Application of the extended Kantorovich method to the vibration of clamped rectangular plates

The transverse vibrations of rectangular plates are analysed by using the extended Kantorovich method of variational calculus. Several combinations of simply supported and clamped boundary conditions are discussed, and closed form solutions are obtained.     

ParaCEST MRI contrast agents capable of derivatization via"click" chemistry

A comprehensive series of lanthanide chelates has been prepared with a tetrapropargyl DOTAM type ligand. The complexes have been characterized by a combination of (1)H NMR, single-crystal X-ray crystallography, CEST and relaxation studies and have also been evaluated for potential use as paramagnetic chemical exchange saturation transfer (ParaCEST) contrast agents in magnetic resonance imaging (MRI). We demonstrate the functionalization of several chelates by means of alkyne-azide "click" chemistry in which a glucosyl azide is used to produce a tetra-substituted carbohydrate-decorated lanthanide complex. The carbohydrate periphery of the chelates has a potent influence on the CEST properties as described herein.     

Cover Picture: Compartmentalization of Chemically Separated Components into Droplets (Angew. Chem. Int. Ed. 15/2009)

Electroosmotic‐flow‐driven droplet generation integrated with capillary electrophoresis (CE) allows the molecular components separated by CE to be compartmentalized into a stream of droplets, as reported by D. T. Chiu and co‐workers in their Communication on page 2719 ff. and illustrated on the cover picture. After separation and droplet compartmentalization, the droplet‐confined bands either can be docked and studied on‐chip or removed off‐chip for a second‐dimension separation and further analysis.

     

The Mathematical Pull of Temptation Revisited

In this paper, we defend and extend a (simple) mathematical model of akrasia.

Infinite Families of Exact Sums of Squares Formulas,Jacobi Elliptic Functions,Continued Fractions,and Schur Functions

In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's 4 and 8 squares identities to 4n ² or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi's special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan's tau function (n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the -function identities of Macdonald. Moreover, the powers 4n(n + 1), 2n ² + n, 2n ² – n that appear in Macdonald's work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing a positive integer by sums of 4n ² or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson's C nonterminating ₆₅ summation theorem, and Andrews' basic hypergeometric series proof of Jacobi's 2, 4, 6, and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n ² and n(n + 1) squares. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Sierpinski (1907), Uspensky (1913, 1925, 1928), Bulygin (1914, 1915), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Bell (1919), Estermann (1936), Rankin (1945, 1962), Lomadze (1948), Walton (1949), Walfisz (1952), Ananda-Rau (1954), van der Pol (1954), Krätzel (1961, 1962), Bhaskaran (1969), Gundlach (1978), Kac and Wakimoto (1994), and, Liu (2001). We list these authors by the years their work appeared.