The Alladi-Gordon identity ∑j k=0(q i−k+1; q)k [ j k ] q(i−k)(j−k) = 1 plays an important role for the Alladi-Gordon generalization of Schur’s partition theorem. By using Joichi-Stanton’s insertion algorithm, we present an overpartition interpretation for the Alladi-Gordon key identity. Based on this interpretation, we further obtain a combinatorial proof of the Alladi-Gordon key identity by es...

around the roots of plants, as Nicole van Dam and Harro Bouwmeester report in a review of this fi eld (Trends Plant Sci. (2016) DOI: http://dx.doi. org/10.1016/j.tplants.2016.01.008). Direct investigation of signalling in the rhizosphere is currently focused on communication between plants and their microbial symbionts, as Vittorio Venturi and Christopher Keel report, although the scope could b...

B asil Gordon, who made major research contributions to number theory, combinatorics, and algebra and who enhanced our understanding of Ramanujan’s remarkable identities, passed away on January 12, 2012, at the age of eighty. Gordon, who was a professor of mathematics at UCLA, was an outstanding teacher at all levels. His legacy will continue, owing to the impact of his fundamental work and to ...

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in this paper, nonlinear klein-gordon equation with quadratic term is solved by means of an analytic technique, namely the homotopy analysis method (ham).comparisons are made between the adomian decomposition method (adm), the exact solution and homotopy analysis method. the results reveal that the proposed method is very effective and simple.

In this work, we successfully construct the new exact traveling wave solutions of the generalized Sinh–Gordon equation by new application of the homogeneous balance method. The idea introduced in this paper can be applied to other nonlinear evolution equations.

Himalayas are one of the greatest reservoirs of medicinal and aromatic plants, which has made people to use them for various health related problems. Due to their high medicinal potential, they are under high demand in national and international market. The ruthless extraction from the wild and illegal trade has contributed to endangerment of the plant. Picrorhiza kurrooa Royle ex Benth, a valu...