Linear Systems of Differential Equations

Linear Systems of Di erential Equations Math 240 First order linear systems Solutions Beyond rst order systems Solutions to homogeneous linear systems As with linear systems, a homogeneous linear system of di erential equations is one in which b(t) = 0. Theorem If A(t) is an n n matrix function that is continuous on the

  System, Linear, Differential, Equations, Linear systems of differential equations, Linear systems

5.3 Planar Graphs and Euler’s Formula

5.3 Planar Graphs and Euler’s Formula Among the most ubiquitous graphs that arise in applications are those that can be drawn in the plane without edges crossing. For example, let’s revisit the example considered in Section 5.1 of the New York City subway system. We considered a graph in which vertices represent subway stops and edges represent

  Graph

Eigenvalues, Eigenvectors, and Diagonalization

Find all of the eigenvalues and eigenvectors of A= 1 1 0 1 : The characteristic polynomial is ( 1)2, so we have a single eigenvalue = 1 with algebraic multiplicity 2. The matrix A I= 0 1 0 0 has a one-dimensional null space spanned by the vector (1;0). Thus, the geometric multiplicity of this eigenvalue is 1.

  Eigenvalue, Eigenvalues and eigenvectors, Eigenvectors, 1 0 0 0, 1 1 1 0

Power series and Taylor series - University of Pennsylvania

1. Geometric and telescoping series The geometric series is X1 n=0 a nr n = a + ar + ar2 + ar3 + = a 1 r provided jrj<1 (when jrj 1 the series diverges). We often use partial fractions to detect telescoping series, for which we can calculate explicitly the partial sums S n. D. DeTurck Math 104 002 2018A: Series 3/42

  Series, Power, Geometric, Power series, Geometric series

Generalized Eigenvectors - University of Pennsylvania

Eigenvectors Math 240 De nition Computation and Properties Chains Facts about generalized eigenvectors The aim of generalized eigenvectors was to enlarge a set of linearly independent eigenvectors to make a basis. Are there always enough generalized eigenvectors to do so? Fact If is an eigenvalue of Awith algebraic multiplicity k, then nullity ...

  Eigenvectors

11 Partial derivatives and multivariable chain rule

the chain rule gives df dx = @f @x + @f @y ·y0. (11.3) The notation really makes a di↵erence here. Both df /dx and @f/@x appear in the equation and they are not the same thing! Derivative along an explicitly parametrized curve One common application of the multivariate chain rule is when a point varies along

  Chain, Partial, Derivatives, Partial derivatives

generatingfunctionology - Penn Math

ating functions can be simple and easy to handle even in cases where exact ... Generating functions can give stunningly quick deriva-tions of various probabilistic aspects of the problem that is repre-sented by your unknown sequence. (d) …

  Quick, Math, Easy, And easy, Generatingfunctionology

Algorithms and Complexity

is the study of computational complexity. Naturally, we would expect that a computing problem for which millions of bits of input data are required would probably take longer than another problem that needs only a few items of input. So the time complexity of a calculation is measured by expressing the running time of the calculation as a ...

  Computational, Complexity, Computational complexity

4 Number Theory I: Prime Numbers - Penn Math

Number theory is filled with questions of patterns and structure in whole numbers. One of the most important subsets of the natural numbers are the prime numbers, to which we now turn our attention. ... make up the matter of our universe. Unlike the periodic table of the elements, however, the list of prime numbers goes on indefinitely.

  Universe, Patterns

Linear Algebra Problems - Penn Math

g) The linear transformation T A: Rn!Rn de ned by Ais onto. h) The rank of Ais n. i) The adjoint, A, is invertible. j) detA6= 0. 14. Call a subset S of a vector space V a spanning set if Span(S) = V. Suppose that T: V !W is a linear map of vector spaces. a) Prove that a linear map T is 1-1 if and only if T sends linearly independent sets

  Linear, Transformation, Linear transformations

18.440: Lecture 18 Uniform random variables

Uniform random variables and percentiles. Toss n = 300 million Americans into a hat and pull one out. uniformly at random. Is the height of the person you choose a uniform random variable? Maybe in an approximate sense? No. Is the percentile of the person I choose uniformly random? In. other words, let p be the fraction of people left in the hat

  Uniform, Variable, Random, Uniform random variables, Uniform random

Chapter 2 The Maximum Likelihood Estimator

Example 2.2.1 (The uniform distribution) Consider the uniform distribution, which has the density f(x; )= 1I [0, ](x). Given the iid uniform random variables {X i} the likelihood (it is easier to study the likelihood rather than the log-likelihood) is L n(X n; )= 1 n Yn i=1 I [0, ](X i). Using L n(X n

  Uniform, Maximum, Variable, Estimator, Random, Likelihood, Maximum likelihood estimator, Uniform random variables

6 Jointly continuous random variables

6.4 Function of two random variables Suppose X and Y are jointly continuous random variables. Let g(x,y) be a function from R2 to R. We define a new random variable by Z = g(X,Y). Recall that we have already seen how to compute the expected value of Z. In this section we will see how to compute the density of Z. The general strategy

  Variable, Random, Random variables

Discrete and Continuous Random Variables

15.063 Summer 2003 1616 Continuous Random Variables A continuous random variable can take any value in some interval Example: X = time a customer spends waiting in line at the store • “Infinite” number of possible values for the random variable.

  Variable, Continuous, Random, Random variables, Continuous random variables

S1 Discrete random variables - PMT

S1 Discrete random variables . PhysicsAndMathsTutor.com (e) Var(X) (3) (Total 10 marks) 14. A fairground game involves trying to hit a moving target with a gunshot.

  Variable, Random, Random variables

Chapter 3 Continuous Random Variables

76 Chapter 3. Continuous Random Variables (LECTURE NOTES 5) with associated standard deviation, ˙= p ˙2. The moment-generating function is M(t) = E 1 etX = Z 1 etXf(x) dx for values of tfor which this integral exists. Expected value, assuming it exists, of a function uof Xis E[u(X)] = Z 1 1 u(x)f(x) dx The (100p)th percentile is a value of ...

  Variable, Random, Random variables

Lecture 15: Order Statistics - Duke University

n iid random variables X k is the kth smallest X, usually called the kth order statistic. X (1) is therefore the smallest X and X (1) = min(X 1;:::;X n) Similarly, X (n) is the largest X and X (n) = max(X 1;:::;X n) Statistics 104 (Colin Rundel) Lecture 15 March 14, 2012 2 / 24 Section 4.6 Order Statistics Notation Detour For a continuous ...

  Variable, Random, Random variables

PROBABILITY AND STATISTICS FOR ECONOMISTS

Preface This textbook is the first in a two-part series covering the core material typically taught in a one-year Ph.D. course in econometrics.