Explicit and Implicit Methods In Solving Differential Equations

Differential equations are equations that involve an unknown function and derivatives. Euler's method are efficient methods to yield fairly accurate approximations of the actual solutions. By manipulating such methods, one can find ways to provide good approximations compared to the exact solution of parabolic partial differential equations and nonlinear parabolic differential equations.

The Devil’s Calculus: Mathematical Models of Civil War

In spite of the movement to turn political science into a real science, various mathematical methods that are now the staples of physics, biology, and even economics are thoroughly uncommon in political science, especially the study of civil war. This study seeks to apply such methods - specifically, ordinary differential equations (ODEs) - to model civil war based on what one might dub the capabilities school of thought, which roughly states that civil wars end only when one side’s ability to make war falls far enough to make peace truly attractive. I construct several different ODE-based models and then test them all to see which best predicts the instantaneous capabilities of both sides of the Sri Lankan civil war in the period from 1990 to 1994 given parameters and initial conditions. The model that the tests declare most accurate gives very accurate predictions of state military capabilities and reasonable short term predictions of cumulative deaths. Analysis of the model reveals the scale of the importance of rebel finances to the sustainability of insurgency, most notably that the number of troops required to put down the Tamil Tigers is reduced by nearly a full order of magnitude when Tiger foreign funding is stopped. The study thus demonstrates that accurate foresight may come of relatively simple dynamical models, and implies the great potential of advanced and currently unconventional non-statistical mathematical methods in political science.

The Differential Requirement of Albumin and Sodium Citrate on the Development of In Vitro Produced Bovine Embryos

In vitro culture for bovine embryos is largely not optimal. Our study was to determine the components necessary for early embryo development. In experiment 1, IVF embryos were cultured for two days in CR1aa medium containing sodium citrate and BSA from two sources (Sigma vs. ICPbio), subsequently for additional five days with cumulus monolayer in 10% FBS CR1aa. We found that supplementation with both Sigma-BSA and sodium citrate significantly increased total blastocyst (BL) development compared with the ICPbio-BSA groups (37% vs. 19-21%), and enhanced the total number of high quality (C1 BL, IETS standard) blastocysts (26% vs. 11-17%) (P < 0.05). In experiment 2 with serum free and/or somatic free culture, we found that CR1aa culture can support a comparable embryo development with a supplement of Sigma BSA. The addition of sodium citrate did not increase blastocyst development in either the Sigma-BSA or the ICPbio-BSA groups. An inferior blastocyst development occurring in ICPbio-BSA culture (1-3%) could be rescued by culture in CRlaa supplemented with 10% FBS (29%), more importantly, by culture in CR1aa with a replacement of Sigma BSA (24%) (P <0.05). C1 blastocysts rescued by FBS and Sigma BSA in ICPbio-BSA culture possessed indistinguishable morphology to embryos developed in a Sigma-BSA, FBS and somatic co-culture system, showing similar cell number/blastocyst (129-180, P > 0.05). Our study found a beneficial effect of sodium citrate and BSA on the in vitro development of bovine IVF embryos during co-culture. We also determined that differential embryotrophic factor(s) contained in BSA and serum, probably not sodium citrate, is necessary for promoting competent morula and blastocyst development in cattle.