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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.
      
There are two well known combinatorial tools in the representation theory ofSLn, the semi-standard Young tableaux and the Gelfand-Tsetlin patterns.
      
We also use known results about canonical bases forUq2 to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmannians), due to Lascoux-Schützenberger and Zelevinsky.
      
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.
      
As a corollary, we obtain a new proof for Roberts' well-known counterexample in dimension seven.
      
The first part of this paper describes the construction of pseudo-Riemannian homogeneous spaces with special curvature properties such as Einstein spaces, using corresponding known compact Riemannian ones.
      
Except for the Borel and some special cases a corresponding result is not known for the semi-centre of the enveloping algebra ofp.
      
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.
      
Previously, only algorithms for linearly reductive groups and for finite groups have been known.
      
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.
      
This setup comprises well known objects such as framed vector bundles, Higgs bundles, and conic bundles.
      
In the examples which have been studied so far, our semistability concept reproduces the known ones.
      
Whether the corresponding results hold in positive characteristic is not known.
      
When the characteristic of k is 0, it is known that the invariants of d vectors, d ≥ n, are obtained from those of n vectors by polarization.
      
Furthermore, the intrinsic definition is just the (now) well-known Crandall-Lions viscosity solution, modified in a natural way to accommodate measurable coefficients.
      
These means are given by some function λ and generalize the well-known Bochner-Riesz means.
      
We provide a direct computational proof of the known inclusion ${\cal H}({\bf R} \times {\bf R}) \subseteq {\cal H}({\bf R}^2),$ where ${\cal H}({\bf R} \times {\bf R})$ is the product Hardy space defined for example by R.
      
It is known thatT (A, D) tiles?n by some subset of?n.
      
It is known [7] that dualizing a form of the Poisson summation formula yields a pair of linear transformations which map a function ? of one variable into a function and its cosine transform in a generalized sense.
      
Implementing this point of view, Poisson Summation Formulas are proved in several spaces including integrable functions of bounded variation (where the result is known) and elements of mixed norm spaces.
      
 

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