Closed form summation of C-finite sequences

We consider sums of the form $$\sum_{j=0}^{n-1}F_1(a_1n+b_1j+c_1)F_2(a_2n+b_2j+c_2)... F_k(a_kn+b_kj+c_k),$$ in which each $\{F_i(n)\}$ is a sequence that satisfies a linear recurrence of degree $D(i)<\infty$, with constant coefficients. We assume further that the $a_i$'s and the $a_i+b_i$'s are all nonnegative integers. We prove that such a sum always has a closed form, in the sense that it evaluates to a linear combination of a finite set of monomials in the values of the sequences $\{F_i(n)\}$ with coefficients that are polynomials in $n$. We explicitly describe two different sets of monomials that will form such a linear combination, and give an algorithm for finding these closed forms, thereby completely automating the solution of this class of summation problems. We exhibit tools for determining when these explicit evaluations are unique of their type, and prove that in a number of interesting cases they are indeed unique. We also discuss some special features of the case of ``indefinite summation," in which $a_1=a_2=... = a_k = 0$

Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media

We analyze frequency conversion and its control among three light waves using a geometric approach that enables the dynamics of the waves to be visualized on a closed surface in three dimensions. It extends the analysis based on the undepleted-pump linearization and provides a simple way to understand the fully nonlinear dynamics. The Poincaré sphere has been used in the same way to visualize polarization dynamics. A geometric understanding of control strategies that enhance energy transfer among interacting waves is introduced, and the quasi-phase-matching strategy that uses microstructured quadratic materials is illustrated in this setting for both type I and II second-harmonic generation and for parametric three-wave interactions

Mars Obliquity History Constrained by Elliptic Crater Orientations

The dynamics of Mars' obliquity are believed to be chaotic, and the historical ~3.5 Gyr (late-Hesperian onward) obliquity probability density function (PDF) is high uncertain and cannot be inferred from direct simulation alone. Obliquity is also a strong control on post-Noachian Martian climate, enhancing the potential for equatorial ice/snow melting and runoff at high obliquities (> 40{\deg}) and enhancing the potential for desiccation of deep aquifers at low obliquities (< 25{\deg}). We developed a new technique using the orientations of elliptical craters to constrain the true late-Hesperian-onward obliquity PDF. To do so, we developed a forward model of the effect of obliquity on elliptic crater orientations using ensembles of simulated Mars impactors and ~3.5 Gyr-long Mars obliquity simulations. In our model, the inclinations and speeds of Mars crossing objects bias the preferred orientation of elliptic craters which are formed by low-angle impacts. Comparison of our simulation predictions with a validated database of elliptic crater orientations allowed us to invert for best-fitting obliquity history. We found that since the onset of the late-Hesperian, Mars' mean obliquity was likely low, between ~10{\deg} and ~30{\deg}, and the fraction of time spent at high obliquities > 40{\deg} was likely < 20%

Selected markets for the essential oils of patchouli and vetiver

The essential oils of patchouli and vetiver, although they differ markedly in their odour and chemical characteristics, share one very important common attribute, namely that they are among the most important naturally-occurring 'base' materials used in the perfumery industry. Although not often used as dominant sources of fragrance in their own right, they are very widely used to give a solid foundation and lasting character to a fragrance, whether it is to be used in a high-class perfume or in cheaper products such as toilet soaps, cosmetic lotions, deodorants and so forth. Both oils have distinctive odour characteristics, among which 'woody'-type notes predominate; neither has yet been, nor seems likely to be, reproduced accurately from synthetic aroma chemical formulations