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In the case of 4-dimensional anticommutative algebras a construction is given that links the associated cubic surface and the 27 lines on it with the structure of subalgebras of the algebra.
      
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.
      
In general the group ring of ann-valued group is not ann-Hopf algebra but it is for ann-coset group constructed from an abelian group.
      
From it, we recover Joseph and Letzter's result by a kind of "quantum duality principle".
      
It is well-known that the ring of invariants associated to a non-modular representation of a finite group is Cohen-Macaulay and hence has depth equal to the dimension of the representation.
      
The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type.
      
It was then observed independentely by Lusztig and Ginzburg-Vasserot (see [L1], [GV]) that this construction admits an affine analogue in terms of periodic flags of lattices.
      
We describe a basis forY, show that it is a polynomial algebra and describe its rank, which we compute explicitly in a number of cases.
      
For this class of Lie Groups we prove the Auslander Conjecture whenever the dimensionn of the group is odd and we show that it is false forn even andn>amp;gt;2.
      
It is also proved that the group of holomorphic automorphisms ofG?υ which commute withG? acts transitively on the set of polynomially convexG-orbits.
      
Although Spin0 () is usually reducible, we show that a Casimir element for always acts scalarly on it.
      
It is injective and its image coincides with the set ofEx-fixed points inI.
      
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.
      
It is possible to replace K? by some conjugate xK?x-1 so that the correspondence is preserved.
      
It is generated by 10 modular forms (5 of weight 1 and 5 of weight 3) and there are 20 relations (5 in weight 5 and 15 in weight 6).
      
This estimate for multiplicities is uniform, i.e., it depends not on G/H, but only on its complexity.
      
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.
      
In this paper we study Freudenburg's counterexample to the fourteenth problem of Hilbert and counterexamples derived from it.
      
It is proved that for any prime $p\geqslant 5$ the group $G_2(p)$ is a quotient of $(2,3,7;2p) = \langle X,Y: X^2=Y^3=(XY)^7 =[X,Y]^{2p}=1 \rangle.$
      
It is also shown that on the nilmanifold $\Gamma\backslash (H^3\times H^3)$ the balanced condition is not stable under small deformations.
      
 

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