Frequency Addition Source of Optical Radiation

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In mathematics, a quasi-polynomial (pseudo-polynomial) is a generalization of polynomials. While the coefficients of a polynomial come from a ring, the coefficients of quasi-polynomials are instead periodic functions with integral period. Quasi-polynomials appear throughout much of combinatorics as the enumerators for various objects.

A quasi-polynomial can be written as $q(k)=c_d(k)k^d+c_{d-1}(k)k^{d-1}+\cdots +c_0(k)$, where $c_i(k)$ is a periodic function with integral period. If $c_d(k)$ is not identically zero, then the degree of q is d. Equivalently, a function $f:\mathbb{\mathbb{N}}\to \mathbb{\mathbb{N}}$ is a quasi-polynomial if there exist polynomials $p_0,\dots ,p_{s-1}$ such that $f(n)=p_i(n)$ when $n\equiv imods$. The polynomials $p_i$ are called the constituents of f.

Examples

• Given a d-dimensional polytope P with rational vertices $v_1,\dots ,v_n$, define tP to be the convex hull of $t{v}_1,\dots ,t{v}_n$. The function $L(P,t)=\#(tP\cap {\mathbb{\mathbb{Z}}}^d)$ is a quasi-polynomial in t of degree d. In this case, L(P,t) is a function $\mathbb{\mathbb{N}}\to \mathbb{\mathbb{N}}$. This is known as the Ehrhart quasi-polynomial, named after Eugène Ehrhart.
• Given two quasi-polynomials F and G, the convolution of F and G is

$(F*G)(k)=\sum _{m=0}^{k}F(m)G(k-m)$

which is a quasi-polynomial with degree $\le \mathrm{deg}F+\mathrm{deg}G+1.$

See also

• Ehrhart polynomial

References

• Stanley, Richard P. (1997). Enumerative Combinatorics, Volume 1. Cambridge University Press. ISBN 0-521-55309-1, 0-521-56069-1.