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We prove versions of James' weak compactness theorem for polynomials and symmetric multilinear forms of finite type. We also show that a Banach space$X$is reflexive if and only if it admits and equivalent norm such that there exists$x$_{0}≠0 in$X$and a weak-^{*}-open subset$A$of the dual space, satisfying that$x$^{*}⊗$x$_{0}attains its numerical radius. for each$x$^{*}in$A$.

In this article the authors present a new solver for the Poisson problem on spherical shells. The method is derived by application of the Fast Fourier Transform (FFT) in all three spherical variables. Through a particular change of variables valid on spherical shells, this new approach avoids a variable coefficient on the radial differential operator, resulting in a constant-coefficient formulation that can be fast-diagonalized via FFT. The authors describe the Fourier expansion in the angular variables and the fast diagonalization of the radial derivative using the FFT, including second-, fourth- and sixth-order finite difference approximations. An algorithmic description of the sixth-order solver is described, and numerical tests that demonstrate the theoretical rate of convergence are presented.

In this paper, we discuss some partitions of affine flag varieties. These partitions include as special cases the partition of affine flag variety into affine Deligne-Lusztig varieties and the affine analogue of the partition of flag varieties into $\cb_w (b) $ introduced by Lusztig in\cite {L1} as part of the definition of character sheaves.

We state a conjecture for the formulas of the depth 4 low-weight rotational eigenvectors and their corresponding eigenvalues for the 3^G subfactor planar algebras. We prove the conjecture in the case when |G| is odd. To do so, we find an action of G on the reduced subfactor planar algebra at f^{(2)}, which is obtained from shading the planar algebra of the even half. We also show that this reduced subfactor planar algebra is a Yang-Baxter planar algebra.