Power Sums Related to Semigroups ${\sl S}(d_1,d_2,d_3)$
The explicit formulas for the sums of positive powers of the integers $s_i$ unrepresentable by the triple of integers $d_1,d_2,d_3\in {\mathbb N}, \gcd(d_1,d_2,d_3)=1$, are derived.
math.NT↗
arXiv subjects
Boris Y. Rubinstein
Publications and source records attributed to Boris Y. Rubinstein.
An Explicit Formula for Restricted Partition Function through Bernoulli Polynomials
Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called 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 polynomials of higher order with periodic coefficients.
math.NT↗
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.
math.NT↗
On Explicit Formula for Restricted Partition Function
A new recursive procedure for calculation of restricted partition function is suggested. An explicit formula for the restricted partition function is found based on this procedure.
math.NT↗
Kinetic Slip Condition, van der Waals Forces, and Dynamic Contact Angle
The profiles of a spreading wetting film are computed taking into account intermolecular forces and introducing a kinetic slip condition at a molecular cut-off distance. This eliminates the stress singularity, so that both "true" and "visible" contact angles are defined unequivocally. The "true" contact angle at the cut-off distance depends on the slip length as well as on the edge propagation speed, but not on gravity or asymptotic inclination angle. These macroscopic factors influence, however, the "visible" contact angle observed in the interval where the actual film profile departs from the intermediate asymptotic curve.
physics.flu-dyn↗
Sylvester Waves in the Coxeter Groups
A new recursive procedure of the calculation of partition numbers function $W(s,{\bf d}^m)$ is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of $W(s,{\bf d}^m)$ and their periods $τ(G)$ for the irreducible Coxeter groups and a list for the first ten symmetric group ${\cal S}_m$ are presented. A {\it least common multiple} ${\cal L}(m)$ of the series of the natural numbers 1,2,..,$m$ plays a role of the period $τ({\cal S}_m)$ of $W(s,{\bf d}^m)$ in ${\cal S}_m$. An asymptotic behaviour of ${\cal L}(m)$ with $m \to \infty$ is found.
math.NT↗