Chapter 1 Geometric setting
... Definition 13. The norm of a located vector XY XY ∥ := ∥Y − X∥. Note that this definition corresponds to our intuition in dimension n = 2 or n = 3 for the length of an arrow. It should also be observed that two equivalent located vectors have the same norm. Definition 14. Let A, B, C, D ∈ Rn . ...
... Definition 13. The norm of a located vector XY XY ∥ := ∥Y − X∥. Note that this definition corresponds to our intuition in dimension n = 2 or n = 3 for the length of an arrow. It should also be observed that two equivalent located vectors have the same norm. Definition 14. Let A, B, C, D ∈ Rn . ...
RELATIVISTIC ADDITION AND GROUP THEORY 1. Introduction
... • There is u ∈ I such that for any x ∈ I, F (x, u) = F (u, x) = x. • For any x ∈ I there is some i(x) ∈ I such that F (x, i(x)) = F (i(x), x) = u. • For x, y, z ∈ I, F (F (x, y), z) = F (x, F (y, z)). Our goal is to prove the following theorem. Theorem 2.1. If F (x, y) = x ∗ y has continuous partial ...
... • There is u ∈ I such that for any x ∈ I, F (x, u) = F (u, x) = x. • For any x ∈ I there is some i(x) ∈ I such that F (x, i(x)) = F (i(x), x) = u. • For x, y, z ∈ I, F (F (x, y), z) = F (x, F (y, z)). Our goal is to prove the following theorem. Theorem 2.1. If F (x, y) = x ∗ y has continuous partial ...
HW #3 Solutions
... absolute maximum and absolute minimum of f(x) on this interval, and also indicate all xvalues at which these two extreme values occur. Solution to #3: We are given a continuous function on a closed, bounded interval, so the Extreme Value Theorem says that () achieves both an absolute maximum and a ...
... absolute maximum and absolute minimum of f(x) on this interval, and also indicate all xvalues at which these two extreme values occur. Solution to #3: We are given a continuous function on a closed, bounded interval, so the Extreme Value Theorem says that () achieves both an absolute maximum and a ...
EE 302: Probabilistic Methods in Electrical Engineering Test II
... is shown to you and contains a value of fX (x), for some unknown value of x. The second card is not shown to you but you are told that it will contain a value X = x. You are asked to select a value X = x, then only you will know what value of X = x the second card contains. What value of X = x would ...
... is shown to you and contains a value of fX (x), for some unknown value of x. The second card is not shown to you but you are told that it will contain a value X = x. You are asked to select a value X = x, then only you will know what value of X = x the second card contains. What value of X = x would ...
Random number generator
... assigning a “head”, and “tail” for the random number which is greater, and less than 0.5, respectively, simulate the behavior of tossing a coin by finding the probability of head or tail in a consecutive N-times tossing test. Plot the results of probability of head or tail with respect to different ...
... assigning a “head”, and “tail” for the random number which is greater, and less than 0.5, respectively, simulate the behavior of tossing a coin by finding the probability of head or tail in a consecutive N-times tossing test. Plot the results of probability of head or tail with respect to different ...
Formal power series
... PRIMITIVE if n>0 and all the partial sums are positive (except for the empty sum and the total sum), and COMPOSITE if n>0 and some partial sum is zero (i.e., the sequence splits up into two or more non-trivial ballot sequences). Let b(n) = # of ballot sequences of length n, p(n) = # number of primit ...
... PRIMITIVE if n>0 and all the partial sums are positive (except for the empty sum and the total sum), and COMPOSITE if n>0 and some partial sum is zero (i.e., the sequence splits up into two or more non-trivial ballot sequences). Let b(n) = # of ballot sequences of length n, p(n) = # number of primit ...
recursive sequences ppt
... Another example; This time write the recursive formula • Briana borrowed $870 from her parents for airfare to Europe. She will pay them back at the rate of $60.00 per month. Let an be the amount she still owes after n months. Find a recursive formula for this sequence. ...
... Another example; This time write the recursive formula • Briana borrowed $870 from her parents for airfare to Europe. She will pay them back at the rate of $60.00 per month. Let an be the amount she still owes after n months. Find a recursive formula for this sequence. ...
The Unexpected Appearance of Pi in Diverse Problems
... The argument used in proving the Theorem above can be modified to give a proof of the fact that there are infinitely many prime numbers. The probability that a randomly picked number from the set {1, 2, , N} is 1 goes to zero as N becomes large. So the product ITp (1 - lip) where P varies over all p ...
... The argument used in proving the Theorem above can be modified to give a proof of the fact that there are infinitely many prime numbers. The probability that a randomly picked number from the set {1, 2, , N} is 1 goes to zero as N becomes large. So the product ITp (1 - lip) where P varies over all p ...
LAWS OF LARGE NUMBERS FOR PRODUCT OF RANDOM
... Example 2. Sources of pollution. Suppose it is known that mn sources of pollution entered a region Cn but the positions of them are unknown. Suppose further that the polluting power of each source is known and that each source damages a circular region around it proportional to its polluting power. ...
... Example 2. Sources of pollution. Suppose it is known that mn sources of pollution entered a region Cn but the positions of them are unknown. Suppose further that the polluting power of each source is known and that each source damages a circular region around it proportional to its polluting power. ...
21 Gaussian spaces and processes
... each other. Thus we may start with an arbitrary map Ψ : I → H from an arbitrary set I to an arbitrary separable Hilbert space H, and use a linear isometry between H and a Gaussian space G for constructing a Gaussian process Ξ : I → G isometric to Ψ in the sense that E Ξ(i1 )Ξ(i2 ) = hΨ(i1 ), Ψ(i2 )i ...
... each other. Thus we may start with an arbitrary map Ψ : I → H from an arbitrary set I to an arbitrary separable Hilbert space H, and use a linear isometry between H and a Gaussian space G for constructing a Gaussian process Ξ : I → G isometric to Ψ in the sense that E Ξ(i1 )Ξ(i2 ) = hΨ(i1 ), Ψ(i2 )i ...
Discrete Random Variables
... • Definition: Mathematically, a random variable (r.v.) on a sample space S is a function1 from S to the real numbers. More informally, a random variable is a numerical quantity that is “random”, in the sense that its value depends on the outcome of a random experiment. • Notation: One commonly uses ...
... • Definition: Mathematically, a random variable (r.v.) on a sample space S is a function1 from S to the real numbers. More informally, a random variable is a numerical quantity that is “random”, in the sense that its value depends on the outcome of a random experiment. • Notation: One commonly uses ...
Means and Variances of Random Variables
... Rules for Means 1) If X is a random variable and a and b are fixed numbers, then : a+bX = a + bX Example : What if Homer tries Nelsons game, but he decides to award $10 for every head that Nelson flips. ...
... Rules for Means 1) If X is a random variable and a and b are fixed numbers, then : a+bX = a + bX Example : What if Homer tries Nelsons game, but he decides to award $10 for every head that Nelson flips. ...
