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We develop methods that give rise to natural monomial bases for such rings over their ground fields and explicitly determine precisely which monomials are zero in the ring of coinvariants.
      
There are natural concepts of the action of ann-valued group on a space and of a representation in an algebra of operators.
      
The natural action ofG onM extends to an action ofG1=G×ZK0.
      
For each compact Lie algebra g and each real representationV of g we construct a two-step nilpotent Lie groupN(g, V), endowed with a natural left-invariant riemannian metric.
      
We also determine all the composition factors of the symmetric tensors of the natural osp(2|2n)-module.
      
It is defined thanks to a natural distance on the space of compact open subgroups of G.
      
These allow natural representations of constant and, sometimes, linear components.
      
Furthermore, the intrinsic definition is just the (now) well-known Crandall-Lions viscosity solution, modified in a natural way to accommodate measurable coefficients.
      
From the overall perspective of the conformal group acting on the classical spaces, it is natural to consider the extension of these methods and questions in the context of SL(2, R), the Heisenberg group, and other Lie groups.
      
The definitions of the ?∞,qα spaces are extended in a natural way to ?∞,∞α and it is proven that this is the same space as ?∞,∞α, which justifies the standard convention in which the two spaces are defined to be equal.
      
Given a dilation matrix A and a natural number r we construct an associated r-regular multiresolution analysis with r-regular wavelet basis.
      
Given a positive locally finite Borel measure μ on R, a natural way to construct
      
We use a different approach which allows to establish that the one-sided Sawyer Ap weights are the natural ones to study the boundedness and convergence of that series in weighted spaces.
      
We prove the natural Fefferman-Stein weak type inequality for the strong maximal function in the plane, under the assumption that the weight satisfies a strong Muckenhoupt condition.
      
They are Polish vector spaces for a natural distance.
      
We also describe natural ways of discretizing those wavelet transforms, or equivalently to obtain atomic decompositions and Banach frames for the corresponding coorbit spaces.
      
Isolation of the natural products from the stems of Jatropha maheshwarii yielded 5 compounds such as Friedelin (0.16%), epi-friedelinol (0.12%), n-octacosanol (0.11%), β-sitosterol (0.2%), and β-sitosterol-3-β-D-glucopyranoside (0.24%).
      
This paper provides a natural condition of the existence of stable positive steady-state solutions for the one predator two prey system.
      
Let P be the transform group on Rn, then P has a natural unitary representation U on L2(Rn).
      
The population dynamics structure is recruitment and natural births with natural deaths.
      
 

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