Restricted Partition Functions as Bernoulli and Euler Polynomials of Higher Order

Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called {\em Sylvester waves}) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The formulas are represented in a form of a finite sum over Bernoulli and Euler polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically independent homogeneous polynomial invariants of the finite groups is discussed.Comment: 15 pages, 2 figures, references added, submitted to The Ramanujan Journa

Strongly nonlinear waves in capillary electrophoresis

In capillary electrophoresis, sample ions migrate along a micro-capillary filled with a background electrolyte under the influence of an applied electric field. If the sample concentration is sufficiently high, the electrical conductivity in the sample zone could differ significantly from the background.Under such conditions, the local migration velocity of sample ions becomes concentration dependent resulting in a nonlinear wave that exhibits shock like features. If the nonlinearity is weak, the sample concentration profile, under certain simplifying assumptions, can be shown to obey Burgers' equation (S. Ghosal and Z. Chen Bull. Math. Biol. 2010, 72(8), pg. 2047) which has an exact analytical solution for arbitrary initial condition.In this paper, we use a numerical method to study the problem in the more general case where the sample concentration is not small in comparison to the concentration of background ions. In the case of low concentrations, the numerical results agree with the weakly nonlinear theory presented earlier, but at high concentrations, the wave evolves in a way that is qualitatively different.Comment: 7 pages, 5 figures, 1 Appendix, 2 videos (supplementary material

Thermal Fluctuations and Rubber Elasticity

The effects of thermal elastic fluctuations in rubber materials are examined. It is shown that, due to an interplay with the incompressibility constraint, these fluctuations qualitatively modify the large-deformation stress-strain relation, compared to that of classical rubber elasticity. To leading order, this mechanism provides a simple and generic explanation for the peak structure of Mooney-Rivlin stress-strain relation, and shows a good agreement with experiments. It also leads to the prediction of a phonon correlation function that depends on the external deformation.Comment: 4 RevTeX pages, 1 figure, submitted to PR