Math 55a Lecture Notes - Evan Chen

X0 g? f 2-f 1 Lemma 2.2 In the above commutative diagram, if gis surjective then f 1 = f 2. Proof. Just use gto get everything equal. Yay. §2.3Groups De nition 2.3. A semi-group is a set Gendowed with an associative1 binary operation: G2!G. Lots of groups work. Example 2.4 (Star sh) Let Gbe an arbitrary set and x a g 0 2G. Then let ab= g 0 for ...

  Lecture, Notes, Math, Math 55a lecture notes

Math Olympiad Hardness Scale (MOHS) - Evan Chen

2 Speci cation Here is what each of the possible ratings means.1 Rating 0M: Sub-IMO. Problems rated 0 are too easy to use at IMO. I can often imagine such a problem could be solved by a strong student in an honors math class, even without olympiad training. Rating 5M: Very easy. This is the easiest rating which could actually appear while

  Math, Easy

Barycentric Coordinates in Olympiad Geometry

problems with a formula sheet beside them. I’ve included Appendix B to facilitate these needs; I also have a version which includes just the formula sheet and the problems, which should be oating around. Happy bashing! 1For ABC counterclockwise, this is positive when P, Q and R are in counterclockwise order, and negative other-wise.

  Sheet, Coordinates, Formula, Formula sheet, Barycentric coordinates, Barycentric

The OTIS Excerpts - Evan Chen

Introduction The book is divided into algebra, combinatorics, and number theory. We do not cover geometry, for which Euclidean Geometry in Mathematical Olympiads [Che16] already serves the role of “comprehensive book”. The twelve main chapters in this book are structured in to four sections.

  Introduction, Excerpt, Otis, Combinatorics, The otis excerpts

Introduction to Functional Equations

Evan Chen (October 18, 2016) Introduction to Functional Equations Remark 2.4. There are of course other approaches. Here is an outline of another one. After showing f is an involution, one can simultaneously let x = f(t), y = f(u) and instead obtain f(t2 + u) = tf(t) + f(u) (check this!). This quickly becomes a \Cauchy equation", see below ...

  Equations

The Incenter/Excenter Lemma - Evan Chen

midpoint of arc BC. Show that L is the center of a circle through I, I A, B, C. A BC I L I A Proof. This is just angle chasing. Let A = \BAC, B = \CBA, C = \ACB, and note that A, I, L are collinear (as L is on the angle bisector). We are going to show that LB = LI, the other cases being similar. First, notice that \LBI = \LBC + \CBI = \LAC ...

  Lemma, Incenter, The incenter excenter lemma, Excenter

Differential Privacy Overview - Apple

Hadamard Count Mean Sketch1 The Hadamard Count Mean–based Sketch technique uses a noise injection method similar to the one used in the Count Mean Sketch technique, but with an important difference: It applies a type of mathematical operation called a Hadamard basis transformation to the hashed encoding before performing the privatization step.

  Privacy, Differential, Apple, Differential privacy, Hadamard

3D Graphics in MATLAB

Hadamard matrix, an orthogonal matrix used in signal processing coding theory. The vectors theta and phi are in the range –Pi <= theta <=Pi , and -Pi/2 <= phi <= Pi/2. Because theta is a row vector and phi is a column vector, the multiplications that produce the matrices X, Y, and Z are vector outer products. k = 5; ...

  Graphics, Matlab, Hadamard, 3d graphics in matlab

Hadamard Matrices and Designs

Hadamard matrix are permuted, the matrix remains Hadamard. It is also true that if any row or column is multiplied by -1, the Hadamard property is retained. [Prove this] Thus, it is always possible to arrange to have the first row and first column of an Hadamard matrix contain only +1 entries. An Hadamard matrix in this form is said to be ...

  Hadamard

ECE 5325/6325: Wireless Communication Systems Lecture ...

ECE 5325/6325: Wireless Communication Systems Lecture Notes, Fall 2011 Prof. Neal Patwari University of Utah Department of Electrical and Computer Engineering

Handbook of Frequency Stability Analysis - NIST

iii Dedication This handbook is dedicated to the memory of Dr. James A. Barnes (1933−2002), a pioneer in the statistics of frequency standards.

  Stability, Frequency, Frequency stability

Introduction to Coding Theory Lecture Notes

Introduction to Coding Theory Lecture Notes∗ YehudaLindell DepartmentofComputerScience Bar-IlanUniversity,Israel January25,2010 Abstract These are lecture notes for an advanced undergraduate (and beginning graduate) course in Coding

  Coding, Theory, Coding theory

7. KOMPLEXE ZAHLEN und die KOMPLEXE e-FUNKTION

Spezialf alle: b) Die Zahlen auf der y-Achse heiˇen die imagin aren Zahlen. Insbesondere heiˇt i= (0;1) die imagin are Einheit. Die Multiplikation von z mit ibewirkt eine Drehung von z um