本スライドは、弊社の有田により2019年8月8日のArithmer Seminarで使用されたものです。
「高度数学」の一端を、社内の非エンジニアにも体感してもらおうと考え、中学校レベルの数学知識のみを前提としていくつかのトピックを紹介しています。
"Arithmer Seminar" is weekly held, where professionals from within and outside our company give lectures on their respective expertise.
The slides are made by the lecturer from outside our company, and shared here with his/her permission.
Arithmer株式会社は東京大学大学院数理科学研究科発の数学の会社です。私達は現代数学を応用して、様々な分野のソリューションに、新しい高度AIシステムを導入しています。AIをいかに上手に使って仕事を効率化するか、そして人々の役に立つ結果を生み出すのか、それを考えるのが私たちの仕事です。
Arithmer began at the University of Tokyo Graduate School of Mathematical Sciences. Today, our research of modern mathematics and AI systems has the capability of providing solutions when dealing with tough complex issues. At Arithmer we believe it is our job to realize the functions of AI through improving work efficiency and producing more useful results for society.
本スライドは、弊社の有田により2019年8月8日のArithmer Seminarで使用されたものです。
「高度数学」の一端を、社内の非エンジニアにも体感してもらおうと考え、中学校レベルの数学知識のみを前提としていくつかのトピックを紹介しています。
"Arithmer Seminar" is weekly held, where professionals from within and outside our company give lectures on their respective expertise.
The slides are made by the lecturer from outside our company, and shared here with his/her permission.
Arithmer株式会社は東京大学大学院数理科学研究科発の数学の会社です。私達は現代数学を応用して、様々な分野のソリューションに、新しい高度AIシステムを導入しています。AIをいかに上手に使って仕事を効率化するか、そして人々の役に立つ結果を生み出すのか、それを考えるのが私たちの仕事です。
Arithmer began at the University of Tokyo Graduate School of Mathematical Sciences. Today, our research of modern mathematics and AI systems has the capability of providing solutions when dealing with tough complex issues. At Arithmer we believe it is our job to realize the functions of AI through improving work efficiency and producing more useful results for society.
This document summarizes Fermat's little theorem and discusses primitive roots modulo n. It states that if p is a prime number, then for any integer a, ap-1 ≡ 1 (mod p). It then defines a primitive root modulo n as a number g such that g generates all numbers from 1 to n-1 when raised to successive powers modulo n. The document provides an example of a primitive root modulo 11 and discusses how primitive roots relate to permutations.
Conditional expectation projection 2018 feb 18 HanpenRobot
The document discusses conditional expectation and orthogonal projection in an inner product space L2. It defines the inner product of random variables X and Y in L2 as their expected value E(XY). It also states that for a subspace L2G of L2, the conditional expectation E(Y|G) is the orthogonal projection of Y onto L2G, and that the expected value of X(Y - E(Y|G)) is 0.
The Laplace transform exhibits a property of duality where the Laplace transform of the nth derivative of a function f(t) is equal to (-1)^n times the nth derivative of the Laplace transform of f(t) with respect to s. This property allows derivatives in the time domain to correspond to derivatives in the frequency domain. As an example, the Laplace transform of t^-3 * e^(-ct) is shown to equal 6/(s+c)^4 by applying this duality property.
The document discusses conjugate cyclic permutations in group theory. It defines a cyclic permutation τ as a bijection from a set of numbers n to itself, where it maps n to n and a number k to the number following it. It then defines the conjugate of a cyclic permutation τ by another cyclic permutation τ-1 as mapping the elements i1, i2, ..., ir of τ to their images under τ-1 in the same order. So the conjugate of τ by τ-1 is obtained by applying τ-1 to each element of τ.
The document discusses the derivative of a quadratic form Q(x) = xTAx. It shows that the gradient of Q(x) is equal to 2Ax. To find the saddle point of Q(x), the derivative with respect to x, which is 2Ax, is set equal to 0. Solving this results in an expression for the coordinates x and y of the saddle point in terms of the coefficients of the quadratic form matrix A.
