Abel's method on summation by parts and terminating well-poised q-series identities

The Abel method on summation by parts is reformulated to present new and elementary proofs of several classical identities of terminating well-poised basic hypergeometric series, mainly discovered by [F H. Jackson, Certain q-identities, Quart. J. Math. Oxford Ser. 12 (1941) 167–172]. This strengthens further our conviction that as a traditional analytical instrument, the revised Abel method on summation by parts is indeed a very natural choice for working with basic hypergeometric series.     

Abel’s lemma on summation by parts and terminating q-series identities

The terminating basic hypergeometric series is investigated through the modified Abel lemma on summation by parts. Numerous known summation and transformation formulae are derived in a unified manner. Several new identities for the terminating quadratic, cubic and quartic series are also established.     

G-frames and g-Riesz bases

G-frames are generalized frames which include ordinary frames, bounded invertible linear operators, as well as many recent generalizations of frames, e.g., bounded quasi-projectors and frames of subspaces. G-frames are natural generalizations of frames and provide more choices on analyzing functions from frame expansion coefficients. We give characterizations of g-frames and prove that g-frames share many useful properties with frames. We also give a generalized version of Riesz bases and orthonormal bases. As an application, we get atomic resolutions for bounded linear operators.     

Sufficient conditions for irregular Gabor frames

Finding general and verifiable conditions which imply that Gabor systems are (resp. cannot be) Gabor frames is among the core problems in Gabor analysis. In their paper on atomic decompositions for coorbit spaces [H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations, and their atomic decomposition, I, J. Funct. Anal. 86 (1989), 307–340], the authors proved that every Gabor system generated with a relatively uniformly discrete and sufficiently dense time-frequency sequence will allow series expansions for a large class of Banach spaces if the window function is nice enough. In particular, such a Gabor system is a frame for the Hilbert space of square integrable functions. However, their proof is based on abstract analysis and does not give direct information on how to determine the density in the sense of directly applicable estimates. It is the goal of this paper to present a constructive version of the proof and to provide quantitative results. Specifically, we give a criterion for the general case and explicit density for some cases. We also study the existence of Gabor frames and show that there is some smooth window function such that the corresponding Gabor system is incomplete for arbitrary time-frequency lattices.     

Stability of g-frames

g-Frames are natural generalizations of frames which cover many other recent generalizations of frames, e.g., bounded quasi-projectors, frames of subspaces, outer frames, oblique frames, pseudo-frames and a class of time-frequency localization operators. Moreover, it is known that g-frames are equivalent to stable space splittings. In this paper, we study the stability of g-frames. We first present some properties for g-Bessel sequences. Then we prove that g-frames are stable under small perturbations. We also study the stability of dual g-frames.     

Density of irregular wavelet frames

We show that if an irregular multi-generated wavelet system forms a frame, then both the time parameters and the logarithms of scale parameters have finite upper Beurling densities, or equivalently, both are relatively uniformly discrete. Moreover, if generating functions are admissible, then the logarithms of scale parameters possess a positive lower Beurling density. However, the lower Beurling density of the time parameters may be zero. Additionally, we prove that there are no frames generated by dilations of a finite number of admissible functions.

     

Stability analysis of thermosolutal convection in a horizontal porous layer using a thermal non-equilibrium model

A stability analysis is carried out to investigate the onset of thermosolutal convection in a horizontal porous layer when the solid and fluid phases are not in a local thermal equilibrium, and the solubility of the dissolved component depends on temperature. To study how the reaction and thermal non-equilibrium affect the double-diffusive convection, the effects of scaled inter-phase heat transfer coefficient H and dimensionless reaction rate k on thermosolutal convection are discussed . The critical Rayleigh number and the corresponding wave number for the stability and overstability convections are obtained. Specially, asymptotic analysis for both small and large values of H and k is presented, and the corresponding asymptotic solutions are compared with numerical results. At last, a nonlinear stability analysis is presented to study how H and k affect the Nusselt number.     

<Emphasis Type="Italic">π</Emphasis>-formulas implied by Dougall’s summation theorem for <Subscript>5</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>-series

The general summation theorem for well-poised ₅ F ₄-series discovered by Dougall (Proc. Edinb. Math. Soc. 25:114–132, 1907) is shown to imply several infinite series of Ramanujan-type for 1/π and 1/π ², including those due to Bauer (J. Reine Angew. Math. 56:101–121, 1859) and Glaisher (Q. J. Math. 37:173–198, 1905) as well as some recent ones by Levrie (Ramanujan J. 22:221–230, 2010).     

Quartic theta hypergeometric series

Four classes of quartic theta hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformations are proved that express the quartic series in terms of well-poised, quadratic and cubic ones. Thirty new summation formulae for terminating quartic theta hypergeometric series are derived consequently.