We define a new function-valued inner product on L2(G), called ?-bracket product, where G is a locally compact abelian group and ? is a topological isomorphism on G. We investigate the notion of ?-orthogonality, Bessel's Inequality and ?-orthonormal bases with respect to this inner product on L2(G).

we define a new function-valued inner product on l2(g), called ?-bracket product, where g is a locally compact abelian group and ? is a topological isomorphism on g. we investigate the notion of ?-orthogonality, bessel's inequality and ?-orthonormal bases with respect to this inner product on l2(g).

This paper is an investigation of $L$-dual frames with respect to a function-valued inner product, the so called $L$-bracket product on $L^{2}(G)$, where G is a locally compact abelian group with a uniform lattice $L$. We show that several well known theorems for dual frames and dual Riesz bases in a Hilbert space remain valid for $L$-dual frames and $L$-dual Riesz bases in $L^{2}(G)$.

1) We identify new parameter branches for the ultra-local boundary Poisson bracket in d spatial dimension with a (d− 1)-dimensional spatial boundary. There exist 2 r-dimensional parameter branches for each d-box, r-row Young tableau. The already known branch (hepth/9912017) corresponds to a vertical 1-column, d-box Young tableau. 2) We consider a local distribution product among the so-called b...

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A Clifford algebra has three major multiplications: inner product, outer product and geometric product. Accordingly, the same Clifford algebra has three versions: Clifford vector algebra, which features on inner products and outer products of multivectors; Clifford bracket algebra, which features on pseudoscalars and inner products of vectors; Clifford geometric algebra, which features on geome...

Aspects of noncanonical Hamiltonian field theory are reviewed. '·1any systems are Hamiltonian in the sense of possessing Poisson bracket structures, yet the equations are not in canonical form. A particular sys tem of thi s type is cons idered, namely reduced magnetohydrodynamics (RllHD) which was derived for tokamak modelling. The notion of a liePoisson bracket is reviewed; these are special P...

Aspects of noncanonical Hamiltonian field theory are reviewed. f·1any systems are Hamiltonian in the sense of possessing Poisson bracket structures. yet the equations are not in canonical form. A particular system of this type is considered. namely reduced magnetohydrodynamics (RflHD) which was derived for tokamak modelling. The notion of a LiePoisson bracket is reviewed; these are special Pois...

Aspects of noncanonical Hamiltonian field theory are reviewed. '·1any systems are Hamiltonian in the sense of possessing Poisson bracket structures, yet the equations are not in canonical form. A particular sys tem of thi s type is cons idered, namely reduced magnetohydrodynamics (RllHD) which was derived for tokamak modelling. The notion of a liePoisson bracket is reviewed; these are special P...