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Continuity of Functions of Several Variables Examples 1

# Continuity of Functions of Several Variables Examples 1

Recall from the Continuity of Functions of Several Variables page that if $z = f(x, y)$ is a two variable real-valued function, then $f$ is said to be continuous at a point $(a, b) \in D(f)$ if $\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)$. Similarly, if $w = f(x, y, z)$ is a three variable real-valued function, then $f$ is said to be continuous at a point $(a, b, c) \in D(f)$ if $\lim_{(x, y, z) \to (a, b, c)} f(x, y, z) = f(a, b, c)$.

We will now look at some more difficult questions regarding the continuity of functions of several variables.

## Example 1

Consider the two variable real-valued function $f(x, y) = \frac{x^2 + y^2 - x^3y^3}{x^2 + y^2}$ for all $(x, y) \in \mathbb{R}^2 \setminus \{ (0, 0) \}$. Determine whether $f$ can be redefined as $\hat{f}$ at the origin such that $\hat{f}$ is continuous for all $(x, y) \in \mathbb{R}^2$.

We first note that $f$ is continuous for all $(x, y) \in \mathbb{R}^2$ such that $x^2 + y^2 \neq 0$ since $f$ is a rational function. In particular, we see that $x^2 + y^2 = 0$ if and only if $(x, y) = (0, 0)$. Hence, the only point of discontinuity occurs at $(0, 0)$. If we can show that $\lim_{(x, y) \to (0, 0)} f(x, y) = L$ for some $L \in \mathbb{R}$ and then define $\hat{f}(0,0) = L$ then we indeed can redefined $f$ at the origin as $\hat{f}$ such that $\hat{f}$ is continuous on all of $\mathbb{R}^2$.

Note that:

(1)
\begin{align} \quad \lim_{(x, y) \to (0,0)} \frac{x^2 + y^2 - x^3y^3}{x^2 + y^2} = \lim_{(x, y) \to (0,0)} \left ( 1 - \frac{x^3y^3}{x^2 + y^2} \right ) \end{align}

Now notice that $\lim_{(x, y) \to (0,0)} \frac{x^3y^3}{x^2 + y^2} = 0$ since:

(2)
\begin{align} \quad 0 ≤ \biggr \rvert \frac{x^3y^3}{x^2 + y^2} \biggr \rvert = \frac{x^2}{x^2 + y^2} \mid x \mid \mid y^3 \mid ≤ \mid x \mid \mid y^3 \mid \end{align}

We have that $\lim_{(x, y) \to (0, 0)} 0 = 0$ and $\lim_{(x, y) \to (0, 0)} \mid x \mid \mid y^3 \mid = 0$, and so by the Squeeze Theorem for multivariable limits we have that $\lim_{(x, y) \to (0,0)} \frac{x^3y^3}{x^2 + y^2} = 0$. Hence:

(3)
\begin{align} \quad \lim_{(x, y) \to (0,0)} \frac{x^2 + y^2 - x^3y^3}{x^2 + y^2} = 1 - 0 = 1 \end{align}

So we will redefine $f$ as follows:

(4)
\begin{align} \quad \hat{f}(x, y) = \left\{\begin{matrix} f(x,y) & \mathrm{if} \: (x, y) \neq (0, 0) \\ 1 & \mathrm{if} \: (x, y) = (0, 0) \end{matrix}\right. \end{align}

Then $\hat{f}$ is continuous for all $(x, y) \in \mathbb{R}^2$. The graph below is of $\hat{f}$.

## Example 2

Let $a$, $b$, and $c$ be nonnegative integers. Prove that $\lim_{(x,y) \to (0,0)} \frac{x^ay^b}{(x^2 + y^2)^c}$ exists if $a + b > 2c$.

To show this, note that $\mid x \mid ≤ \sqrt{x^2 + y^2}$ and $\mid y \mid ≤ \sqrt{x^2 + y^2}$. Now applying the Squeeze Theorem for multivariable functions we have that:

(5)
\begin{align} \quad 0 ≤ \biggr \rvert \frac{x^a y^b}{(x^2 + y^2)^c} = \frac{\mid x \mid^a \mid y \mid^b}{(x^2 + y^2)^c} ≤ \frac{\sqrt{x^2 + y^2}^a \sqrt{x^2 + y^2}^b}{(x^2 + y^2)^c} = \frac{(x^2 + y^2)^{\frac{a + b}{2}}}{(x^2 + y^2)^c} = (x^2 + y^2)^{\frac{a + b}{2} - c} \end{align}

Now $\lim_{(x, y) \to (0, 0)} 0 = 0$ and $\lim_{(x, y) \to (0, 0)} (x^2 + y^2)^{\frac{a + b}{2} - c}$ if $\frac{a+b}{2} - c > 0$, that is $a + b > 2c$. So if $a + b > 2c$, we can invoke the Squeeze Theorem and get that:

(6)
\begin{align} \quad \lim_{(x,y) \to (0,0)} \frac{x^ay^b}{(x^2 + y^2)^c} = 0 \end{align}


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# objective c programming: the big nerd ranch guide

3.0.3913.0. eval(ez_write_tag([[300,250],'calculator_academy-medrectangle-4','ezslot_13',107,'0','0']));eval(ez_write_tag([[300,250],'calculator_academy-medrectangle-4','ezslot_14',107,'0','1']));eval(ez_write_tag([[300,250],'calculator_academy-medrectangle-4','ezslot_15',107,'0','2'])); To view this video please enable JavaScript, and consider upgrading to a Result. In other words if you have a problem, say log base 2 of 4, the answer is 2 because 2^2=4. Find the logarithm with base 10 of number 100. lg(100) = 2. Supposez que nous ayons la puissance , alors le logarithme de N dans la base a est égal à x, ou Learn more about log rules, or explore hundreds of other calculators addressing topics such as math, finance, health, and fitness, among others. Please note that the base of log number b must be greater than 0 and must not be equal to 1. $$\mathbf{Now} a^x \times a^y = MN \mathbf{ or}, a^{x + y} = MN$$ Therefore from definition, we have, $$\log_a(MN) = x + y = \log_a M + \log_a N$$ [putting the values of x and y]. The logarithm is the inverse operation to the exponentiation (power function). and it is widely used in computer science and programming languages. Let's say it's 100. There are a few specific types of logarithms. Les cookies nous permettent de personnaliser le contenu et les annonces, d'offrir des fonctionnalités relatives This is how you can write a log. In other words, a logarithm can be represented as the following: with b being the base, x being a real number, and y being an exponent. Unfortunately, not all logarithms can be calculated that easily. (i) Putting M = a on both sides of the change of base rule formula [$$\log_a M = \log_b M \times \log_b$$] we get, $$\log_a a = \log_b a \times \log_a b = 1$$ [since, loga a = 1], $$\log_b M = \dfrac{\log_a M}{\log_a b}$$, It’s a very useful calculator for measuring log. Base de 10. Une vidéo sur youtube explique ce point avec plus de détails et de manière très claire. After using logarithm calculator, we can find out that. Principe de calcul du logarithme. This is an example of a simple logarithm as it basically counts the number of multiplications of the same factor - in this case 2. By … The notation is logbx or logb(x) where bis the base and x is … Il calcule également le logarithme naturel et le logarithme décimal. Finally, we need to set up the equation above to solve for X. log2 (x + 1) = log3 (27) ln (x + 2) − ln (x + 1) = 1 ln (x) + ln (x − 1) = ln (3x + 12) 4 + log3 (7x) = 10 ln = Calculate × Reset. Log base 2: an example. This log equation solver is a free to use tool to facilitate mathematicians, engineers, physicists and students in particular. Log calculator is a key that gives great opportunity to work with logarithms. Logarithm is considered to be one of the basic concepts in mathematics. Natural Logarithm Calculator. content_copy. In the simplest case, the log sums the number of occurrences of the same factor in repetitive multiplication; e.g., since $$10000 = 10 \times 10 \times 10 \times 10 = 104$$, the "logarithm to base 10" of 10000 is 4. What this implies is that the log of a given number, say x, is exponent to which another static number, the base (say b), must be elevated, to reproduce the number x. Le logarithme en base de 10 peut s'écrire sous cette forme log10 ou juste log. Il existe plusieurs logarithmes selon la base utilisée. $$\mathbf{Let:} \log_a M = x \Rightarrow a^x = M^n$$  , $$\mathbf{and} \log_a b = z \Rightarrow a^z = b$$, $$\mathbf{Now} a^x = M = b^y - (a^z)y = a^yz$$, $$\mathbf{Therefore} x = yz \space \mathbf{or} \space \log_a M = \log_b M \times \log_a b$$ [putting the values of x ,y and z]. For example, to get 16, it is necessary to raise two to the fourth power. Its just a matter of choosing. Some people confuse ln function with antilog. In Math, the log is the reverse function of exponentiation. This is an operation in mathematics, known as exponentiation. The logarithm of a number x with respect to base b is the exponent to which b has to be raised to yield x. The natural logarithm has the constant e (approximately equal to 2.718281828) as its base. In the simplest case, the log sums the number of occurrences of the same factor in repetitive multiplication; e.g., since $$10000 = 10 \times 10 \times 10 \times 10 = 104$$, the "logarithm to base 10" of 10000 is 4. En appuyant sur RESET vous pouvez effacer les résultats. About Log Base 2 Calculator The Log Base 2 Calculator is used to calculate the log base 2 of a number x, which is generally written as lb (x) or log 2 (x). Cette calculatrice calcule rapidement la decimal d'un logarithme en base de 10. In other words if you have a problem, say log base … The natural logarithm of x is the base e logarithm of x: ln x = log e x = y. The Log (Logarithm) Calculator is used to calculate the logarithm log b x for a base b and a number x. Logarithm. En d'autre termes, on doit résoudre l'équation suivante: 10x = 100. For example, utilizing the “Log” function on the number 10 would disclose that there is a need to increase the base number of 10 by itself to equal the number 10. In general, the natural logarithm of x is written as lnx, or if the basis e is implied, it is written as log x. The user-interface of this ln calculator ensures that even this is not necessary but we will list the steps just in case. See also. For example, 23 = 8 ⇒ log2 8 = 3 (the logarithm of 8 to base 2 is equal to 3, because 23 = 8). Therefore, logarithm is the exponent to which it is necessary to raise a fixed number (which is called the base), to get the number y. Pour les garder et les exporter au format CSV appuyez sur SAUVEGARDE. you can contact us anytime. Ainsi log(100) signifie que l'on cherche le résultat du logarithme en base de 10. Binary logarithm of value a is the power x raised to number 2. And the number (x) which we are calculating log base of (b) must be a positive real number. Le nombre ne pourra jamais être négatif. The logarithm for an user-defined base can be calculated as well as all inverse calculations. avec d'autres informations que vous leur avez fournies ou qu'ils ont collectées lors de votre utilisation de Le but du présent outil est de connaître le logarithme décimal en base de 10. x: is real number, x>0, logb(x) = y, and x = logb(bx) and it has a huge number of applications in engineering, scientific research, technology, etc. You can calculate anything on Calculators.tech. You have to follow very simple steps to do the calculation. This is how you can write a log. Par conséquent, le résultat de log10(100) = 2. Conceptually, the log of a number y with base be is equal to the exponential value the base needs to be raised to in order to equal y. For this example we will take a base of 5. The logarithm of a number x with respect to base b is the exponent to which b has to be raised to yield x. Finally, so called natural logarithm uses the number e (which is approximately equal to 2.71828) as its base, The logarithm to base 10 is usually referred to as the common logarithm, $$\mathbf{Let:} \log_a M = x \Rightarrow x^a = M$$ $$\mathbf{ and } N = y \Rightarrow a^y = N$$. and other precise sciences. Logarithms being special ways to make clear what exponent needed to multiply into a specific number. The logarithm (log) operation in mathematics is the inverse to exponentiation, meaning that the log of a number is the exponent to which another fixed number called a "base" was raised to produce the number. JavaScript has to be enabled to use the calculator. Fill the calculator form and click on Calculate button to get result here. And to get a 64, you need to raise two to the sixth power. Let's assume you want to use this tool as a log base 2 calculator. supports HTML5 video, Calculator Academy© - All Rights Reserved 2020, how to calculate log base 2 without calculator, how to calculate log base 2 in calculator, how to calculate log base 2 with calculator, how to calculate log base 2 in scientific calculator, how to find log base 2 in scientific calculator, how to take log base 2 in scientific calculator, how to use log base 2 in scientific calculator, how to put log base 2 in scientific calculator, log to the base 2 in scientific calculator, how to solve log base 2 in scientific calculator, how to find log to the base 2 in calculator, how to find log base 2 value in scientific calculator, how to calculate log base 2 using scientific calculator, how to calculate log2 in scientific calculator, how to solve log base 2 without calculator.

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## Problem 488

Let $G$ be a finite group and let $S$ be a non-empty set.
Suppose that $G$ acts on $S$ freely and transitively.
Prove that $|G|=|S|$. That is, the number of elements in $G$ and $S$ are the same.

## Problem 487

Let
$P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}$ be an ideal of the ring
$\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.$ Then determine the quotient ring $\Z[\sqrt{10}]/P$.
Is $P$ a prime ideal? Is $P$ a maximal ideal?

## Problem 486

Determine whether there exists a nonsingular matrix $A$ if
$A^4=ABA^2+2A^3,$ where $B$ is the following matrix.
$B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.$

If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 485

Let
$A=\begin{bmatrix} 1 & -14 & 4 \\ -1 &6 &-2 \\ -2 & 24 & -7 \end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix} 4 \\ -1 \\ -7 \end{bmatrix}.$ Find $A^{10}\mathbf{v}$.

You may use the following information without proving it.
The eigenvalues of $A$ are $-1, 0, 1$. The eigenspaces are given by
$E_{-1}=\Span\left\{\, \begin{bmatrix} 3 \\ -1 \\ -5 \end{bmatrix} \,\right\}, \quad E_{0}=\Span\left\{\, \begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix} \,\right\}, \quad E_{1}=\Span\left\{\, \begin{bmatrix} -4 \\ 2 \\ 7 \end{bmatrix} \,\right\}.$

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 484

Let $A$ be a square matrix and its characteristic polynomial is give by
$p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).$ Find the rank of $A$.

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 483

Diagonalize the matrix
$A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &1 &1 \\ 1 & 1 & 1 \end{bmatrix}.$ Namely, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 482

For which values of constants $a, b$ and $c$ is the matrix
$A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}$ diagonalizable?

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 481

Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less.
Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where
\begin{align*}
\end{align*}

(a) Find a basis of $P_2$ among the vectors of $S$. (Explain why it is a basis of $P_2$.)

(b) Let $B’$ be the basis you obtained in part (a).
For each vector of $S$ which is not in $B’$, find the coordinate vector of it with respect to the basis $B’$.

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 480

(a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying
$2x+4y+3z+7w+1=0.$ Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(b) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying
$2x+4y+3z+7w=0.$ Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(These two problems look similar but note that the equations are different.)

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 479

Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix.

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 478

Let $T:\R^2 \to \R^3$ be a linear transformation given by
$T\left(\, \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \,\right) = \begin{bmatrix} x_1-x_2 \\ x_2 \\ x_1+ x_2 \end{bmatrix}.$ Find an orthonormal basis of the range of $T$.

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 477

Determine whether the matrix
$A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}$ is diagonalizable.

If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 476

Let
$A=\begin{bmatrix} 1 & 2 & 1 \\ -1 &4 &1 \\ 2 & -4 & 0 \end{bmatrix}.$ The matrix $A$ has an eigenvalue $2$.
Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$.

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 475

Find all the eigenvalues of the matrix
$A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 &0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}.$

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 474

Prove that every finite group of order $72$ is not a simple group.

## Problem 473

Prove that the determinant of an $n\times n$ skew-symmetric matrix is zero if $n$ is odd.

## Problem 472

Let $T:\R^2 \to \R^2$ be a linear transformation and let $A$ be the matrix representation of $T$ with respect to the standard basis of $\R^2$.

Prove that the following two statements are equivalent.

(a) There are exactly two distinct lines $L_1, L_2$ in $\R^2$ passing through the origin that are mapped onto themselves:
$T(L_1)=L_1 \text{ and } T(L_2)=L_2.$

(b) The matrix $A$ has two distinct nonzero real eigenvalues.

## Problem 471

Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.

(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.

(b) Let
$A^{100}=aA^2+bA+cI,$ where $I$ is the $3\times 3$ identity matrix.
Using the Cayley-Hamilton theorem, determine $a, b, c$.

(Kyushu University, Linear Algebra Exam Problem)

## Problem 470

Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer.
Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$.
Then prove that $H$ is a normal subgroup of $G$.

(Michigan State University, Abstract Algebra Qualifying Exam)

## Problem 469

Let $H$ be a subgroup of a group $G$.
Suppose that for each element $x\in G$, we have $x^2\in H$.

Then prove that $H$ is a normal subgroup of $G$.

(Purdue University, Abstract Algebra Qualifying Exam)


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TL;DR

On with TASK #2 from The Weekly Challenge #159. Enjoy!

The challenge

You are given a positive number $n. Write a script to generate the Moebius Number for the given number. Please refer to wikipedia page for more informations. Example 1: Input:$n = 5
Output: -1

Example 2:

Input: $n = 10 Output: 1  Example 3: Input:$n = 20
Output: 0

The questions

This challenge is a sibling of PWC150 - Square-free Integer, with the exception that this time we’re told that we have to consider only positive numbers. I mean… integer numbers, right?

The solution

We’re adapting the solution from PWC150 - Square-free Integer, just to remain on the lazy side:

#!/usr/bin/env perl
use v5.24;
use warnings;
use experimental 'signatures';
no warnings 'experimental::signatures';

say $_, ' ', moebius_number($_) for (@ARGV ? @ARGV : (1 .. 10));

sub moebius_number ($n) { return 0 unless$n % 4;
($n, my$n_divisors) = $n % 2 ? ($n, 0) : ($n / 2, 1); my$divisor = 3;
while ($n >=$divisor) {
if ($n %$divisor == 0) {
++$n_divisors;$n /= $divisor; return 0 unless$n % $divisor; }$divisor += 2; # go through odd candidates only
}
return 1 - 2 * ($n_divisors % 2); }  And the same goes for Raku: #!/usr/bin/env raku use v6; sub MAIN (*@args) { put$_, ' ', möbius-number($_) for @args} sub möbius-number ($n is copy) {
return 0 if $n %% 4; ($n, my $n-divisors) =$n %% 2 ?? (($n / 2).Int, 1) !! ($n, 0);
my $divisor = 3; while$n >= $divisor { if$n %% $divisor { ++$n-divisors;
$n = ($n / $divisor).Int; return 0 if$n %% $divisor; }$divisor += 2; # go through odd candidates only
}
return 1 - 2 * ($n-divisors % 2); }  The case for divisibility by 2 is handled specially just to be able to increment the $divisor variable by 2 instead of by 1. Apart from this, we’re sticking to the definition in the Wikipedia page, returning 0 whenever we find a square in the divisors, or using the even/odd count of prime factors to figure out the right value to return.

Stay safe folks!


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# 4 The Collection in Photos

At present, the collection contains a total of 294 slide rules, with 266 linear slide rules, 23 circular rules, as well as 2 and 3 spiral and helical rules, respectively. The “rule” for entrance into the collection is that there must be a logarithmic scale as part of the device. Hence sectors and Gunter’s rules are eligible, for instance, though they are not technically “slide” rules. However, items such as Addiators, Curtas, and other mechanical computational tools are not included.

Perhaps the best way to scan through the slide rule collection is by way of the images in this chapter. The entries are ordered according to Year and sub-ordered according to Maker and Model. Some of the dates for the slide rules are accurately known, but for many the dates are simply honest guesses. For a few further details of each rule of the Collection as well as other information regarding manufacturers, see Slide Rule Makers. Looking for a particular rule, or set of slide rules? A Searchable Table can be used for a quick scan of the entire collection.

The concept of a logarithm was developed by John Napier starting at the end of the 16th century, with his major publication on the subject in 1614. Napier’s tables of logarithms, soon improved upon by his colleague Henry Briggs, could be used to more easily perform multiplication and division of numbers used in astronomy and navigation, for example. By 1620 Edmund Gunter created a scale on a rule that had distances to the numbers on the scale proportional to the logarithms of those numbers, thus allowing quick multiplication without the use of tables. He used calipers or “compasses” to measure distances along the scales and do the additions. Gunter’s scales were of great use in navigation, where bearings were made and trigonometric calculations performed to map out a course. “Gunter’s Rules” had logarithmic scales from 1 to 10 or 1 to 100, as well as logarithmic scales of trigonometric functions. Another instrument often used for navigational calculations, available since the time of Galileo (perhaps its inventor) in the 1500s, was the sector. It had scales that could be used for multiplication and division, finding roots and squares and trigonometric values, and was also used with calipers or compasses. English sectors in the mid-1600s began to include Gunter’s scales on them as well.

Within a short time, William Oughtred showed how two Gunter rules could be slid back and forth next to each other to perform the same task without the use of calipers and the “slide rule” was born. The earliest examples of these instruments are now found in museums, while the proliferation of slide rules within the general public did not begin to take hold until perhaps the middle of the 19th century.

Continue to Slide Rule Makers


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# How to consider gravity in the design of a compensator?

I have to solve an exercise for the Digital Control System course (using MATLAB software) which stands:

"A ball is suspended inside a vertical tube by airflow 'u' and connected via a spring of stiffness K to the bottom of the tube. The ball is subjected to gravity and a viscous friction with coefficient 'B'. The force 'F' exerted on the ball by the airflow is proportional to the airflow 'u' via the constant G; airflow can only be positive (entering the tube)."

I have also all the data needed to solve the problem numerically, but this is not important for the question. What I need to do is: "Write the system equations in state space form with airflow as input and the ball vertical position 'z' as output.Then, select a sampling time and design a digital control system that regulates the ball position by acting on the airflow to the following specifications:

• Zero steady-state error (in response to the desired altitude step input).

• Max overshoot: 30%;

• Settling time at 5% less than 8 seconds."

After this, we have to compute the transfer function of the plant and put it in unitary feedback with the compensator.

I usually write first the system dynamics equations, then from these I choose a suitable set of state variables and I write the matrices A, B, C and D according to the state variables (I use the 'ss' function). The problem is that I don't know how to consider the gravity in this case because it comes in the system dynamics as a constant term (-g*m). For example by considering the state variables as [z' z] I obtained the following matrices:

A = [-B/m -K/m; 1 0];

B = [G/m; 0];

C = [0 1];

D = 0;

I tried to design the compensator (a simple PID) without considering the gravity and by adding it later in the Simulink model used to test the system (after designing the compensator we have to build a Simulink model in which the discrete time compensator is tested with the transfer function of the continuous time system) but of course the system output is no more able to meet the requirements. For the gravity transfer function I considered to have the mass as input and the position as output

Am I wrong in not considering the gravity when designing the compensator? Or perhaps, if correctly implemented the gravity should not affect the system output?

Why don't you use pole placement method to meet your control objective. Since you already have a state space model it is much easier. You have to just check the controllability of your system and then based on control requirement decide on the poles. You can use cmu's Controll tutorial. http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction&section=ControlStateSpace

At the same time it doesn't not make sense to me to model the whole system just to make a PID controller. With a good enough system model you can make better controllers such lqr etc.

However if you are hell bent on using a PID controller, you can still improve your PID controllers performance by making use of anti-windup and thereby making integral term much more aggressive which will increasing the response time. PID controllers don't require gravity compensators (PD controller does), but you can assist it by providing feed forward and you can always tune it to meet your requirement. At the same time you can use bode plot to check whether it is theoretically possible to attain such a bandwidth of the system without making it unstable.

The presence of gravity does impact on how fast you can achieve steady-state with zero error, because the integral part of your PID compensator needs to charge in order to counteract the term $mg$. As a result, you won't meet the requirement on the settling time, with the same setting of the gains.

It's still possible to follow your design steps if you consider adding eventually a feed-forward term $u_{ff}=mg/G$ to sum up at the output of your PID.

Therefore, $$u=u_{PID}+u_{ff}$$

• Thank you for the answer, but from a theoretical point of view what I did to consider the effect of the gravity on the system is correct? Because I also thought that I could simply compute the equilibrium point of the system (z0 by which we have Kz0 = mg) and then subtract that position from the system transfer function output, is it right? – PaulRox Jan 18 '17 at 9:28
• Hi Paul, from a theoretical standpoint, gravity acts as a disturbance, thus you should study the sensitivity function that accounts for the effect of the disturbance on the output. Here some notes on that. – Ugo Pattacini Jan 18 '17 at 17:00


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# The Minimax Learning Rates of Normal and Ising Undirected Graphical Models

18 Jun 2018Luc DevroyeAbbas MehrabianTommy Reddad

Let $G$ be an undirected graph with $m$ edges and $d$ vertices. We show that $d$-dimensional Ising models on $G$ can be learned from $n$ i.i.d... (read more)

PDF Abstract


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Sales Toll Free No: 1-855-666-7446

Conservation of Linear Momentum

Top
 Sub Topics In physics, there are a number of conservation laws like linear momentum, angular momentum etc. These are fundamental and symmetric in nature and are verified by experiments. The law of conservation of linear momentum was proposed by Newton three centuries ago that became a basic law in physics. Let us know more about it.

What is Linear Momentum?

Momentum in general is the measure of the quantity of motion. The linear momentum is the momentum in linear path. It is also defined as a body's translation motion and is measure of the product of mass m and velocity v, given as
p = mv

The SI unit of linear momentum is kilogram meters per second (kgms-1).

Principle of Conservation of Linear Momentum

The linear momentum is the product of the mass of an object and the velocity of an object. Let take any two objects moving with some velocities towards each other. Then the momentum will be conserved. Hence,

the principle of conservation of linear momentum states that the sum of linear momentum in a closed system is constant. If p1, p2, p3,.... are the momentum in a closed system then principle of conservation of momentum states that the total momentum is constant.
p1 + p2 + p3 + p4 + ....... = constant.

Conservation of Linear Momentum Equation

We know that the linear momentum is given as
$\vec{p}$ = m $\vec{v}$

Using it we can rewrite newtons second law as
$\vec{F}$ = m$\vec{a}$ = m $\frac{d \vec{v}}{dt}$ = $\frac{d\vec{p}}{dt}$
A special case is when $\vec{F}$ = $\vec{0}$ => $\vec{p}$ is constant.

Let us consider the system of particles then the momentum for the system of particles is given as
$\vec{P}$ = $\sum_i$ $\vec{p_i}$ = $\sum_i$ mi $\vec{v_i}$

To get the linear momentum of individual particles we go for the larger momentum $\vec{P}$ in a many particle system.
$\frac{d\vec{P}}{dt}$ = $\sum_i$ $\frac{d \vec{p}}{dt}$ = $\sum_i$ Fi = $\vec{F_{net}}$
If no net force acts on the system of particles then we get
$\vec{F_{net}}$ = $\vec{0}$ => P is a constant

Thus, the linear momentum of the system of particles is conserved when no net force acts on the system.

Law of Conservation of Linear Momentum

Let us consider two bodies of masses m1 and m2 moving towards each other having initial velocities u1 and u2. Both collide with each other for time t. After the collision, they move with the velocities v1 and v2.

From newtons second law, Force applied by A on B is given by

FAB = $\frac{m_2 v_2 - m_2 u_2}{t}$

Similarly, Force applied by B on A
FBA = $\frac{m_1 v_1 - m_1 u_1}{t}$

From Newton’s third law we have
FAB = -FBA

Hence,
$\frac{m_2v_2 - m_2u_2}{t}$ = $\frac{-(m_1 v_1 - m_1u_1)}{t}$

Or

m2v2-m2u2 = -m1v1+m1u1

Or

m1u1 + m2u2 = m1v1 + m2v2

This means the total momentum before collision is equal to total momentum after collision.

Conservation of Linear Momentum Example

Lets go through some examples on linear momentum:

Bullet and gun : when a bullet is fired from the total linear momentum of the system remains conserved. On firing the gun the bullet moves forward whereas gun recoils backward.

Motion of a rocket : The upward motion of rocket can be explained using law of conservation of linear momentum. In a rocket the burnt gases comes out of the nozzle in downward direction with tremendous velocity. As a result the rocket gains an equal linear momentum to move in opposite direction that makes the rocket move with greater speed.

Boy and boat : A boy sitting in a boat suddenly jumps into the water that makes the boat move in opposite direction. Hence momentum is conserved.

Lets go through some sample numerical examples on conservation of linear momentum:

Solved Examples

Question 1: A body has mass of 2 kg moves with velocity 3ms-1. Calculate its linear momentum.
Solution:

Given :
Mass m = 2 kg, Velocity v = 3 ms-1

The linear momentum is given by
p = mv
= 2 kg $\times$ 3 ms-1
= 6 kgms-1.

Question 2: A bullet of mass 50 g moves with a velocity of 200 ms-1 collides with a freely suspended bag of mass 1 kg and gets struck in it. Calculate the velocity the sand bag acquires.
Solution:

Given:
Mass of bullet m1 = 50 g
= $\frac{50}{1000}$
= 0.050 kg

Velocity of bullet v1 = 200 ms-1

Therefore, Linear momentum of bullet = m1v1
= (0.050 kg) $\times$ (200 ms-1)
= 10 kg ms-1

As the bullet gets struck into the sand bag (mass M = (0.050 + 1)kg = 1.050 kg) the linear momentum after collision = 1.05 $\times$ v2 kg ms-1

By law of conservation of momentum, linear momentum before collision = Momentum after collision
10 kgms-1 = 1.050 v2 kgms-1

Final Velocity v2 = $\frac{10\ kgms^{-1}}{1.050 kg}$
= 9.52 ms-1


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# Naturally Occurring Uranium in Groundwater in Northeastern Washington State

May 27, 2020

Uranium is a radioactive element (radionuclide) that occurs naturally in rock, soil, and water – usually in low concentrations. Radionuclides are unstable atoms with excess energy and as radionuclides decay, they emit radiation. The uranium decay sequence also includes other radionuclides of concern such as radium and radon (DeSimone and others, 2014).


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# Can this problem be written as a linear program?

 0 I have this objective function: $$\text{minimize} \quad \frac{\sum^n_{i=1} {c_i x_i}}{1 - \sum^n_{i=1} c_i x_i} + \sum^n_{i=1} t_i c_i x_i$$ subject to: $$\sum^n_{i=1} {c_i x_i} \leq 1$$ $$0 \le x \le 1$$ where $$r$$ and $$c$$ are continuous given vectors. Is it possible to convert it into a linear program? Thank you. asked 28 Sep '17, 08:27 ftam 11●2 accept rate: 0% fbahr ♦ 4.6k●7●16 Is this a homework problem? (28 Sep '17, 09:52) Mark L Stone Mark L Stone, No, it is a model for solving a trading problem with nonlinear optimization, but it took too long for large problems. do you have a suggestion? (28 Sep '17, 09:56) ftam 1 https://en.wikipedia.org/wiki/Linear-fractional_programming (28 Sep '17, 10:03) Mark L Stone Thank you Mark, my problem is how to deal with the second term if I reformulate it as a linear fractional program? (28 Sep '17, 10:07) ftam I leave that for you an an exercise to work out, including whether it can be. (28 Sep '17, 10:10) Mark L Stone 2 Mark, you need to start putting your answers in answer blocks, rather than comments, so that you can accumulate karma credits. They might be useful after the impending apocalypse. (28 Sep '17, 16:28) Paul Rubin ♦♦ How strong is OR Exchange's cyber security? Even if I acquire a large number of karma points, it will all be for naught if Russia, China, Iran, or North Korea hack the site, reducing the USD value of the cache from zip to zilch. (28 Sep '17, 19:45) Mark L Stone 3 Based on the number of spammer accounts, I'd say our security is NP-hard (as in, "Not Particularly-hard"). (28 Sep '17, 23:11) Paul Rubin ♦♦ @Mark L Stone, So that is called a linear plus linear fractional program, it is clearly not as easy as the linear fractional programming. (13 Oct '17, 05:03) ftam showing 5 of 9 show 4 more comments
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• sasogeek
$$\large (a^2-b^2)$$ is proven to be equal to $$\large (a+b)(a-b)$$ (difference of two squares) . what then is $$\large (a^2+b^2)$$ and hence expand $$\large (a^3+b^3)$$ to the same form without the exponents. P.S. this is my own personal question. I need some understanding
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Accelerating the pace of engineering and science

# rarx

Estimate parameters of ARX or AR models recursively

## Description

thm = rarx(z,nn,adm,adg) estimates the parameters thm of single-output ARX model from input-output data z and model orders nn using the algorithm specified by adm and adg. If z is a time series y and nn = na, rarx estimates the parameters of a single-output AR model.

[thm,yhat,P,phi] = rarx(z,nn,adm,adg,th0,P0,phi0) estimates the parameters thm, the predicted output yhat, final values of the scaled covariance matrix of the parameters P, and final values of the data vector phi of single-output ARX model from input-output data z and model orders nn using the algorithm specified by adm and adg. If z is a time series y and nn = na, rarx estimates the parameters of a single-output AR model.

## Definitions

The general ARX model structure is:

$A\left(q\right)y\left(t\right)=B\left(q\right)u\left(t-nk\right)+e\left(t\right)$

The orders of the ARX model are:

Models with several inputs are defined, as follows:

A(q)y(t) = B1(q)u1(tnk1)+...+Bnuunu(tnknu)+e(t)

## Input Arguments

z

Name of the matrix iddata object that represents the input-output data or a matrix z = [y u], where y and u are column vectors.

For multiple-input models, the u matrix contains each input as a column vector:

```u = [u1 ... unu]
```
nn

For input-output models, specifies the structure of the ARX model as:

```nn = [na nb nk]
```

where na and nb are the orders of the ARX model, and nk is the delay.

For multiple-input models, nb and nk are row vectors that define orders and delays for each input.

For time-series models, nn = na, where na is the order of the AR model.

 Note:   The delay nk must be larger than 0. If you want nk = 0, shift the input sequence appropriately and use nk = 1 (see nkshift).

adm = 'ff' and adg = lam specify the forgetting factor algorithm with the forgetting factor λ=lam. This algorithm is also known as recursive least squares (RLS). In this case, the matrix P has the following interpretation: R2/2 * P is approximately equal to the covariance matrix of the estimated parameters.R2 is the variance of the innovations (the true prediction errors e(t)).

adm ='ug' and adg = gam specify the unnormalized gradient algorithm with gain gamma = gam. This algorithm is also known as the normalized least mean squares (LMS).

adm ='ng' and adg = gam specify the normalized gradient or normalized least mean squares (NLMS) algorithm. In these cases, P is not applicable.

adm ='kf' and adg =R1 specify the Kalman filter based algorithm with R2=1 and R1 = R1. If the variance of the innovations e(t) is not unity but R2; then R2* P is the covariance matrix of the parameter estimates, while R1 = R1 /R2 is the covariance matrix of the parameter changes.

th0

Initial value of the parameters in a row vector, consistent with the rows of thm.

Default: All zeros.

P0

Initial values of the scaled covariance matrix of the parameters.

Default: 104 times the identity matrix.

phi0

The argument phi0 contains the initial values of the data vector:

φ(t) = [y(t–1),...,y(tna),u(t–1),...,u(tnbnk+1)]

If z = [y(1),u(1); ... ;y(N),u(N)], phi0 = φ(1) and phi = φ(N). For online use of rarx, use phi0, th0, and P0 as the previous outputs phi, thm (last row), and P.

Default: All zeros.

## Output Arguments

thm

Estimated parameters of the model. The kth row of thm contains the parameters associated with time k; that is, the estimate parameters are based on the data in rows up to and including row k in z. Each row of thm contains the estimated parameters in the following order:

```thm(k,:) = [a1,a2,...,ana,b1,...,bnb]
```

For a multiple-input model, the b are grouped by input. For example, the b parameters associated with the first input are listed first, and the b parameters associated with the second input are listed next.

yhat

Predicted value of the output, according to the current model; that is, row k of yhat contains the predicted value of y(k) based on all past data.

P

Final values of the scaled covariance matrix of the parameters.

phi

phi contains the final values of the data vector:

φ(t) = [y(t–1),...,y(tna),u(t–1),...,u(tnbnk+1)]

## Examples

Adaptive noise canceling: The signal y contains a component that originates from a known signal r. Remove this component by recursively estimating the system that relates r to y using a sixth-order FIR model and the NLMS algorithm.

```z = [y r];
[thm,noise] = rarx(z,[0 6 1],'ng',0.1);
% noise is the adaptive estimate of the noise
% component of y
plot(y-noise)
```

If this is an online application, you can plot the best estimate of the signal y - noise at the same time as the data y and u become available, use the following code:

```phi = zeros(6,1);
P=1000*eye(6);
th = zeros(1,6);
axis([0 100 -2 2]);
plot(0,0,'*'), hold on
% Use a while loop
while ~abort
[th,ns,P,phi] = rarx([y r],'ff',0.98,th,P,phi);
plot(time,y-ns,'*')
time = time + Dt
end
```

This example uses a forgetting factor algorithm with a forgetting factor of 0.98. readAD is a function that reads the value of an A/D converter at the indicated time instant.


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Matlab Wavelet

MATLAB has a library of wavelet functions, showing their "continuous forms" as well as the the decomposition and reconstruction filters. Learn more about image processing, imfuse Image Processing Toolbox, MATLAB, Image Acquisition Toolbox, Wavelet Toolbox. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. One-Dimensional Continuous Wavelet Analysis This section takes you through the features of continuous wavelet analysis using the MATLAB Wavelet Toolbox. This example shows how the variable time-frequency resolution of the continuous wavelet transform can help you obtain a sharp time-frequency representation. ECG Signal Analysis Using Wavelet Transforms Figure 1. The output decomposition structure consists of the wavelet decomposition vector c and the bookkeeping vector l, which contains the number of coefficients by level. Principal component analysis in Matlab. Multilevel 2-D wavelet decomposition. Like the Fourier transform, the continuous wavelet transform (CWT) uses inner products to measure the similarity between a signal and an analyzing function. Wound Segmentation using Wavelet Transform in OpenCV. Over 100 Matlab projects and wavelet techniques provide the latest applications of DSP, including image processing, games, filters, transforms, networking, parallel processing, and sound. The normal ECG waveform. For any further information please email us. Hi, Thanks for A2A. The toolbox supports the manuscript: Wavelet Density Estimation for Stratified Size-Biased Sample, by Pepa Ramirez and Brani Vidakovic. DISCRETE WAVELET TRANSFORM The discrete wavelet transform (DWT) is one of the three forms of wavelet transform. Generate MATLAB Code for 1-D Decimated Wavelet Denoising and Compression Wavelet 1-D Denoising. Torrence and G. here is the tree view of what i said earlier. As the forward wavelet transform - finding the transform coefficients and from a given function - can be implemented by the analysis filter bank, the inverse wavelet transform - reconstructing the function from the coefficients and - can be implemented by the synthesis filter bank. Lustig, EECS UC Berkeley Announcements! •Ofﬁce hours TA 1-2pm •Midterm 10/12. % % % INPUTS: % % Y = the time series of length N. Thank you all. Like the Fourier transform, the continuous wavelet transform (CWT) uses inner products to measure the similarity between a signal and an analyzing function. DISCRETE WAVELET TRANSFORM The discrete wavelet transform (DWT) is one of the three forms of wavelet transform. In this video, we will discuss how to use MATLAB to denoise a signal using the discrete wavelet transform. Climatologists develop drought prediction models using machine learning and wavelet transforms. Generating time-frequency representations for use in deep CNNs is a powerful approach for signal classification. Orthonormal dyadic discrete wavelets are associated with scaling functions φ(t). Understanding Images in the MATLAB Environment. i try to for the answer but i dint get the answer. Wavelet Packets. The application of the Morlet wavelet analysis is also used to discriminate abnormal heartbeat behavior in the electrocardiogram (ECG). Contribute to chris-torrence/wavelets development by creating an account on GitHub. The basic idea behind wavelet denoising, or wavelet thresholding, is that the wavelet transform leads to a sparse representation for many real-world signals and images. 97) Uvi_Wave 2. Here is the matlab documentation for the command to do this. This introductory video covers what wavelets are and how you can use them to explore your data in MATLAB®. wpviewcf(T,CMODE) plots the colored coefficients for the terminal nodes of the tree T. The Morlet wavelet transform method is applied to music transcription. The output decomposition structure consists of the wavelet decomposition vector c and the bookkeeping vector l, which contains the number of coefficients by level. It should be noted that much of the work on this third edition was done by Gabriel Peyre. There are two signals here. Matlab's tours - A Numerical Tour of Data Science Toggle navigation Home. The book contains MATLAB source and chapter 13, starting on page 211, is "Wavelets in MATLAB". WaveLab is a library of MATLAB routines for wavelet analysis, wavelet- packet analysis, cosine-packet analysis and matching pursuit. In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. We used a standard matlab wavelet code package, WAVELAB 802, to perform the transforms. 1 Matlab Implementation of Wavelet-Based Denoising using the Separable DWT. Torrence and G. What this means is that the wavelet transform concentrates signal and image features in a few large-magnitude wavelet coefficients. Over 100 Matlab projects and wavelet techniques provide the latest applications of DSP, including image processing, games, filters, transforms, networking, parallel processing, and sound. speech, pictures, etc. approach in which the MATLAB wavelet transform toolbox is adopted not only to detect power quality problems but also to classify them. • The wavelet coefficients measure how closely correlated the wavelet is with each section of the signal • For compact representation, choose a wavelet that matches the shape of the image components –Example: Haar wavelet for black and white drawings ³ f W M ( s,W) f ( x )\ s ,W ( x ) dx. Matlab code for 2DWT (forward) Matlab code for 2DWT (inverse) Ruye Wang 2008-12-16. If x is real-valued, cfs is a 2-D matrix, where each row corresponds to one scale. The example also compares the wavelet coherence and cross-spectrum against their Fourier counterparts. X = wpcoef(T,N) returns the coefficients associated with the node N of the wavelet packet tree T. I would suggest doing these steps yourself, which are easy to write in matlab. All figures and tables in the accompanying paper:. Taswell, Stanford) Matlab Wavelet Toolbox (Rice Univ. About Wavelet Packet Analysis. Don't show me this again. In our implementation, the main function calls the algorithm as a function. Use in music. Legendre wavelets can be easily loaded into the MATLAB wavelet toolbox—The m-files to allow the computation of Legendre wavelet transform, details and filter are (freeware) available. One gets information on both the amplitude of any "periodic" signals within the series, and how this amplitude varies with time. It can typically be visualized as a "brief oscillation" like one recorded by a seismograph or heart monitor. Contains functions for computing and plotting discrete wavelet transforms (DWT) and maximal overlap discrete wavelet transforms (MODWT), as well as their inverses. The MODWT is an undecimated wavelet transform over dyadic (powers of two) scales, which is frequently used with financial data. Execution Analysis of Lynn Wavelet Filter Algorithms for Removal of Low Frequency Noise from ECG Signal P. 1st generation wavelets using filter banks (periodic and orthogonal). As signal description is not given here you can use any of 1. Generally, 8 to 10 levels is sufficient to get an accurate representation of the scaling function and wavelet. Wavelet thresholding properites were investigated in a series of papers by Donoho and Johnstone, see for instance. 19 ms, which is the same time step as in my reflectivity data. Related Data and Programs: HAAR, a MATLAB library which computes the Haar transform of data. Example of wavelet transform. The basic idea behind wavelet denoising, or wavelet thresholding, is that the wavelet transform leads to a sparse representation for many real-world signals and images. cwtstruct = cwtft(sig) returns the continuous wavelet transform (CWT) of the 1–D input signal sig. Matlab Code (Latest Release) Instructions: Download Spherical Wavelets version 1. Introduction to Wavelet Families. Bretherton Winter 2015 Refs: Matlab Wavelet Toolbox help. Wavelet packet atoms are waveforms indexed by three naturally interpreted parameters: position, scale (as in wavelet decomposition), and frequency. Most of the wavelet decomposition algorithms usually decompose the original image into a sequence of new images (usually called wavelet planes) of decreasing size. PDF Reference Paper We include a PDF reference technique to help you understand our project Technique Support Our dedicated support team is here to answer any questions you may have about your project. The toolbox also includes apps and functions for decimated and nondecimated discrete wavelet analysis of signals and images, including wavelet packets and dual-tree transforms. Wavelet ♥A small wave Wavelet Transforms ♥Convert a signal into a series of wavelets ♥Provide a way for analyzing waveforms, bounded in both frequency and duration ♥Allow signals to be stored more efficiently than by Fourier transform ♥Be able to better approximate real-world signals. The toolbox includes algorithms for continuous wavelet analysis, wavelet coherence, synchrosqueezing, and data-adaptive time-frequency analysis. Cambridge: Cambridge University Press. If x is real-valued, cfs is a 2-D matrix, where each row corresponds to one scale. Can be used to remove 60 Hz and harmonic noise. The Matlab Signal toolbox is not required. The structure is organized as in this level-3 decomposition diagram. It relies on averaging and differentiating values in an image matrix to produce a matrix which is sparse or nearly sparse. Audio compression using wavelet transform in MATLAB + Post New Thread. This MATLAB function returns the time-domain and centered wavelets corresponding to the wavelet passband filters in the discrete wavelet transform (DWT) filter bank fb. In this article I provide an application that uses discrete wavelet transforms to explore one dimensional signals. Details of the SGWT are in the paper "Wavelets on Graphs via Spectral Graph Theory" David Hammond, Pierre Vandergheynst, Remi Gribonval,. Aslak Grinsted, John Moore and Svetlana Jevrejeva. Yes, you can create your own wavelet and use it in matlab. Wavelet decomposition filters associated with an orthogonal or biorthogonal wavelet, specified as even-length real-valued vectors. Its results are compatible with MATLAB Wavelet Toolbox. 3 Materials and Methods 3. What follows is an introduction to some wavelet families. To denoise it, you set coefficients based on some threshold to zero. 5 or higher. It denotes a univariate function ψ(multivariate wavelets exist as well and will be discussed subsequently), deﬁned on R, which, when subjected to the fundamental operations of shifts (i. Walden (2000) Wavelet Methods for Time Series Analysis. Multilevel 2-D wavelet decomposition. for those who hate preambles. They are usually presented in mathematical formulae, but can actually be un-derstood in terms of simple comparisons or correlations with the signal being ana-lyzed. Contribute to gpeyre/matlab-toolboxes development by creating an account on GitHub. Next: Applications Up: wavelets Previous: Fast Wavelet Transform (FWT) 2D DWT. There is no scaling function associated with this wavelet. I recently stumbled upon a bothering fact when using the pywt. Use in music. This example shows how the variable time-frequency resolution of the continuous wavelet transform can help you obtain a sharp time-frequency representation. Matlab code for Integer Wavelet transform? Unanswered Questions. There are other ways to add wavelet functionality to Java, such as employing Matlab/Scilab-Wavelab wrappers or open source libraries. Base your routine on the discussion of separable wavelets and two-dimensional wavelet transforms in Section 7. Orthonormal dyadic discrete wavelets are associated with scaling functions φ(t). The output decomposition structure consists of the wavelet decomposition vector c and the bookkeeping vector l, which contains the number of coefficients by level. This is the basic wavelet function, but we now need some way to change the overall size as well as slide the entire wavelet along in time. I have tried to use the function mexihat in MatLab and tuned the lb, ub and n parameters to solve my problem, but I can´t figure it out. These functions can be used to define wavelets. Generate MATLAB Code for 1-D Decimated Wavelet Denoising and Compression Wavelet 1-D Denoising. Wavelet analysis is similar to Fourier analysis in the sense that it breaks a signal down into its constituent parts for analysis. Contains functions for computing and plotting discrete wavelet transforms (DWT) and maximal overlap discrete wavelet transforms (MODWT), as well as their inverses. To determine if the Wavelet Toolbox™ software is already installed on your system, check for a subfolder named wavelet within the main toolbox folder. Matlab Code for Integer Wavelet Transform. ECG Signal Analysis Using Wavelet Transforms Figure 1. There are two signals here. Analyzing wavelet used to the compute the single-level DWT, specified as a character vector or string scalar. Note has width of order , and is centered about (see2# 5# 45 4 4 diagram earlier). This introductory video covers what wavelets are and how you can use them to explore your data in MATLAB®. 1-D Wavelet Packet Analysis. It describes the types of supported images and how the MATLAB ® environment represents them, as well as techniques for analyzing color images. " If the wavelet can be selected to resemble components of the image, then a compact representation results. Pentru aceste abordari ale prelucrarii imaginii cu ajutorul trans- formatei wavelet, se implementeaza rutinele Matlab necesare pentru testarea si experimentarea diferitelor caracteristici si performante ale acestor structuri. Generate Code to Denoise a Signal. MATLAB implementation of Legendre wavelets. zip performs debiasing and estimattes density by smoothed linear projection wavelet esimator. I just download toolbox_wavelets and try to use in MatLab 2015b. Digital Signal Processing Using Matlab. Torrence & Compo Wavelet Analysis Software. What this means is that the wavelet transform concentrates signal and image features in a few large-magnitude wavelet coefficients. Image compression using wavelet transform *wavelet transform give better information about non-stationary signals in time domain. The structure is organized as in this level-3 decomposition diagram. It is very close to what you will do in this tutorial. 892 lines. Please acknowledge the use of this software in any publications:. The backbone of the multi-scale analysis is provided by an isotropic band-limited wavelet pyramid, and the detection of directional features is provided by coupling the pyramid with a generalized Riesz transform. Legendre wavelets can be easily loaded into the MATLAB wavelet toolbox—The m-files to allow the computation of Legendre wavelet transform, details and filter are (freeware) available. The wavelet that meets this criteria is the Mexican hat, or Ricker, wavelet. The book contains MATLAB source and chapter 13, starting on page 211, is "Wavelets in MATLAB". [email protected] Periodic boundary conditions are applied to the signal which must be of size 2^N. 7k 8 8 gold badges 67 67 silver badges 85 85 bronze badges. txt) or read online for free. bmp’); %Red Component of Colour Image Red_Input_Image=Input_Image(:,:,1); %Green Component of Colour. The WTMM algorithm finds singularities in a signal by determining maxima. Integer Wavelet Transform Using Matlab Codes and Scripts Downloads Free. This wavelet has [-4 4] as effective support. The toolbox supports the manuscript: Wavelet Density Estimation for Stratified Size-Biased Sample, by Pepa Ramirez and Brani Vidakovic. Analyzing wavelet used to the compute the single-level DWT, specified as a character vector or string scalar. To install this toolbox on your computer, see the appropriate platform-specific MATLAB ® installation guide. In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. Most notably, it is used for signal coding, to represent a discrete signal in a more redundant form, often as a preconditioning for data compression. Use in music. Start the Wavelet Signal Denoiser app by choosing it from the Apps tab on the MATLAB® Toolstrip. Complexity of FWT. There are a number of different variations of the wavelet transform. Load and plot a cusp signal. Since the variation of the abnormal heartbeat is a non-stationary signal, this signal is suitable for wavelet-based analysis. % % % INPUTS: % % Y = the time series of length N. Output arguments are the wavelet function PSI computed on the grid X, and the grid X. X = wpcoef(T,N) returns the coefficients associated with the node N of the wavelet packet tree T. Generating time-frequency representations for use in deep CNNs is a powerful approach for signal classification. There are two signals here. X = wprcoef(T) is equivalent to X = wprcoef(T,0). Matlab GUI project. Select a Web Site. The so-called first generation wavelets and scaling functions are dyadic dilations and translates of a single function. A sparse matrix is a matrix in which a large portion of its entries are 0. sig can be a vector, a structure array, or a cell array. m implements Haar wavelet and AudioCompression2. 2-svn601 folder. When we use the default "symmetric" padding, the inverse wavelet transform is not the adjoint of the wavelet transform (and it's not the mathematical inverse of the wavelet transform, "only in one direction"). Since this is a Matlab tutorial, it is assumed that you have Matlab running on your computer with version 5 or later. Compo, 1998: A Practical Guide to Wavelet Analysis. Thank you all. If I have the wavelet $\psi_{s,u}$, which is represented by psi in MATLAB, and I want to compute the above integral. Introduction to Wavelet Families. -- on to Wavelets 2--. Wavelet Transforms | A Quick Study Ivan W. Posted 2 weeks ago. Wavelet Packets. For example, the MODWT coefficients are norm-preserving in the sense that if you add up the energy in the wavelet coefficients and the final-level scaling coefficients, you get back the same value as the energy (L2 norm squared) of the original signal. Load a signal or image Perform a wavelet packet analysis of a signal or image Compress a signal Remove noise from a signal Compress an image Show statistics and. Aslak Grinsted, John Moore and Svetlana Jevrejeva. We will use the CWT function in MATLAB to compute the Continuous Wavelet Transform. Carrying out the statistical test in the wavelet domain require an inverse wavelet transform. However, apart from Matlab build-in function,I've found out another way to produce psi and phi function during investigation of Wavelet application to mechanical vibration. Hello All, I need to ask if anybody had managed to successfully generate C code from MATLAB code which uses Wavelet toolbox functions like wptree. Generate Code to Denoise a Signal. Generating Daubechies wavelet function (psi) and wavelet scaling (phi) using Matlab Posted on December 30, 2011 by Jonny Latuny Matlab provides a function to generate wavelet function values and wavelet scaling values via build-in function called wavefun. Haar wavelet compression. The library supports real and complex calculations with single and double precision. In this case look for the folder of matlab that contains the startup and path definition files and use it instead. dwt2 computes the single-level 2-D wavelet decomposition. Wavelet transforms co-localize in both frequency domain and time domain and are effective for non stationary signals such as neurological tremor. on Signal Processing. Functions, with associated limitations, that support code generation. In mathematics, the continuous wavelet transform (CWT) is a formal (i. The book also provides the mathematical processes and techniques needed to ensure an understanding of DSP theory. The wavelet packet method is a generalization of wavelet decomposition that offers a richer signal analysis. zip (In linux, type "unzip SWv1. It describes the types of supported images and how the MATLAB ® environment represents them, as well as techniques for analyzing color images. Aslak Grinsted, John Moore and Svetlana Jevrejeva. I use Matlab r2018a and it comes with a function "wcoherence" that returns the magnitude-squared wavelet coherence between two equal length signals. The term “wavelet” (originally called wavelet of constant shape) was introduced by J. Unlike sinusoids, which extend to infinity, a wavelet exists for a finite duration. MATLAB Research on ECG Signal Analysis. 892 lines. This wavelet has [-4 4] as effective support. In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. Load an example image. The availability of a wide range of wavelets is a key strength of wavelet analysis. X = wprcoef(T) is equivalent to X = wprcoef(T,0). Selesnick Polytechnic University Brooklyn, NY September 27, 2007 This is an expanded version of the 'Quick Study' in Physics Today magazine, October, 2007. It can typically be visualized as a "brief oscillation" like one recorded by a seismograph or heart monitor. To determine if the Wavelet Toolbox™ software is already installed on your system, check for a subfolder named wavelet within the main toolbox folder. Bretherton Winter 2015 Refs: Matlab Wavelet Toolbox help. I would like the wavelet to have a dominant frequency of about 70Hz and the time step to be 0. Wavelet analysis attempts to solve these problems by decomposing a timeseries into time/frequency space simultaneously. Plot the resulting quaternary wavelet packet tree. Integer Wavelet Transform Using Matlab Codes and Scripts Downloads Free. Single-level discrete 2-D wavelet transform 3. So if you idwt either approx or details, by replacing the other by zeros, you almost get your coefficient in order: on the top, the original signal, the two low frequencies in the second plot, the high frequencies in the third plot. The wavelet family short name can be for a user-defined wavelet (see wavemngr for more information) or one of the values listed here. m implements Haar wavelet and AudioCompression2. Wavelet Transform Matlab Code Wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. First of all name your wavelet and study its response on different types of signals and compare with existing wavelets, this can be done. Climatologists develop drought prediction models using machine learning and wavelet transforms. The signal S is real, the wavelet can be real or complex. [c,l] = wavedec(x,n,wname) returns the wavelet decomposition of the 1-D signal x at level n using the wavelet wname. If I have the wavelet $\psi_{s,u}$, which is represented by psi in MATLAB, and I want to compute the above integral. However, you can calculate corresponding frequencies (i. We aim to identify outliers that may be caused by poor calibration of the machine or other outliers. We use cookies to offer you a better experience, personalize content, tailor advertising, provide social media features, and better understand the use of our services. Wavelet decomposition filters associated with an orthogonal or biorthogonal wavelet, specified as even-length real-valued vectors. dwtmode(mode) sets the signal or image extension mode for both discrete wavelet and wavelet packet transforms to mode. 5e and f, respectively, as generated from the Matlab ® wavelet coherence function 'wcoher'. Learn more about signal processing, wavelet, fft Wavelet Toolbox. from Matlab help page on wavelets. Aslak Grinsted, John Moore and Svetlana Jevrejeva. Add to Cart. Output arguments are the wavelet function PSI computed on the grid X, and the grid X. One gets information on both the amplitude of any "periodic" signals within the series, and how this amplitude varies with time. The 1D and 2D wavelet transforms can be implemented as a filter bank. WaveLab is a library of MATLAB routines for wavelet analysis, wavelet- packet analysis, cosine-packet analysis and matching pursuit. Analyze a signal with wavelet packets using the Wavelet Analyzer app. Results 1 to 1 of 1 Audio compression using wavelet transform in MATLAB. For an existing figure F produced by a previous call to the drawtree function, drawtree(T,F) draws the wavelet packet tree T in the figure whose handle is F. Haar wavelet compression. e = wentropy(x,'user','ABC'). The paper is organized as follows: Section I, depicts about Ultrasound images, speckle noise. All 10 sets of coefficients are 512X512. Specifically, the algorithm starts with a single wavelet or scaling coefficient at the coarsest resolution level and reconstructs the wavelet or scaling function to the specified finest resolution level. Launch Matlab (v 6. Choose a web site to get translated content where available and see local events and offers. I can 't find it. The paper is organized as follows: Section I, depicts about Ultrasound images, speckle noise. cwtstruct = cwtft(sig) returns the continuous wavelet transform (CWT) of the 1–D input signal sig. The basic idea behind wavelet denoising, or wavelet thresholding, is that the wavelet transform leads to a sparse representation for many real-world signals and images. Description [PSI,X] = morlet(LB,UB,N) returns values of the Morlet wavelet on an N point regular grid in the interval [LB,UB]. This introductory video covers what wavelets are and how you can use them to explore your data in MATLAB®. pdf), Text File (. Select a Web Site. [C,S] = wavedec2(X,N,wname) returns the wavelet decomposition of the matrix X at level N using the wavelet wname. WAVELET is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version. Installing Wavelet Toolbox Software. Obtain the continuous wavelet transform (CWT) of a signal or image, construct signal approximations with the inverse CWT, compare time-varying patterns in two signals using wavelet coherence, visualize wavelet bandpass filters, and obtain high resolution time-frequency representations using wavelet synchrosqueezing. Contains functions for computing and plotting discrete wavelet transforms (DWT) and maximal overlap discrete wavelet transforms (MODWT), as well as their inverses. The toolbox also includes apps and functions for decimated and nondecimated discrete wavelet analysis of signals and images, including wavelet packets and dual-tree transforms. matlab feature-extraction dwt wavelet kmeans-clustering discrete-wavelet-transformation. Torrence & Compo Wavelet Analysis Software. Signal Denoising with Wavelets. I have read the wavelet toolbox manual, but it doesn't provide everything I need in terms of an explanation or instructions. Scilab Wavelet Toolbox is designed to work with any Scilab Image Processing Toolbox like SIP or SIVP for displaying 2-D results. The wavelets used for analysis can have many vanishing moments. To compute the real part of the complex wavelet, we set all coefficients to zero, except for one coefficient in the upper DWT, and then compute the inverse transform. Wavelet packet atoms are waveforms indexed by three naturally interpreted parameters: position, scale (as in wavelet decomposition), and frequency. Torrence & Compo Wavelet Analysis Software. Colorado School of Mines Image and Multidimensional Signal Processing Continuous Wavelet Transform •Wavelets represent the scale of features in an image, as well as their position. With each wavelet type of this class, there is a scaling function (called the father wavelet) which generates an orthogonal multiresolution analysis. The basic idea behind wavelets is to express a signal as a linear combination of given sets of functions (wavelet transform). cwtstruct = cwtft(sig) returns the continuous wavelet transform (CWT) of the 1-D input signal sig. Using discrete wavelet transform for feature extraction of CT medical images. Jevrejeva (2004), Application of the cross wavelet transform and wavelet coherence to geophysical time series, Nonlin. Abstract—The wavelet technique is used to analyze aircraft noises during takeoff and landing at Tripoli international airport (Libya). drawtree(T) draws the wavelet packet tree T, and F = drawtree(T) also returns the figure's handle. X = wprcoef(T,N) computes reconstructed coefficients of the node N of the wavelet packet tree T. In this example, Windows XP Startup. The application of the CWT to two. The decomposition is done with respect to either a particular wavelet (see wfilters for more information) or particular wavelet decomposition filters. The package includes discrete wavelet transforms, column-wise discrete wavelet transforms, and wavelet packet transforms. Introduction to Wavelet Families. For example, the MODWT coefficients are norm-preserving in the sense that if you add up the energy in the wavelet coefficients and the final-level scaling coefficients, you get back the same value as the energy (L2 norm squared) of the original signal. MATLAB小波变换工具箱 Wavelet Toolbox 实际操作与训练 12-12 阅读数 1万+ 这篇本章是对MATLAB中小波变换工具箱运用的一个简单总结. zip performs debiasing and estimattes density by smoothed linear projection wavelet esimator. Introduction to Wavelets in Image Processing. The data array is further processed in MATLAB platform using Daubechies wavelet transform (dB 6, level 9) to cross check the results of interpretation of one‘s awareness level as obtained from LabVIEW environment. We will now look at two types of wavelet transforms: the Continuous Wavelet Transform and the Discrete Wavelet. Matlab's tours - A Numerical Tour of Data Science Toggle navigation Home. Matlab, Fortran Code. Versions are provided for Macintosh, UNIX and Windows machines. Description. fundamentals and principles of the wavelets theory. Lifting Method for Constructing Wavelets. The implementation of the denoising algorithm is similar to the separable DWT case. More Views. -- on to Wavelets 2--. com) Stay Current In Your Field • Broaden Your Knowledge • Increase Productivity 349 Berkshire Drive • Riva, Maryland 21140. Princomp can be used in the following way:. Note that you have to redo this step every time you restart matlab, unless you ask matlab to save the. You can generate MATLAB ® code to reproduce app-based 1-D wavelet denoising at the command line. LoD is the lowpass decomposition filter, and HiD is the highpass decomposition filter. 701 is a library of MATLAB routines for wavelet analysis, wavelet- packet analysis, cosine-packet analysis and matching pursuit. The basic idea behind wavelet denoising, or wavelet thresholding, is that the wavelet transform leads to a sparse representation for many real-world signals and images. The toolbox includes many wavelet transforms that use wavelet frame representations, such as continuous, discrete, nondecimated, and stationary wavelet transforms. Does anybody have trials in this track ? - and i need to know if this kind of code generation is supported in the MATLAB (i mean the usage of the Wavelet toolbox functions). Taswell, Stanford) Matlab Wavelet Toolbox (Rice Univ. After a pause for computation, the tool displays the two analyses. Single-level discrete 2-D wavelet transform 3. MATLAB Wavelet denoising; MATLAB Wavelet transform; Super MATLAB Wavelet analysis to study the manual - program code; MATLAB Wavelet feature extraction of texture feature; MATLAB training program (call MATLAB c/c + +) MATLAB training program (called MATLAB c/c + +) MATLAB seismic data is read and normalized; MATLAB training programs (two. Filters some input data using wavelets. There are a number of different variations of the wavelet transform. Wavelet family short name, specified as a character vector or string scalar. We will use the CWT function in MATLAB to compute the Continuous Wavelet Transform. i try to for the answer but i dint get the answer. 1 Introduction Wavelets are an e cient tool for analyzing data that varies on a wide range of scales, especially when the data is statistically non-stationary, e. A matrix is similar but contains multiple rows and columns of numbers. However, apart from Matlab build-in function,I've found out another way to produce psi and phi function during investigation of Wavelet application to mechanical vibration. Princomp can be used in the following way:. sig can be a vector, a structure array, or a cell array. The application of the CWT to two. The output decomposition structure consists of the wavelet decomposition vector C and the bookkeeping matrix S, which contains the number of coefficients by level and orientation. Matlab code for Integer Wavelet transform? Unanswered Questions. Colorado School of Mines Image and Multidimensional Signal Processing Continuous Wavelet Transform •Wavelets represent the scale of features in an image, as well as their position. txt) or read online for free. When we use the default "symmetric" padding, the inverse wavelet transform is not the adjoint of the wavelet transform (and it's not the mathematical inverse of the wavelet transform, "only in one direction"). The column size of cfs is equal to the length of x. pdf), Text File (. Designed to be incremental in difficulty. We will now look at two types of wavelet transforms: the Continuous Wavelet Transform and the Discrete Wavelet.


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# calculate vector between 2 vectors in 3d sphere

I have a sphere with known radius $R$ and 2 vectors $\vec a$ and $\vec b$ within this sphere both starting from $(0,0,0)$ and ending on the edges of this sphere so $\left\lVert\vec a \right\rVert = R$ and $\left\lVert\vec b \right\rVert = R$.

The angle between these 2 vectors is $\alpha = 90^o$ but their coordinates are unspecified( known but they can be anywhere in the sphere)

How can I calculate vector $\vec c$ coordinates if this vector also starts at $(0,0,0)$, its length is also equal to spheres radius : $\left\lVert\vec c \right\rVert = R$ , and is somewhere between $\vec a$ and $\vec b$ specified by the known angle of $\beta$ that is an angle between $\vec a$ and $\vec c$ and is $0^o \lt\beta \lt 90^o$

If $\vec{c}$ lies between $\vec{a}$ and $\vec{b}$, it lies in the plane spanned by them. So there are $s,t$ so that $\vec{c}=s\vec{a}+t\vec{b}$. If we know the angle between $\vec{c}$ and $\vec{a}$ is $\beta$, we know that $$tR^2 = \vec{c} \cdot \vec{a} = R^2\cos{\beta},$$ so $t=\cos{\beta}$. But then the angle between $\vec{b}$ and $\vec{c}$ is $90^{\circ}-\beta$, so $$sR^2 = \vec{c} \cdot \vec{b} = \cos{(90^{\circ}-\beta)} = \sin{\beta},$$ so $$\vec{c} = \vec{a}\cos{\beta}+\vec{b}\sin{\beta}.$$ We can verify that $\vec{c} \cdot \vec{c} = \vec{a} \cdot \vec{a} \cos^2{\beta} + \vec{a} \cdot \vec{b} \sin{\beta}\cos{\beta} + \vec{b} \cdot \vec{b} \sin^2{\beta} = R^2$ since $\vec{a} \cdot \vec{b} = 0$.


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# What is the Principle of Maximum Conformality?

I'm trying to understand this article about an advance in the theoretical understanding of QCD which centers on the Principal of Maximum Conformality. What is this Principle? In other words, what is being maximized and what does this tell us about the structure of QCD? Also, is this a new principle or a new application of an old principal?

Here's the full paper on the principle's application to top physics.

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It is an approach to perturbative QCD which resolves ambiguities regarding the renormalization scale of the theory. It is done by summing terms for which the $\beta$-function is non-zero into the running coupling. In this sense, the remaining terms are now "maximally conformal" due to $\beta=0$. This results in predictions independent of the renormalization scheme. This approach seems to be relatively new, see http://arxiv.org/abs/1107.0338 for a detailed treatment.


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207 articles – 1938 Notices  [english version]
 HAL : in2p3-00145475, version 1
 1st International Conference on Molecular Imaging Technology - EuroMedIm 2006, Marseille : France
 Nuclear medical imaging using $\beta+\gamma$ coincidences from $^{44}$Sc radio-nuclide with liquid xenon as detection medium
 C. Grignon1, J. Barbet2, M. Bardiès2, T. Carlier2, J.F. Chatal2, O. Couturier2, Jean-Pierre Cussonneau1, A. Faivre2, L. Ferrer2, S. Girault1, T. Haruyama, P. Le Ray1, L. Luquin1, S. Lupone1, V. Métivier1, E. Morteau1, N. Servagent1, D. Thers1
 (2007)
 We report on a new nuclear medical imaging technique based on the measurement of the emitter location in the three dimensions with a few mm spatial resolution using β+γ emitters. Such measurement could be realized thanks to a new kind of radio-nuclides which emit a γ-ray quasi-simultaneously with the β+ decay. The most interesting radio-nuclide candidate, namely 44Sc, will be potentially produced at the Nantes cyclotron ARRONAX. The principle is to reconstruct the intersection of the classical line of response (obtained with a standard PET camera) with the direction cone defined by the third γ-ray. The emission angle measurement of this additional γ-ray involves the use of a Compton telescope for which a new generation of camera based on a liquid xenon (LXe) time projection chamber is considered. GEANT3 simulations of a large acceptance LXe Compton telescope combined with a commercial micro-PET (LSO crystals) have been performed and the obtained results will be presented. They demonstrate that a good image can be obtained from the accumulation of each three-dimensional measured position. A spatial resolution of 2.3 mm has been reached with an injected activity of 0.5 MBq for a 44Sc point source emitter.
 1 : SUBATECH - Laboratoire SUBATECH Nantes 2 : Recherches en cancérologie
 Thème(s) : Sciences du Vivant/Ingénierie biomédicale/ImageriePhysique/Physique/Physique MédicaleSciences du Vivant/Ingénierie biomédicale/Médecine nucléaire
 Mot(s)-clé(s) : Liquid xenon – Medical imaging – Scandium
 in2p3-00145475, version 1 http://hal.in2p3.fr/in2p3-00145475 oai:hal.in2p3.fr:in2p3-00145475 Contributeur : Dominique Girod <> Soumis le : Jeudi 10 Mai 2007, 14:11:57 Dernière modification le : Jeudi 10 Mai 2007, 14:18:05


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# Operator norm directly from phase space representation of photonic quantum operator

I'm interested in calculating the operator norm of a Hermitian operator, say $B$, acting on the Hilbert space of square integrable functions. The context is I have an optical system in all its infinite dimensional glory. To be clear, by operator norm I mean the largest eigenvalue, or $||B||_{\infty}=\sup \{ \langle \psi \vert B \vert \psi \rangle | \langle \psi \vert \psi \rangle =1 \}$.

I have a neat expression for $B$ in terms of its Characteristic function / Wigner function. My question is whether anyone knows a neat and easy way of calculating the operator norm within the phase space picture. Of course, I could truncate the operator to small Fock numbers (assuming that this is a good approximation), work out the matrix in the Fock basis and crunch the numbers. However, I'm just curious whether there is a more elegant solution. I have in mind such equalities as $tr ( A ) = \int W_{A}(r) dr$ and $tr ( A^{\dagger} B ) = \pi \int W_{A}(r)W_{B}(r)dr$, which give a useful correspondence between Wigner functions and the operators themselves.... are these any such expressions for norms?

Being pessimistic I guess the answer might simply be NO! However, I thought it would be worth consulting the collective wisdom of TPSE.

• For sure you have $||B||_{\infty} \leq \pi \ \sup_{r} |W_B(r)|$. – Piotr Migdal Nov 15 '11 at 17:09
• @PiotrMigdal I did actually spot that bound myself after I posted, but I would want something much tighter. e.g. tight enough to give the right answer for a thermal Gaussian. – Earl Nov 16 '11 at 10:57

It is not an complete answer, but (hopefully) it may help:
$$\text{tr} \left( A B \right) = \pi \int W_A(r) W_B(r) dr = \pi \int P_A(r) Q_B(r) dr,$$ where $A$ and $B$ are self-adjoint operators, $r=(p,q)$ is a point of the phase space, $W(r)$ is a Winger function, $P(r)$ - Glauber-Sudarshan P-representation and $Q(r)$ - a Husimi Q representation.

The second equality holds, as $Q(r)$ is a Gaussian-smeared $W(r)$, and $P(r)$ is a Gaussian-sharpened $W(r)$ (to see it explicitly, use the convolution theorem).

Then you can use advantages of each of the mentioned representations:

• $P_A(r)$: is sometimes easier to handle than its Wigner counterpart,
• $Q_B(r)$: is easily obtainable numerically (or sometimes - even analytically) from $W_B(r)$; for a non-negatively definite operator it is nonnegative, i.e. $Q_B(r) \geq 0$.

In particular for $P_A(r)\geq 0$ and $Q_B(r)\geq 0$ one gets a bound $$\int P_A(r) Q_B(r) dr \leq \left( \sup_r\ Q_b(r) \right) \int P_A(r)dr = \sup_r\ Q_b(r).$$ In other words, when one restricts to finding maximum over a mixture of coherent states, then $\pi\ \sup_r\ Q_b(r)$ is the upper bound, which is tight.

However, the state achieving maximum need not to be coherent, so in general one gets $$||B||_\infty \geq \pi\ \sup_\alpha \langle \alpha | B | \alpha \rangle = \pi \ \sup_r\ Q_B(r).$$

Additionally, for some applications it may help that: $$\sup_r\ Q_B(r)\leq \sup_r\ W_B(r)\leq \sup_r\ |W_B(r)|.$$

• Thanks for your great answer. I didn't think to use the Husimi Q representation, but it seems to give quite a good bound! Though the bound is not always tight, it is tight for some important cases. For example, your bound would be tight for a thermal Gaussian, and displacements of a thermal Gaussian. I think this might be tight enough to be useful to me. – Earl Nov 16 '11 at 11:01
• Actually, I'm not quite sure I understand why this is an upper bound on $||B||_{\infty}$ and not lower, and why there isn't a $\pi$ involved. So, via wikipedia, we have a $Q$ function: $Q_{B}(\alpha) = \frac{1}{\pi} \langle \alpha \vert B \vert \alpha \rangle$. Taking the sup gives, $\sup_{\alpha} Q_{B} ( \alpha ) = \sup_{\alpha} \langle \alpha \vert B \vert \alpha \rangle / \pi \leq \sup_{\psi} \langle \psi \vert B \vert \psi \rangle / \pi = ||B||_{\infty} / \pi$. Have I misunderstood something here? – Earl Nov 16 '11 at 11:14
• @Earl: Haste makes waste - sorry for my omission. I lost $\pi$ and it wasn't on purpose. When it comes to the inequality (this one with $||B||_\infty$) - now I see it is not right. I wanted to use $\int P(r) Q(r) dr \leq \int P(r) (\sup_{R} Q(R)) dr$ but I've forgotten that $P(r)$ is not positive. I am fixing my post in a sec. – Piotr Migdal Nov 16 '11 at 12:23
• Cool. But maybe we can also have both bounds such that $\pi \sup_{r} |Q_{B}(r)| \leq ||B||_{\infty} \leq \pi \sup_{r} |P_{B}(r)|$. For the following reason. $||B||_{\infty} = \pi | \sup_{\psi} \int P_{B}(r) Q_{\psi}(r) dr | \leq \sup_{\psi} \int | P_{B}(r) |.| Q_{\psi}(r) | dr$, where $Q_{\psi}$ is for some $\vert \psi \rangle$. Which is simply $||B||_{\infty} \leq \pi (\sup_{r}|P(r)|)(\int |Q_{\psi}|(r))$, but since Q is positive and integrates to 1, we have $||B||_{\infty}\leq \pi \sup_{r}|P(r)|$. This gives the other bound, which happens to be the one I prefer to have. – Earl Nov 16 '11 at 13:35
• Ignore my last comment, as I think the Wigner function gives a better upper bound than the P-function. – Earl Nov 16 '11 at 14:18

In the case when $B$ is a compact operator, a limiting process can be applied to get the operator's norm to any arbitrarily small precision. This process is a Hilbert space generalization of the linear algebra relation for a Hermitian matrix $A$.

$||A||_{\infty} = \lim_{n\rightarrow \infty}|tr(A^n)|^{\frac{1}{n}}$.

For the data given in the question the operator composition is to be performed by means of the star product:

$W_{AB}(r) =(W_A \star W_B)(r) =\int W_A(r_1)W_B(r_2) k(r, r_1, r_2) dr$

Thus one needs to know the star product kernel $k$. (Of course, in the ordinary phase space case, the star product is the Moyal star product).

Thus formally for the operator case:

$||B||_{\infty} = \lim_{n\rightarrow \infty}|tr(\star^{n} B)|^{\frac{1}{n}}$.

The limit exists when $B$ is compact.

• A small tweak in the transcription of Cayley's method: $||B||_{\infty} = \lim_{n\rightarrow \infty}|\int dr (\star B)^{n}|^{\frac{1}{n}}$. It may be easier to parse out this way. – Cosmas Zachos Sep 16 '15 at 17:04
• Actually, the abs value in the above power iteration algorithm produces the spectral radius, but it is assumed the reader appreciates the trivial connection to the norm sought. – Cosmas Zachos Sep 16 '15 at 19:40
• @Cosmas Zachos Thank you for your remarks Prof. Zachos. It is a great pleasure to have you on physics S.E. – David Bar Moshe Sep 17 '15 at 7:32
• Thanks. Actually a practical and instructive illustration of this method is using [the Weyl transform of a Gaussian of operators][1] to compute the norm of the bounded Boltzmann operator $B\equiv\exp(-\frac{\hat{p}^2+\hat{x}^2}{2kT})$. [1]: physics.stackexchange.com/questions/113604/… – Cosmas Zachos Sep 23 '15 at 19:51

There is an obvious property of the Wigner function when one takes the limit $p\rightarrow\infty$. By the Riemann's lemma, the Wigner function is expected to go to zero in this limit but it does driven by the largest eigenvalue. Your question can be stated in the following way. The Wigner function for a generic operator $A$ can be defined as

$$W_A(x,p)=\frac{1}{2\pi\hbar}\int_{-\infty}^{+\infty}dy e^{-\frac{i}{\hbar}py}\langle x-y|A|x+y\rangle.$$

Now, assume that you have diagonalized $A$ so that $A|a_n\rangle=a_n|a_n\rangle$, you can write immediately

$$W_A(x,p)=\frac{1}{2\pi\hbar}\sum_n a_n\int_{-\infty}^{+\infty}dy e^{-\frac{i}{\hbar}py}\phi_n^*(x-y)\phi_n(x+y)=\frac{1}{2\pi\hbar}\sum_n a_n W_n(x,p)$$

being $\phi_n(x)=\langle a_n|x\rangle$. Of course, if the spectrum of $A$ will not run to infinity, taking the limit of increasing $p$ should grant the work done. This can also be argued from the unbounded case of the Hamiltonian of the harmonic oscillator. In this case you will get

$$W_A(x,p)=\sum_n E_n W_n^{H.O.}(x,p)$$

being now

$$W_n^{H.O.}(x,p)=\frac{(-1)^n}{\pi\hbar}e^{-\frac{p^2}{\hbar^2\kappa^2}-\kappa^2x^2}L_n\left(2\frac{p^2}{\hbar^2\kappa^2}+2\kappa^2x^2\right)$$

and $L_n(x)$ the Laguerre polynomials and $\kappa^2=m\omega/\hbar$. In the limit $p\rightarrow\infty$, being $x$ fixed, the Laguerre polynomial will give the driving contribution $p^n$. In this particular case, it is the "energy" of the oscillator to drive to zero with the power of $n$.


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## TuringTest 3 years ago Let f be a function from the set of integers to the set of positive integers. Suppose that, for any two integers m and n, the difference$f(m)−f(n)$is divisible by$f(m−n)$Prove that, for all integers m and n with$f(m)≤f(n)$the number$f(n)$ is divisible by $f(m)$

1. moneybird

$\frac{f(m) - f(n)}{f(m-n)}=k$Setting n = 0, $f(m) | f(0)$Setting m = 0, $f(-n)| f(n)$Setting m = 0, n = -n, $f(n)| f(-n)$$f(n) = f(-n)$Setting m = 2n, $f(n) |f(2n)$$f(2n) \ge f(n)$Setting n as -n $f(m+n)|(f(m) - f(n))$$f(m) - f(n) \ge f(m+n)$ Assume $f(m+n) > f(m)$, then$f(m+n) > f(m) - f(n)$However, this contradicts to $f(m) - f(n) \ge f(m+n)$, which means $f(m+n)$ is either smaller or equal to f(m). Assume $f(m+n) < f(m)$then setting m = n, $f(2n) < f(n)$, which contradicts to $f(2n) \ge f(n)$, Thus $f(m+n) = f(m)$ Substitute this back to the original equation$\frac{f(m+n)-f(n)}{f(m)}$Thus, we proved $f(m) | f(n)$

2. TuringTest

Wow moneybird this looks really great! I'll have to give it special attention on a non-holiday though, I don't think I'm smart enough to really evaluate it. Certainly not right now at least. I'll be sure to get others to look at this, this was an IMO problem, yeah?

3. moneybird

thanks

4. TuringTest

Having had a bit more time to digest this I can say I follow the logic and see no mistakes. That said I don't think I'm qualified to give you the all clear myself. I'll be curious to hear other opinions. Again, excellent job!

5. Zarkon

I don't see a problem

6. Zarkon

though I would add more detail.

7. TuringTest

like? what cases are not covered that should be?

8. Zarkon

all the cases are covered...just more words to explain what is going on

9. TuringTest

well he still deserves a medal then :)

10. Zarkon

I don't like this sub forum...I'm only level 2 here :)

11. TuringTest

there ya go! on the way to guru again.

12. JamesJ

Interesting proof, because it seems you've proven something stronger. When you get to the statement $f(m+n) = f(m)$ m and n are arbitrary, and hence f is constant.

13. asnaseer

I'm not clear on how setting m=0 leads to:$f(-n)| f(n)$on the 5th line of the proof. same goes for the 7th line of the proof where setting m=0 and n=-n seems to lead to:$f(n)| f(-n)$how do we reach these conclusions?

14. moneybird

Actually I missed one step, which is setting m = -n and n equal to zero.$\frac{f(-n)-f(0)}{f(-n)} \implies f(-n)| f(0)$

15. asnaseer

is that a legal step? you are (in the same step) setting m=-n AND n=0. isn't that equivalent to setting m=0?

16. moneybird

This means that f(0) is divisible by f(-n). w When m = 0, $\frac{f(0)-f(n))}{f(-n)} \implies f(-n) | f(n)$

17. Zarkon

I understand the 1st step where you showed $$f(m)|f(0)$$, but I am afraid I don't follow the rest of the arguments. this is probably beyond my expertise as I canot see how the next steps are deduced?


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# What do we mean by $X>Y$, where $X$ and $Y$ are random variables?

Let $X$ and $Y$ be any random variables defined in the same sample space. What do we mean exactly by $X>Y$?

Also, what do we mean $X>c$, where $c$ is a constant?

Thank you very much!

• How are you defining a random variable? – anomaly Feb 11 '17 at 13:00
• Thank you. Let $X$ and $Y$ be any random variables defined in the same sample space. – Probability is wonderful Feb 11 '17 at 13:16
• I was asking whether you're defining a random variable as something vague like an "event" (which is standard for probability courses that don't talk about measure theory), or as a measurable function $f:\Omega \to \mathbb{R}$ (or $\mathbb{C}$, etc.) as below. – anomaly Feb 11 '17 at 13:40
• I see. Sorry for my inaccuracy. I'd like to define a random variable as a measure function, like what you just said – Probability is wonderful Feb 11 '17 at 14:08

Random variables are measurable functions from a measurable space $(\Omega, \mathcal A)$ to another measurable space $(G, \mathcal G)$. Usually the latter space is $(\mathbb R, \mathcal B(\mathbb R)$. So saying $X > Y$ means that, for every $\omega \in \Omega$, we have $$X(\omega) > Y(\omega)$$
Of course since $X(\omega), Y(\omega) \in G$, it is implied that the relation "$>$" is defined in $G$
• I see. Thank you very much! So, $X>Y$ implies that $P(X>Y)=1$ , right? – Probability is wonderful Feb 11 '17 at 14:10
While @Ant's answer is correct there can be another meaning. "$X>Y$" can refer to some event, which can be formalized as $$\{\omega:X(\omega)>Y(\omega)\}.$$ In which case $P(X>Y)$ can have any value.
You can understand $Y>c$ in the same way by considering $c$ a a (constant) random variable $(c(\omega)=c\ \forall \omega)$.
A simple example: throw two dice, observe the scores $X$ and $Y$. You can compute $P(X>Y)=5/12$.