Representations of twisted Yangians associated with skew Young diagrams

Let $G_M$ be either the orthogonal group $O_M$ or the symplectic group $Sp_M$ over the complex field; in the latter case the non-negative integer $M$ has to be even. Classically, the irreducible polynomial representations of the group $G_M$ are labeled by partitions $\mu=(\mu_{1},\mu_{2},\,\ldots)$ such that $\mu^{\prime}_1+\mu^{\prime}_2\le M$ in the case $G_M=O_M$, or $2\mu^{\prime}_1\le M$ in the case $G_M=Sp_M$. Here $\mu^{\prime}=(\mu^{\prime}_{1},\mu^{\prime}_{2},\,\ldots)$ is the partition conjugate to $\mu$. Let $W_\mu$ be the irreducible polynomial representation of the group $G_M$ corresponding to $\mu$. Regard $G_N\times G_M$ as a subgroup of $G_{N+M}$. Then take any irreducible polynomial representation $W_\lambda$ of the group $G_{N+M}$. The vector space $W_{\lambda}(\mu)={\rm Hom}_{\,G_M}( W_\mu, W_\lambda)$ comes with a natural action of the group $G_N$. Put $n=\lambda_1-\mu_1+\lambda_2-\mu_2+\ldots\,$. In this article, for any standard Young tableau $\varOmega$ of skew shape $\lm$ we give a realization of $W_{\lambda}(\mu)$ as a subspace in the $n$-fold tensor product $(\mathbb{C}^N)^{\bigotimes n}$, compatible with the action of the group $G_N$. This subspace is determined as the image of a certain linear operator $F_\varOmega (M)$ on $(\mathbb{C}^N)^{\bigotimes n}$, given by an explicit formula. When $M=0$ and $W_{\lambda}(\mu)=W_\lambda$ is an irreducible representation of the group $G_N$, we recover the classical realization of $W_\lambda$ as a subspace in the space of all traceless tensors in $(\mathbb{C}^N)^{\bigotimes n}$. Then the operator $F_\varOmega\(0)$ may be regarded as the analogue for $G_N$ of the Young symmetrizer, corresponding to the standard tableau $\varOmega$ of shape $\lambda$. This symmetrizer is a certain linear operator on $\CNn$$(\mathbb{C}^N)^{\bigotimes n} $ with the image equivalent to the irreducible polynomial representation of the complex general linear group $GL_N$, corresponding to the partition $\lambda$. Even in the case $M=0$, our formula for the operator $F_\varOmega(M)$ is new. Our results are applications of the representation theory of the twisted Yangian, corresponding to the subgroup $G_N$ of $GL_N$. This twisted Yangian is a certain one-sided coideal subalgebra of the Yangian corresponding to $GL_N$. In particular, $F_\varOmega(M)$ is an intertwining operator between certain representations of the twisted Yangian in $(\mathbb{C}^N)^{\bigotimes n}$.     

Logarithmic derivatives of solutions to linear differential equations

Given an ordinary differential field of characteristic zero, it is known that if and satisfy linear differential equations with coefficients in , then is algebraic over . We present a new short proof of this fact using Gröbner basis techniques and give a direct method for finding a polynomial over that satisfies. Moreover, we provide explicit degree bounds and extend the result to fields with positive characteristic. Finally, we give an application of our method to a class of nonlinear differential equations.

     

Oxygen plasma and high pressure H<Subscript>2</Subscript>O vapor heat treatments used to fabricate polycrystalline silicon thin film transistors

Oxygen plasma and high pressure H₂O vapor heat treatment were applied to fabrication of n-channel polycrystalline silicon thin film transistors (poly-Si TFTs). 13.56 MHz-oxygen-plasma treatment at 250 °C, 100 W for 5 min effectively reduced defect states of 25-nm-thick silicon films crystallized by 30 ns-pulsed XeCl excimer laser irradiation. 1.3×10⁶-Pa-H₂O vapor heat treatment at 260 °C for 3 h was carried out in order to improve electrical properties of SiO_x gate insulators and SiO_x/Si interfaces. A carrier mobility of 470 cm²/V s and a low threshold voltage of 1.8 V were achieved for TFTs fabricated with crystallization at 285 mJ/cm². Received: 18 November 2002 / Accepted: 25 November 2002 / Published online: 11 April 2003 RID="*" ID="*"Corresponding author. Fax: +81-42/388-7109, E-mail: tsamesim@cc.tuat.ac.jp     

On tame weakly symmetric algebras having only periodic modules

We classify (up to Morita equivalence) all tame weakly symmetric finite dimensional algebras over an algebraically closed field having simply connected Galois coverings, nonsingular Cartan matrices and the stable Auslander-Reiten quivers consisting only of tubes. In particular, we prove that these algebras have at most four simple modules.Received: 25 February 2002     

A simple proof of a theorem on quasi-Baer rings

We give a simple proof of a theorem, due to Birkenmeier, Kim and Park, which states that if $R[x, x^{-1}]$ or $R[[x, x^{-1}]]$ is a quasi-Baer ring then R is a quasi-Baer ring. Received: 8 April 2002     

Hypersurfaces of the Euclidean sphere with nonnegative Ricci curvature

In this paper we prove that a compact oriented hypersurface of a Euclidean sphere with nonnegative Ricci curvature and infinite fundamental group is isometric to an H(r)-torus with constant mean curvature. Furthermore, we generalize, whithout any hypothesis about the mean curvature, a characterization of Clifford torus due to Hasanis and Vlachos. Received: 19 March 2002     

Composite Julia sets generated by infinite polynomial arrays

A definition of a generalized filled-in Julia set generated by an infinite array of proper polynomial mappings in  is introduced. It is shown that such Julia sets depend analytically on the defining polynomial mappings.     

Blow-up and pattern formation in hyperbolic models for chemotaxis in 1-D

In this paper we study finite time blow-up of solutions of a hyperbolic model for chemotaxis. Using appropriate scaling this hyperbolic model leads to a parabolic model as studied by Othmer and Stevens (1997) and Levine and Sleeman (1997). In the latter paper, explicit solutions which blow-up in finite time were constructed. Here, we adapt their method to construct a corresponding blow-up solution of the hyperbolic model. This construction enables us to compare the blow-up times of the corresponding models. We find that the hyperbolic blow-up is always later than the parabolic blow-up. Moreover, we show that solutions of the hyperbolic problem become negative near blow-up. We calculate the zero-turning-rate time explicitly and we show that this time can be either larger or smaller than the parabolic blow-up time. The blow-up models as discussed here and elsewhere are limiting cases of more realistic models for chemotaxis. At the end of the paper we discuss the relevance to biology and exhibit numerical solutions of more realistic models.     

Precise rate in the law of iterated logarithm for <Emphasis Type="Italic">ρ</Emphasis>-mixing sequence

Let {X, X _n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set . Suppose lim _n→∞ and , where d=2, if −1<b<0 and d>2(b+1), if b≥0. It is proved that, for any b>−1,