‎Highest weight modules of the double affine Lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎Singular vectors of Verma modules are‎ ‎determined using a similar condition with horizontal affine Lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also show that the subdivision of ladder graph Ln with m-pendant S(Ln)  mk1 is odd grace...

In a recent paper, the authors gave combinatorial formulas for Laurent expansions of super λ -lengths in marked disk, generalizing Schiffler's T -path formula. present we give an alternate expression these terms double dimer covers on snake graphs. This generalizes Musiker, Schiffler, and Williams.

We develop a numerical scheme to determine which planar snake motions are optimal for locomotory efficiency, across a wide range of frictional parameter space. For a large coefficient of transverse friction, we show that retrograde traveling waves are optimal. We give an asymptotic analysis showing that the optimal wave amplitude decays as the -1/4 power of the coefficient of transverse frictio...

This paper depicts e paraphernalia of an alchemist who believed in imitating creation and generating a soul thereby. The magic wands of creation are a compass and a triangular carpenter's square. They can produce the dual natured soul Ruh or Spirit symbolized as Cock and Nafs of "the soul" as snake: The real creative energy by nature is ultrasonic energy, characterizing the word of command of t...

Three types of double-triangular mechanism are introduced. These contain planar, spherical or spatial triangle pairs that move with respect to each other while remaining in line contact. Using planar, spherical and spatial trigonometry, the three pertinent direct kinematic problems are solved and typical numerical examples are presented.