Determining the algebraic K-theory of rings of integers in number fields has been the goal of much research. In [10] D. Quillen showed that the Hurewicz map h : Q0(S ) → BGL(Z) (see 1.1 for the notation) induces an interesting map on homotopy groups from the stable homotopy groups of spheres to the algebraic K-theory of the ring Z of rational integers. Quillen observed that if ` is an odd prime...

Let X be a smooth quasi-projective variety over the algebraic closure of the rational number field. We show that the cycle map of the higher Chow group to Deligne cohomology is injective and the higher Hodge cycles are generated by the image of the cycle map as conjectured by Beilinson and Jannsen, if the cycle map to Deligne cohomology is injective and the Hodge conjecture is true for certain ...

The Virasoro master equation (VME) describes the general affine-Virasoro construction T = LJaJb + iD ∂Ja in the operator algebra of the WZW model, where L ab is the inverse inertia tensor and D is the improvement vector. In this paper, we generalize this construction to find the general (one-loop) Virasoro construction in the operator algebra of the general nonlinear sigma model. The result is ...

Let G be a connected reductive algebraic group over an algebraic closed field. We define a (surjective) map from the set of conjugacy classes in the Weyl group to the set of unipotent classes in G.

We analyse extensions $\Sigma$ of groupoids G by bundles A abelian groups. describe a pushout construction for such extensions, and use it to the extension group given groupoid bundle A. There is natural action Sigma on dual A, yielding corresponding transformation groupoid. The this map from fibre product with its Cartesian circle twist over resulting prove that full C*-algebra isomorphic $\Si...

Set $K=\mathbb{Q}(i)$ and suppose that $c\in \mathbb{Z}[i]$ is a square-free algebraic integer with $c\equiv 1 \imod{\langle16\rangle}$. Let $L(s,\chi_{c})$ denote the Hecke $L$-function associated quartic residue character modulo $c$. For $\sigma>1/2$, we prove an asymptotic distribution function $F_{\sigma}$ for values of logarithm \begin{equation*} L_c(s)= L(s,\chi_c)L(s,\overline{\chi}_{c})...

In addition, define certain lattices and subsets: L∨ = X∗(T ) = group of algebraic homomorphisms Gm → T L = Hom(X∗(T ),Z), the dual of L ∨ X(T ) = group of algebraic homomorphisms T → Gm, the weights of T . There is a canonical map from L toX(T )—to λ corresponds the multiplicative character e defined by the formula e(μ(x)) = x 〉 , which makes sense because all algebraic homomorphisms Gm → Gm a...

Let f : CP 99K CP be a rational map with algebraic and topological degrees both equal to d ≥ 2. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms T : CP → CP, the perturbed map T ◦f admits exactly two ergodic measures of maximal entropy log d, one of saddle and one of repelling type. Neither measure is supported in...

let $g$ be a simple graph‎, ‎and $g^{sigma}$‎ ‎be an oriented graph of $g$ with the orientation ‎$sigma$ and skew-adjacency matrix $s(g^{sigma})$‎. ‎the $k-$th skew spectral‎ ‎moment of $g^{sigma}$‎, ‎denoted by‎ ‎$t_k(g^{sigma})$‎, ‎is defined as $sum_{i=1}^{n}( ‎‎‎lambda_{i})^{k}$‎, ‎where $lambda_{1}‎, ‎lambda_{2},cdots‎, ‎lambda_{n}$ are the eigenvalues of $g^{sigma}$‎. ‎suppose‎ ‎$g^{sigma...

By frame of reference transformations, an input variable in one coordinate system is transformed into an output variable in a different coordinate system depending on another input variable. If the variables are represented as neural population codes, then a sigma–pi network is a natural way of coding this transformation. By multiplying two inputs it detects coactivations of input units, and by...