On representations of finite type

Representations of the (infinite) canonical anticommutation relations and the associated operator algebra, the fermion algebra, are studied. A "coupling constant" (in (0,1]) is defined for primary states of "finite type" of that algebra. Primary, faithful states of finite type with arbitrary coupling are constructed and classified. Their physical significance for quantum thermodynamical systems at high temperatures is discussed. The scope of this study is broadened to include a large class of operator algebras sharing some of the structural properties of the fermion algebra.     

Selection response in finite populations

Formulae were derived to predict genetic response under various selection schemes assuming an infinitesimal model. Account was taken of genetic drift, gametic (linkage) disequilibrium (Bulmer effect), inbreeding depression, common environmental variance, and both initial segregating variance within families (sigma AW02) and mutational (sigma M2) variance. The cumulative response to selection until generation t(CRt) can be approximated as [equation: see text] where Ne is the effective population size, sigma AW infinity 2 = Ne sigma M2 is the genetic variance within families at the steady state (or one-half the genic variance, which is unaffected by selection), and D is the inbreeding depression per unit of inbreeding. R0 is the selection response at generation 0 assuming preselection so that the linkage disequilibrium effect has stabilized. beta is the derivative of the logarithm of the asymptotic response with respect to the logarithm of the within-family genetic variance, i.e., their relative rate of change. R0 is the major determinant of the short term selection response, but sigma Me2 Ne and beta are also important for the long term. A selection method of high accuracy using family information gives a small Ne and will lead to a larger response in the short term and a smaller response in the long term, utilizing mutation less efficiently.     

有限伪反射群的二维不变式

设F是特征数为0的域,V是F上的n维向量空间,G是作用在n维向量空间V上的有限伪反射群,F[V*]G是由n个代数无关的齐次不变式f01,f2…,fn在F上生成的多项式代数.在有限伪反射群的一般不变式理论的基础上,求出了G的二维不变式环F[2V*]G的一组基本不变式,f1(x1,x2,…,xn),f2(x1,x2,…,xn),…,fn(x1,x2,…,xn),f1(y1,y2,…,yn),f2(y1,y2,…,yn),…,fn(yl,y2,…,yn),这里F[2V*]=F[x1,x2,…,xn;y1,y2,…yn].并给出了F[2V*]G的基本不变式和有限伪反射群G之间的关系.