Restriction of Stable Rank Two Vector Bundles in Arbitrary Characteristic

Let X be a smooth variety defined over an algebraically closed field of arbitrary characteristic and 𝒪 _X (H) be a very ample line bundle on X. We show that for a semistable X-bundle E of rank two, there exists an integer m depending only on Δ(E) · H ^dim(X)?2 and H ^dim(X) such that the restriction of E to a general divisor in |mH| is again semistable. As corollaries, we obtain boundedness results, and weak versions of Bogomolov's Theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.     

Stable Finiteness of Group Rings in Arbitrary Characteristic

We show that every (discrete) group ring D[G] of a free-by-amenable group G over a division ring D of arbitrary characteristic is stably finite, in the sense that one-sided inverses in all matrix rings over D[G] are two-sided. Our methods use Sylvester rank functions and the translation ring of an amenable group.     

循环群$Z_{2}$的模向量不变式

Let G be the finite cyclic group Z_2 and V be a vector space of dimension 2n with basis x_1,...,x_n,y_1,...,y_n over the field F with characteristic 2.If σ denotes a generator of G,we may assume that σ（x_i）= ayi,σ（y_i）= a～-1x_i,where a ∈ F.In this paper,we describe the explicit generator of the ring of modular vector invariants of F[V]～G.We prove that F[V]～G = F[l_i = x_i ＋ ay_i,q_i = x_iy_i,1 ≤ i ≤ n,M_I = X_I ＋ a～-I-Y_I],where I∈An = {1,2,...,n},2 ≤-I-≤ n.     

二次特征值问题中特征值和特征向量的可微性

二次特征值问题中特征值和特征向量的可微性,给出了其导数的计算公式及其算法.     

Polynomial Invariants of Certain Pseudo-Symplectic Groups Over Finite Fields of Characteristic Two

Let F _q be a finite field of characteristic two, S be a nonsingular non-alternate symmetric matrix over F _q and Ps _n (F _q , S) be the associated pseudo-symplectic group. Let Ps _n (F _q , S) act linearly on the polynomial ring F _q [x ₁,…, x _n ]. In this note, we find an explicit set of generators of the ring of invariants of Ps _n (F _q , S) for n = 2, 4 and 2ν +1. In particular, the results assert that the ring of invariants of Ps ₄(F _q , S) is not a polynomial algebra but is an example of hypersurface and the ring of invariants of Ps _2ν+1(F _q , S) is a complete intersection.     

Properties of Characteristic Frequencies of Linear Equations of Arbitrary Order

New characteristics of oscillation properties of solutions are suggested, namely, the frequencies equal to the upper or the lower time-average of the null-points, sign alteration points, or roots (their multiplicity being taken into account). We examine several definitions of the principal values of frequencies of a linear homogeneous equation; for an equation with constant coefficients, all these values coincide with the moduli of the imaginary parts of the roots of the corresponding characteristic polynomial. It is shown that for equations of an arbitrary (> 2) order, these definitions yield different values, which are, in general, unstable with respect to uniformly small, or even infinitesimal, perturbations of coefficients, and the extreme values belong to precisely the second Baire class, both in the uniform and the compact-open topologies.     

A Metric Characteristic of Minimal Surfaces on Arbitrary Carnot Groups

Mathematical Notes -     

A New Prehomogeneous Vector Space of Characteristic p

Let K be an algebraically closed field of characteristic 3. Let \Lambda_1,\Lambda_2,\Laambda_3 denote all: the fundamental dominant weights of GL(4). Then the K-dimension of the irreducible GL(4)- module V with the highest weight \Lambda_1 +\Lambda_2 is equal to 16, and it is denoted , by V(16). In this paper, the following results are proved: (1) (GL(4),\Lambda_1 +\Lambda_2 ,V(16)) is a regular irreducible prehomogeneous vector space.The degree of its irreducible relative invariant is 8, the associated character is X(g) = (det g)^6. (2) There exist only one 6-dimensional GL (4) -orbit and one 9-dimensional GL(4)- orbit in V(16). When m=7, 8 or l\leq m\leq 5, there are no m-dimensional GL(4)-orbits.     

A birational anabelian reconstruction theorem for curves over algebraically closed fields in Arbitrary Characteristic

The aim of Bogomolov’s programme is to prove birational anabelian conjectures for function fields K|k of varieties of dimension ≥ 2 over algebraically closed fields. The present article is concerned with the 1-dimensional case. While it is impossible to recover K|k from its absolute Galois group alone, we prove that it can be recovered from the pair (Aut(\(\bar K\)|k), Aut(\(\bar K\)|K)), consisting of the absolute Galois group of K and the larger group of field automorphisms fixing only the base field.     

Dickson-Mui Invariants

Let V be a finite-dimensional vector space over a finite field.The general and special linear groups, GL(V) and SL(V), acton the exterior algebras ^*V and ^*V^* of V and its dual V^*, andon the symmetric algebra S^*V. The subring of SL(V)-invariantsof ^*VS^*V was determined by Dickson and Mui. This paper describesthe equivalent, but simpler, calculation of the invariant subringof ^*VS^*V as a representation of GL(V)/SL(V). 2000 MathematicsSubject Classification 13A50.     

三维流形不变量θ(Ωp)(M3)

本文利用Jones-Kauffman模和Kirby技巧等给出了某些三维流形不变量的一般计算方法.     

The Cauchy Problem on a Characteristic Cone for the Einstein Equations in Arbitrary Dimensions

We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary space–time dimensions n + 1 ≥ 3. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.     

有限群在特征为零的任意域上的表示的一些注记

Let G be a finite group and K a field of characteristic zero.It is well-known that if K is a splitting field for G,then G is abelian if and only if any irreducible representation of G has degree 1.In this paper,we generalize this result to the case that K is an arbitrary field of characteristic zero（that is,K need not be a splitting field for G）,and we also obtain the orthogonality relations of irreducible K-characters of G in this case.Our results generalize some well-known theorems.     

一些特殊变换群的联合不变量和联合微分不变量

The theory of moving frames developed by Peter J Olver and M Fels has importaut applications to geometry,classical invariant theory.We will use this theory to classify joint invariants and joint differential invariants of some transformation groups.     

Feedback Differential Invariants

The problem of feedback equivalence for control systems is considered. An algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found.     

Near-Rings of Invariants

For a group S acting on a group G we let I(S,G) = {?: G → G ? σ(x) = ? (x), ∈ G, σ ∈ S}, the near-ring of invariants of (S,G). In this paper we investigate the transfer of information between the ideal structure of I(S,G) and the group action (S, G).