The syntomic regulator for the K-theory of fields

We define complexes analogous to Goncharov's complexes for the K-theory of discrete valuation rings of characteristic zero. Under suitable assumptions in K-theory, there is a map from the cohomology of those complexes to the K-theory of the ring under consideration. In case the ring is a localization of the ring of integers in a number field, there are no assumptions necessary. We compute the composition of our map to the K-theory with the syntomic regulator. The result can be described in terms of a p-adic polylogarithm. Finally, we apply our theory in order to compute the regulator to syntomic cohomology on Beilinson's cyclotomic elements. The result is again given by the p-adic polylogarithm. This last result is related to one by Somekawa and generalizes work by Gros.     

On the finite dimensionality of a K3 surface

For a smooth projective surface X the finite dimensionality of the Chow motive h(X), as conjectured by Kimura, has several geometric consequences. For a complex surface of general type with p _g = 0 it is equivalent to Bloch’s conjecture. The conjecture is still open for a K3 surface X which is not a Kummer surface. In this paper we prove some results on Kimura’s conjecture for complex K3 surfaces. If X has a large Picard number ρ = ρ(X), i.e. ρ = 19,20, then the motive of X is finite dimensional. If X has a non-symplectic group acting trivially on algebraic cycles then the motive of X is finite dimensional. If X has a symplectic involution i, i.e. a Nikulin involution, then the finite dimensionality of h(X) implies ${h(X) \simeq h(Y)}$ , where Y is a desingularization of the quotient surface ${X/\langle i \rangle }$ . We give several examples of K3 surfaces with a Nikulin involution such that the isomorphism ${h(X) \simeq h(Y)}$ holds, so giving some evidence to Kimura’s conjecture in this case.     

On the maximal number of du Val singularities for a K3 surface

A complex K3 surface or an algebraic K3 surface in characteristics distinct from 2 cannot have more than 16 disjoint nodal curves.

     

On the brauer group of a designularization of a normal surface 1

If ‰ : ? S → is a desingularization of the norm3 surface S, then it is shown that the induced map H² _et:(S, G_m) → H^2: _et(?, G_m) is surjective. It

follows that if all of the singularities of S are rational, the Brauer group

map B(S) → B(?) is surjective. An example is given to show that this

property fails if the dimension of S ≥ 3.     

On K1-theory of the Euclidean space

The algebraic functor K₁ of the ring of continuous functions of three variables is computed.     

On the nonlinear regulator problem

A new method is presented to obtain a state feedback form solution to an optimal control problem with nonlinear dynamics and a quadratic performance index. The method is based on solving an integral equation equivalent to the two-point boundary-value problem related to the optimization problem by applying an inverse theorem concerning analytic nonlinear operators. Compared with the previous methods, this one is straightforward, more generally applicable, and gives important additional knowledge about the solution. An example is presented to illustrate the use of the method.     

On the rational (K + 1,K + 1)-type difference equation

In this paper we investigate the boundedness character of the positive solutions of the rational difference equation of the form $$x_{n + 1} = \frac{{a_0 + \sum\nolimits_{j = 1}^k {a_j x_{n - j + 1} } }}{{b_0 + \sum\nolimits_{j = 1}^k {b_j x_{n - j + 1} } }}, n = 0,1,...$$ where k ε N, andaj,bj, j = 0,1,…, k, are nonnegative numbers such thatb ₀+∑ _j=1 ^k b _j x _n-j+1>0 for everyn ∈N ∪{0}. In passing we confirm several conjectures recently posed in the paper: E. Camouzis, G. Ladas and E. P. Quinn, On third order rational difference equations (part 6).     

On the regulator problem for linear degenerate control systems

In this paper, we study the finite-time regulator problem for linear, autonomous, degenerate systems, i.e., systems described by the differential equation with det(K)=0. Two investigations of the regulator problem are presented, which depend on the quadratic cost associated with the differential equation.This work was performed under the auspices of the National Research Council of Italy, Gruppo Nazionale per l'Analisi Funzionale e le Sue Applicazioni, Rome, Italy.     

半完全环的K1群

在一定的条件下，半完全环R的K1群可以通过R／J（R）的直和分解得到。     

On the kernel of the regulator map

By using the infinitesimal methods due to Bloch, Green, and Griffiths in [1], [4], we construct an infinitesimal form of the regulator map and verify that its kernel is ${\mathrm{\Omega }}_{\mathbb{C}/\mathbb{Q}}^1$, which suggests that Question 1.1 seems reasonable at the infinitesimal level.     

On Indecomposable Elements of K1 of a Product of Elliptic Curves

Let E₁, E₂ be elliptic curves with good reduction over a local field k of residue characteristic p. Let X be the smooth projective model of E₁ × E₂ over the ring of integers of k. We show that KerCH²(X) CH² (E₁ × E₂)) is a finite p-group, by giving a new construction of indecomposable elements of H¹ _Zar(E₁ × E₂, K₂). As an application we show that the prime to p part of the torsion subgroup of CH²(E₁ × E₂) is finite.     

On K_(1,k)-factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n,which partition the set of edges of λKm,n.In this paper,it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n,whenever k is any positive integer,is that(1) m ≤ kn,(2) n ≤ km,(3) km-n ≡ kn-m ≡ 0(mod(k2-1)) and(4) λ(km-n)(kn-m) ≡ 0(mod k(k -1)(k2 -1)(m n)).