety resulting in changes in social interaction. Here we test trpc4 gene knockout rats in a model of social anxiety/interaction. We found that the Trpc4 knockout animals spent significantly less time exploring a juvenile intruder rat compared to their wild-type counterparts and Sprague-Dawley (SD) rats. Furthermore, Trpc4 wild-type (Fisher 344) rats explored the juvenile significantly less than the SD rats. These findings indicate that the _trpc4_ gene plays a role in modulating cellular excitability in specific regions of the brain associated sociality and/or anxiety