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 $L^{2}(G)$.

In the present paper, a three dimensional annular developing incompressible laminar flow through 270- degree curved pipe is numerically simulated. The dimensionless governing equations of continuity, momentums and energy are driven in toroidal coordinates. The governing equations are discretized by projection algorithm using forward difference in time and central difference in space. A three-di...

A novel and rather simple method is presented to describe the physics of space-charge region in a planar diode. The method deals with the issue in the time domain and as a consequence transient time behavior can be achieved. Potential distributions and currents obtained using this technique, supposing zero initial velocity for electrons, reveal absolute agreement with Child's results. Moreover,...

We study the vector-valued spectrum $\mathcal {M}_{u,\infty }(B_{\ell _2},B_{\ell _2})$, which is set of non-zero algebra homomorphisms from {A}_u(B_{\ell _2})$ (the uniformly continuous holomorphic functions on $B_{\ell _2}$) to