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Reductive group actions on affine quadrics with 1-dimensional quotient: Linearization when a linear model exists
      
Such an action is called linearizable if it is equivalent to the restriction of a linear orthogonal action in the ambient affine space of the quadric.
      
A linear model for a given action is a linear orthogonal action with the same orbit types and equivalent slice representations.
      
We prove that if a reductive group action on an affine quadric with a 1-dimensional quotient has a linear model, then the action is linearizable.
      
We find presentations for the irreducible crystallographic complex reflection groupsW whose linear part is not the complexification of a real reflection group.
      
As in the case of affine Weyl groups, they can be obtained by adding a further node to the diagram for the linear part.
      
Using these monomial bases we prove that the image of the transfer for a general linear group over a finite field is a principal ideal in the ring of invariants.
      
Methods are developed for the calssification of homogeneous Riemannian hypersurfaces and the classification of linear transitive reductive algebraic group actions on pseudo-Riemannian hypersurfaces.
      
The finite irreducible linear groups with polynomial ring of invariants
      
We prove the following result: LetG be a finite irreducible linear group.
      
This allows us to obtain a complete list of all irreducible linear groups with a polynomial ring of invariants.
      
The aim of this paper is to discuss a construction of a class of linear isomorphisms σ:S(g)→U(g) which commute with the adjoint representation.
      
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.
      
A varietyQ??19 is explicitely constructed as the union of 22 disjoint irreducible components which are either points or open subsets of linear spaces.
      
Finally the cubic surfaces of each component ofQ are studied in details by determining their stabilizers, their rational representations and whether they can be expressed as the determinant of a 3×3 matrix of linear forms.
      
We introduce and study the notion of essential dimension for linear algebraic groups defined over an algebraically closed fields of characteristic zero.
      
Invariant linear connections on homogeneous symplectic varieties
      
We find all homogeneous symplectic varieties of connected semisimple Lie groups that admit an invariant linear connection.
      
LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V).
      
In the algebraic setting we prove that if a complex linear group G acts on a Kahler manifold in a symplectically asystatic fashion, then the G-orbits are spherical.
      
 

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