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The theory is applied to the case of cubic hypersurfaces, which is the one most relevant to special geometry, obtaining the solution of the two classification problems and the description of the corresponding homogeneous special K?hler manifolds.
      
Well known wonderfulG-varieties are those of rank zero, namely the generalized flag varietiesG/P, those of rank one, classified in [A], and certain complete symmetric varieties described in [DP] such as the famous space of complete conics.
      
[M]) for the symplectic case and Berndt and Vanhecke [BV1] for the rank-one case.
      
In the present article we propose a more detailed proof of this fact than the one given by Varagnolo and Vasserot.
      
The first one is a conjecture of Ian Hughes which states that iff1, ..., fn are primary invariants of a finite linear groupG, then the least common multiple of the degrees of thefi is a multiple of the exponent ofG.
      
More precisely, each orbit of the above action intersects one componentX ofQ in a finite number of points and the action of PGL4 restricted on each componentX is equivalent to the action of a finite groupGX onX which can be explicitely computed.
      
We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.
      
In the course of the proof we show that one can reduce the study of generating semi-invariants to the case when the quiver has no oriented paths of length greater than one.
      
In the special case whenFn is the projective spaceRPn, one also obtains the upper bound.
      
Thus we confirm a conjecture of Brundan for one more case.
      
Strong multiplicity one theorems for affine Hecke algebras of type A
      
We describe two opposite direction functors between Kronecker webs and integrable bihamiltonian structures: one is left inverse to the other.
      
We also prove that iff(X,Y) is a polynomial overC with one place at infinity, then for every λ∈C,f-λ also has one place at infinity.
      
Recently one of the authors obtained a classification of simple infinite dimensional Lie superalgebras of vector fields which extends the well known classification of E.
      
This map corresponds geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple group G.
      
It is known [M4] that K?-orbits S and G?-orbits S' on a complex flag manifold are in one-to-one correspondence by the condition that S ∩ S' is nonempty and compact.
      
Then one can associate to every vector bundle $E$ of rank $r$ over $X$ a vector bundle $E_\rho$ with fibre $V$.
      
We construct a one-parameter family of flat connections ?κ on h with values in any finite-dimensional g-module V and simple poles on the root hyperplanes.
      
Can one factor the classical adjoint of a generic matrix
      
For char k = 0, it is shown that if n is odd, adj(X) is not the product of two noninvertible n × n matrices over k[xij], while for n even, only one special sort of factorization occurs.
      
 

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