Into the Unknown
Fourier Series Sometimes complicated motions can actually be composed of motions of many different frequencies. The type of mathematical analysis that’s useful is called Fourier series. Fourier series is a way of representing a function – an infinite series of cosine and sines. You can view these as a linear superposition of waves of various frequencies. We use orthogonality…
Systems of Homogeneous Linear First-order ODEs The system of linear first order homogeneous equations can be written in matrix form. To solve it we’re going to use An ansatz X(t) = V e位t . We’re going to look for solutions of the ansatz. The principle of superposition. When we find solutions of the ansatz, we’ll multiply…
Corporate & Commercial Law II: Business Forms, Financing & Governmental RegulationUniversity of Illinois at Urbana-Champaign University of Illinois offered lots of amazing business courses and this is one of them. This course first introduces various forms of business, among which LLC (Limited Liability Company) interests me a lot, which is quite modern and essentially a…
Multidimensional Integration In Vector Calculus, we have to worry about integrating over two or three variables: double integrals or triple integrals. Double integral can be interpreted as a volume, like a single integral is the area under the curve. When the limits don’t depend on any variable, and we could do integrals separately. Polar coordinates Polar coordinates…
In some cases, variables must take integer values, or binary values. Formulating and solving the models with integer variables is integer programming. There are mainly 2 methods to solve it: The branch-and-bound algorithm finds an optimal solution for an integer program. Essentially, it decomposes the integer program into multiple linear programs, then solves all of…
Laplace transform Laplace transform is a technique for solving differential equations. By using the Laplace transform you can convert a differential equation into another space where the equation is easier to solve. Laplace transform is a linear transformation and it is invertible. The strength of the Laplace transform technique, is to solve the differential equations where you…
Understanding and Visualizing Data with PythonUniversity of Michigan This is a fantastic statistics course taught in the setting of Python programming language. It astonishes me that the professor did a cartwheel at the beginning, very impressive. The course mainly focus on the fundamentals of statistics. After the lecture of concepts, there are hands-on videos tutoring…
Partial derivatives Partial derivative is to differentiate functions of multiple variables. Assume a function f = f(x, y), you are differentiating f with respect x, that is the usual definition of a derivative of a function of one variable, but y is held constant. There is something called mix partial. It does not depend on…
Propositional Logic A proposition (often denoted by capital letters P, Q, etc) is a statement, that can be either true or false. Its negation 卢P is obtained by prepending “it is not true that …” to P. If you negate twice, you get P back again: 卢(卢P) = P . Two propositions are equivalent P…
The general form of inhomogenous linear differential equations is: We need 2 initial conditions x(0) = x0, x'(0) = u0 to have a unique solution, there is a 3-step method to solve it: Solve the homogeneous equation, where g(t) = 0, to find the general solution to the homogeneous equation:xh(t) = C1 x1(t) + C2…
The simplex method is the most fundamental tool in linear programming, it is a single algorithm that is able to solve any kinds of linear program regardless of its format, number of variables and constraints. We firstly transform the linear program to a standard form, then we focus on the standard form. The idea of…
Gross Income Gross income is all income from whatever source defined, it includes income realized in any form whether in money, property or services. If receive property or services, use Fair Market Value. Taxpayer recognize gross income when: They receive an economic benefit Increase in net worth (assets minus liabilities) Borrowed funds represent a liability,…
Six Sigma PrinciplesUniversity System of Georgia There are criticisms of Six Sigma, sure is that it is not suitable for every company and industry. Some people like it, some don’t. But still, there’s no doubt how ubiquitous and popular it is. To those who work in a gigantic organization, Six Sigma shall not be unfamiliar….
Linear homogeneous differential equation is the second-order ODEs that has the form below, It is important that the function p and q do not depend on x and any of its derivatives: A second-order homogeneous ODE with constant coefficients takes the form below, where a, b and c are constants. Euler Method for Higher-Order ODEs…
Euler Method Euler method is a very simple numerical method that can be used to solve differential equations. For example dy / dx = f(x, y) , there is usually no such a thing as an analytical solution, so one has to solve numerically with an initial condition y(x0) = y0 . Numerical methods approximate…