Stochastic RR matrix for Uq(An(1))U_q(A^{(1)}_n)

We show that the quantum $R$ matrix for symmetric tensor representations of $U_q(A^{(1)}_n)$ satisfies the sum rule required for its stochastic interpretation under a suitable gauge. Its matrix elements at a special point of the spectral parameter are found to factorize into the form that naturally extends Povolotsky's local transition rate in the $q$-Hahn process for $n=1$. Based on these results we formulate new discrete and continuous time integrable Markov processes on a one-dimensional chain in terms of $n$ species of particles obeying asymmetric stochastic dynamics. Bethe ansatz eigenvalues of the Markov matrices are also given.Comment: 21 pages. Remark 9 added, Typos in Appendix A fixe

Survey on invariant quasimorphisms and stable mixed commutator length

In this survey, we review the history and recent developments of invariant quasimorphisms and stable mixed commutator length.Comment: 26 pages, 1 figure; minor revisio

The space of non-extendable quasimorphisms

For a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the space of quasimorphisms and quasi-cocycles on $N$ non-extendable to $G$. To treat this space, we establish the five-term exact sequence of cohomology relative to the bounded subcomplex. As its application, we study the spaces associated with the kernel of the (volume) flux homomorphism, the IA-automorphism group of a free group, and certain normal subgroups of Gromov hyperbolic groups. Furthermore, we employ this space to prove that the stable commutator length is equivalent to the stable mixed commutator length for certain pairs of a group and its normal subgroup.Comment: 58 pages, 1 figure. Major revision. Theorem 1.12 in v3 has been generalized to Theorem 1.2 in the current version: this new theorem treats hyperbolic mapping tori in general cases, and it serves as a leading application of our main theore

Coarse group theoretic study on stable mixed commutator length

Let $G$ be a group and $N$ a normal subgroup of $G$. We study the large scale behavior, not the exact values themselves, of the stable mixed commutator length $scl_{G,N}$ on the mixed commutator subgroup $[G,N]$; when $N=G$, $scl_{G,N}$ equals the stable commutator length $scl_G$ on the commutator subgroup $[G,G]$. For this purpose, we regard $scl_{G,N}$ not only as a function from $[G,N]$ to $\mathbb{R}_{\geq 0}$, but as a bi-invariant metric function $d^+_{scl_{G,N}}$ from $[G,N]\times [G,N]$ to $\mathbb{R}_{\geq 0}$. Our main focus is coarse group theoretic structures of $([G,N],d^+_{scl_{G,N}})$. Our preliminary result (the absolute version) connects, via the Bavard duality, $([G,N],d^+_{scl_{G,N}})$ and the quotient vector space of the space of $G$-invariant quasimorphisms on $N$ over one of such homomorphisms. In particular, we prove that the dimension of this vector space equals the asymptotic dimension of $([G,N],d^+_{scl_{G,N}})$. Our main result is the comparative version: we connect the coarse kernel, formulated by Leitner and Vigolo, of the coarse homomorphism $\iota_{G,N}\colon ([G,N],d^+_{scl_{G,N}})\to ([G,N],d^+_{scl_{G}})$; $y\mapsto y$, and a certain quotient vector space $W(G,N)$ of the space of invariant quasimorphisms. Assume that $N=[G,G]$ and that $W(G,N)$ is finite dimensional with dimension $\ell$. Then we prove that the coarse kernel of $\iota_{G,N}$ is isomorphic to $\mathbb{Z}^{\ell}$ as a coarse group. In contrast to the absolute version, the space $W(G,N)$ is finite dimensional in many cases, including all $(G,N)$ with finitely generated $G$ and nilpotent $G/N$. As an application of our result, given a group homomorphism $\varphi\colon G\to H$ between finitely generated groups, we define an $\mathbb{R}$-linear map `inside' the groups, which is dual to the naturally defined $\mathbb{R}$-linear map from $W(H,[H,H])$ to $W(G,[G,G])$ induced by $\varphi$.Comment: 69 pages, no figure. Minor revision (v2): some symbols change

Identification and targeted disruption of the mouse gene encoding ESG1 (PH34/ECAT2/DPPA5)

BACKGROUND: Embryonic stem cell-specific gene (ESG) 1, which encodes a KH-domain containing protein, is specifically expressed in early embryos, germ cells, and embryonic stem (ES) cells. Previous studies identified genomic clones containing the mouse ESG1 gene and five pseudogenes. However, their chromosomal localizations or physiological functions have not been determined. RESULTS: A Blast search of mouse genomic databases failed to locate the ESG1 gene. We identified several bacterial artificial clones containing the mouse ESG1 gene and an additional ESG1-like sequence with a similar gene structure from chromosome 9. The ESG1-like sequence contained a multiple critical mutations, indicating that it was a duplicated pseudogene. The 5' flanking region of the ESG1 gene, but not that of the pseudogene, exhibited strong enhancer and promoter activity in undifferentiated ES cells by luciferase reporter assay. To study the physiological functions of the ESG1 gene, we replaced this sequence in ES cells with a β-geo cassette by homologous recombination. Despite specific expression in early embryos and germ cells, ESG1(-/- )mice developed normally and were fertile. We also generated ESG1(-/- )ES cells both by a second independent homologous recombination and directly from blastocysts derived from heterozygous intercrosses. Northern blot and western blot analyses confirmed the absence of ESG1 in these cells. These ES cells demonstrated normal morphology, proliferation, and differentiation. CONCLUSION: The mouse ESG1 gene, together with a duplicated pseudogene, is located on chromosome 9. Despite its specific expression in pluripotent cells and germ cells, ESG1 is dispensable for self-renewal of ES cells and establishment of germcells

BATTLE: Genetically Engineered Strategies for Split-Tunable Allocation of Multiple Transgenes in the Nervous System

Elucidating fine architectures and functions of cellular and synaptic connections requires development of new flexible methods. Here, we created a concept called the “battle of transgenes,” based on which we generated strategies using genetically engineered battles of multiple recombinases. The strategies enabled split-tunable allocation of multiple transgenes. We demonstrated the versatility of these strategies and technologies in inducing strong and multi-sparse allocations of multiple transgenes. Furthermore, the combination of our transgenic strategy and expansion microscopy enabled three-dimensional high-resolution imaging of whole synaptic structures in the hippocampus with simultaneous visualizations of endogenous synaptic proteins. These strategies and technologies based on the battle of genes may accelerate the analysis of whole synaptic and cellular connections in diverse life science fields