A rigorous construction of the field of real numbers: the unique (up to isomorphism) completely ordered field with the least upper bound property; along with various formulations of completeness and with a postlude on the measure of sets.
Template for the Electrical and Computer Engineering Department of Northeastern University. Communications, Control, and Signal Processing Qualifying Exam.
Stručný přehled vzorců a vztahů z předmětu PST. Neobsahuje všechny triviální věci (pravděpodobnost jevu) a některé komplikovanější, které se nedají snadno stručně shrnout (CLT, Markova nerovnost etc.). Veškerou většinu ostatní potřebné výbavy na zkoušku ale ano.
A detailed report of findings on the altitudes which can be reached by super pressure balloons and how various factors and considerations affect this. Superpressure balloons are deployed and researched by various organisations including NASA, to solve technical limitations such as cell tower coverage as well as advancing fields of research. Balloons are used in planetary exploration, and weather prediction to teaching primary school physics. The versatile yet simple aerostat has been a valuable tool in many areas of engineering and their altitude ceiling is of great scientific interest. To solve the problem without the ability to physically reproduce the scenario, required mathematical models to be created as a means of simulating the effects of real world physics. A degree great enough to output an accurate and hence useful result without becoming too complex to be computable is the fine balance attempted to be created by this paper.
I made a template for Math 3E & 3F at Berkeley City College. This will work well for most typed problem sets for most classes. This also will work for UC Berkeley's DiffEQ class
In this article we will discuss a new type of notation for homogenous polynomials of $3$ variables, and its applications in solving Olympiad inequalities using the AM-GM inequality, Muirhead’s Inequality, and Schur’s Inequality. I suggest reading [1] first for a clearer explantion on the mechanics of this notation. This article is more focused on the applications to Olympiad inequalities
The density of solid water, unlike most molecules, is less than that of its liquid form. Its precise value is of use in many applications. Freezing a spherical droplet of water and analyzing the changed shape from a sphere to a sphere with a slight peak in order to find the density of ice. We find the density of ice to be at 0.90 ± 1.66 · 106 g/mL. The precision of our measurement was limited by uncertainty in the angle measurements of the peak of the droplet.