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https://thirdspacelearning.com/gcse-maths/number/dividing-fractions/ | GCSE Maths Number FDP Fractions
Dividing Fractions
# Dividing Fractions
Here we will learn about dividing fractions including how to divide fractions, divide fractions by whole numbers, and divide mixed fractions.
There are also dividing fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
## What is dividing fractions?
Dividing fractions is where we find the reciprocal of (flip) the second fraction, change the divide sign to a multiply and then multiply the fractions together.
For example:
There are 6 pizzas eaten at a party. Each person eats half a pizza. How many people ate pizzas?
We need to find how many halves there are in 6 pizzas.
Let’s illustrate this with a diagram:
There are 12 halves so there must have been 12 people at the party.
So,
$6 \div \frac{1}{2}=12$
We notice that this is the same as 6 × 2 which equals 12.
So to divide two fractions we flip the second fraction and multiply them.
$\begin{array}{l} 6 \div \frac{1}{2} \\\\ 6 \times \frac{2}{1} \\\\ =12 \text { people } \end{array}$
## How to divide fractions
In order to divide fractions:
1. Flip the second fraction (find its reciprocal)
2. Change the divide sign to multiplication
3. Multiply the fractions together
4. Simplify if possible
Example: Divide two proper fractions
$\frac{1}{2} \div \frac{1}{3}$
Flip the second fraction:
$\frac{1}{2} \bigcirc\frac{3}{1}$
Multiply:
$\frac{1}{2} \times \frac{3}{1}=\frac{3}{2}$
Simplify if possible:
$\frac{3}{2}$
Example: Divide a fraction by a whole number
$\frac{1}{2} \div 4$
Put the whole number over 1:
$\frac{1}{2} \div \frac{4}{1}$
Flip the second fraction:
$\frac{1}{2} \bigcirc \frac{1}{4}$
Multiply:
$\frac{1}{2} \times \frac{1}{4}=\frac{1}{8}$
Simplify if possible:
$\frac{1}{8}$
Example: Divide two mixed number fractions
$1 \frac{1}{2} \div 2 \frac{1}{4}$
Change the mixed fractions into improper fractions:
$\frac{3}{2} \div \frac{9}{4}$
Flip the second fraction:
$\frac{3}{2} \bigcirc \frac{4}{9}$
Multiply:
$\frac{3}{2} \times \frac{4}{9}=\frac{12}{18}$
Simplify if possible:
$\frac{2}{3}$
## What is a reciprocal?
The reciprocal of a number is a number which, when multiplied by the number itself, equals 1.
For example:
$\frac{2}{1} \times \frac{1}{2}= 1$
So,
$\frac{1}{2}$
is the reciprocal of
$\frac{2}{1}$
## Dividing fractions examples
### Example 1: dividing two proper fractions
Divide the fractions below:
$\frac{1}{7} \div \frac{1}{2}$
1. Flip the second fraction (find its reciprocal)
$\frac{1}{7} \bigcirc \frac{2}{1}$
Turn the second fraction upside down. The numerator becomes the denominator and vice versa.
2Change the divide sign to multiplication
$\frac{1}{7} \times \frac{2}{1}$
3Multiply the fractions together
$\frac{1}{7} \times \frac{2}{1}$
Remember: To multiply two fractions together multiply the numerators together and the denominators together.
$\frac{1 \times 2}{7 \times 1}=\frac{2}{7}$
4Simplify if possible
The fraction cannot be simplified so we are left with
$\frac{2}{7}$
### Example 2: dividing three proper fractions
Divide the fractions below:
$\frac{1}{5} \div \frac{1}{2} \div \frac{2}{3}$
In this case we have three fractions so flip the third fraction as well (find its reciprocal).
$\frac{1}{5} \bigcirc \frac{2}{1} \bigcirc \frac{3}{2}$
Turn the second and third fractions upside down. The numerator becomes the denominator and vice versa.
$\frac{1}{5} \times \frac{2}{1} \times \frac{3}{2}$
$\frac{1}{5} \times \frac{2}{1} \times \frac{3}{2}$
Remember: To multiply two fractions together multiply the numerators together and the denominators together.
$\frac{1 \times 2 \times 3}{5 \times 1 \times 2}=\frac{6}{10}$
$\frac{6}{10}$
can be simplified to
$\frac{6 \div 2}{10 \div 2}=\frac{3}{5}$
## How to divide fractions by whole numbers
In order to divide fractions by whole numbers:
1. Put the whole number over 1
2. Flip the second fraction (find its reciprocal)
3. Change the divide sign to multiplication
4. Multiply the fractions together
5. Simplify if possible
## Dividing fractions by whole numbers examples
### Example 3: dividing a fraction by a whole number
Divide the fractions below:
$\frac{1}{3} \div \ 4$
$\frac{1}{3} \div \frac{4}{1}$
$\frac{1}{3} \bigcirc \frac{1}{4}$
Turn the second fraction upside down. The numerator becomes the denominator and vice versa.
$\frac{1}{3} \times \frac{1}{4}$
$\frac{1}{3} \times \frac{1}{4}$
Remember: To multiply two fractions together multiply the numerators together and the denominators together.
$\frac{1 \times 1}{3 \times 4}=\frac{1}{12}$
The fraction cannot be simplified so we are left with
$\frac{1}{12}$
### Example 4: worded question dividing with a whole number
There are 12 pies eaten at a party. Each person ate a quarter of a pie. How many people ate pies?
First we need to create an equation to model the problem
Since 12 pies are eaten and each person ate a quarter, if we find out how many quarters there are in 12, it will tell us how many people ate pies.
We need to divide \frac{1}{4} into 12.
As an equation:
$12 \div \frac{1}{4}$
Now we are ready to perform the calculation:
$\frac{12}{1} \div \frac{1}{4}$
$\frac{12}{1} \bigcirc \frac{4}{1}$
Turn the second fraction upside down. The numerator becomes the denominator and vice versa.
$\frac{12}{1} \times \frac{4}{1}$
$\frac{12}{1} \times \frac{4}{1}$
Remember: To multiply two fractions together multiply the numerators together and the denominators together.
$\frac{12 \times 4}{1 \times 1}=\frac{48}{1}$
We are left with 48.
48 people ate pies.
## How to divide mixed fractions
In order to divide mixed fractions:
1. Change mixed fraction(s) to improper fraction(s)
2. Flip the second fraction (find its reciprocal)
3. Change the divide sign to multiplication
4. Multiply the fractions together
5. Simplify and convert back to a mixed number if possible
## Dividing mixed fractions examples
### Example 5: dividing mixed fractions
Divide the fractions below:
$1 \frac{1}{3} \div 2 \frac{1}{2}$
Remember: To convert a mixed number into an improper fraction multiply the denominator by the whole number and then add it to the numerator.
$1 \frac{+1}{\times3}=\frac{4}{3}$
$2 \frac{+1}{\times2}=\frac{5}{2}$
$\frac{4}{3} \bigcirc \frac{2}{5}$
Turn the second fraction upside down. The numerator becomes the denominator and vice versa.
$\frac{4}{3} \times \frac{2}{5}$
$\frac{4}{3} \times \frac{2}{5}$
Remember: To multiply two fractions together multiply the numerators together and the denominators together.
$\frac{4 \times 2}{3 \times 5}=\frac{8}{15}$
The fraction cannot be simplified so we are left with
$\frac{8}{15}$
### Example 6: worded question with dividing mixed fractions
A rectangle has an area of 1 \frac{1}{4} m^{2} and a width of \frac{1}{5} m . What is the length of the rectangle?
First we need to create an equation to model the problem
If we divide the width into the total area we will be able to calculate the length.
$l=1 \frac{1}{4} \div \frac{1}{5}$
Now we are ready to perform the calculation:
Remember: To convert a mixed number into an improper fraction multiply the denominator by the whole number and then add it to the numerator.
$1 \frac{+1}{\times4}=\frac{5}{4}$
$\frac{5}{4} \bigcirc \frac{5}{1}$
Turn the second fraction upside down. The numerator becomes the denominator and vice versa.
$\frac{5}{4} \times \frac{5}{1}$
$\frac{5}{4} \times \frac{5}{1}$
Remember: To multiply two fractions together multiply the numerators together and the denominators together.
$\frac{5 \times 5}{4 \times 1}=\frac{25}{4}$
The fraction cannot be simplified so we are left with a length of
$\frac{25}{4}m= 6 \frac{1}{4} \mathrm{m}$
### Common misconceptions
• Thinking that dividing by \frac{1}{2} is the same as halving a number
E.g.
$4 \div \frac{1}{2} \neq 2$
• Flipping the first fraction instead of the second fraction
A common error is to flip the first fraction instead of the second
E.g.
$\frac{9}{10} \div \frac{3}{5}$
$\frac{10}{9} \div \frac{3}{5}$
This is incorrect.
Dividing fractions is part of our series of lessons to support revision on fractions. You may find it helpful to start with the main fractions lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
### Practice dividing fractions questions
1. \frac{4}{11} \div \frac{1}{7}
\frac{4}{77}
\frac{17}{77}
\frac{28}{11}
\frac{81}{11}
Flip the second fraction and change divide to multiply:
\begin{aligned} \frac{4}{11} \div \frac{1}{7} &= \frac{4}{11} \times \frac{7}{1}\\\\ &=\frac{28}{11} \end{aligned}
2. \frac{2}{3} \div \frac{6}{10}
\frac{10}{9}
\frac{12}{18}
\frac{21}{9}
\frac{9}{10}
Flip the second fraction and change divide to multiply:
\begin{aligned} \frac{2}{3} \div \frac{6}{10} &= \frac{2}{3} \times \frac{10}{6}\\\\ &= \frac{20}{18} \end{aligned}
Simplify the fraction: \frac{20}{18} = \frac{10}{9}
3. \frac{2}{9} \div \ 5
\frac{0.4}{1.8}
\frac{10}{9}
\frac{10}{45}
\frac{2}{45}
First we need to rewrite 5 as \frac{5}{1} .
We need to calculate \frac{2}{9} \div \frac{5}{1} .
Flip the second fraction and change divide to multiply:
\begin{aligned} \frac{2}{9} \div \frac{5}{1} &= \frac{2}{9} \times \frac{1}{5}\\\\ &=\frac{2}{45} \end{aligned}
4. A seamstress has 15 yards of fabric on the roll. She needs \frac{1}{3} yard of fabric to sew each garment. How many garments can she make?
5
45
18
12
We need to calculate 15 \div \frac{1}{3} .
First we rewrite 15 as \frac{15}{1} and then flip the second fraction and change the divide to multiply.
\begin{aligned} \frac{15}{1} \div \frac{1}{3} &= \frac{15}{1} \times \frac{3}{1}\\\\ &= \frac{45}{1} \end{aligned}
\frac{45}{1} is 45 so she can make 45 garments.
5. 3 \frac{2}{3} \div 1 \frac{1}{5}
3 \frac{1}{18}
6 \frac{1}{3}
3 \frac{10}{3}
4 \frac{6}{15}
First we need to write the mixed numbers as improper fractions.
3 \frac{2}{3} = \frac{11}{3}\\
1 \frac{1}{5} = \frac{6}{5}
We need to calculate \frac{11}{3} \div \frac{6}{5} .
Flip the second fraction and change the divide to multiply:
\begin{aligned} \frac{11}{3} \div \frac{6}{5} &= \frac{11}{3} \times \frac{5}{6}\\\\ &= \frac{55}{18} \end{aligned}
Convert the fraction back to a mixed number: \frac{55}{18} = 3 \frac{1}{18}
6. There are 5 \frac{1}{3} pizzas at a party. Each person ate 1 \frac{1}{3} pizzas. How many people ate pizzas?
6
5
4
3
We need to calculate 5 \frac{1}{3} \div 1 \frac{1}{3} .
First we write the improper fractions as mixed numbers:
5 \frac{1}{3} = \frac{16}{3}
1 \frac{1}{3} = \frac{4}{3}
We need to calculate \frac{16}{3} \div \frac{4}{3} .
Flip the second fraction and change the divide to multiply:
\begin{aligned} \frac{16}{3} \div \frac{4}{3} &= \frac{16}{3} \times \frac{3}{4}\\\\ &= \frac{48}{12} \end{aligned}
Simplify the fraction: \frac{48}{12}=4
### Dividing fractions GCSE questions
1. Work out \frac{3}{5} \div \frac{1}{7}
(3 marks)
\frac{7}{1}
Evidence of flipping second fraction
(1)
Evidence of multiplication (multiplication sign seen)
(1)
\frac{21}{5}
(1)
2. Work out 1 \frac{1}{4} \div 2 \frac{2}{5} . Express your answer in the simplest form.
(4 marks)
\frac{5}{4} or \frac{12}{5} seen (converts to improper fraction)
(1)
Evidence of flipping the second fraction
(1)
Evidence of multiplication
(1)
\frac{25}{48}
(1)
3. (a) There are 24 pizzas eaten at a party. Each friend eats \frac{2}{3} of a pizza. How many friends were at the party?
(4 marks)
24 \div \frac{2}{3}
(1)
Evidence of finding the reciprocal of the second fraction \frac{3}{2} AND changing the divide to a multiply.
(1)
36 friends
(1)
## Learning checklist
You have now learned how to:
• Use the multiplication operation, including formal written methods, applied to proper and improper fractions and mixed numbers
## Still stuck?
Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.
Find out more about our GCSE maths revision programme. | 2023-04-02 02:36:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9972782135009766, "perplexity": 2923.153063531493}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296950373.88/warc/CC-MAIN-20230402012805-20230402042805-00131.warc.gz"} |
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http://stackoverflow.com/questions/6400413/android-xml-drawable-transparent-gradient/6400729 | # Android XML drawable transparent gradient
I would like some clarification on what the code in front of the HTML color code is called and how it functions.
1) I'm assuming, in the example below, the endColor of #00000000 with the two preceding 00 tells the color to be generated more transparent than say FF.
2) But what's the scale?
3) Is there some kind of hex scale that equates to certain percentages?
I'm really confused and can find no documentation because I'm not even sure of the the terminology I should be searching for other than 'xml transparent gradient' which doesn't tell me what I want to know.
Any/All help is appreciated. Thanks
<?xml version="1.0" encoding="utf-8"?>
<shape xmlns:android="http://schemas.android.com/apk/res/android"
android:shape="rectangle">
android:startColor="#DD63594A"
android:endColor="#00000000"
android:angle="90"/>
android:top="1dp"
android:right="4dp"
android:bottom="1dp" /> | 2015-01-27 12:36:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17955145239830017, "perplexity": 1490.0865904308696}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422120842874.46/warc/CC-MAIN-20150124173402-00230-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://plainmath.net/1307/solve-the-equation-3-plus-sin-theta-equal-frac-6-sqrt-2-2 | # Solve the equation 3+sin theta=frac{6-sqrt{2}}{2}
Solve the equation $3+\mathrm{sin}\theta =\frac{6-\sqrt{2}}{2}$
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Separate the right side
$3+\mathrm{sin}\theta =3-\frac{\sqrt{2}}{2}$
Substruct 3 from both sides
$\mathrm{sin}\theta =-\frac{\sqrt{2}}{2}$
The reference angle is ${\theta }^{\prime }=45°$ since $45°=\frac{\sqrt{2}}{2}$. Recall that sine is negative on QIII and QIV.
The QIII solution is
$\theta =180°+{\theta }^{\prime }=180°+45°$
$\theta =225°$
The QIV solution is
$\theta =360°-{\theta }^{\prime }=360°-45°$
$\theta =315$° | 2022-08-12 09:58:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 27, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9232685565948486, "perplexity": 4021.13899114566}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571597.73/warc/CC-MAIN-20220812075544-20220812105544-00089.warc.gz"} |
https://physics.stackexchange.com/questions/276131/finding-focal-point-of-concave-lens-using-concave-and-convex-lenses | # finding focal point of concave lens using concave and convex lenses
I was doing this experiment. Let's look at this image:
Suppose we get the imaginary object at dv by convergence lens. And this object is like a real object for the concave lens. Then how do we get a real object on the screen? It is known that only virtual objects can be made on the screen by convex lenses, and the image should be before the real object and after the lens.
According to TLE (Thin lens equation) we can find the focal point, however i'm confused about what is real and imaginary here.
Also, I had tried hard to draw ray diagram for this one. No success in forming an imaginary image in dv or real image at dr.
If the concave lens was not there and the object was at a distance greater that its focal length the convex lens would form a real image.
Introducing a concave lens results in the incoming rays from the convex lens being refracted as shown in the diagram above.
The refracted rays are still convergent and so form an image $I$ at a distance $q_2$, your $d_v$, from the the concave lens.
After passing through the convex lens where the rays would have met is at a distance $p_2$, your $d_R$, from the concave lens and that can be thought of as a virtual object for the concave lens.
• Thank you, only 2 things - I think you confused dv and dr. And also, I think the object for the concave lens is acctually I. And the virtual image is at point p2. Aug 24, 2016 at 12:42
• @TheCapacitor You can certainly look at it your way but my way is equally valid because of the reversibility of light. Try it both ways and you should get the same value for the focal length of the concave lens. Aug 24, 2016 at 14:23 | 2022-08-20 02:55:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7906149625778198, "perplexity": 385.31098115621035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573876.92/warc/CC-MAIN-20220820012448-20220820042448-00234.warc.gz"} |
https://mathoverflow.net/questions/194986/integrability-conditions-of-lax-pairs/309869 | # Integrability - conditions of lax pairs
I'm trying to understand what the conditions are for the Lax pairs for the zero-curvature representation:
$$\partial_t U - \partial_x V + [U,V]=0$$
where $U=U(x,t,\lambda)$ and $V=V(x,t,\lambda)$ are matrix-valued functions and $\lambda$ is a parameter.
The motivation behind this question is that the Lax pairs for the KdV equation:
$$u_t + 6uu_x - u_{xxx} = 0$$
is given by:
$$U = \begin{pmatrix} 0 & 1 \\ \lambda + u & 0 \end{pmatrix} \text{ and } V = \begin{pmatrix} u_x & 4 \lambda - 2u \\ 4 \lambda^2 + 2 \lambda u + u_{xx} - 2u^2 & - u_x \end{pmatrix}$$
Now, it is not too difficult to verify that this indeed satisfies the zero-curvature representation, but I'm trying to figure out why we cannot use the Lax pairs:
$$U = \begin{pmatrix} 0 & 0 \\ \lambda + u & 0 \end{pmatrix} \text{ and } V = \begin{pmatrix} 0 & 0 \\ \lambda + 3 u^2 - u_{xx} & 0 \end{pmatrix}$$
These matrices clearly satisfy the zero-curvature representation, but for some reason none of the notes I've been reading use them. What is the reason that they are not a valid Lax pair for the KdV equation?
I've also asked this question here (I hope that's ok): http://www.physicsoverflow.org/26475/integrability-conditions-of-lax-pairs
One way to see this, is that you want the zero-curvature representation to be useful and tell you something you didn't know before. Your representation has the problem of being singular, in the sense that the Lax matrices have zero determinant. It would be more eveident if we were really speaking of the Lax representation: $$L_t=[M,L]$$ where all the usefulness comes from having the possibility to say that the eigenvalues (or the coefficients of the characteristic polynomial) of $L$ do not evolve (and hence are integrals of motion). You see that, if you take an $L$ for which the coefficients of the characteristic polynomials are identically zero, you don't achieve much. This very same argument applies to the zero curvature representation if you remember how to pass from it to the Lax representation via the monodromy matrix. In any case, even if you ignore for the time being this problem and try to go through inverse scattering, you very soon hit the same wall.
In addition to the answer by @issoloroap, the article Prolongation structures of nonlinear evolution equations by Allan Fordy (here is the Mathscinet link and here is its first page on Google books) explains why "good" zero-curvature representations should live in (semi)simple Lie algebras.
Another important point is that the spectral parameter should be essential, i.e., it should not be removable by gauge transformations, cf. e.g. this paper and references therein, and for the discussion of related issues in the case of dispersionless sysems in more than two independent variables see e.g. this article, and this one, and references therein. | 2021-10-20 04:43:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.883171796798706, "perplexity": 208.2939218749697}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585302.56/warc/CC-MAIN-20211020024111-20211020054111-00064.warc.gz"} |
http://mysciencephysics.com/chemistry/city-air-pollution-drastically-enhances-formation-of-pure-aerosols-over-the-amazon-rainforest/ | # City air pollution drastically enhances formation of pure aerosols over the Amazon rainforest
### Code availability
We used the neighborhood regional Climate Analysis and Forecasting Mannequin coupled to chemistry (WRF-Chem model three.5.139,40) for producing modeling outcomes on this Manuscript. WRF-Chem is a neighborhood mannequin and is accessible to customers. Particular WRF-Chem configurations and modifications to gas- and particle-phase chemistry parameterizations used to generate outcomes on this examine are described under.
### WRF-Chem setup
We use the regional WRF-Chem mannequin39,40 at cloud-, chemistry, and emissions-resolving scales i.e. at 2 km grid spacing, which is at a a lot increased decision than that utilized in earlier international modeling research (usually ~100’s of km)62. Since high-resolution simulations explicitly resolve options in clouds, emissions, and chemistry, they don’t undergo from uncertainties in parameterizations wanted to signify these options in coarser decision international fashions. Hint gases, aerosols, and clouds are simulated concurrently with meteorology40. Biogenic VOC emissions are predicted utilizing the Mannequin of Emissions of Gases and Aerosols from Nature (MEGAN v 2.1)51, which is coupled to the Group Land Mannequin (CLM). CLM is run on the identical grid spacing as WRF-Chem.
We use a nested grid configuration with an outer 10 km grid spacing area protecting 1500 × 1000 km and an interior 2 km grid spacing area protecting 450 × 300 km centered over Manaus Metropolis. Meteorological and chemical boundary circumstances, land-surface scheme, and radiation scheme used for configuring the WRF-Chem runs used on this work are listed in Supplementary Desk 1. The land floor information and emissions of hint gases and aerosols used for the simulations have been the most effective accessible merchandise for South America. The floor albedo, vegetation, and inexperienced fraction used on this examine are documented in Beck et al.63. All mannequin predictions analysed on this examine are for the high-resolution interior area that higher resolves emissions, chemistry, and clouds in comparison with the outer area. Additionally, the two km grid spacing interior area explicitly resolves deep convective clouds, so no convective cloud parameterization is used for the interior area.
The Nationwide Facilities for Environmental Prediction (NCEP) Local weather Forecast System Model 2 (CFSv2) reanalysis information (CFSR)64 offers the meteorological preliminary and boundary circumstances. Meteorological circumstances have been spun-up for 24 h, adopted by 72 h of simulation, whereas the hint fuel and aerosol species from the earlier simulation have been used as preliminary circumstances. We performed concatenated Four-day simulations, following the strategy of Medeiros et al.41 for this area. The chemical boundary circumstances for hint gases and aerosols over the outer area are supplied by a quasi-global WRF-Chem simulation in 201465, whereas the interior area acquired boundary circumstances from the outer area.
### Meteorological fields
Supplementary Determine 6 exhibits that the mannequin moderately simulates the multi-day variations of a number of meteorological fields with measurements, together with floor temperature, particular humidity, wind speeds, boundary layer top, downwelling photo voltaic radiation, and floor latent warmth flux. The floor temperature, particular humidity, and wind speeds are averaged from three websites round Manaus and downwind areas (T1-Manaus, T2, and T3 websites), boundary layer top and downwelling photo voltaic radiation are taken from the T3 web site, and floor latent warmth flux is taken from the T0k web site. The mannequin is randomly sampled for 1000 grid factors over land inside 50 km radius centered at T3 and Manaus for all of the meteorological fields aside from latent warmth flux. For latent warmth flux, the mannequin randomly sampled 200 grid factors inside 30 km radius of T0k web site the place latent warmth flux above forest cover from a tower measurement is accessible. A random sampling technique to the mannequin output is chosen to imitate massive spatial variability from just a few single level observations through the comparatively brief examine interval. Floor meteorology measurements at T3 are from ARM MET datastream66, floor radiative flux measurements are from the ARM RADFLUX product67. Boundary layer top on the T3 web site was derived utilizing the vertical velocity statistics from the ARM DLPROFWSTATS4NEWS product68. The tactic follows Tucker et al.69 through the use of profiles of Doppler Lidar measured vertical velocity variance as a measure of the turbulence inside the boundary layer. Ranging from the floor, the primary vertical top degree the place the vertical velocity variance drops under zero.04 m2 s−2 is designated because the boundary layer top.
### Emissions of hint gases and aerosols
Since our WRF-Chem simulations are performed at excessive decision, together with emissions of hint gases and aerosols have been difficult for the Amazon, since detailed high-resolution emission inventories are scarce for this area. We mix a number of emissions inventories from completely different sources to get affordable estimates of hint gases and aerosol emissions.
We embody main emissions of gases akin to CO, non-methane unstable natural compounds (NMVOC), sulfur dioxide (SO2), ammonia (NH3), and oxides of nitrogen (NOx) and aerosols, together with natural carbon (OC), black carbon (BC) and sulfate (SO4) from anthropogenic and biomass burning sources. Emissions of aerosols and gases from the site visitors sector have been included from an in depth excessive decision 2 km × 2 km gridded emissions stock developed for this area primarily based on the methodology described in a earlier examine70. We additionally included emissions of CO, NOx, SO2, VOCs, and particulate OC, BC, and SO4 from energy vegetation over the Manaus area together with a mixture of gasoline oil, diesel, and pure fuel utilized in 2014 for electrical energy era and emissions from a big oil refinery primarily based on a latest examine41. Emissions of CO, NOx, SO2, VOCs, and particulate OC, BC, and SO4 from these level sources have been included. Extra anthropogenic SO2 and SO4 space emissions have been additionally included primarily based on VOCA (http://bio.cgrer.uiowa.edu/VOCA_emis/) and the Emissions Database for International Atmospheric Analysis (EDGAR v4.1), respectively. NH3 emissions from trade, vitality, residential, and agriculture are from the Hemispheric Transport of Air Air pollution (HTAP_v2.2) 2010 emissions stock71.
### Biogenic and biomass burning emissions
We included biomass burning emissions together with each gases and aerosols from the 2007 Fireplace Stock from NCAR (FINN07)72. FINN07 particulate emissions embody natural carbon (transformed to OA utilizing an OA/OC ratio of 1.Four), black carbon, PM2.5, and PM10. NMVOC emissions from each anthropogenic and biomass burning sources are speciated in line with the SAPRC-99 mechanism.
We additionally embody emissions of biogenic unstable natural compounds (BVOC). BVOC emissions are derived from the newest model of Mannequin of Emissions of Gases and Aerosols from Nature (MEGAN v2.1) that has been lately coupled inside the land floor scheme CLM4 (Group Land Mannequin model Four.zero) in WRF-Chem73. The 138 biogenic species from MEGAN are lumped into three biogenic VOC courses: isoprene (ISOP), terpenes (TERP), and sesquiterpenes (SESQ).
### Unspeciated natural emissions
Unspeciated natural emissions are historically not included in emission inventories, however are vital for anthropogenic SOA formation74,75,76. About 10–20% of whole non-methane natural fuel (NMOG) emissions usually are not routinely included in emissions inventories74. These unspeciated emissions have important potential to type SOA since they’re semi-volatile or intermediate volatility organics (SIVOCs). We signify all unspeciated NMOG emissions as an intermediate volatility species (i.e. C* = 104 µg m−three) for biomass burning and fossil-fuel sources known as a gas-phase species, IV-POA (g). Emissions of IV-POA (g) are assumed to be 20% of the entire non-methane natural fuel (NMOG) emissions for each biomass burning and fossil-fuel sources primarily based on unspeciated fraction of NMOG emissions reported in Jathar et al.74. As well as, in our mannequin, we assume that 50% of the emitted POA evaporates instantaneously and contributes to IV-POA (g), in step with Jathar et al.74, whereas the remaining 50% is assumed to be non-volatile. This reduces the variety of POA tracers that should be advected within the mannequin and will increase computational effectivity since our focus is principally on SOA formation. Oxidation of the evaporated POA additionally contributes to anthropogenic SOA formation, as described later.
### Results of soil NO emissions
We included sources of NO emissions from soils inside WRF-Chem. Earlier research counsel soil NO emissions for tropical forests within the vary 20–60 µg NO m−2 h−177,78,79. Nevertheless, a lot of this NO reacts inside the cover with ozone and doesn’t enter the above-canopy ambiance. This in-canopy discount of NO reduces the efficient flux of NO within the above-canopy ambiance by ~75%. We select the higher sure of soil NO emissions and scale back it by 75% to acquire an efficient NO emissions flux of 15 µg NO m−2 h−1 (eight.three × 109 molecules cm−2 s−1). This worth is near the soil NOx emissions vary steered by discipline measurements over Amazon rainforests (1.2 to 7.zero × 109 molecules cm−2 s−1), as mentioned by Liu et al.36. Below background Amazonian circumstances, Liu et al.36 steered that the relative response fee of isoprene peroxy radicals (ISOPOO) with HO2 to that with NO is roughly unity. Certainly, our WRF-Chem simulations present that the ratio of reactions charges of ISOPOO with NO to that with HO2 is unity beneath background circumstances. This will increase confidence within the capability of the mannequin to simulate the relative response charges of isoprene peroxy radicals. In distinction, a earlier examine utilizing the worldwide mannequin GEOS-Chem predicted a a lot smaller relative response fee of ISOPOO with NO in comparison with HO2, which was attributed to its order of magnitude decrease soil NOx emissions in comparison with measurements36.
### Background sources of sulfate within the Amazon
Along with soil NOx emissions, we additionally included emissions of dimethyl sulfide (DMS) of zero.eight ng m−2 s−1 from native soil and plant emissions inside the Amazon rainforest primarily based on a latest examine80. DMS can also be advected from the oceans inside our modeling area. Oxidation of DMS leads to the formation of SO2, which is a background sulfate supply. Nevertheless, mannequin simulations point out that native DMS emissions are a minor supply of sulfate, whereas the Manaus plume is a serious supply, which impacts each in-plume and background sulfate concentrations. Simulated background sulfate of ~zero.1 µg m−three agrees with plane measurements (e.g. on 13 March). The mannequin simulates the rising developments of sulfate inside plumes in comparison with the background (not proven). Nevertheless, in-plume sulfate simulated by the mannequin is an element of two increased than the noticed sulfate, which is inside the anticipated uncertainties of sulfate emissions sources inside the Amazon.
### Simulating SOA utilizing the VBS strategy
Simulated SOA from pure gas-phase chemistry pathway is represented within the mannequin utilizing a volatility foundation set (VBS) strategy. The VBS strategy represents a number of generations of oxidation of biogenic VOCs that embody isoprene, monoterpene, and sesquiterpene compound courses, and anthropogenic and biomass burning precursors utilizing a lumped set of compounds. Preliminary yields are decided by becoming environmental chamber measurements and usually range with VOC, NOx, and oxidants (Supplementary Desk 2). We modified the VBS strategy to incorporate additional getting older of organics at longer-timescale getting older past that noticed in environmental chambers, as described later on this part.
### Simulating anthropogenic SOA from unspeciated NMOG emissions
Oxidation of anthropogenic IV-POA (g) by OH radicals leads to the formation of semi-volatile SOA species that may be represented by becoming environmental chamber measurements utilizing a VBS strategy. Semi-volatile SOA formation yields as a result of oxidation of anthropogenic IV-POA (g) emissions have been assumed to be the identical as these reported for on- and off-road diesel car sources and biomass burning/wooden burning from Desk S3 in Jathar et al.74 as proven under:
$$startlleft( proper) + mathrm,_ + mathrm,_ mathrm,_mathrmthreemathrm,_mathrmfinish.$$
(1)
SVOC1, SVOC2, SVOC3, and SVOC4 signify lumped VBS species with C* of zero.1, 1, 10 and 100 μg m−three, respectively. These preliminary yields signify the primary few generations of chemistry measured in chamber experiments. The sum of particle-phase concentrations of SVOC1, SVOC2, SVOC3, and SVOC4 includes anthropogenic SOA (Supplementary Determine 5e) in our examine.
### Simulating pure biogenic SOA
Yields range primarily based on precursor sort, oxidants (OH, ozone or nitrate i.e. NO3 radicals) and in addition NOx ranges through the measurements. The general NOx-dependent yield is calculated as a sum of excessive and low NOx yields weighted by NOx branching ratio87 at every mannequin grid level and time.
We embody further reactions for the VBS bins inside the SAPRC-99 mechanism:
$$,,, to mathop limits_^Four ,$$
(2)
$$a_ = a_ + a_left( proper),$$
(three)
the place BVOC(g) are the first biogenic fuel species (isoprene, terpene, or sesquiterpene), BVSOA(g)i represents SOA precursor species shaped after photochemical oxidation of the BVOC(g), ‘i’ is the volatility bin (i = 1, …, Four equivalent to C* = zero.1, 1, 10, and 100 µg m−three), ai is the general NOx-dependent molar yield calculated from eq. (2), ai,excessive and ai,low are the molar yields beneath excessive and low NOx circumstances, respectively, as proven in Supplementary Desk 2, and B is the NOx branching ratio as outlined by Lane et al.87. On this work, we additionally included additional NOx-dependent multigenerational getting older of each biogenic SOA and anthropogenic organics as described under.
### Additional getting older of VBS organics
Within the ambiance, longer-timescale getting older (past that noticed in chambers) can change SOA yields in comparison with these decided from chamber measurements. Multigenerational getting older leads to each functionalization (lowering volatility) and fragmentation (rising volatility) reactions. In our earlier research88,89, we confirmed that gas-phase fragmentation processes, which are sometimes uncared for in chemical transport SOA modeling parameterizations, might have massive results on each regional and international SOA loadings. As well as, the branching ratio between fragmentation and functionalization is reported to range with the relative response charges between NOx, HO2, and RO2 radicals. Gasoline-phase fragmentation is reportedly extra prevalent beneath high-NOx in comparison with low NOx circumstances90,91. On this examine, we assume that the chance of fragmentation equals the branching ratio between peroxy-NO radicals response charges to the sum of all peroxy radical reactions charges (together with peroxy-peroxy and peroxy-NOx reactions). Nevertheless, we assign an higher restrict of 75% fragmentation primarily based on our earlier sensitivity research that different this branching ratio (however with out an express NOx dependence)88,89. Every era of getting older of the VBS SOA species leads to each functionalization and fragmentation reactions as a operate of peroxy-NOx branching ratio, calculated at every WRF-Chem grid and time-step. As well as, we assume small fraction of organics fragment to species of a lot increased volatility and usually are not tracked. The utmost fraction of organics that’s moved outdoors the VBS vary is assumed as 10% by mass equivalent to the utmost fragmentation branching of 75%88,89. A sensitivity simulation, which turned off this extra getting older confirmed a minor lower in simulated mass concentrations of SOA within the background over the Amazon in comparison with the default simulation (not proven). We count on that the impact of NOx-dependent multigenerational getting older is much less pronounced over the Amazon in comparison with extra polluted areas (such because the continental United States) seemingly as a result of smaller background oxidant concentrations over the Amazon. Thus, the added multigenerational getting older doesn’t have an effect on the primary outcomes and conclusions of this examine.
### Aerosol remedies in MOSAIC module
The condensation of low volatility gases (H2SO4 and CH3SO3H) and the dynamic partitioning of semi-volatile inorganic gases (HNO3, HCl, and NH3) to size-distributed liquid, mixed-phase, and stable atmospheric aerosols are represented by the Mannequin For Simulating Aerosol Interplay and Chemistry (MOSAIC) aerosol module43. On this examine, the aerosol species simulated in MOSAIC embody sulfate, nitrate, ammonium, different inorganics (OIN), elemental carbon, natural carbon and aerosol water. We represented aerosols by Four-size sections with dry particle diameter ranges of zero.039–zero.156, zero.156–zero.624, zero.624–2.5, and a pair of.5–10.zero µm. Each interstitial and activated (cloud-borne) species equivalent to all aerosol chemical elements are included and advected. Additionally, every simulated dimension bin consists of each particle quantity and mass. The MOSAIC aerosol module consists of remedies of nucleation, coagulation, and condensation as described in earlier research43. The dimensions-dependent dry deposition of particles (each quantity and mass) relies on the strategy of Zhang et al.92. As well as, each in-cloud and below-cloud moist removing of hint gases and aerosols are simulated following Easter et al.93.
### Gasoline-phase chemistry
Gasoline-phase chemistry on this examine relies on the Statewide Air Air pollution Analysis Heart (SAPRC-99) mechanism94, which incorporates 211 reactions of 56 gases and 18 free radicals. This mechanism is up to date to incorporate gas-phase photochemical oxidation of gas-phase natural species to type SOA particles. We embody SOA shaped as a result of oxidation of semi-volatile and intermediate volatility natural compounds (S/IVOC) emitted from anthropogenic and biomass burning sources (SI-SOA) and conventional SOA (V-SOA) shaped as a result of oxidation of unstable natural compounds (VOC) precursors from biogenic emissions. We additionally prolonged this gas-phase chemistry mechanism to incorporate isoprene epoxydiol (IEPOX) formation (Supplementary Desk 5). VOC oxidation and catalytic results of NOx on the oxidant cycle sustains the atmospheric oxidation capability95. NO is critical for HOx biking and formation of ozone and OH radicals. Extra OH recycling mechanisms have been steered within the literature44, nonetheless, these recycling mechanisms typically trigger substantial overestimation of noticed OH45. Due to this fact, on this examine, we don’t embody further OH recycling mechanisms within the mannequin apart from reactions between HO2 and NO.
### Multi-phase IEPOX chemistry
Multiphase SOA formation from isoprene oxidation is simulated utilizing new aqueous chemistry modules that we added inside WRF-Chem primarily based on the simpleGAMMA mannequin42. These aqueous chemistry modules are coupled to the mannequin for simulating aerosol interactions and chemistry (MOSAIC), which simulates key inorganic species like sulfate, nitrate, ammonium ions, particle acidity, and water wanted by the simpleGamma mannequin43. The uptake of IEPOX inside aqueous aerosols is set by its solubility (Henry’s regulation fixed, HIEPOX), adopted by its response within the particle part42. Right here, we set HIEPOX as 1.7 × 108 M atm−1 following Gaston et al.24, which represents the upper finish of HIEPOX values steered within the literature24,96,97,98,99,100. Thus, IEPOX-SOA simulated on this examine, most probably represents an higher sure estimate. Solely a fraction of the epoxide reactively taken up by particles contributes to IEPOX-SOA formation101. The fraction of low volatility accretion merchandise of IEPOX-SOA might range considerably at completely different areas as a result of variable chemistry and partitioning. On this examine, following measurements throughout GoAmazon2014/5 by Isaacman et al.26, we constrained this fraction to zero.Four i.e. solely 40% of IEPOX-SOA merchandise persist within the particle-phase as a result of their low volatility in our simulations. Merchandise of IEPOX reactive uptake which might be semi-volatile evaporate from particles, leaving solely low volatility accretion merchandise as IEPOX-SOA and organosulfates27.
### Key elements affecting computational value of simulations
Our simulations use detailed SOA parameterizations represented by the VBS strategy, and a lot of gas- and particle-phase VBS species should be replicated for various supply classes, together with anthropogenic and biogenic courses (isoprene, terpene, sesquiterpenes courses) to resolve their particular person contributions. Particle-phase species additionally multiply with variety of dimension bins and in addition should be replicated for interstitial and cloud-borne species which might be advected within the mannequin. Thus, a lot of gas- and particle-phase species (whole of 420) are advected within the mannequin, drastically rising the computational value in comparison with chemistry packages with out SOA inside WRF-Chem. As well as, the excessive decision nested grid configuration (2 km grid spacing) additionally will increase WRF-Chem computational prices in comparison with international modeling research that use a lot coarser grid spacings (~100–200 km grid spacings).
### Simulations with Manaus emissions on/off
We examine WRF-Chem simulations with Manaus emissions on/off to quantify how Manaus emissions amplify oxidant biking and biogenic SOA formation over the Amazon. Plume areas simulated by the mannequin could be shifted in comparison with observations as a result of minor errors in simulated wind route and dispersion. We conduct a cautious evaluation to determine the shifts in model-simulated plume in comparison with plane measurements. Determine 1 and Supplementary Figures three and Four present that OA, ozone, CO, and NOy concentrations alongside measured and simulated flight transects can be utilized to precisely diagnose the shifts in simulated plume in comparison with measurements. The simulated CO baseline has some uncertainty relying on the boundary circumstances (from international WRF simulations) and was adjusted by a relentless worth of ~30 ppb for higher visible comparability with measurements. The important thing right here is in-plume CO values are considerably bigger than the background. Over the Amazon, NOy sharply will increase inside city plumes by greater than an order of magnitude in comparison with background areas and is used to determine the shifted plume within the mannequin in comparison with observations. Our evaluation on this examine focuses on plane transects ~500 m altitude since they’re inside the blended boundary layer through the daytime. Plane measurements signify a snapshot of modifications that happen in biogenic SOA as a result of anthropogenic emissions over the in any other case pristine wet-season Amazon. | 2019-10-21 05:01:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5932630300521851, "perplexity": 11728.483071684672}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987756350.80/warc/CC-MAIN-20191021043233-20191021070733-00524.warc.gz"} |
https://mne.tools/stable/generated/mne.minimum_norm.apply_inverse_raw.html | # mne.minimum_norm.apply_inverse_raw¶
mne.minimum_norm.apply_inverse_raw(raw, inverse_operator, lambda2, method='dSPM', label=None, start=None, stop=None, nave=1, time_func=None, pick_ori=None, buffer_size=None, prepared=False, method_params=None, verbose=None)[source]
Apply inverse operator to Raw data.
Parameters
rawRaw object
Raw data.
inverse_operatordict
Inverse operator.
lambda2float
The regularization parameter.
method“MNE” | “dSPM” | “sLORETA” | “eLORETA”
Use minimum norm, dSPM (default), sLORETA, or eLORETA.
label
Restricts the source estimates to a given label. If None, source estimates will be computed for the entire source space.
startint
Index of first time sample (index not time is seconds).
stopint
Index of first time sample not to include (index not time is seconds).
naveint
Number of averages used to regularize the solution. Set to 1 on raw data.
time_funccallable()
Linear function applied to sensor space time series.
pick_oriNone | “normal” | “vector”
If “normal”, rather than pooling the orientations by taking the norm, only the radial component is kept. This is only implemented when working with loose orientations. If “vector”, no pooling of the orientations is done and the vector result will be returned in the form of a mne.VectorSourceEstimate object. This does not work when using an inverse operator with fixed orientations.
buffer_size
If not None, the computation of the inverse and the combination of the current components is performed in segments of length buffer_size samples. While slightly slower, this is useful for long datasets as it reduces the memory requirements by approx. a factor of 3 (assuming buffer_size << data length). Note that this setting has no effect for fixed-orientation inverse operators.
preparedbool
If True, do not call prepare_inverse_operator().
method_params
Additional options for eLORETA. See Notes of apply_inverse().
New in version 0.16.
verbose
If not None, override default verbose level (see mne.verbose() and Logging documentation for more).
Returns
stc
The source estimates.
apply_inverse_epochs
apply_inverse | 2019-11-18 05:41:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43314647674560547, "perplexity": 6811.705195361485}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669454.33/warc/CC-MAIN-20191118053441-20191118081441-00530.warc.gz"} |
https://openclassrooms.com/en/courses/6389626-train-a-supervised-machine-learning-model/6405931-understand-the-linear-regression-algorithm | Last updated on 8/5/21
## Understand the Linear Regression Algorithm
In machine learning, you use regression to predict continuous values such as temperature, stock price, and life expectancy. Regression is a statistical technique that finds relationships between features.
We will be looking at using sklearn's linear regression functionality in a subsequent chapter. You don't need to fully understand the math behind the algorithm to use it, but you should understand the concepts covered in this chapter as they are the foundation for a lot of machine learning techniques.
Let's consider a simplified version of the income versus happiness scatter plot and plot just 10 points:
The big idea with linear regression is to fit a line to the data:
Make some predictions, such as here where we can predict that a country with an average income of \$70k will have a happiness score of 5.51:
The line becomes a way to estimate the happiness from the income, effectively a machine learning model!
But is the line above the best fitting line? How can you tell?
If you measure the distance from the points to the fitted line, you have a measure of how close the fitted line is to the actual, real-life data points:
The distance between the fitted line and an actual point is called a residual. The residuals for all the points are shown by the red dotted lines above. Here are the residuals (i.e., the lengths of the red dotted lines):
-1.03
1.78
-0.72
0.9
-0.74
1.06
-0.65
-0.95
-0.99
1.33
The length of any one residual line tells you how good the model is estimating the happiness value for that sample point. So naturally, the total length of all the residual lines tells you how good the model is at estimating the happiness value from all the sample points. But, if you add up all the numbers above, the negative and positive values will tend to cancel each other out.
-1.03 + 1.78 + -0.72 + 0.9 + -0.74 + 1.06 + -0.65 + -0.95 + -0.99 + 1.33 =
Total: 0.01
So, a better way would be to take the absolute values and sum those:
1.03
1.78
0.72
0.9
0.74
1.06
0.65
0.95
0.99
1.33
Total: 10.15
In this way, you are just looking at the magnitude of the residual and ignoring whether it is above or below the line. After all, an error of 1.03 is as significant, whether it's an overestimate or an underestimate.
Another way to eliminate the sign is to square each value and sum the squares. Let's square the original signed residuals and sum them:
1.0609
3.1684
0.5184
0.81
0.5476
1.1236
0.4225
0.9025
0.9801
1.7689
Total: 11.30
In general, use the squaring method rather than the absolute value method. This is the measure of how good a fit your line is. It's called the sum of squared residuals, or SSR. It's also sometimes called the sum of squared errors, or SSE.
So, now you have a way of measuring the first attempt at fitting a line, let's try out a few others:
Much better!
Much worse!!
Now, you are probably thinking that there must be a better way to find the best fitting line than trying them out one-by-one. You're right! It's called the ordinary least squares method (OLS method).
You may remember from your school math lessons the formula for a straight line:
y = mx + c
where m is the slope of the line, and c is the intercept on the y axis.
If m= 0.5 and c = 20, you get the following function:
y = 0.5x + 20
Let's plot that function:
The line crosses the y-axis at y=20. This is the intercept c in our formula and the slope is 1 in 2. This is the slope, m is the formula for a line.
To define any line, we need to specify the right values for m and c. Looking back to our set of points:
Find the values of m and c, which minimizes the sum of squared residuals to these points. Easier said than done! Fortunately, some formulae do just that. Here's how to compute the slope:
And here's how to compute the y intercept:
Here is a simple Python function that implements these formulae, given a set of points in the arrays x and y:
def ols(x, y):
xmean = x.mean()
ymean = y.mean()
xvariance = sum([(x - xmean)**2 for x in x])
xycovariance = 0
for i in range(len(x)):
xycovariance += (x[i] - xmean) * (y[i] - ymean)
m = xycovariance / xvariance
c = ymean - m * xmean
return m, c
Plugging the five sample points into this function, you get:
m = 0.05442586953454801
c = 2.283356593384597
And plugging m and c into the general formula for a straight line, you get:
y= 0.05442586953454801 * x + 2.283356593384597
Let's plot the line:
This is the line of best fit using the OLS method.
Just to prove this is better than the other lines, let's plot the residuals and calculate the SSR:
This beats our previous best SSR of 6.23.
The whole point of this exercise is to build a model to make predictions. Now that we have a formula for the line, we have a formula for predicting happiness from income!
happiness = 0.05442586953454801 * income + 2.283356593384597
To predict happiness from an income of 70:
happiness = 0.05442586953454801 * 70 + 2.283356593384597 = 6.09
And here is the graphical representation of the above:
So far, in this chapter, we have only discussed situations where we are predicting a value from another single feature. To do this, we fitted a line using the formula:
y = mx + c
But what if there are two or more input features?
For two input features, the formula looks like this:
Instead of describing a line, it describes a two-dimensional plane. We need this formula to predict happiness from both income and employment levels.
The regression task changes from "fitting a line that best predicts happiness from income," to "fitting a 2D plane that best predicts happiness from income and employment levels:"
For three input features, the formula looks like this:
We are fitting a 3D plane to the data!
In other words, for each new feature you add, you need to compute an additional slope m, and we increase the dimensionality of the fitted plane.
• Linear regression works by fitting a line to a set of data points.
• You can measure the distance from the line to your actual data points. These are called residuals.
• Find your total error by summing the squares of the residuals for all data points. This is called the sum of squared residuals, or SSR.
• The ordinary least squares (OLS) method finds the optimal fitting line by minimizing the SSR.
• Use the line computed using the OLS method to predict one feature (the y-axis) from the other feature (the x-axis).
• Extend the approach to use multiple input features. This is called multiple regression.
In the next chapter, we will take a look at the KNN algorithm. | 2021-10-26 11:43:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6365662217140198, "perplexity": 666.992235720493}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587877.85/warc/CC-MAIN-20211026103840-20211026133840-00156.warc.gz"} |
https://laurentperrinet.github.io/publication/ladret-19-sfn/ | # Orientation selectivity to synthetic natural patterns in a cortical-like model of the cat primary visual cortex
### Abstract
A key property of the neurons in the primary visual cortex (V1) is their selectivity to oriented stimuli in the visual field. Orientation selectivity allows the segmentation of objects in natural visual scenes, which is the first step in building integrative representations from retinal inputs. As such, V1 has always been of central interest in creating artificial neural networks and the recent years have seen a growing interest in the creation of explainable yet robust and adaptive models of cortical visual processes, for fundamental or applied purposes. One notable challenge for those models is to behave reliably in generic natural environments, where information is usually hidden in noise, while most models are typically studied with oriented gratings. Here we show that a simple biologically inspired neural network accounts for orientation selectivity to natural-like textures in the cat’s primary visual cortex. Our spiking neural network (SNN) is made of point neurons organized in recurrent and hierarchical layers based on the structure of cortical layers IV and II/III. We found that Spike-timing plasticity and synaptic recurrence allowed the SNN to self-organize its connections weights and reproduce the activity of neurons recorded with laminar probes in cortical areas 17 and 18 of cats, notably orientation tuning responses. After less than 5 seconds of stimulus presentation, the SNN displays narrow orientation selectivity (bandwidth = 10$,^∘$) characteristic of sparse representations, removes noise from the input and learns the structure of natural pattern repetitions. Our results support the use of natural stimuli to study theoretical and experimental cortical dynamics. Furthermore, this model encourages using SNNs to reduce complexity in cortical networks as a method to understand the separate contribution of different components in the laminar organization of the cortex. From an applied perspective, the computations this network performs could also be used as an alternative to classical blackbox Deep Learning models used in artificial vision.
Type
Publication
Proceedings of the Society for Neuroscience conference | 2019-08-18 19:31:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22912408411502838, "perplexity": 1923.8843069882757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313996.39/warc/CC-MAIN-20190818185421-20190818211421-00267.warc.gz"} |
https://punchingkitty.com/2010/05/19/mardy-gilyard-got-robbed/ | # Mardy Gilyard Got Robbed
Newly drafted Rams wide receiver Mardy Gilyard got jacked.
“I am OK. Just upset, you know, more than anything,” Gilyard said Tuesday from his Northern Kentucky home. “It’s part of living in the city. I know from my experiences in the city when it warms up – as soon as it warms up – the grimy cats in the city come out. I wasn’t paying attention to my surroundings as much as I should have been.”
Well that can happen. Sucks though especially since everything we’ve heard about Gilyard points to him being a great guy. Big draft pick like him should probably get a body guard.
Two men armed with a gun approached Gilyard and his bodyguard, Terry Hobbs, and robbed them of $300 in cash and$1,000 in jewelry about 9:30 p.m., a police report shows. No injuries were reported.
Oh.
Maybe get a good body guard and not just the gentle giant you’ve known since middle school. What’s the point of a bodyguard if you can still get jacked in the street. Maybe it was more of a Ben Roethlisberger-style “I Don’t Think It’s Rape If You Hold Her Down For Me” kinda bodyguard? If it is, don’t waste your money Mardy, plenty of jersey-chasers around here.
We also really love that Gilyard lied to the mugger.
“We see your chains! Give me your chains!” Gilyard recalled the robber yelling at him. The man also demanded the keys to Gilyard’s new car. Gilyard , who had the keys in his pants pocket, told him the car had a keyless ignition.
“I’m like, ‘brother, I ain’t got the keys,” Gilyard recalled. “So then it was, ‘well, forget the keys! Give me the money! What about the money!’ That’s when he put a .38 revolver in my face. I gave him the pocket change I had on me.”
Dumb ass mugger. Wait, crap. Here’s hoping he doesn’t read the Cincinnati Enquirer.
Good thing, if he’s already been getting jacked: Should fit in just fine here in St. Louis.
via KSDK and The Cincinnati Enquirer | 2019-04-22 02:22:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2955853343009949, "perplexity": 8773.494632653837}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578533774.1/warc/CC-MAIN-20190422015736-20190422041736-00485.warc.gz"} |
http://en.wikipedia.org/wiki/Talk:Confluence_(abstract_rewriting) | # Talk:Confluence (abstract rewriting)
WikiProject Computer science (Rated Start-class)
This article is within the scope of WikiProject Computer science, a collaborative effort to improve the coverage of Computer science related articles on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Start This article has been rated as Start-Class on the project's quality scale.
## I am not a Wiki-LaTeX expert, but this code may work for the article
I am not an expert in all bells and whistles, but this code may make more easy to embed the diagrams compared to upload a picture:
$\begin{array}{ccccc} {}&{}&{(11+9)\times(2+4)}&{}&{}\\ {}&{\swarrow}&{}&{\searrow}&{}\\ {20\times(2+4)}&{}&{}&{}&{(11+9)\times 6}\\ {}&{\searrow}&{}&{\swarrow}&{}\\ {}&{}&{20+6}&{}&{}\\ \end{array}$
If someone knows how to use \xymatrix, in wiki, it should look better.
Please feel free to use it (or \xymatrix if possible) to draw the examples of semi-confluent and local-confluent diagrams.
Is there any link with lattices or with domains in type theory?
## suggestion
IMHO the motivating example given in the article is not term rewriting. A better example would be
$(11+9) \times 2 + (11+9) \times 4$
versus
$11 \times (2+4) + 9 \times (2+4)$
91.23.255.99 (talk) 19:57, 25 November 2007 (UTC)
I don't agree; the original example is certainly term rewriting and is preferable because it is simpler. --Daira Hopwood ⚥ (talk) 16:28, 8 September 2013 (UTC)
### What is the difference in the shape of the corresponding diagrams?
Do you mean it is better because $20\times 6$ is reductible?
I do not see the difference in the diagram of both reductions. Could you please be more explicit in why it is a better example.
Or are you thinking that it is better to first present some rules like $a\to bc, b\to c, c\to d, \ldots$ or something alike before presenting the diagram?
## self-motivating variations
The article said "A number of variations on the idea of confluence exist. These are important since they enable us to draw equivalences between confluence in the sense above and these variations." It doesn't make much sense for the variations to be important because they enable us to do something that we would have had no reason to want to do if they hadn't existed in the first place. Unfortunately I don't know the real reason for their importance; this should perhaps be added. Joriki 03:53, 8 July 2006 (UTC)
Well, perhaps a good illustration of local confluence's important is in Newman's lemma: if we have strongly normalizing elements and local confluence we have confluence. I'm sure I remember reading elsewhere that the concept is important and why, but I'll need to go hit the books and get back to you on that one. Dysprosia 08:36, 8 July 2006 (UTC)
Local confluence is also important in the Knuth-Bendix procedure, since "local confluence" is decidable, whereas "confluence" is not. 133.6.205.147 (talk) 02:24, 8 April 2008 (UTC)
Local confluence is a weaker property than confluence. Thus for a given relation it at most as hard to prove/decide local confluence as to prove confluence. Proving local confluence is indeed often much easier. If we know in addition that the given relation is terminating, then we can use Newman's Lemma to show that the relation is not only locally confluent, but in fact confluent.Hermel (talk) 13:25, 13 July 2009 (UTC)
## Confusion between defintions and equivalence of notions
The Church-Rosser property and confluence are equivalent, but in order to have some equivalence to show, the definitions must differ. In Terese the authors define them to be the same (and make note of it on p.11) from the way more common definitions of Church-Rosser and confluence that appear in all other sources cited in the abstract rewriting system article.
In general, it's not a good idea to follow Terese for definitions here because of the way complicated fashion in which they define an ARS, i.e. with indexed relations; that generates way more notions than needed in this presentation. Those notions are actually useful later in the book for defining say rewriting modulo an equivalence relation, but none of that stuff appears in this wiki article. Terese also uses unusual notations, like double-headed arrows for the reflexive transitive closure, and I don't see that convention followed in this article. Pcap ping 00:57, 12 August 2009 (UTC)
## Wrong diamond diagram
The diamond diagram is wrong in that it does not follow (let alone explain) the convention used universally in this area of using dashed arrows for existential quantification and solid arrows only for universal quantification. Pcap ping 00:57, 12 August 2009 (UTC) | 2014-07-31 23:46:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7078923583030701, "perplexity": 951.3953272887999}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510273766.32/warc/CC-MAIN-20140728011753-00202-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/34266/frame-experiment-in-repeated-measures-anova-lens | # Frame experiment in Repeated Measures ANOVA lens
Someone is conducting the following hypothetical experiment and they've told me that a repeated measures ANOVA is required. I'm having trouble framing it like I've been taught in the texts. That is, identifying my factors, levels, indep/dep variables, etc etc.
Multiple participants are asked to look at a red square, blue square and green square as we measure their pupil opaqueness as a continuous time series for 1000ms (1 second) per trial. In each trial we show one square. We collect 100 trials looking at red, 100 at blue, and 100 at green all randomly shown. We do this for 50 participants. So our data looks like this
50 (participants) * 3 (red blue or green groups) * 100 (trials per group) * 1000 (ms samples per trial)
We then have a hypothesis that opaqueness is greater when looking at red then either blue or green (blue or green aren't really different from each other).
One way repeated measures anova
Could you help me formalize this a little more on both the group level and single subject level? Clearly identify why and how a repeated measures ANOVA is appropriate, what my independent variable is, what my dependent variable is, my factors, levels, etc?
What I think is the following: Based on the information found here:
The measurement of the dependent variable is repeated. It is not possible to use the standard ANOVA in such a case as such data violates the assumption of independence of data and as such it will not be able to model the correlation between the repeated measures.
However, it must be noted that a repeated measures design is very much different from a multivariate design.
For both, samples are measured on several occasions, or trials, but in the repeated measures design, each trial represents the measurement of the same characteristic under a different condition.
So repeated measures is appropriate because we are measuring the same participants opaqueness under different conditions, where the different conditions are either showing the red, blue or green squares? Someone also said I could compare across subjects and/or across colors? There is also the issue of performing three separate repeated measures anova, one to look at red vs blue, red vs green, and finally blue vs green. Does this make sense?
Two Way repeated measures anova
To add insult to injury, lets now say that I wanted to measure opaqueness in three different locations in someone's eye for every trial. Meaning every trail would now have 3 measurements of opaqueness. How does that fit in above? Someone told me this would be a two way repeated measures anova where factor 1: condition factor 2: measurement location.
EDIT
I updated the question to show that I am in interested in looking at a group level and at an individual level. How can I do this at an individual level? Would it involve looking at time? I'm thinking that I could 1. average across trials to generate one specific value of opaquness per time point. Then place them going down on the first column in a table similar to the one shown here. The second column would be the average value of all trials for that time point. OR I could also average across time, and place my trials there. What do you think?
## 1 Answer
You are on the right track. A repeated measures ANOVA will begin to address your hypothesis. Between subjects ANOVA assumes independence of observations. This is violated in your study because many observations are taken of single participant. In terms of sorting out your variables recall that in an experiment your dependent variable is the one that you measure. It's value depends on the variables you manipulate, your independent variables. Thus, pupil opaqueness is your dependent and colour and location are your independent variables or factors. A handy way to summarize all of this information is as follows: You are doing a 3 (colour, red/green/blue) x 3 (location on eye) within subjects ANOVA. Two factors each with 3 levels. The 'x' indicates you are also interesting in testing interactions between location and colour. Will some locations be more opaque with certain colours? Incidentally, you would not be able to test for such possible interactions if you performed 3 one-way ANOVAs. In addition, you would inflate your familywise error rate. Stay away from that option. Finally, note that ANOVA tests for equality of all conditions. Thus, follow up t-tests or a linear contrast would be required to answer your original hypothesis following a significant ANOVA.
• Could I conduct a one way repeated measures anova (not considering location of measurement) on a per subject level? I measure opaqueness as a continuous time series for 300 seconds per trial Aug 16 '12 at 22:35
• Can you please expand on what you mean by 'per subject level'. Are you interested in individual differences? Aug 17 '12 at 14:14
• I'm interested in looking at whether there was a difference between red blue and green squares on an individual rather than just a group. For example, say if the person had red have a large difference (blue and green didn;t) then I would give that person a special filter for their glasses. I want to be able to make statements per individual as well, not just as an entire group Aug 17 '12 at 17:29
• What is the difference between a categorical predictor and a factor? Aug 17 '12 at 21:20
• Categorical predictor and factor have the same meaning. You would tend to hear the first in the context of regression and the second in the ANOVA framework but they are identical. Aug 20 '12 at 15:39 | 2021-09-20 14:59:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4388148784637451, "perplexity": 845.2806137367008}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057039.7/warc/CC-MAIN-20210920131052-20210920161052-00443.warc.gz"} |
https://www.physicsforums.com/threads/is-this-dirac-ket-bra-correct.419551/ | # Is this Dirac Ket Bra correct?
1. Aug 1, 2010
### PRB147
A B are two-body and one-body operators respectively.
Is the following equation correct? If so, Would you give me the proof in real space?
$$\sum\limits_{ijklm}\langle ij|A|km \rangle \langle m |B |l\rangle= \sum\limits_{ijkl}\langle ij|A B |k l\rangle$$
2. Aug 1, 2010
### Fredrik
Staff Emeritus
I assume |km> is a tensor product state. $|km\rangle=|k\rangle\otimes|m\rangle$.
If $A=A_1\otimes A_2$, then $\langle ij|A|km\rangle=\langle i|A_1|k\rangle\langle j|A_2|m\rangle$, so when you do the sum over m, you get
$$\sum_{ijkl}\langle i|A_1|k\rangle\langle j|A_2B|l\rangle$$
If A2 is the identity operator, this simplifies to
$$\sum_{ijkl}\langle i|A_1|k\rangle\langle j|B|l\rangle=\sum_{ijkl}(\langle i|\otimes\langle j|)(A_1|k\rangle\otimes B|l\rangle)=\sum_{ijkl}\langle ij|(A_1\otimes B)|kl\rangle$$
Notational abuse is common when dealing with tensor products. If you write $A_1=A$, i.e. $A=A\otimes I$, and $AB=A\otimes B$, even though this doesn't really make sense, you get an expression that looks the way you want it to. But is it really the same? You need to think about how your A and B are defined, and in particular if they're defined the same way at different places in your equation.
I don't know what you mean by "real space".
Last edited: Aug 1, 2010
3. Aug 1, 2010
### PRB147
Yes, the state can be represented by the tensor product, but the operator A is not so.
Here , A is coulomb interaction and is not separable, B is the kinetic term.
Is the above equation correct?
4. Aug 1, 2010
### Fredrik
Staff Emeritus
This would be a good time for you to post your own attempt to prove that it is. (It's sort of the custom around here. If you want help with something, show us what you've done so far).
5. Aug 2, 2010
### qbert
what's the rule for matrix multiplication for a 4x4 matrix and a 2x2 matrix?
as operators they act on completely different spaces! there is no way to
compose them (if AB is "do B then A").
what you have written is wrong.
exactly!
6. Aug 2, 2010
### Hurkyl
Staff Emeritus
Every operator is a limit of linear combinations of separable operators.
7. Aug 29, 2010
### PRB147
Thank You all!
I can not prove, then I ask for help. | 2018-06-21 03:11:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8084461092948914, "perplexity": 851.2063756147461}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864019.29/warc/CC-MAIN-20180621020632-20180621040632-00077.warc.gz"} |
https://crypto.stackexchange.com/questions/81246/how-to-construct-a-prg | # How to construct a prg?
Does the adversary know the random seed s? If he does not, isn't one possible construction simply the random seed itself? if the output needs to be bigger than random seed s cant I just add a hardcore bit to the random seed s? Also, regardless of the number of leading bits of the output that are revealed (even n-1 where n is length of output), will the output be computationally indistinguishable from a truly random string?
Does the adversary know the random seed s?
No. Let $$G$$ be an arbitrary PRG. Given the seed $$s$$ and a string $$y$$, it is trivial to check whether $$y = G(s)$$, thus the output of a PRG cannot be indistinguishable from random if the seed is known. Therefore, this definition would be uninstantiable and thus not useful.
If he does not, isn't one possible construction simply the random seed itself?
A family of functions $$G : \{0,1\}^{i(\lambda)} \to \{0,1\}^{o(\lambda)}$$ needs to have two properties to be called a PRG.
1. It needs to be expanding, i.e. it must hold that $$o(\lambda)>i(\lambda)$$.
2. It must hold that or a uniformly chosen seed $$s\gets\{0,1\}^{i(\lambda)}$$, the value $$G(s)$$ is computationally indistinguishable from a uniformly chosen string $$y \gets \{0,1\}^{o(\lambda)}$$.
The PRG you are suggesting is the identity function, which does indeed fulfill property 2 from, above, but not property 1.
if the output needs to be bigger than random seed s cant I just add a hardcore bit to the random seed s?
Well, the identity function does not have any hardcore predicates, so this does not work.
Also, regardless of the number of leading bits of the output that are revealed (even n-1 where n is length of output), will the output be computationally indistinguishable from a truly random string?
By definition of a secure PRG, the entire output is computationally indistinguishable from a truly random string. So by a trivial reduction, so is any prefix of the output.
• Thank you so much! What is a prefix of the seed is revealed? Will that always result in a non-secure prg? – Daniel Jun 9 at 15:11
• Would it be possible to give an example of a conjectured prg? I cant think of one and have looked everywhere on the internet. Also, does the adversary know the prg function? – Daniel Jun 9 at 15:12
• You can come up with contrived examples of PRGs where leaking a prefix of the seed does not break security. E.g. you can construct a PRG that simply ignores the prefix. Then, leaking it makes no difference. For more meaningful examples you would need to look for some kind of leakage resilience. – Maeher Jun 9 at 16:29
• What kind of example are you looking for? Practical examples are a secure stream cipher, such as AES in CTR mode or ChaCha20. Theoretically the question is which assumptions you are willing to make. There's the famous HILL construction from any one-way function. From OWP there's a much simpler construction using Goldreich-Levin's hardcore predicate. – Maeher Jun 9 at 16:36
• Thanks, understood! Appreciate the help! – Daniel Jun 10 at 16:36
If the output needs to be bigger than random seed $$s$$ [...]
The output does need to be bigger than the seed.
[...] can't I just add a hardcore bit to the random seed $$s$$?
This is similar to the construction in Theorem 7.19 in Katz and Lindell's textbook (2nd ed., p. 258):
Theorem 7.19: Let $$f$$ be a one-way permutation with hard-core predicate $$\mathsf{hc}$$. Then algorithm $$G$$ defined by $$G(s) = f(s) \mathbin\| \mathsf{hc}(s)$$ is a pseudorandom generator with expansion factor $$\ell(n) = n + 1$$.
But note that this talks of the hardcore bit $$\mathsf{hc}$$ of a one-way permutation $$f$$. What you're actually proposing is this:
$$G'(s) = s \mathbin\| \mathsf{hc}(s)$$
...which is trivially distinguishable because your output discloses $$s$$ itself and therefore the adversary can just trivially compute $$\mathsf{hc}(s)$$. That's precisely the role that the one-way permutation $$f$$ plays—it conceals the seed $$s$$.
• Thank you! Would the prg in 7.19 be indistinguishable if the first x bits of the seed were leaked? also, does the prg function? does he know that g'(s) = s|| hc(s) – Daniel Jun 10 at 16:36 | 2020-12-02 06:18:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7337357997894287, "perplexity": 793.5367778912388}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141692985.63/warc/CC-MAIN-20201202052413-20201202082413-00610.warc.gz"} |
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Zbl 0651.60080
Ricciardi, Luigi M.; Sato, Shunsuke
First-passage-time density and moments of the Ornstein-Uhlenbeck process.
(English)
[J] J. Appl. Probab. 25, No.1, 43-57 (1988). ISSN 0021-9002
Formulas are derived expressing the first-passage-time probability density function $g(t,S\vert x\sb 0)$ through the boundary S and the n th moments $$t\sb n(S\vert x\sb 0)=\int\sp{\infty}\sb{0}t\sp ng(t,S\vert x\sb 0)dt,\quad n=1,2,...,$$ for the Ornstein-Uhlenbeck process with drift -x/$\theta$ $(\theta >0)$ and infinitesimal variance $\mu$. It is proved that for $\theta =1$, $\mu =2$ and large S $$g(t,S\vert x\sb 0)\sim g(S)\exp (-g(S)t),\quad and\quad t\sb n(S\vert x\sb 0)\sim n![g(S)]\sp n,\quad n=1,2,...,$$ where $g(z)=2(2\pi)\sp{-1/2}\exp (-z\sp 2/2)$.
[B.Grigelionis]
MSC 2000:
*60J60 Diffusion processes
Keywords: first-passage-time; Ornstein-Uhlenbeck process
Cited in: Zbl 0954.60067
Highlights
Master Server | 2013-05-21 13:21:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8674286007881165, "perplexity": 6730.325414150389}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368700014987/warc/CC-MAIN-20130516102654-00007-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-factor-8x-3-4x-2-18x-9 | # How do you factor 8x^3+4x^2-18x-9?
Jun 22, 2016
$8 {x}^{3} + 4 {x}^{2} - 18 x - 9 = \left(2 x - 3\right) \left(2 x + 3\right) \left(2 x + 1\right)$
#### Explanation:
Notice that the ratio between the first and second terms is the same as that between the third and fourth terms. So we can factor this cubic by grouping:
$8 {x}^{3} + 4 {x}^{2} - 18 x - 9$
$= \left(8 {x}^{3} + 4 {x}^{2}\right) - \left(18 x + 9\right)$
$= 4 {x}^{2} \left(2 x + 1\right) - 9 \left(2 x + 1\right)$
$= \left(4 {x}^{2} - 9\right) \left(2 x + 1\right)$
$= \left({\left(2 x\right)}^{2} - {3}^{2}\right) \left(2 x + 1\right)$
$= \left(2 x - 3\right) \left(2 x + 3\right) \left(2 x + 1\right)$
Note that we also used the difference of squares identity:
${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$
with $a = 2 x$ and $b = 3$ | 2020-01-20 23:13:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 10, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.764507532119751, "perplexity": 248.3170887592919}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250601040.47/warc/CC-MAIN-20200120224950-20200121013950-00370.warc.gz"} |
https://math.stackexchange.com/questions/159836/int-0-infty-frace-x-sinxx-dx-evaluate-integral | # $\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$ Evaluate Integral
Compute the following integral:
$$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$$
Any hint, suggestion is welcome.
• this looks like something that could be done with complex contour integration. en.wikipedia.org/wiki/Methods_of_contour_integration – Dennis Gulko Jun 18 '12 at 10:08
• http://www.wolframalpha.com/input/?i=Integrate[%28e^-x*Sin[x]%2Fx%29%2C{x%2C0%2CInfinity} Ans = Pi/4=0.785398. In case you don't want to use limits and make it indefinite than use "Integrate[(e^-x*Sin[x]/x)]". This is not step by step solution obviously but may be helpful to you. – Rorschach Jun 18 '12 at 10:17
• Consider $I(p)=\int_{0}^{\infty}\frac{e^{-px}\sin(x)}{x}dx$,then $\frac{d}{dp}I(p)=-\int_{0}^{\infty}e^{-px}\sin(x)dx$.$I(p)$ tends to zero when $p$ tends to infinity.Solve this differential equation and work out $I(1)$.That is the answer. – zy_ Jun 18 '12 at 10:21
• @yzhao Would you consider writing that as an answer? – anon Jun 18 '12 at 10:38
Yet a different approach: parametric integration. Let $$F(\lambda)=\int_{0}^{\infty} \frac{e^{-\lambda x} \sin(x)}{x}\,dx,\qquad\lambda>0.$$ Then $$F'(\lambda)=-\int_{0}^{\infty} e^{-\lambda x} \sin(x)\,dx=-\frac{1}{1+\lambda^2}.$$ Integrating and taking into account that $\lim_{\lambda\to\infty}F(\lambda)=0$ we have $$F(\lambda)=\frac\pi2-\arctan\lambda$$ and $$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x}\,dx=F(1)=\frac\pi4.$$
• @Chris:I'm sorry for just noticing you comment,but Julian gives a detailed description of my idea. – zy_ Jun 18 '12 at 11:21
• Nice answer (+1). I enjoy seeing a variety of solutions! – robjohn Jun 18 '12 at 11:44
Using Laplace Transform, $$\mathcal{L}(\sin(x)) = \frac{1}{s^2 + 1}$$ $$\mathcal{L}\left(\frac{\sin(x)}{x}\right) = \int_r^\infty \frac{1}{s^2 + 1} ds = \frac{\pi}{2} - \arctan(r)$$ Therefore, $$\int_0^\infty e^{-rx} \frac{\sin(x)}{x} dx = \frac{\pi}{2} - \arctan(r)$$ Substituting r = 1, $$\int_0^\infty e^{-x} \frac{\sin(x)}{x} dx = \frac{\pi}{4}$$
• thanks for your delicate solution! – user 1357113 Jun 18 '12 at 10:57
Another approach: $$\begin{eqnarray*} \int_{0}^{\infty} dx\, \frac{e^{-x} \sin(x)}{x} &=& \int_{0}^{\infty}dx\, \frac{e^{-x}}{x} \sum_{k=0}^\infty \frac{(-1)^k x^{2k+1}}{(2k+1)!} \\ &=& \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} \int_{0}^{\infty}dx\, x^{2k} e^{-x} \\ &=& \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}(2k)! \\ &=& \sum_{k=0}^\infty \frac{(-1)^k}{2k+1} \hspace{5ex} \textrm{(Leibniz series for \pi)}\\ &=& \frac{\pi}{4}. \end{eqnarray*}$$
• this is another magic shot! Nice job! Thanks! :-) – user 1357113 Jun 19 '12 at 5:52
• @Chris: Thanks, Chris. Another good question! – user26872 Jun 19 '12 at 7:40
Write this as $$\lim_{\epsilon\to0}\int_\epsilon^{1/\epsilon}\frac{e^{-(1-i)x}-e^{-(1+i)x}}{2ix}\,\mathrm{d}x\tag{1}$$ and then consider the path integral $$\frac{1}{2i}\int_{\gamma_\epsilon} e^{-z}\,\frac{\mathrm{d}z}{z}\tag{2}$$ where $\gamma_\epsilon$ comes in along the line $(1+i)x$, makes a quarter circle clockwise along $|z|=\epsilon$, goes out along the line $(1-i)x$ and then back a quarter circle counter-clockwise along $|z|=1/\epsilon$. There are no poles inside this path, so the integral in $(2)$ is $0$.
The part along $|z|=1/\epsilon$ dies away exponentially as $\epsilon\to0$. The two parts along the lines sum to our integral, $(1)$, and the part along $|z|=\epsilon$ tends to $\frac14$ of the integral of $\frac{1}{2iz}$ clockwise around the origin; that is, $-\pi/4$. Since the sum of these parts is $0$, the limit in $(1)$ must be $\pi/4$. That is, $$\int_0^\infty\frac{e^{-x}\sin(x)}{x}\mathrm{d}x=\frac{\pi}{4}\tag{3}$$
• @robjohn that's a nice solution! But I think that you should also justify that principal value of the integral equals integral itself. – qoqosz Jun 18 '12 at 11:08
• @qoqosz: I am not using a principal value since $\frac{\sin(x)}{x}$ is a nice bounded function. I use the $\epsilon$ and $1/\epsilon$ to connect with the contour integral. Perhaps I am missing your concern. – robjohn Jun 18 '12 at 11:17
• @robjohn $\frac{x}{1+x^2}$ is also bounded but when integrating over $\mathbb{R}$ it exists only in P.V. sense. Well, maybe it's not a 'nice' function ;). – qoqosz Jun 18 '12 at 11:21
• @qoqosz: $e^{-x}\frac{\sin(x)}{x}$ is a nice bounded function that dies away exponentially. It converges absolutely over $[0,\infty)$. In fact, all of the integrals above converge absolutely. What am I missing? – robjohn Jun 18 '12 at 11:31
• @qoqosz: Ah, I just saw the improper-integral tag. In the Riemann integral sense, it is improper because the domain of integration is not bounded, but it is not improper in the Lebesgue sense. – robjohn Jun 18 '12 at 11:36
If you know a bit about Fourier theory. You could Parseval's theorem $$\int \!dx \,f(x) g(x)^* = \int \!d\xi\,\hat f(\xi) \hat g(\xi)^*$$ with $f(x) = \sin(x)/x$, $g(x) = \Theta(x) e^{-x}$ and $\hat{f}$, $\hat{g}$ their Fourier transforms and $\Theta(x)$ the Heaviside step function.
Hint: $\hat{f}(\xi) = \tfrac12\sqrt{\frac{\pi}{2}} [\Theta(1-\xi) + \Theta(1+\xi)] =\sqrt{\frac{\pi}{2}} \mathop{\rm rect}(\xi)$.
Let
$$f(z) = \frac{e^{-z+iz}}{z}$$
and let $C$ be the contour that travels along $0$ to $R$, makes a quarter of a circle around to $iR$ and back to $0$, properly indented around $0$ with a quarter circle of radius $\delta$ to avoid the pole.
As $R \to \infty$, the integral over rounded part of the contour tends to $0$ and the part around $0$ tends to $-i\frac{\pi}{2}$ (N.B. this is $-i\frac{\pi}{2}$ of the residue at $z=0$) as $\delta \to 0$. Then by Cauchy's theorem:
$$0=\oint_C f(z)\,dz =\\ \int_0^\infty \frac{e^{-z+iz}}{z}\,dz -\int_0^{\infty} \frac{e^{-iz-z}}{z}\,dz - i\frac{\pi}{2}$$
And upon taking imaginary parts and solving:
$$\frac{\pi}{4}=\int_0^\infty \frac{e^{-z}\sin(z)}{z}\,dx$$
• hehe (+1). Glad to have the opportunity to learn more on complex analysis. :-) – user 1357113 Jul 4 '13 at 16:06
• @Chris'swisesister I thought you may be interested in a slightly different contour for variety – Argon Jul 4 '13 at 16:06
• sure. I'm always interested in various ways. Thanks! :-) – user 1357113 Jul 4 '13 at 16:07
• Could you also add the contour (a picture) in case you have one ready? – user 1357113 Jul 4 '13 at 16:09
• @Chris'swisesister I could try and draw one, I will update this soon. – Argon Jul 4 '13 at 16:11
$$\int_0^{\infty} \frac{e^{-x}\sin x}{x}\,dx=\int_0^{\infty}\int_0^{\infty} e^{-x}\sin x \,e^{-xy}\,dy\,dx=\int_0^{\infty} \int_0^{\infty} e^{-x(1+y)}\sin x\,dx\,dy$$ From integration by parts or otherwise, one can show that: $$\int_0^{\infty}e^{-x(1+y)}\sin x\,dx=\frac{1}{1+(1+y)^2}$$ Hence $$\int_0^{\infty} \int_0^{\infty} e^{-x(1+y)}\sin x\,dx\,dy=\int_0^{\infty} \frac{dy}{1+(1+y)^2}=\left(\arctan(1+y)\right|_0^{\infty}=\boxed{\dfrac{\pi}{4}}$$
• Neat, but isn't that more or less the same as Julián Aguirre's answer? – IAmNoOne May 9 '14 at 0:12
| 2019-09-23 13:43:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9676538705825806, "perplexity": 655.3298104564963}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514576965.71/warc/CC-MAIN-20190923125729-20190923151729-00473.warc.gz"} |
https://math.stackexchange.com/questions/3726749/advantages-of-using-cubic-splines-to-expand-functions | # Advantages of using cubic splines to expand functions
I've recently determined the deuteron's binding energy using cubic B-Splines to expand the system of coupled differential equations I obtained for my problem. This method of expanding functions using such basis was suggested by a professor of mine and I arrived to quite satisfying results.
However I would like to understand better the advantages of these particular splines.
For example, I've read something about the fact these splines have "local minimal support". What exactly is that? Does it relate to the fact that I've defined the splines piecewise?
• A spline of degree $2$ or less is in general less exact than a spline of degree $3$ , this is obvious. But taking in account too many nodes, the interpolating polynomial can heavily oscillate and moreover the determination of the interpolation polynomial can be numerically very instable.The best compromis is a (piecewise) cubic spline. – Peter Jun 19 at 18:41
• Another possibility to avoid the Runge-phenomenon is Chebycheff-interpolating. The nodes are not equidistant in this method. – Peter Jun 19 at 18:46
• @Peter Nonetheless, even by defining the cubic spline piecewise, it still amounts for some instability correct? – RicardoP Jun 19 at 18:59
• I am not an expert concerning the stability, but I can barely imagine an example where a cubic spline fails due to instability. But I guess there are cases pathological enough, maybe someone can point out such a case (if it exists at all). – Peter Jun 19 at 19:04
• @Peter I see! In my case particularly, I noticed that small variations of some physical parameters which were included in my spline definition would lead to great changes on my final results and I was wondering if that was something related to some inherent instability of the method. Thank you so much for the answers! – RicardoP Jun 19 at 21:18 | 2020-10-21 02:43:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5874168276786804, "perplexity": 555.9630313124002}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107874637.23/warc/CC-MAIN-20201021010156-20201021040156-00494.warc.gz"} |
https://www.physicsforums.com/threads/google-gives-0-0-1.559811/ | 1. Dec 13, 2011
EternityMech
they also gave 2 instead 288 on the infamous equation. who works in the math department for google?
2. Dec 13, 2011
arabianights
when i was young, i use scratch book instead of calculator to do math
and now people use google instead of calculator to do math
3. Dec 13, 2011
KingNothing
when i was young, i use calculator instead of google to do math
and now people use wolfram alpha instead of google to do math
4. Dec 13, 2011
What is the "infamous" equation?
5. Dec 13, 2011
He probably means 48÷2(9+3). We shouldn't talk about it, though.
6. Dec 13, 2011
neyzenyelda
I am as idiot as google. I thought zero to the power zero equaled one, too.
7. Dec 13, 2011
Jack21222
I could be very very wrong about this, but as I understand it, while 0^0 is technically undefined, it's often defined as 1 to simplify certain problems.
8. Dec 13, 2011
It's 2.
9. Dec 13, 2011
micromass
No, it's not.
Please do not let this thread turn in another debate about 48÷2(9+3) or this thread will be locked.
See here for the "infamous equation": https://www.physicsforums.com/showthread.php?t=494675
See here for $0^0$: https://www.physicsforums.com/showthread.php?t=530207 [Broken]
Last edited by a moderator: May 5, 2017
10. Dec 13, 2011
dextercioby
Wolframalpha is the the best resource out there for a lazy person who doesn't care about his math skills.
Yes, google can't be trusted.
Last edited by a moderator: Dec 13, 2011
11. Dec 13, 2011
arabianights
has anyone read 'Stories of your life and others' by Ted Chiang?
one of the stories deals with 0/0
12. Dec 13, 2011
D H
Staff Emeritus
It has so many values for physical constants that are outdated or flat-out wrong. Avogadro's number, the astronomical unit, Newton's gravitational constant, pick one: It's probably wrong to some degree or another. For example, the google calculator value for the AU differs from the published value by 129 kilometers. The uncertainty in the published value is 3 meters.
13. Dec 13, 2011
dlgoff
I'd rather use my [STRIKE]slid[/STRIKE] slide rule after hearing all of this. :)
Last edited: Dec 13, 2011
14. Dec 13, 2011
jhae2.718
Not this crap again!
15. Dec 13, 2011
lisab
Staff Emeritus
*wonders of slid rule is the past tense of slide rule*
16. Dec 13, 2011
Jimmy Snyder
*wonders if of is future tense of if*
17. Dec 13, 2011
lisab
Staff Emeritus
18. Dec 13, 2011
BobG
My slide rule won't calculate 0^0, so it must be undefined.
19. Dec 13, 2011
dlgoff
Dang. I've been looking for that Zero all day. Go figure.
20. Dec 13, 2011
dlgoff
Notice my reason for editing. I just slid it back into its sheath. | 2018-03-21 03:48:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6435427665710449, "perplexity": 5854.905535885538}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647567.36/warc/CC-MAIN-20180321023951-20180321043951-00042.warc.gz"} |
https://cloud.learningu.org/teach/teachers/merrick/bio.html | # Splash Biography
## MERRICK CAI, ESP Teacher
Major: Teacher
College/Employer: Not available.
Not Available.
## Past Classes
(Clicking a class title will bring you to the course's section of the corresponding course catalog)
M70: Will the bird get home? and other stories in random walks in Rainstorm Spring 2020 (May. 30 - 31, 2020)
We'll have a fun introduction to random walks! Specifically, we'll discuss what happens when you take a random walk on $$\mathbb{Z}$$ (the number line), $$\mathbb{Z}^2$$ (the lattice in the coordinate plane), and $$\mathbb{Z}^3$$ (3D space). What happens when a very confused man leaves his house and starts wandering aimlessly? Will he ever return home? What about a bird? Sign up for this class to find out! (It will not be super rigorous, so enjoy the ideas!)
M71: Hidden Markov Models: Predicting the Market with Warren Buffett's Breakfasts in Rainstorm Spring 2020 (May. 30 - 31, 2020)
This course will provide an introduction to Markov Chains and Hidden Markov Models, so we may predict the daily market performance based entirely on Warren Buffett's daily eating habits! He once famously said “$3.17 is a bacon, egg and cheese biscuit, but if the market’s down this morning, I’ll pass up the$3.17 and go with the \$2.95.” | 2021-03-05 01:07:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18777059018611908, "perplexity": 3941.455888697295}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178369553.75/warc/CC-MAIN-20210304235759-20210305025759-00256.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-using-the-quadratic-formula-x-4-26x-2-25 | # How do you solve using the quadratic formula x^4 - 26x^2 +25?
Apr 30, 2015
In this way:
let $t = {x}^{2}$, so:
${t}^{2} - 26 t + 25 = 0 \Rightarrow$
$\Delta = {b}^{2} - 4 a c = {26}^{2} - 4 \cdot 1 \cdot 25 = 676 - 100 = 576 = {24}^{2}$
${t}_{1 , 2} = \frac{- b \pm \sqrt{\Delta}}{2 a} = \frac{26 \pm 24}{2}$
${t}_{1} = \frac{26 - 24}{2} = 1$
${t}_{2} = \frac{26 + 24}{2} = \frac{50}{2} = 25$.
So:
${x}^{2} = 1 \Rightarrow x = \pm 1$
${x}^{2} = 25 \Rightarrow x = \pm 5$.
Since the coefficient $b$ of the quadratic equation is even, we could use also the reducted formula.
$\frac{\Delta}{4} = {\left(\frac{b}{2}\right)}^{2} - a c = 169 - 25 = 144 = {12}^{2}$
${t}_{1 , 2} = \frac{- \frac{b}{2} \pm \sqrt{\frac{\Delta}{2}}}{a} = \frac{13 \pm 12}{1}$
with, obviously, the same two solutions! | 2019-10-19 04:21:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 11, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.748590350151062, "perplexity": 741.1319800341554}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986688826.38/warc/CC-MAIN-20191019040458-20191019063958-00004.warc.gz"} |
https://infoscience.epfl.ch/record/88895 | Formats
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### Abstract
The dielectric response of the relaxer ferroelectric PbMg1/3Nb2/3O3 (PMN) is found to be a non-analytical function of the ac field, with the absolute value of the non-linear component of the polarization given by \P-nl\ alpha E-m(gamma(omega,T)). The dependence on the temperature, T, and frequency, omega, of the exponent gamma manifests itself in the form of a crossover in gamma(T) from approximately 2 to 3 upon cooling, whose position depends upon the frequency of the applied ac held. From the comparison of y(w, T) with the linear complex dielectric permittivity, epsilon(l)* (omega, T), of PMN, the following can be correlated: (i) gamma approximate to 2 in the regime of the quasi-static response, and y starts deviating from 2 simultaneously with the onset of the frequency dispersion; and (ii) gamma approximate to 3 when the spectrum of relaxation times in PMN becomes flat. These findings can be understood in terms of the change in the type of the motion of the interphase boundaries of the microscopic polar regions existing in PMN, namely in the change of the scale on which the interphase boundaries can move. | 2021-06-16 01:57:41 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8597056865692139, "perplexity": 1248.0868631600442}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487621699.22/warc/CC-MAIN-20210616001810-20210616031810-00139.warc.gz"} |
https://www.studysmarter.us/textbooks/physics/physics-for-scientists-and-engineers-a-strategic-approach-with-modern-physics-4th/relativity/q-33-a-particle-has-momentum-what-is-the-particles-speed-in-/ | Q. 33
Expert-verified
Found in: Page 1060
Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
Book edition 4th
Author(s) Randall D. Knight
Pages 1240 pages
ISBN 9780133942651
A particle has momentum . What is the particle’s speed in m/s?
See the step by step solution
Step 1: Given Information
We have given that a particle has momentum .
We have to find the particle’s speed in .
Step 2: Simplify
The momentum in relativity is given by
Here
and
So, the equation becomes | 2022-12-06 07:05:45 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9279996752738953, "perplexity": 4396.643469356315}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711074.68/warc/CC-MAIN-20221206060908-20221206090908-00555.warc.gz"} |
https://www.ris-ai.com/book-recommendation-system-with-machine-learning | • +91-9872993883
• +91-8283824812
• info@ris-ai.com
# Book Recommendation System with Machine Learning ¶
Recommendation systems are used to predict the Rating or Preference that a user would give to an item.
Almost every major company has applied them in some form or the other: Amazon uses it to suggest products to customers, YouTube uses it to decide which video to play next on auto play, and Facebook uses it to recommend pages to like and people to follow.
In this project, you will see how to build a Book Recommendation System model using Machine Learning Techniques.
Lets import the libraries and read the datasets :
In [1]:
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
books.columns = ['ISBN', 'bookTitle', 'bookAuthor', 'yearOfPublication', 'publisher', 'imageUrlS', 'imageUrlM', 'imageUrlL']
users.columns = ['userID', 'Location', 'Age']
ratings.columns = ['userID', 'ISBN', 'bookRating']
print(ratings.shape)
print(list(ratings.columns))
b'Skipping line 6452: expected 8 fields, saw 9\nSkipping line 43667: expected 8 fields, saw 10\nSkipping line 51751: expected 8 fields, saw 9\n'
b'Skipping line 92038: expected 8 fields, saw 9\nSkipping line 104319: expected 8 fields, saw 9\nSkipping line 121768: expected 8 fields, saw 9\n'
b'Skipping line 144058: expected 8 fields, saw 9\nSkipping line 150789: expected 8 fields, saw 9\nSkipping line 157128: expected 8 fields, saw 9\nSkipping line 180189: expected 8 fields, saw 9\nSkipping line 185738: expected 8 fields, saw 9\n'
b'Skipping line 209388: expected 8 fields, saw 9\nSkipping line 220626: expected 8 fields, saw 9\nSkipping line 227933: expected 8 fields, saw 11\nSkipping line 228957: expected 8 fields, saw 10\nSkipping line 245933: expected 8 fields, saw 9\nSkipping line 251296: expected 8 fields, saw 9\nSkipping line 259941: expected 8 fields, saw 9\nSkipping line 261529: expected 8 fields, saw 9\n'
/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py:2718: DtypeWarning: Columns (3) have mixed types.Specify dtype option on import or set low_memory=False.
interactivity=interactivity, compiler=compiler, result=result)
(1149780, 3)
['userID', 'ISBN', 'bookRating']
Plotting the rating distribution :
In [2]:
plt.rc("font", size=15)
ratings.bookRating.value_counts(sort=False).plot(kind='bar')
plt.title('Rating Distribution\n')
plt.xlabel('Rating')
plt.ylabel('Count')
plt.show()
In [3]:
print(books.shape)
print(list(books.columns))
(271360, 8)
['ISBN', 'bookTitle', 'bookAuthor', 'yearOfPublication', 'publisher', 'imageUrlS', 'imageUrlM', 'imageUrlL']
In [4]:
print(users.shape)
print(list(users.columns))
(278858, 3)
['userID', 'Location', 'Age']
Plotting the age distribution :
In [5]:
users.Age.hist(bins=[0, 10, 20, 30, 40, 50, 100])
plt.title('Age Distribution\n')
plt.xlabel('Age')
plt.ylabel('Count')
plt.show()
To ensure statistical significance, users with less than 200 ratings, and books with less than 100 ratings are excluded.
In [6]:
counts1 = ratings['userID'].value_counts()
ratings = ratings[ratings['userID'].isin(counts1[counts1 >= 200].index)]
counts = ratings['bookRating'].value_counts()
ratings = ratings[ratings['bookRating'].isin(counts[counts >= 100].index)]
## Collaborative Filtering Using k-Nearest Neighbors (kNN) ¶
kNN is a Machine Learning Algorithm to find clusters of similar users based on common book ratings, and make predictions using the average rating of top-k nearest neighbors.
For example, we first present ratings in a matrix with the matrix having one row for each item (book) and one column for each user.
In [7]:
combine_book_rating = pd.merge(ratings, books, on='ISBN')
columns = ['yearOfPublication', 'publisher', 'bookAuthor', 'imageUrlS', 'imageUrlM', 'imageUrlL']
combine_book_rating = combine_book_rating.drop(columns, axis=1)
userID ISBN bookRating \
0 277427 002542730X 10
1 3363 002542730X 0
2 11676 002542730X 6
3 12538 002542730X 10
4 13552 002542730X 0
bookTitle
0 Politically Correct Bedtime Stories: Modern Ta...
1 Politically Correct Bedtime Stories: Modern Ta...
2 Politically Correct Bedtime Stories: Modern Ta...
3 Politically Correct Bedtime Stories: Modern Ta...
4 Politically Correct Bedtime Stories: Modern Ta...
Now we will group by book titles and create a new column for total rating count.
In [8]:
combine_book_rating = combine_book_rating.dropna(axis = 0, subset = ['bookTitle'])
book_ratingCount = (combine_book_rating.
groupby(by = ['bookTitle'])['bookRating'].
count().
reset_index().
rename(columns = {'bookRating': 'totalRatingCount'})
[['bookTitle', 'totalRatingCount']]
)
bookTitle totalRatingCount
0 A Light in the Storm: The Civil War Diary of ... 2
1 Always Have Popsicles 1
2 Apple Magic (The Collector's series) 1
3 Beyond IBM: Leadership Marketing and Finance ... 1
4 Clifford Visita El Hospital (Clifford El Gran... 1
Now we will combine the rating data with the total rating count data, this gives us exactly what we need to find out which books are popular and filter out lesser-known books.
In [9]:
rating_with_totalRatingCount = combine_book_rating.merge(book_ratingCount, left_on = 'bookTitle', right_on = 'bookTitle', how = 'left')
pd.set_option('display.float_format', lambda x: '%.3f' % x)
print(book_ratingCount['totalRatingCount'].describe())
userID ISBN bookRating \
0 277427 002542730X 10
1 3363 002542730X 0
2 11676 002542730X 6
3 12538 002542730X 10
4 13552 002542730X 0
bookTitle totalRatingCount
0 Politically Correct Bedtime Stories: Modern Ta... 82
1 Politically Correct Bedtime Stories: Modern Ta... 82
2 Politically Correct Bedtime Stories: Modern Ta... 82
3 Politically Correct Bedtime Stories: Modern Ta... 82
4 Politically Correct Bedtime Stories: Modern Ta... 82
count 160576.000
mean 3.044
std 7.428
min 1.000
25% 1.000
50% 1.000
75% 2.000
max 365.000
Name: totalRatingCount, dtype: float64
In [10]:
pd.set_option('display.float_format', lambda x: '%.3f' % x)
print(book_ratingCount['totalRatingCount'].describe())
count 160576.000
mean 3.044
std 7.428
min 1.000
25% 1.000
50% 1.000
75% 2.000
max 365.000
Name: totalRatingCount, dtype: float64
In [11]:
print(book_ratingCount['totalRatingCount'].quantile(np.arange(.9, 1, .01)))
0.900 5.000
0.910 6.000
0.920 7.000
0.930 7.000
0.940 8.000
0.950 10.000
0.960 11.000
0.970 14.000
0.980 19.000
0.990 31.000
Name: totalRatingCount, dtype: float64
In [12]:
popularity_threshold = 50
rating_popular_book = rating_with_totalRatingCount.query('totalRatingCount >= @popularity_threshold')
userID ISBN bookRating \
0 277427 002542730X 10
1 3363 002542730X 0
2 11676 002542730X 6
3 12538 002542730X 10
4 13552 002542730X 0
bookTitle totalRatingCount
0 Politically Correct Bedtime Stories: Modern Ta... 82
1 Politically Correct Bedtime Stories: Modern Ta... 82
2 Politically Correct Bedtime Stories: Modern Ta... 82
3 Politically Correct Bedtime Stories: Modern Ta... 82
4 Politically Correct Bedtime Stories: Modern Ta... 82
### Filter to users in US and Canada only¶
In [13]:
combined = rating_popular_book.merge(users, left_on = 'userID', right_on = 'userID', how = 'left')
userID ISBN bookRating \
0 277427 002542730X 10
1 3363 002542730X 0
3 12538 002542730X 10
4 13552 002542730X 0
5 16795 002542730X 0
bookTitle totalRatingCount \
0 Politically Correct Bedtime Stories: Modern Ta... 82
1 Politically Correct Bedtime Stories: Modern Ta... 82
3 Politically Correct Bedtime Stories: Modern Ta... 82
4 Politically Correct Bedtime Stories: Modern Ta... 82
5 Politically Correct Bedtime Stories: Modern Ta... 82
Location
0 gilbert, arizona, usa
1 knoxville, tennessee, usa
3 byron, minnesota, usa
4 cordova, tennessee, usa
5 mechanicsville, maryland, usa
#### Implementing kNN¶
We convert our table to a 2D matrix, and fill the missing values with zeros (since we will calculate distances between rating vectors).
We then transform the values(ratings) of the matrix dataframe into a scipy sparse matrix for more efficient calculations.
In [14]:
from scipy.sparse import csr_matrix
from sklearn.neighbors import NearestNeighbors
model_knn = NearestNeighbors(metric = 'cosine', algorithm = 'brute')
print(model_knn)
NearestNeighbors(algorithm='brute', metric='cosine')
In [17]:
query_index = np.random.choice(us_canada_user_rating_pivot.shape[0])
print(query_index)
distances, indices = model_knn.kneighbors(us_canada_user_rating_pivot.iloc[query_index,:].values.reshape(1, -1), n_neighbors = 6)
458
[[0. 0. 0. 0. 0. 0. 0. 0. 0. 4. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 8. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 5. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
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0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]]
Out[17]:
'Silent Witness'
In [18]:
for i in range(0, len(distances.flatten())):
if i == 0:
else:
print('{0}: {1}, with distance of {2}:'.format(i, us_canada_user_rating_pivot.index[indices.flatten()[i]], distances.flatten()[i]))
Recommendations for Silent Witness:
1: No Safe Place, with distance of 0.5749435974563907:
2: Fatal Cure, with distance of 0.6202526181022707:
3: The Drawing of the Three (The Dark Tower, Book 2), with distance of 0.6319151434622534:
4: Tell Me Your Dreams, with distance of 0.6983953804487513:
5: Dust to Dust, with distance of 0.702137110734639:
Here above are the recommendations for the Silent Witness. I hope you find this article helpful . | 2022-12-08 05:38:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7511575222015381, "perplexity": 1135.5577488102053}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711278.74/warc/CC-MAIN-20221208050236-20221208080236-00793.warc.gz"} |
https://www.bartleby.com/solution-answer/chapter-1-problem-19re-single-variable-calculus-concepts-and-contexts-enhanced-edition-4th-edition/9781337687805/6dcdd62f-5562-11e9-8385-02ee952b546e | # The function f ∘ g and its domain.
### Single Variable Calculus: Concepts...
4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
### Single Variable Calculus: Concepts...
4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
#### Solutions
Chapter 1, Problem 19RE
(a)
To determine
Expert Solution
## Answer to Problem 19RE
Solution:
The function (fg)(x)=ln(x29) and its domain is (,3)(3,).
### Explanation of Solution
Given:
The functions are f(x)=lnx,g(x)=x29.
Calculation:
The composite function (fg)(x) is defined as follows.
(fg)(x)=f(g(x))=f(x29)
Substitute x29 for x in f(x),
f(x29)=ln(x29)
Thus, the composite function (fg)(x)=ln(x29).
Note that the function f(x) is defined on all positive real numbers and hence the domain of f(x) is (0,).
The function g(x) is defined on real numbers and hence the domain of the function g(x) is (,).
To find the domain of (fg)(x)=ln(x29) set the expression x29>0 and solve.
x29>0x2>9|x|>3
Thus, the required domain is (,3)(3,).
(b)
To determine
Expert Solution
## Answer to Problem 19RE
Solution:
The function (gf)(x)=(lnx)29 and its domain is (0,).
### Explanation of Solution
Given:
The functions are f(x)=lnx,g(x)=x29.
Calculation:
The composite function (gf)(x) is defined as follows.
(gf)(x)=g(f(x))=g(lnx)
Substitute lnx for x in g(x).
g(lnx)=(lnx)29
Thus, the composite function (gf)(x)=(lnx)29.
Since the function (gf)(x) is defined on positive real numbers, the domain of the composite function (gf)(x) is (0,).
(c)
To determine
Expert Solution
## Answer to Problem 19RE
Solution:
The composite function (ff)(x)=ln(lnx) and its domain is (,).
### Explanation of Solution
Given:
The function is f(x)=lnx.
The composite function (ff)(x) is defined as follows.
(ff)(x)=f(f(x))=f(lnx)
Substitute lnx for x in f(x).
f(lnx)=ln(lnx)
Thus, the composite function (ff)(x)=ln(lnx).
Since, the function f(x) is defined on positive real numbers, the domain of the function f(x) is (0,).
Find the domain of the function (ff)(x)=ln(lnx) as follows.
Set the expression, lnx>0.
Take exponential on both sides,
elnx>e0x>1
Therefore, the domain of the composite function (ff)(x) is (1,).
(d)
To determine
Expert Solution
## Answer to Problem 19RE
Solution:
The composite function (gg)(x)=x418x2+72 and its domain is (,).
### Explanation of Solution
Given:
The function is g(x)=x29.
Calculation:
The composite function (gg)(x) is defined as follows.
(gg)(x)=g(g(x))=g(x29)
Substitute x29 for x in g(x),
g(x29)=(x29)29=x42x2(9)+819=x418x2+72
Thus, the composite function (gg)(x)=x418x2+72.
Since the function (gg)(x) is defined on all real numbers, the domain of the composite function (gg)(x) is (,).
### Have a homework question?
Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers! | 2021-07-29 20:54:08 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8118801116943359, "perplexity": 5404.446743124408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153897.89/warc/CC-MAIN-20210729203133-20210729233133-00275.warc.gz"} |
https://www.bartleby.com/solution-answer/chapter-3-problem-19p-single-variable-calculus-concepts-and-contexts-enhanced-edition-4th-edition/9781337687805/48ac8b12-5563-11e9-8385-02ee952b546e | # The change in point R as P is taken closer and closer to the x -axis.
### Single Variable Calculus: Concepts...
4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
### Single Variable Calculus: Concepts...
4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
#### Solutions
Chapter 3, Problem 19P
To determine
## The change in point R as P is taken closer and closer to the x-axis.
Expert Solution
The point R approaches the midpoint of radius AO when P is taken closer and closer to the x-axis.
### Explanation of Solution
Draw the vertical line from P to O in the given graph as shown below in Figure 1.
From Figure 1, the angle PQO=OQR=θ.
The triangle QOR is isosceles.
Thus, |QR|=|RO|.
Let |QR|=x and |QO|=r.
The cosine law of the triangle QOR is,
|RO|2=|QR|2+|QO|22|QR||QO|cosθ
Since |QR|=|RO|,
x2=x2+r22xrcosθ2xrcosθ=r2x=r22rcosθx=r2cosθ
From Figure 1, assume that the line |PO|=y.
sinθ=yr
The point P is closer and closer to the x-axis, that is, |PO|=y is approaching to 0, then sinθ=yr approach zero and the value θ also approaches to zero.
If θ approaches to zero, then x=r2cosθ approaches to r2.
Since AO is a radius of the circle, |AO|=r.
If the point P is closer and closer to the x-axis, then x approach r2.
That is, R approach AO2.
Therefore, the point R approaches the midpoint of radius AO.
### Have a homework question?
Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers! | 2021-07-26 23:58:30 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9296032190322876, "perplexity": 3916.533889006769}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046152156.49/warc/CC-MAIN-20210726215020-20210727005020-00357.warc.gz"} |
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# In a workshop 10 machines working simultaneously can finish a work of
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Joined: 08 Jul 2010
Posts: 1955
Location: India
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In a workshop 10 machines working simultaneously can finish a work of [#permalink]
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15 May 2017, 02:23
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Question Stats:
58% (01:43) correct 43% (02:05) wrong based on 41 sessions
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In a workshop, 10 machines working simultaneously can finish a work of manufacturing 3000 components in 10 days while working 9 hours a day. What is the minimum number of additional machines required at the same workshop to finish the work of manufacturing 5000 similar components if work needs to be delivered in 6 days and maximum number of hours that machines can operate in a day is 10?
A) 4
B) 5
C) 10
D) 15
E) 25
SOURCE: http://www.GMATinsight.com
[Reveal] Spoiler: OA
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Last edited by GMATinsight on 15 May 2017, 03:55, edited 2 times in total.
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Joined: 02 Aug 2009
Posts: 5660
In a workshop 10 machines working simultaneously can finish a work of [#permalink]
### Show Tags
15 May 2017, 03:38
GMATinsight wrote:
If 10 machines working simultaneously can finish a work of manufacturing 3000 components in 10 days while working 9 hours a day. Then what is the minimum number of additional machines required to finish the work of manufacturing 5000 similar components if work needs to be delivered in 6 days and maximum number of hours that machines can operate in a day is 6.
A) 4
B) 5
C) 10
D) 15
E) 25
SOURCE: http://www.GMATinsight.com
Hi
When you say addl machines, does it mean these 10 are working for 9 hrs as earlier or these also wk for 6hrs.
Irrespective of either 6 or 9 ans cannot be D..
Let's take all for 9 hrs, so these 10 machine will cater for days 10 to 6 by 10*10/6..
Now to cater for increase in components from 3000 to 5000.... 10*10/6*5000/3000=500/18~28..
So ATLEAST 18 even if we take time as 9 hrs for all...
1) If we take all to be in 6 hrs, 500/18 * 9/6=1500/36~42 so addl 42-10=32..
2) if these 10 continue for 9 hrs, we can see how many these 10 make and rest can be seen with 6h.
After the necessary amendments...
To cater for change from 3000 to 5000 components
10 machines will become $$\frac{10*5000}{3000}$$
This will further cater for change of 10days to 6days..... Machines required $$\frac{10*5000*10}{3000*6}$$
Finally to cater for extra 1hr from 9 to 10hr...... Machines becomes $$\frac{10*5000*10*9}{3000*6*10}$$
=25..
D
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Re: In a workshop 10 machines working simultaneously can finish a work of [#permalink]
### Show Tags
15 May 2017, 03:45
chetan2u wrote:
GMATinsight wrote:
If 10 machines working simultaneously can finish a work of manufacturing 3000 components in 10 days while working 9 hours a day. Then what is the minimum number of additional machines required to finish the work of manufacturing 5000 similar components if work needs to be delivered in 6 days and maximum number of hours that machines can operate in a day is 6.
A) 4
B) 5
C) 10
D) 15
E) 25
SOURCE: http://www.GMATinsight.com
Hi
When you say addl machines, does it mean these 10 are working for 9 hrs as earlier or these also wk for 6hrs.
Irrespective of either 6 or 9 ans cannot be D..
Let's take all for 9 hrs, so these 10 machine will cater for days 10 to 6 by 10*10/6..
Now to cater for increase in components from 3000 to 5000.... 10*10/6*5000/3000=500/18~28..
So ATLEAST 18 even if we take time as 9 hrs for all...
1) If we take all to be in 6 hrs, 500/18 * 9/6=1500/36~42 so addl 42-10=32..
2) if these 10 continue for 9 hrs, we can see how many these 10 make and rest can be seen with 6h.
Last number is 10 hours. Sorry for a typo
Posted from my mobile device
_________________
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Joined: 28 Mar 2017
Posts: 891
Re: In a workshop 10 machines working simultaneously can finish a work of [#permalink]
### Show Tags
15 May 2017, 03:47
GMATinsight wrote:
If 10 machines working simultaneously can finish a work of manufacturing 3000 components in 10 days while working 9 hours a day. Then what is the minimum number of additional machines required to finish the work of manufacturing 5000 similar components if work needs to be delivered in 6 days and maximum number of hours that machines can operate in a day is 10
A) 4
B) 5
C) 10
D) 15
E) 25
SOURCE: http://www.GMATinsight.com
As per above information, i am getting 32 as the answer.
_________________
Director
Joined: 28 Mar 2017
Posts: 891
Re: In a workshop 10 machines working simultaneously can finish a work of [#permalink]
### Show Tags
15 May 2017, 03:51
GMATinsight wrote:
chetan2u wrote:
GMATinsight wrote:
If 10 machines working simultaneously can finish a work of manufacturing 3000 components in 10 days while working 9 hours a day. Then what is the minimum number of additional machines required to finish the work of manufacturing 5000 similar components if work needs to be delivered in 6 days and maximum number of hours that machines can operate in a day is 6.
A) 4
B) 5
C) 10
D) 15
E) 25
SOURCE: http://www.GMATinsight.com
Hi
When you say addl machines, does it mean these 10 are working for 9 hrs as earlier or these also wk for 6hrs.
Irrespective of either 6 or 9 ans cannot be D..
Let's take all for 9 hrs, so these 10 machine will cater for days 10 to 6 by 10*10/6..
Now to cater for increase in components from 3000 to 5000.... 10*10/6*5000/3000=500/18~28..
So ATLEAST 18 even if we take time as 9 hrs for all...
1) If we take all to be in 6 hrs, 500/18 * 9/6=1500/36~42 so addl 42-10=32..
2) if these 10 continue for 9 hrs, we can see how many these 10 make and rest can be seen with 6h.
Last number is 10 hours. Sorry for a typo
Posted from my mobile device
Posted from my mobile device
Taking this information into account, I am getting 15 as the answer. Is it correct?
_________________
SVP
Joined: 08 Jul 2010
Posts: 1955
Location: India
GMAT: INSIGHT
WE: Education (Education)
Re: In a workshop 10 machines working simultaneously can finish a work of [#permalink]
### Show Tags
15 May 2017, 04:03
1
KUDOS
Expert's post
2
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BOOKMARKED
GMATinsight wrote:
In a workshop, 10 machines working simultaneously can finish a work of manufacturing 3000 components in 10 days while working 9 hours a day. What is the minimum number of additional machines required at the same workshop to finish the work of manufacturing 5000 similar components if work needs to be delivered in 6 days and maximum number of hours that machines can operate in a day is 10?
A) 4
B) 5
C) 10
D) 15
E) 25
SOURCE: http://www.GMATinsight.com
OE
CONCEPT: $$\frac{(Machine_Power * Time)}{Work} = Constant$$
i.e. $$\frac{(M_1 * T_1)}{W_1} = \frac{(M_2 * T_2)}{W_2}$$
i.e. $$\frac{(10 * (10*9)}{3000} = \frac{(M_2 * (6*10))}{5000}$$
i.e. $$T_2 = 25$$
i.e. Additional Machines required = 25-10 = 15
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Re: In a workshop 10 machines working simultaneously can finish a work of [#permalink]
### Show Tags
17 May 2017, 12:56
GMATinsight wrote:
In a workshop, 10 machines working simultaneously can finish a work of manufacturing 3000 components in 10 days while working 9 hours a day. What is the minimum number of additional machines required at the same workshop to finish the work of manufacturing 5000 similar components if work needs to be delivered in 6 days and maximum number of hours that machines can operate in a day is 10?
A) 4
B) 5
C) 10
D) 15
E) 25
SOURCE: http://www.GMATinsight.com
Nice question! For these questions, I almost always find the individual machine or worker rate, using a slightly changed version of the RTW table. Just add one column for "Number of workers/machines," and it's easy:
Attachment:
Revised Work Formula Table.jpg [ 40.42 KiB | Viewed 641 times ]
Revised formula: (# of workers)*(rate)*(time) = Work
In this case, you have to use HOURS for units of time.
Calculate first row: 10 days * 9 hrs/day = 90 hours total
Calculate second row: 6 days * 10 hrs/day = 60 hours total
1. Find individual machine rate ==>
(10)*(R)*(90) = 3000
R = $$\frac{3000}{900}$$ =$$\frac{10}{3}$$
2. Use that rate in second row to find TOTAL number of machines needed for new task ==>
(# of machines TOTAL)*($$\frac{10}{3}$$)*(60) = 5000
# of machines = $$\frac{5000}{200}$$ = 25 TOTAL
Question asks, how many more machines needed for new task. 25 (need) - 10 (have) = 15.
[Reveal] Spoiler:
D
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Re: In a workshop 10 machines working simultaneously can finish a work of [#permalink] 17 May 2017, 12:56
Display posts from previous: Sort by | 2018-02-22 08:50:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6951044201850891, "perplexity": 3063.2796467128583}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814079.59/warc/CC-MAIN-20180222081525-20180222101525-00475.warc.gz"} |
https://www.r-bloggers.com/2020/08/annotating-spc-plots-using-annotate-with-ggplot/ | Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
Annotating SPC plots using annotate with ggplot
Statistical Process Control (SPC) charts are widely used in healthcare analytics to examine how the metric varies over time and whether this variation is abnormal. Christopher Reading has already published a blog on SPC Charts and you also can read more about SPC charts here or here.
Here is a simple example of annotting points and text on SPC plots using ggplot2 package. We won’t explain all the parameters in the annotate function. Instead we see this as a short show and tell piece with signsposts at end of the blog.
So let’s get started and generate some dummy data from a normal distribution with a mean of 0 and and a standard deviation of 1.
set.seed(2020) # set the random number seed to ensure you can replicate this example
y <- rnorm(30, 0, 1) # generate 30 random numbers for the y-axis
y <- c(y, rep(NA, 10)) # add 10 NA's to extend the plot (see later)
x <- 1: length(y) # generate the x-axis
df <- tibble(x=x, y=y) # store as a tibble data frame for convenience
Now we can plot the data using ggplot function.
fig1 <-
ggplot(df, aes(x,y)) +
geom_point(size=2) +
geom_line() +
ylim(-4,4) # increase the y-axis range to aid visualisation
fig1 # plot the data
One of the main features of SPC charts are upper and lower control limits. We can now plot this as an SPC chart with lower and upper control limits set at 3 standard deviations from the mean. Although in practice the calculation of control limits differs from this demo, for simplicity we imply control limits and a mean as set numbers. Alternatively, you could use qicharts2 package to do SPC calculations and then use the generated ggplot2 object and keep following our steps.
fig1 <-
fig1 +
geom_hline(yintercept = c(3,0,-3), linetype='dashed') + # adds the upper, mean and lower lines
annotate("label", x=c(35,35,35), y=c(3,0,-3), color='darkgrey', label= c("Upper control limit", "Average line", "Lower control limit"), size=3) # adds the annotations
fig1 # plot the SPC
Remarkably we see a point below the lower control limit even though the data are purely pseudo-random. A nice reminder that control limits are guidelines not hard and fast tests of non-randomness. We can now highlight this remarkable special cause data point which is clearly a false signal also known as special cause variation.
fig1 <-
fig1 +
annotate("point", x=18, y=df$y[18], color='orange', size=4) + annotate("point", x=18, y=df$y[18])
fig1 # plot the SPC with annotations
We can now add a label for the special cause data point. You can play around with the vjust value (eg try -1, 0, 1) to get a feel for what it is doing to the vertical position of the label. There is also a hjust which operates on the horizontal plane.
fig1 <-
fig1 +
annotate("label", x=18, y=df\$y[18], vjust=1.5, label = "Special cause variation", size=3)
fig1 # plot the SPC with more annotations | 2021-10-22 07:50:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4786938428878784, "perplexity": 2801.397321734631}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585460.87/warc/CC-MAIN-20211022052742-20211022082742-00214.warc.gz"} |
https://www.mail-archive.com/lyx-users@lists.lyx.org/msg75703.html | # Re: changing margins in article class when using Beamer
```On 08/11/2009 03:27 PM, Graham M Smith wrote:
```
I'm using Beamer, and producing a presentation handout by switching to the beamer article class, which defaults to wide margins.
```
```
If I change the margins in documents settings, this then prevents the beamer slides being produced (I get a compile error). I assume I can put something in the preamble that will switch the margins for beamer article only, but don't know what this might be.
```
```
If you can find some macro that is defined in this case, but not in the other case, then all you need is:
``` \ifx\thatmacro\undefined\relax\else | 2021-05-08 17:38:59 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9439369440078735, "perplexity": 1748.486376612388}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988882.94/warc/CC-MAIN-20210508151721-20210508181721-00596.warc.gz"} |
http://mathhelpforum.com/pre-calculus/213429-pre-cal-matrices-system-equations.html | Math Help - Pre-Cal/Matrices/System of Equations
1. Pre-Cal/Matrices/System of Equations
A bank teller is counting the total amount of money in a cash register at the end of the day. There is a total of $2600 in denominations of$1, $5,$10 and $20 dollar bills. The total number of paper bills is$235. The number of twenty dollar bills is twice the number of $1 dollar bills and the number of$5 dollar bills is ten more than the number of \$1 dollar bills. What is the system of equations to repersent the situation. Then use matrices to find the number of bills for each denomination.
I seriously dont know where to begin here.
Thank you
2. Re: Pre-Cal/Matrices/System of Equations
I think you mean that there is a total of 235 bills
3. Re: Pre-Cal/Matrices/System of Equations
A + b + c + d = 235
d = 2a
a + 10 = b
4. Re: Pre-Cal/Matrices/System of Equations
a + 5b + 10c + 20d = 2600
5. Re: Pre-Cal/Matrices/System of Equations
What does the b stand for in a+10=b? and how would I create a matrix out of this equation? | 2014-07-10 16:41:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4634493589401245, "perplexity": 1068.8637751374692}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1404776420123.16/warc/CC-MAIN-20140707234020-00086-ip-10-180-212-248.ec2.internal.warc.gz"} |
http://nrich.maths.org/public/leg.php?code=-99&cl=2&cldcmpid=1222 | # Search by Topic
#### Resources tagged with Working systematically similar to Low Go:
Filter by: Content type:
Stage:
Challenge level:
### There are 325 results
Broad Topics > Using, Applying and Reasoning about Mathematics > Working systematically
### First Connect Three
##### Stage: 2 and 3 Challenge Level:
The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?
### Tetrahedra Tester
##### Stage: 3 Challenge Level:
An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?
### Magic Potting Sheds
##### Stage: 3 Challenge Level:
Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?
### Fault-free Rectangles
##### Stage: 2 Challenge Level:
Find out what a "fault-free" rectangle is and try to make some of your own.
### Triangles to Tetrahedra
##### Stage: 3 Challenge Level:
Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.
### First Connect Three for Two
##### Stage: 2 and 3 Challenge Level:
First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.
### Maths Trails
##### Stage: 2 and 3
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
### Colour Islands Sudoku
##### Stage: 3 Challenge Level:
An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.
### Games Related to Nim
##### Stage: 1, 2, 3 and 4
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
### Colour in the Square
##### Stage: 2 Challenge Level:
Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?
### Seven Flipped
##### Stage: 2 Challenge Level:
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
### Consecutive Negative Numbers
##### Stage: 3 Challenge Level:
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
### More Magic Potting Sheds
##### Stage: 3 Challenge Level:
The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?
### Combining Cuisenaire
##### Stage: 2 Challenge Level:
Can you find all the different ways of lining up these Cuisenaire rods?
### Broken Toaster
##### Stage: 2 Short Challenge Level:
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
### Two and Two
##### Stage: 2 and 3 Challenge Level:
How many solutions can you find to this sum? Each of the different letters stands for a different number.
### Number Daisy
##### Stage: 3 Challenge Level:
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
### Cayley
##### Stage: 3 Challenge Level:
The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?
### Ones Only
##### Stage: 3 Challenge Level:
Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.
### Diagonal Sums Sudoku
##### Stage: 2, 3 and 4 Challenge Level:
Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.
### Neighbours
##### Stage: 2 Challenge Level:
In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?
### Sticky Numbers
##### Stage: 3 Challenge Level:
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
### A Square of Numbers
##### Stage: 2 Challenge Level:
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
### Twinkle Twinkle
##### Stage: 2 Challenge Level:
A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.
### Page Numbers
##### Stage: 2 Short Challenge Level:
Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?
### Red Even
##### Stage: 2 Challenge Level:
You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?
### M, M and M
##### Stage: 3 Challenge Level:
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
### Nine-pin Triangles
##### Stage: 2 Challenge Level:
How many different triangles can you make on a circular pegboard that has nine pegs?
### Counting on Letters
##### Stage: 3 Challenge Level:
The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?
### Factor Lines
##### Stage: 2 Challenge Level:
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
##### Stage: 3 Challenge Level:
Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".
### Counting Cards
##### Stage: 2 Challenge Level:
A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?
### 1 to 8
##### Stage: 2 Challenge Level:
Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.
### Multiples Grid
##### Stage: 2 Challenge Level:
What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?
### Weights
##### Stage: 3 Challenge Level:
Different combinations of the weights available allow you to make different totals. Which totals can you make?
### Special Numbers
##### Stage: 3 Challenge Level:
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
### Summing Consecutive Numbers
##### Stage: 3 Challenge Level:
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
### You Owe Me Five Farthings, Say the Bells of St Martin's
##### Stage: 3 Challenge Level:
Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?
### Teddy Town
##### Stage: 1 and 2 Challenge Level:
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
### American Billions
##### Stage: 3 Challenge Level:
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
### Chocoholics
##### Stage: 2 Challenge Level:
George and Jim want to buy a chocolate bar. George needs 2p more and Jim need 50p more to buy it. How much is the chocolate bar?
### Isosceles Triangles
##### Stage: 3 Challenge Level:
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
### Masterclass Ideas: Working Systematically
##### Stage: 2 and 3 Challenge Level:
A package contains a set of resources designed to develop students’ mathematical thinking. This package places a particular emphasis on “being systematic” and is designed to meet. . . .
### More Transformations on a Pegboard
##### Stage: 2 Challenge Level:
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
### More on Mazes
##### Stage: 2 and 3
There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.
### Code Breaker
##### Stage: 2 Challenge Level:
This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?
### One to Fifteen
##### Stage: 2 Challenge Level:
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
### Counters
##### Stage: 2 Challenge Level:
Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win? | 2014-11-26 05:33:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34067219495773315, "perplexity": 2041.5170486619108}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931005799.21/warc/CC-MAIN-20141125155645-00239-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://www.finedictionary.com/argue.html | # argue
## Definitions
• A fox without a tail arguing with other foxes
• WordNet 3.6
• v argue present reasons and arguments
• v argue give evidence of "The evidence argues for your claim","The results indicate the need for more work"
• v argue have an argument about something
• ***
Webster's Revised Unabridged Dictionary
• Interesting fact: Two in every three car buyers pays the sticker price without arguing.
• Argue To blame; to accuse; to charge with. "Thoughts and expressions . . . which can be truly argued of obscenity, profaneness, or immorality.""Men of many words sometimes argue for the sake of talking; men of ready tongues frequently dispute for the sake of victory; men in public life often debate for the sake of opposing the ruling party, or from any other motive than the love of truth.""Unskilled to argue , in dispute yet loud,
Bold without caution, without honors proud."
"Betwixt the dearest friends to raise debate ."
• Argue To contend in argument; to dispute; to reason; -- followed by with; as, you may argue with your friend without convincing him.
• Argue To debate or discuss; to treat by reasoning; as, the counsel argued the cause before a full court; the cause was well argued.
• Argue To invent and offer reasons to support or overthrow a proposition, opinion, or measure; to use arguments; to reason. "I argue not
Against Heaven's hand or will."
• Argue To persuade by reasons; as, to argue a man into a different opinion.
• Argue To prove or evince; too manifest or exhibit by inference, deduction, or reasoning. "So many laws argue so many sins."
• ***
Century Dictionary and Cyclopedia
• argue To bring forward reasons to support or to overthrow a proposition, an opinion, or a measure; use arguments; reason: as, A argues in favor of a measure, B argues against it.
• argue To contend in argument; dispute: as, you may argue with your friend a week without convincing him.
• argue To debate or discuss; treat by reasoning; state the reasons for or against: as, the counsel argued the cause before the Supreme Court; the cause was well argued.
• argue To evince; render inferable or deducible; show; imply: as, the order visible in the universe argues a divine cause.
• argue To affect in any way by argument; induce a change in the mind of, or in regard to, by persuasion or reasoning: as, to argue one out of his purpose; to argue away a false impression.
• argue To accuse or charge; impeach or convict: used with of.
• argue Synonyms Argue, Dispute, Debate, Discuss, plead, expostulate, remonstrate. To argue is to defend one's opinion, or to exhibit reasons or proofs in favor of some assertion or principle; it implies a process of detailed proof by one or more persons. To dispute may be to call in question the statements or arguments of an opposing party: as, to dispute about an award. It often means the alternate giving of reasons, especially by two persons. It is often applied to mere bickering, and is in general less dignified than the other words. To debate is to interchange arguments in a somewhat formal manner, as in debating societies and legislative bodies. To discuss is, by derivation, to shake or knock a subject to pieces in order to find the truth, or the best thing to be done. A debate, therefore, may be viewed as a discussion, or a discussion as a debate. Strictly, a discussion is an amicable presentation of opinions, not limited, like the others, to affirmative and negative sides of a proposition, and with the expectation on the part of all that the conclusion will be the adoption of no one person's opinion or plan unmodified. To argue a point, to dispute a position, to dispute with a neighbor, to debate a motion, to discuss a subject or a plan.
• ***
Chambers's Twentieth Century Dictionary
• v.t Argue ärg′ū prove or evince: to prove by argument: to discuss:
• v.i Argue to offer reasons: to dispute (with against, for, with, about):—pr.p. arg′ūing; pa.p. arg′ūed
• v.t Argue ärg′ū (obs.) to accuse
• ***
## Quotations
• Anacharsis
“Wise men argue cases, fools decide them.”
• Pierre De Beaumarchais
“It is not necessary to understand things in order to argue about them.”
• Thomas Carlyle
“A man lives by believing something: not by debating and arguing about many things.”
• Gilbert K. Chesterton
“People generally quarrel because they cannot argue.”
• Marie Ebner-Eschenbach
“Fear not those who argue but those who dodge.”
• Oliver Goldsmith
“There is no arguing with him, for if his pistol misses fire, he knocks you down with the butt end of it.”
## Idioms
Argue the toss - (UK) If you argue the toss, you refuse to accept a decision and argue about it.
***
## Etymology
Webster's Revised Unabridged Dictionary
OE. arguen, F. arguer, fr. L. argutare, freq. of arguere, to make clear; from the same root as E. argent,
Chambers's Twentieth Century Dictionary
O. Fr. arguer—L. argutāre, freq. of arguĕre, to prove.
## Usage
### In literature:
Cousin Charley argued if they did not see the show come in they'd miss one of the big sights of the day: they had plenty of time.
"Watch Yourself Go By" by Al. G. Field
She doesn't trouble to argue; she begins to laugh, and raises her eyebrows.
"Marriage à la mode" by Mrs. Humphry Ward
You'll have to argue it out by yourself later.
"Highways in Hiding" by George Oliver Smith
Another letter describes a great intellectual riddle, which was argued for four days at the School of Logic at Louvaine.
"Short Studies on Great Subjects" by James Anthony Froude
He would still be able to argue with his father on terms not too unequal, he hoped.
"Clayhanger" by Arnold Bennett
Against Miss Pettigrew's tacit approval of the word there was no arguing.
"Priscilla's Spies" by George A. Birmingham
If you please, I have laid down the proposition, and we will now argue the point.
"Mr. Midshipman Easy" by Captain Frederick Marryat
Some suffragettes have argued, in this matter, that in political crises men also have acted just as badly or worse.
"The Task of Social Hygiene" by Havelock Ellis
But, I submit, where all is plain there is nothing to be argued.
"Masterpieces of Negro Eloquence" by Various
I don't argue, and I'm not afraid.
"Erik Dorn" by Ben Hecht
Even arguing that a whirlwind may stand still axially, it discharges tangentially.
"The Book of the Damned" by Charles Fort
After a great deal of arguing I quieted them and got them to lay down their weapons.
"Across Unknown South America" by Arnold Henry Savage Landor
Mr. Cahoon argued no more.
"Fair Harbor" by Joseph Crosby Lincoln
But I am not going to argue with you, sir.
"Orley Farm" by Anthony Trollope
Therefore, he argued, it was spontaneously generated.
"Natural Law in the Spiritual World" by Henry Drummond
It was argued in the House of Commons that no steamship could ever cross the Atlantic with steam, alone, as a propelling power.
"Little Journeys to the Homes of the Great, Volume 11 (of 14)" by Elbert Hubbard
He disdained to argue.
"Shining Ferry" by Sir Arthur Thomas Quiller-Couch
I could argue the matter no more and fell back upon a last plan.
"Lalage's Lovers" by George A. Birmingham
When they start to argue, my motto is, theyre sold.
"Greener Than You Think" by Ward Moore
But first they would argue.
"Eight Keys to Eden" by Mark Irvin Clifton
***
### In poetry:
I used to sit with him, and smoke,
And talk of your blue eyes,
And argue how I best might act
To make your heart my prize.
"To Kate. (In Lieu Of A Valentine)" by Ellis Parker Butler
Long nights argued away
In meeting halls
Back of interminable stairways - In Roumanian wine-shops
And little Russian tea-rooms…
"The Ghetto" by Lola Ridge
He could in every action show
Some sin, and nobody could doubt him.
He argued high, he argued low,
He also argued round about him.
"Sir Macklin" by William Schwenck Gilbert
To "Twenty-firstly" on they go,
The lads do not attempt to scout him;
He argued high, he argued low,
He also argued round about him.
"Sir Macklin" by William Schwenck Gilbert
Yet faith itself could never doubt;
For, as the sacred volume saith,
Much doubting argues little faith.
"The Believer's Principles : Chap. IV." by Ralph Erskine
In lavish stream his accents flow,
TOM, BOB, and BILLY dare not flout him;
He argued high, he argued low,
He also argued round about him.
"Sir Macklin" by William Schwenck Gilbert
### In news:
Mouthpiece for Assemblymember Diane Gordon argues entrapment .
Proponents of eradication argue that it would be terrible to waste the $9 billion already spent, and a new analysis concluded that eradication , if successful, would save up to$50 billion by 2035.
Esteemed lawyer Paul Clement's next challenge is arguing against health-care law.
The Ethical Debate, Overall argues that people should be thinking much less about themselves and much more about society at large when deciding to have kids.
In it, he argued that liberalism eventually leads to totalitarianism.
Taking a hard-line after seizing an election victory, the White House on Wednesday argued that the election results validated President Obama's view that the tax cuts benefiting wealthy Americans must expire .
Some may say that Stefan Karlsson's assertion that an increasing British trade deficit is a sign of an overvalued pound, arguing instead that it reflects the euro area debt crisis.
James Madison argued that we needn't fear political parties.
Li argues the core elitist faction is the "taizidang," or so-called "princelings" -- the offspring of former revolutionary leaders and high-ranking officials.
Craig argues that in an increasingly automated world, playout operators need stimulation and a break from routine.
Senior baseball analyst for ESPN Insider and former major league GM Jim Bowden argues that no matter what the Rangers do before the trade deadline, there's no way they can match the Angels.
For years, the military has been held in high regard — to the point where some argue it became less accountable to civilian authorities.
Farm worker advocates argued that the new rules would protect young people, who suffer higher injury rates on farms than in other industries.
Agriculture groups argued that the rules would hinder the ability of some teenagers to work on family farms in New York state.
***
### In science:
It is argued that pointer states are selected by the interaction of quantum systems with the environment, and are not based on any measurement by a conscious observer.
Comment on "Hidden assumptions in decoherence theory"
From these results is argued that in general the n-point Green’s functions for Yang-Mills theories can have nonperturbative pieces which can not be represented as the sum of Feynman diagrams.
A String Approximation for Cooper Pair in High-T$_{\bf c}$ superconductivity
Though one could argue that the magnetization plateaus are connected with the quantum nature of the spins we found for special parameter sets even in the classical chain plateaus in the dependence −mz versus Ω0 (compare dashed curves in Figs. 8a, 8b and in Figs. 2b, 4b).
Thermodynamic properties of the periodic nonuniform spin-1/2 isotropic XY chains in a transverse field
This configuration is argued to be stable in the large N limit6 .
AdS/CFT Correspondence and Type 0 String Theory
As we will discuss below, this approach leads to a simple physical picture for the Hall constant and it might be argued that at least for certain cases, for example for a system of finite size in the y−direction, it is indeed the right one.
Reactive Hall response
Thereby, under the circumstances considered (string-scale unification plus small tan β ), gravitational corrections are expected not only to spoil the SU (5) quark-lepton mass predictions for the d and s quarks, as argued in , but also for the b quark.
From Prototype SU(5) to Realistic SU(7) SUSY GUT
We argue that the statistical properties of these eigenvalues are universal and can be described by a random matrix theory with the global symmetries of the QCD partition function.
Random Matrix Theory and Chiral Symmetry in QCD
We argue in this review that the complexity of the QCD vacuum leads to a low-energy description that is completely dictated by the global symmetries of QCD.
Random Matrix Theory and Chiral Symmetry in QCD
We have argued that spontaneous breaking of chiral symmetry means that a small quark mass leads to a macroscopic realignment of the QCD vacuum.
Random Matrix Theory and Chiral Symmetry in QCD
This yields Z eff (θ , m) = Xν Below, we argue that the weight factors can be ignored for light quarks but that they are necessary when one considers the quenched theory (which corresponds to Nf = 0 or, equivalently, to the limit of very heavy quarks).
Random Matrix Theory and Chiral Symmetry in QCD
We argued in Sec. 1.1 that chiral symmetry breaking can be understood in terms of the stiffness of the Dirac spectrum resulting from interactions of the strong color force.
Random Matrix Theory and Chiral Symmetry in QCD
Parisi argued that localized states can only occur in quenched systems [236].
Random Matrix Theory and Chiral Symmetry in QCD
They wish to argue that since the pumped state is out of equilibrium, we would assign a nonzero level of complexity to this state.
Response to Comments on "Simple Measure for Complexity"
This idea is further explored in (Denecker, Marek, & Truszczynski 1998), where the authors argue that the well-founded semantics for logic programming implements a generalized principle of non-monotone induction.
SLDNFA-system
The second assertion is well known in the theory of partitions, but we argue probabilistically.
Random matrix theory over finite fields: a survey
*** | 2020-01-19 04:04:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3236325681209564, "perplexity": 6671.272455135643}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250594209.12/warc/CC-MAIN-20200119035851-20200119063851-00231.warc.gz"} |
https://physics.stackexchange.com/questions/554986/why-front-part-of-a-body-undergoing-rolling-pushes-the-surface-a-bit-more | Why Front part of a body undergoing rolling pushes the surface a “bit more”?
Original Post : here
On the accepted answer , it was said that the Normal Force is more on the right side of the centre of mass which provides an anti-torque to the rotation of the body which slows down the rolling.
I also found some similar explanations on "Why a rolling Body Slows Down" in the book "Concepts of Physics by HC Verma"
In the second picture , you can see that it is written that the Normal Force is shifted Right of the center of mass because the front part pushes the surface a bit more . Here it is :
In fact, when the sphere rolls on the table, both the sphere and the surface deform near the contact. The contact is not at a single point as we normally assume, rather there is an area of contact.The front part pushes the table a bit more strongly than the back part. As a result the normal force doesnt pass through the center, it is shifted towards the right. This force then has an anticlockwise torque. The net torque causes an angular deceleration.
But it is not explicitly explained(neither in the book , nor in the answer of the above mentioned post) why the front side pushes it a "bit more" than the back side.
Why does this happen?
• The front part pushes the table a bit more strongly than the back part What the hell,-What's the reason for that ? If mass of wheel is distributed uniformly, then normal forces should be symmetric in both sides. Also if it is like author says,- then wheel made hole in ground must be asymmetric too and not circular one. Is there any proof for that ? – Agnius Vasiliauskas May 26 '20 at 16:22
• @AgniusVasiliauskas , Currently I don't have a proof for that. In the OP question also the accepted answer had a similar explanation for anti clockwise torque but it didn't explain the reason for asymmetrical distribution of the Normal Force. – Noah J. Standerson May 26 '20 at 16:31
• @AgniusVasiliauskas In this case we are not assuming rigid bodies/surfaces – BioPhysicist May 26 '20 at 16:41
• @BioPhysicist ok, then explain how follows from continuum mechanics that normal forces are asymmetric and why forces depicted asymmetrically, author depicts wheel made hole in symmetrical way? It doesn't make sense – Agnius Vasiliauskas May 26 '20 at 16:48
• @AgniusVasiliauskas , I have added a possible explanation below. Please see if it has some error. – Noah J. Standerson May 27 '20 at 15:18
But it is not explicitly explained(neither in the book , nor in the answer of the above mentioned post) why the front side pushes it a "bit more" than the back side.
It is due to the viscoelastic behavior of the contacting materials.
For purely elastic materials the relationship between stress and strain is linear so that the loading and unloading (compressing and uncompressing) forces are equal. See the diagram at the left below.
Viscoelastic materials behave like elastic materials in that both eventually recover from deformation when the load is removed. See diagram to the right below. However, the viscous behavior of a viscoelastic material is such that the stress (force) during unloading is less than that during loading for the same amount of deformation giving the material a strain rate dependent on time. The area in red between the loading and unloading curves represents the hysteresis heat loss. In contrast with ideal elastic behavior, the deformation when the material is viscoelastic does not recover right after the load is removed. In other words, there is a time delay for the material strain to fully recover, which is not shown in the diagram to the right.
In terms of say a tire rolling, the above means the forces acting on the leading portion of the tire (in the direction of motion) in contact with the road under compression (loading) are greater than the forces acting on the trailing portion of the tire in contact with the road under decompression (unloading). The overall result is the difference between the compression and decompression forces results in a net torque counter to the rotation of the tire.
Hope this helps.
I clicked some photos of circular duct tape . I intentionally pressed the duct tape hard so that the deformation can be seen.
At the Normal position :
Now , In the next infinitesimal time $$dt$$ , lets say the tape covers a small distance $$dx$$ .
Here is a picture of it :
As you can see , in the small time interval $$dt$$ , the back part of the tape was still deformed due to which when the tape was rotated , the point(s) of contact somewhat shifted towards the right ( points of contact was more on the right side of the centre). That might be the same reason why a fully inflated football rolls for a longer time than the one which is partially inflated.
Due to this (I think) , the Normal force is shifted "a bit" right
Note : Since this is only an observation and I do not have any mathematical proof for this , if you feel like there is some error in the observation , then comment below.
• first picture is Ok, because deformation is symmetric as it should be, but Im not sure about second. Where does this asymmetry comes from ? I could believe in such asymmetry at $t_0$ time, because body generates acceleration and thus due to acceleration forward part of wheel can experience bigger load. But as it starts to move in constant speed - highest load should return back under COM of wheel (first picture). Besides it unreasonable that contact area after $dt$ time passes gets smaller - area should be the same, just it will be shifting along movement path. Sorry, but unconvincable. – Agnius Vasiliauskas May 27 '20 at 16:13
• @AgniusVasiliauskas it's not an asymmetry of deformation, it's a tilt of the not yet re-deformed object. – Ruslan May 27 '20 at 16:34
• @AgniusVasiliauskas , Thanks for the comment! . I have replaced the second image so that the symmetry is visible – Noah J. Standerson May 27 '20 at 16:34
• @AgniusVasiliauskas , I believe the asymmetry is caused because the initially "deformed" part has not yet "reformed" in the time $dt$ . Due to this , the deformed part is not in contact of the surface. and hence experiences no Normal Force – Noah J. Standerson May 27 '20 at 16:39
• @NoahJ.Standerson. I made my own test of rolling toilet paper, check this out here. As you see no any assymetry or tilting, deformed area just slides along movement direction. In last frames you can however see some tilting, but this is just due to changed my elbow angle, thus projected pushing force vector changes and some tilting arises in the end. But normally if you push wheel straight to COM and forward horizontally - no any tilting/asymmetry seen. Convincing ? – Agnius Vasiliauskas May 27 '20 at 16:53
The front of the sphere is moving toward the ground, the back away (as per clockwise rotation). Therefore, momentum from a tiny piece of sphere there is more downward force on the front side than the back, deforming the ball/ground more.
Suppose the ball is moving in the vacuum, the floor surface is perfectly horizontal and ball and floor are made from hard material.
When an horizontal force is applied to start the movement, the ball initially slides, until the torque mentioned in the previous post, caused by the friction force, transforms the sliding in rotating movement.
In that process the ball loses translational kinetic energy (the friction force opposes the velocity), but acquires rotational kinetic energy.
Once the ball is rotating without slip, there is no friction force opposing the velocity.
The only process that can take energy from the ball is elastic deformation in the region of contact, that transform it from a point into an area. I believe it is illustrated in the reference of your post. But that effect is relevant in a soccer ball for example, and very small in a bowling ball. | 2021-02-28 10:41:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.606377363204956, "perplexity": 651.6016172992711}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178360745.35/warc/CC-MAIN-20210228084740-20210228114740-00295.warc.gz"} |
https://www.physicsforums.com/threads/mean-and-variance-of-a-data-set.883294/ | # Homework Help: Mean and Variance of a data set
Tags:
1. Aug 27, 2016
### DeldotB
1. The problem statement, all variables and given/known data
In this problem we will be generating and analyzing lists of normally distributed random numbers. The distribution we are sampling has true mean 0 and standard deviation 1.
1. If we sample this distribution N=5 times, what do we expect the mean to be? How about the standard deviation? Whats the error on the mean?
2. Relevant equations
$\bar{x}= \Sigma \frac{x_i}{N}$
$s^2= \Sigma \frac{(x_i- \mu)^2}{N}$
3. The attempt at a solution
Im not sure where to go here. What does it mean to have a true mean of zero? What is meant by "true" mean - I havent seen this this phrase used before. I read that if a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean, but how does this help me when I want to sample this distribution? Any help would be appreciated! I have never taken a statistics class.
2. Aug 27, 2016
### Ray Vickson
Suppose one run of your experiment consists of taking a random sample of size N = 5 from a standard normal distribution (mean = 0, variance = 1). In any run of your experiment, the computed mean of your data set is $\bar{x} = \frac{1}{5}(x_1 + x_2 + x_3 + x_4 + x_5)$, where the $x_i$ constitute your sample of 5 numbers. Note that $\bar{x}$ is itself a sample point from a random variable $\bar{X}$: in one experiment it might = 1.7, in another experiment it might = -0.83, etc., etc. So, $\bar{X}$ itself has some true mean and some true variance; these would be well approximated by repeating the experiment 100,000 times and taking the average and sample variance of your 100,000 $\bar{x}$ values. Remember, however, that for any particular experiment the computed $\bar{x}$ and the computed sample variance $s^2(x)$ will very likely differ at least a bit from the true values of 0 and 1 respectively.
$$s^2= \frac{1}{N} \sum_{i=1}^N (x_i- \mu)^2)$$
is correct only if you pretend you know $\mu$; it is NOT what we usually call the "sample variance". The usual definition of sample variance is that we also estimate $\mu$ from the data as well, so we are dealing with
$$\text{sample variance} = \frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x})^2,$$
$$\bar{x} = \frac{1}{N} \sum_{i=1}^N x_i .$$
Note that we divide by $N-1$ instead of $N$; the need for doing that arises because we have already "used up" one piece of information when we computed $\bar{x}$, so have left only $N-1$ extra pieces of information that can be used when estimating variance. Theoretically, the true mean of the random variable
$$S = \frac{1}{N-1} \sum_{i=1}^N (x_i -\bar{x})^2$$
is 1, which is the true value of the variance. Had we divided by N instead we would have a random variable with mean $(N-1)/N = 1 - (1/N)$, instead of the true value 1. Of course, for large $N$ it makes hardly any noticeable difference. | 2018-05-21 22:53:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7934649586677551, "perplexity": 208.17004688195007}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864558.8/warc/CC-MAIN-20180521220041-20180522000041-00186.warc.gz"} |
https://joxivoqaxazopop.michellemadsenpoet.com/velocities-of-zero-book-17120xw.php | Last edited by Shasar
Tuesday, July 7, 2020 | History
4 edition of Velocities of zero found in the catalog.
Velocities of zero
Marwan Hassan
# Velocities of zero
## by Marwan Hassan
Written in English
Edition Notes
Classifications The Physical Object Statement Marwan Hassan. LC Classifications JV Pagination 200 p. ; Number of Pages 200 Open Library OL22518551M ISBN 10 1894770021
operation; low speed or damper velocities zero to mm/sec ( in/sec), mid-speed or damper velocities mm/sec (2 to 8 in/sec) and high speed, above mm/sec (8 in/sec). The rebound and compression valves (piston and compression piston) all work the same. Below are descriptions of what controls each valving stage, followed byFile Size: 1MB. The acceleration has to be either zero at that point or positive. Suppose the acceleration is positive at the moment that the velocity is zero. Plainly it was negative before the velocity became zero; we could not have slowed down to zero from a positive velocity if the acceleration was positive or zero.
ZERO SUM GAME Best of Lists: * Best Books of the Month at The Verge, Book Riot, Unbound Worlds, SYFY, & Kirkus * The Mary Sue Book Club Pick * Library Journal Best Debuts of Fall and Winter. A blockbuster near-future thriller, S.L. Huang's Zero Sum Game introduces a math-genius mercenary who finds herself being manipulated by someone possessing unimaginable power Brand: Tom Doherty Associates. To read Zero Zero Zero is to wallow in a depravity and barbarism that only humankind seems able to produce. To describe the murderers and torturers as bestial is to do animals a great disservice. It is an immensely sad book and Saviano appears to be a very sad author/5().
Suppose the velocities $\dot q_1$, $\dot q_2$, $\dot q_3$ are 5, 9 and Now with this much of information I can predict the future motion of my system only if the the accelerations in the 3 directions are ZERO. If a force is acting on the system then definitely acceleration in the 3 directions are not zero. We can write these constraints as a matrix dependent on the configuration theta times the joint velocities theta-dot equal to zero. If we call this matrix A of theta, we can write the velocity constraints as A of theta times theta-dot equals zero, where the A matrix has k rows and n columns.
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Malign Velocities steps in and registers the futurist thrill of those theorists who would arrive at communism via an advanced, high tech capitalism - and registers the often disastrous results of these 'accelerations', which took us more often to Stalinism or neoliberalism than to utopia.
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Its always a great experience when you go into a book with no expectations and it turns out you really enjoy it. Thats what happened to me with Zeroes.I hadnt read a single review beforehand, hadnt read anything by Scott Westerfeld (though I think I will remedy that now) and have never even heard of the other two authors.
The reason I picked up this book was the premise/5. By book’s end, Velocities of zero book reader will dispute Seife’s claim that zero is among the most fertile—and therefore most dangerous—ideas that humanity has devised Seife’s prose provides readers who struggled through math and science courses a clear window for seeing both the powerful techniques of calculus and the conundrums of modern Cited by: SWARM is the second book in the ZEROES trilogy centered on a group of teens in Cambria, California that have unusual powers.
This second book was faster paced than the first and readers got to know the super-powered teens better. We find that some of the powers are changeable and find there are other teens born in the year that also have 4/5(38).
Consider the following: (i) the book is at rest, (ii) the book is moving at a constant velocity, (iii) the book is moving Velocities of zero book constant acceleration. Under which of these. Praise For Zero Limits "This riveting book can awaken humanity.
It reveals the simple power of four phrases to transform your life. It's all based in love by an author spreading love. You should get ten copies of itone for you and nine to give away.
It's that good."4/5. Book of Velocities () Siwan () Book of Velocities is a solo album by pianist Jon Balke recorded in and released on the ECM label in Reception. The Allmusic review by Thom Jurek awarded the album 3½ stars stating "nothing could have prepared the listener for this gorgeous set of unedited, unprocessed, and undubbed piano Genre: Jazz.
The way I see it, the book appears to blend prodigious research with conjecture. It ranges freely over decades and continents, offering a dizzying catalog of vivid characters and horrible acts. Indeed, to derive the equations, one may first change the frame of reference so that one of the known velocities is zero, determine the unknown velocities in the new frame of reference, and convert back to the original frame of reference.
Examples Ball 1: mass = 3 kg, velocity = 4 m/s Ball 2: mass = 5 kg, velocity = −6 m/s. Then I'll run a charge test in.5 grain increments to that Max recording velocity and any pressure signs. Here is what I used to get a feel for "book max" in gr target bullets using H My velocities are pretty much in line with published.
Hodgdon, MK, gr H, MV Nosler, custom comp, HSC, MV. Their velocities may be zero. Complete the following statement: Momentum will be conserved in a two-body collision only if the net external force acting on the two-body system is zero.
Zero is a wonderful number book that looks at not only the concept of counting, it tackles the social issue of self acceptance and diversity. I liked how Otoshi used feeling words that most children of three or four year old would not know and gives opportunity to extend their vocabulary/5().
LeHew, J, Guala, M & McKeon, BJA study of convection velocities in a zero pressure gradient turbulent boundary layer. in 40th AIAA Fluid Dynamics Conference.,40th AIAA Fluid Dynamics Conference, Chicago, IL, United States, 6/28/Cited by: 4.
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The rate of change of distance is called the velocity. It is denoted by expression for the velocity is as follows: Here, dx is the change in the distance and dt is the time. Velocity is a physical vector quantity; both magnitude and direction are needed to define scalar absolute value of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI (metric system) as metres per second (m/s) or as the SI base unit of (m⋅s −1).For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a SI base units: m/s.
I'm not sure I understand your question. Acceleration is defined as the change in velocity. So the acceleration being 0 simply means that there was no change in velocity.
In other words, it didn't get faster or slower — or change directions. Howev. what is zero velocity. i am doing this thing for physics and i have the meaning but i need the word of the meaning.
i need help asap. Answer Save. 7 Answers. Relevance. Thomas. 1 decade ago. Favorite Answer. Velocity is speed with direction. So having zero velocity is having no speed (being stopped) and not changing direction. In each of these situations, an object has a velocity relative to a medium (such as a river) and that medium has a velocity relative to an observer on solid ground.
The velocity of the object relative to the observer is the sum of these velocity vectors, as indicated in Figure and Figure These situations are only two of many in which it is useful to add velocities.
Well in this reference frame, the center of mass velocity, by definition, is zero. And therefore by eqn. the total momentum is also zero. I'll notate all variables in this new system the same as the old, but just to remind ourselves that we're in this new frame I'll also add " ' " to them.Zero: The Biography of a Dangerous Idea is an enjoyable memory trip.
Without realizing it you are being reminded of the history you knew about zero from Pythagoras and Aristotle to Babylonia up to today ; the importance of zero's inclusion in our number system, the leadership and presence off the base number system even in to-day's world /5().There are many approximate models of friction, and you can add your favorite model of friction torque, replacing the zero joint torques by joint torques that depend on the joint velocities.
One advantage of having zero friction and zero joint torques, however, is that we know that no energy is dissipated. | 2020-09-23 08:51:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44765451550483704, "perplexity": 2297.1434140537913}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400210616.36/warc/CC-MAIN-20200923081833-20200923111833-00632.warc.gz"} |
http://mathematica.stackexchange.com/tags/arithmetic/hot?filter=all | # Tag Info
26
Oleksandr is correct about the way evaluation works. a/b seems to be interpreted (parsed) directly as Times[a, Power[b,-1]], or more readably: $a\times b^{-1}$. Divide[a,b] is interpreted as is. Evaluation then proceeds from these forms, and the arithmetic is carried out differently for the two cases: either $a\times (1/b)$ or $a/b$. Here are some ...
24
In a sense described below, this answer finds $422716$ distinct solutions. The innovations presented here are using postfix operators to eliminate problems with parentheses; avoiding having to deal with unary negation; initially computing "too many" solutions, some of which make no sense, and eliminating them at the end (rather than writing more ...
15
EDIT: As @Rojo points out in the comments, my code doesn't really find all solutions. For example, a term of the form a * (b * c + d) can't be represented with "precedence plus/minus" operators. I'm not sure if it is salvageable, but as it is, the code below does not find all solutions. A very simple solution would be to define two new operators $\oplus$ ...
14
While it would've been nice if the package handled it automatically, it can be fixed with a simple overloading of Quantity: Unprotect@Quantity; Quantity /: (0 | 0.) Quantity[_, unit_] := Quantity[0, unit] Protect@Quantity; You can add this to your init.m, so that you don't have to define it each time. You can test your examples with this: 0. Quantity[1, ...
11
You can input numbers in any base up to 36 using the notation base^^digits. Digits over 9 are represented using a, b, c, ... You can print numbers in any base up to 36 using BaseForm. Thus, In[1]:= a=2^^0.10101 Out[1]= 0.65625 In[2]:= BaseForm[a^2,2] Out[2]//BaseForm= Subscript[0.0110111001, 2] Note that the internal representation of numbers doesn't ...
10
10
Here's one idea. Hold the expression unevaluated and go up the expression tree from (near) the bottom, level by level, and evaluate. expr = HoldForm[1/((a + 2 b)/c^2)] /. {a -> 1, b -> 2, c -> 3} out = ToExpression@ToString[FullForm@#] & /@ (ReplacePart[expr, # -> Extract[expr, #] & /@ #] & /@ ...
9
Another option: RootApproximant[0.1845095405274387, 1]
9
There isn't really such a thing as binary arithmetic (at least in Mathematica). Numbers can be represented in any base, and this user-visible representation is completely independent from how arithmetic is done. Try this: BaseForm[(2^^1010101011)*(2^^1111101110), 2] Things to look up: BaseForm Digits in numbers
8
Calculating eigenvalues involves solving for the roots of the characteristic polynomial, which is of degree equal to the order of the size of the matrix. When you input real numbers, it can search for the roots of the polynomial using numerical techniques. When you input exact integers (or rationals, probably) it tries to find exact answers for the roots of ...
8
Someone certainly had to write this recursive one: Clear[f] f[x_Integer, y_Integer] := x + f[x, y - 1] f[x_Integer, 0] := 0 f[x_Integer] := f[x, x] + f[x - 1] f[0] := 0
8
Still another possibility: Last[Convergents[0.1845095405274387]]
8
Try this : s[n_] := Total[ Range[n]^2] to check how it works, e.g. : s[5] // Trace There is also a purely symbolic approach, e.g. : $\quad n^2$ ~ Sum ~$n\quad$ (see Infix notation) : (n^2) ~ Sum ~ n 1/6 (-1 + n) n (-1 + 2 n) Note : Sum[ n^2, n] returns the same as Sum[ i^2, {i, n-1}] does, i.e. indefinite sums starts at 0 while ...
8
An extended comment. I'm not sure if this has been realized, please correct me if it has. The result of the Divide[a,b] operation is not the same as the first 3 which are identical. {a, b} = List @@ RandomReal[{-50, 50}, {2, 1*^7}]; x1 = a/b; x2 = a b^-1; x3 = a/b; x4 = Divide[a, b]; Now... Tally[x1 - x2] Tally[x2 - x3] Both give 10^7 zeros. ...
7
Total@Flatten[ConstantArray[#, #] & /@ Range[9]] I think this exercise is somewhat of a Rorschach test… I don't know what the above says about me, though :)
7
A Mathematica minded answer: HarmonicNumber[n, -2] So: Simplify[Sum[i^2, {i, n}] == HarmonicNumber[n, -2]] (* True *)
7
Figuring out what the following snippet does is left as an exercise for the reader: With[{n = 9}, s = t = 0; j = 1; Do[ t += j; s += t; j += 2, {n}]; s ]
7
Here's a possibility nextops = HoldForm /@ {Plus, Times, Divide, Subtract}; (nextOp[#1] = #2) & @@@ Most@Transpose@{nextops, RotateLeft@nextops}; nextchildren = True; SetAttributes[{nextPlus, nextTimes}, Flat]; next[{i_}] := False; next[l_List] := HoldForm[Plus][{l[[1]]}, l[[2 ;;]]]; next[op_[arg1_, arg2_]] /; nextchildren := With[{res = ...
7
A couple of bits of code for your consideration: FromDigits@#/10^(Length@# - #2) & @@ RealDigits[0.1845095405274387] Rationalize[0.1845095405274387, \$MachineEpsilon]
6
You can enter a number in an arbitrary base using base^^digits: alpha = 2^^0.10101; BaseForm[alpha, 2] BaseForm[alpha^2, 2]
6
I feel like I should be prefacing this answer with three confessions, considering that this is an arithmetic question. First, I had a hard time with the multiplication tables until I was nine years old. Second, even after I finally got the hang of multiplication, I was never a fan of multiplying from right-to-left; I preferred going left-to-right. (Arthur ...
5
Not a full answer since I need to sleep :) but more of an observation, which might help. It seems to have to do with the fact that 0 and 0. are not the same in Mathematica. This simple example shows it UnitConvert[0. + Quantity[5, "Meters"], "Inches"] (*--> UnitConvert[0. + Quantity[5, "Meters"], "Inches"] *) while UnitConvert[0 + ...
4
Independently I arrived at something similar to Michael's answer, yet different. I borrowed his formatting function after seeing it as it works better than what I had. Perhaps this will also be of use: evalFromBottom[expr_, lv_: 1] := If[lv > Depth@expr, expr, With[{ev = Replace[expr, x_ :> RuleCondition[x], {-lv}]}, If[expr === ev, ...
4
Yes, it is possible. Since multiplication of positive integers is repeated addition, we can repeatedly add instead of multiply: n = 10; sum = 0; Do[ Do[ sum = sum + i, {j, 1, i}], {i, 1, n}]; Print[sum]
4
An oddball one using a recursive, memoizing function for the square. Clear[sq]; sq[n_Integer] := sq[n] = sq[n - 1] + n + (n - 1) sq[1] = 1; Sum[sq[n], {n, 6}] 91 It's not something I would directly use for such a goal, but you asked for something without explicit multiplications and you got it. Alternatively, if we don't interpret Dot as some ...
4
Ten ways to beat a dead horse Sunday afternoon on an airplane without wifi and this was the problem I remembered reading at breakfast. Forgive me for the time I had on my hands. All because @Aky resurrected this dead horse. (Thanks by the way. I'm glad to have figured out the CellularAutomaton one, but we landed before I could come up with a good MapAll ...
4
Here's a way to brute-force search for numbers that have the property that the sum of the digits raised to an integer power is equal to the number itself. list = {}; Do[If[Total[IntegerDigits[b^e]] == b, AppendTo[list, {b, e}]], {b, 2, 800}, {e, 2, 100}]; This returns a list of the numbers and powers (here's just the first 50), ordered so that they ...
4
As rasher and the documentation both say, Equal has a certain level of fuzziness. The same is true of SameQ, though it has a more stringent tolerance. The following computations are all done with machine precision numbers. Similar things should hold with arbitrary precision numbers but the analysis might be trickier. (* 12 zeros, difference = ...
4
Please tell me if this simplified function does what you want: f[x_, n_] := Round[x, 10^(1 - n + ⌊ Log10 @ Abs @ x ⌋)] ~SetPrecision~ n Test: Table[f[x*Pi, 4], {x, {1/100, 1/10, 1, 10, 100}}] % // FullForm {0.03142, 0.3142, 3.142, 31.42, 314.2} List[0.031424., 0.31424., 3.1424., 31.424., 314.24.] Update The OP wrote: I understand that ...
3
I'll join the bandwagon with SparseArray: sq[n_Integer] := Tr@SparseArray[{{i_, i_} :> i^2}, n]
Only top voted, non community-wiki answers of a minimum length are eligible | 2016-02-13 07:02:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4203970432281494, "perplexity": 2598.808411835814}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701166222.10/warc/CC-MAIN-20160205193926-00333-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://openstudy.com/updates/55705ef7e4b02626b39546c1 | anonymous one year ago 6x - 2y = 50 x + 3/2y = 12 Solve Using Substition @Jamierox4ev3r
• This Question is Open
1. Jamierox4ev3r
alright :) so first, manipulate the 2nd equation so that it equals x
2. Jamierox4ev3r
like so x+3/2y=12 -3/2y -3/2y $$x=\Large\frac{-3}{2}y$$
3. Jamierox4ev3r
i mean ** $$x=\large\frac{-3}{-2}y + 12$$
4. Jamierox4ev3r
so then we plug that in to the second equation $$6(\large\frac{-3}{2}+12)$$-2y=50
5. anonymous
Ok
6. Jamierox4ev3r
so then distribute :) $$6(\large\frac{-3}{2}+12)-2y=50$$ -9+72-2y=50
7. Jamierox4ev3r
from here, add like terms -9+72-2y=50 63-2y=50
8. Jamierox4ev3r
then subtract 63 from both sides 63-2y=50 -63 -63 -------- -2y=-13
9. Jamierox4ev3r
lastly, divide -13 by -2y. then you'll be able to solve for y :)
10. Jamierox4ev3r
@Ami22 I hope you've found this helpful! I will tell you how to solve for x in a little bit, alright? my mom wants me to go pick up a few things from the store. In the meantime, could you medal me? All you have to do is click on the blue best response button next to my name. Thank you :)
11. anonymous
ok | 2017-01-16 20:03:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48681309819221497, "perplexity": 3277.275666897249}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560279248.16/warc/CC-MAIN-20170116095119-00523-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2990376/a-equiv-b-mod-p-implies-apn-equiv-bpn-mod-pn | # $a \equiv b$ (mod $p$) implies $a^{p^n} \equiv b^{p^n}$ (mod $p^n$)?
Let $$p$$ be a prime number. If $$a \equiv b$$ (mod $$p$$), does that imply $$a^{p^n} \equiv b^{p^n}$$ (mod $$p^n$$)?
I think the answer will be yes, and I suspect that the way of proving it will involve writing $$a^{p^n}-b^{p^n}$$ as a multiple of $$(a-b)^n$$.
I also noticed that $$n, and I'm wondering if this will make the proof easier or not.
## 2 Answers
$$a=b+kp$$ where $$k$$ is an integer
$$(b+kp)^{p^n}=b^{p^n}+\binom{p^n}1b^{p^n-1}kp+\binom{p^n}2b^{p^n-2}(kp)^{2}+\cdots+(kp)^{p^n}$$
$$\equiv b^{p^n}\pmod{p^{n+1}}$$
• So simple yet so useful! Thank you! – Pascal's Wager Nov 8 '18 at 23:45
There is following theorem (Lifting The Exponent Lemma). We will use notation $$p^{\alpha}||n$$ for positive integer $$n$$, nonnegative integer $$\alpha$$ and prime $$p$$, which equivalent to $$p^{\alpha}|n$$ and $$p^{\alpha+1}\nmid n$$.
Theorem. Let $$a,b,n$$ be a positive integers and $$p^{\alpha}||a-b$$, $$p^{\beta}||n$$. Then:
• if $$p>2$$ and $$\alpha\geq 1$$ then $$p^{\alpha+\beta}||a^n-b^n$$;
• if $$p=2$$ and $$\alpha\geq 2$$ then $$2^{\alpha+\beta}||a^n-b^n$$.
Note. If $$p=2$$ and $$\alpha=1$$ then $$2^{\beta+1}||a^n-b^n$$.
Your statement is a consequence of this theorem.
Some links about LTE lemma:
What can I do with the lifting the exponent lemma?
https://brilliant.org/wiki/lifting-the-exponent/ | 2019-04-18 13:01:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 30, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9788867831230164, "perplexity": 426.70723334553986}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578517639.17/warc/CC-MAIN-20190418121317-20190418143317-00283.warc.gz"} |
http://mathhelpforum.com/calculus/1359-cone-edge.html | # Math Help - cone edge
1. ## cone edge
Hi ,
I dont know how to get the edge of the cone in cylindrical coordinates.
we have a cone starting at the origin, of heigth 2 and the top is a circle of radius 1 (center at the origin).
we want to calculate the volume.
I set the integral
(|(a,b) means integral from a to b)
for theta : |(0,2*pi)
for r: |(0,1)
for z: |(X,2) I dont know how to find the lower bound X
We see that the inferior limit of Z is 0. But if I plug X=0 in your integration bounds we will end up with the volume of the cylinder which has the same circle (base) as the cone and the same height.
Because what you did was taking theta from 0 to 2pi with r from 0 to 1 (so you calculated the area of the circle) and then if you do z from 0 to 2 it is the same as if you multiplied the area of the circle by the height (2) (since all 3 parts of your triple integral are independent) so it is really the volume of the cylindre we find (the one which has the same base and height as the cone).
So we must use something else for the bounds.
Theta and r are fine since they help us find the area of the base which is the same in cylinder and cone. For z :
The line representing the edge of the cone does not depend on theta. So for any theta we will have the line z=2-2r. So z goes from 0 to the line 2-2r. We can understand that z does not go from 0 to 2 for any r since z is stopped by the line 2-2r. For example : when r=0, z goes until 2-2*0 = 2 (it is the top of the cone) and when r=1 (i.e. at the edge of the circle) z=2-2*1=0 (we can see that because the edge of the circle corresponds with the edge of the upper part of the cone).
If that is ok, then your intergal is as shown in the figure. You can verify the result since you know V(cone) = Area(base)*Height /3 = pi r^2 * Height /3 = pi *2/3.
Ok. I hope it helps you. (the integral drawing is done by OpenOffice.org 2.0 Math)
3. Originally Posted by hemza
We see that the inferior limit of Z is 0. But if I plug X=0 in your integration bounds we will end up with the volume of the cylinder which has the same circle (base) as the cone and the same height.
Because what you did was taking theta from 0 to 2pi with r from 0 to 1 (so you calculated the area of the circle) and then if you do z from 0 to 2 it is the same as if you multiplied the area of the circle by the height (2) (since all 3 parts of your triple integral are independent) so it is really the volume of the cylindre we find (the one which has the same base and height as the cone).
So we must use something else for the bounds.
Theta and r are fine since they help us find the area of the base which is the same in cylinder and cone. For z :
The line representing the edge of the cone does not depend on theta. So for any theta we will have the line z=2-2r. So z goes from 0 to the line 2-2r. We can understand that z does not go from 0 to 2 for any r since z is stopped by the line 2-2r. For example : when r=0, z goes until 2-2*0 = 2 (it is the top of the cone) and when r=1 (i.e. at the edge of the circle) z=2-2*1=0 (we can see that because the edge of the circle corresponds with the edge of the upper part of the cone).
If that is ok, then your intergal is as shown in the figure. You can verify the result since you know V(cone) = Area(base)*Height /3 = pi r^2 * Height /3 = pi *2/3.
Ok. I hope it helps you. (the integral drawing is done by OpenOffice.org 2.0 Math)
Thank you so much for the clear explanation.
I understand now what is going on in the multiple integrals . I will continue to practice using your explanation.
B | 2014-12-22 15:35:31 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8933534026145935, "perplexity": 330.5440602708637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802775394.157/warc/CC-MAIN-20141217075255-00153-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://gamedev.stackexchange.com/questions/92112/access-entities-components-via-the-entity-that-holds-them-or-via-a-separate-sys | Access Entities components via the Entity that holds them, or via a separate System?
Now, I'm implementing a component based game engine and I came to a thought: Which way should I access my components?
• Have a list of Entities, which have a list of Components, and access them by going trough each entity, and updating/drawing/whatever on each component.
or
• Have separate systems for each component type and acces them by going trough that systems components.
Or does it really matter? I think that the second way would seem faster...
I would recommend having both.
Entity/Component Graph
This defines which entities are attached to which components. In my engine, there is no such thing as an "entity", but any component can have child components. This graph is not used to draw or update components, but is only used for logical queries and events that are supposed to effect component sub-trees in the graph. The graph is just made of weak references to components (in C++, pointers. In other languages, unique identifiers), and doesn't contain the components itself.
For example, I have an "onDeath" event that uses the component graph to tell which components need to be destroyed when a parent component has died.
Component Lists and Systems
Parallel to the component graph is a set of component lists. For instance, there is a list of all the Physics components in the game, a list of all the Sprite components, and so on. These lists are aggregated into a big map of component type to component list called the ComponentManager.
ComponentManager is responsible for updating and drawing all components, one component type at a time. This is far faster than iterating through the component graph, because it has much better cache performance, and allows you to batch state of components before drawing/updating them. | 2020-05-31 21:53:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.392070472240448, "perplexity": 927.7234062488616}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347413786.46/warc/CC-MAIN-20200531213917-20200601003917-00559.warc.gz"} |
https://sob5050.com/pnpfbmg/article.php?id=619bcf-frobenius-norm-python | # frobenius norm python
For values of ord < 1, the result is, strictly speaking, not a tutorial-like examples and some informal chatting on C/C++/Java/Python software development (and more) Pages. For example, the following code sums a list of three expressions: expr_list = [expr1, expr2, expr3] expr_sum = sum (expr_list) Version 0.6.0. A Frobenius matrix is a special kind of square matrix from numerical mathematics.A matrix is a Frobenius matrix if it has the following three properties: all entries on the main diagonal are ones; the entries below the main diagonal of at most one column are arbitrary SLUG = "more-matrix-math-in-python… numpy.linalg.norm Notes The condition number of x is defined as the norm of x times the norm of the inverse of x [R37] ; the norm can be the usual L2-norm (root-of-sum-of-squares) or one of a number of other matrix norms. The goal of this tutorial is to enter mathematics for data science by coding with Python/Numpy. If axis is None then either a vector norm (when x Norms are any functions that are characterized by the following properties: 1- Norms are non-negative values. Related. Example Codes: numpy.linalg.norm() We will use this function to find the norm … I can find the value of frobenius norm is a scalar. Default is 'euclidean' which is equivalent to Frobenius norm if tensor is a matrix and equivalent to 2-norm for vectors. Below is an example where we use NMF to produce 3 topics and we showed 3 bigrams/trigrams in each topic. Input array. 2-norm ... ints, 2-list of python:ints, optional) – If dim is an int, vector norm will be calculated over the specified dimension. The Frobenius norm is an extension of the Euclidean norm to × and comes from the Frobenius inner product on the space of all matrices. 1.1 Frobenius norm The Frobenius norm of a matrix Xis a measure of the \length" of a matrix. The cond() function is capable of returning the condition number using one of … I have been studying about norms and for a given matrix A, I haven't been able to understand the difference between Frobenius norm $||A||_F$ and operator-2 norm $|||A|||_2$. It’s written: jjXjj F = sX ij X2 ij; where iand jrange over all entries in the matrix X. If you think of the norms as a length, you easily see why it can’t be negative. If this is set to True, the axes which are normed over are left in the purposes. G. H. Golub and C. F. Van Loan, Matrix Computations, These are the top rated real world Python examples of scipylinalg.norm extracted from open source projects. Cichocki, Andrzej, and P. H. A. N. Anh-Huy. The built-in Python sum should be used to add together a list of expressions. Cichocki, Andrzej, and P. H. A. N. Anh-Huy. In particular, the Euclidean and Frobenius norms are related to each other by the following inequalities. Is there any fast way to compute the exact Frobenius norm of the matrix or its accurate approximation (perhaps, via Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Using the axis argument to compute vector norms: Using the axis argument to compute matrix norms: {non-zero int, inf, -inf, ‘fro’, ‘nuc’}, optional, array([ 1.41421356, 2.23606798, 5. ]). are computed. ... Matrix norms: the Frobenius norm. The default is None. The Frobenius Norm of a matrix is defined as the square root of the sum of the squares of the elements of the matrix. The function norm(X, "fro") is called the Frobenius norm and norm(X, "nuc") the nuclear norm. Actual number of iterations. To calculate the norm of the array you have to use the numpy.linalg.norm() method. If axis is None, x must be 1-D or 2-D. ord : {non-zero int, inf, -inf, ‘fro’}, optional. The default method optimizes the distance between the original matrix and WH, i.e., the Frobenius norm. Derivative of squared Frobenius norm of a matrix with penalty term associated with projection operator. If both axis and ord are None, the 2-norm of are computed. compute the vector norms. numpy.linalg.norm¶ numpy.linalg.norm(x, ord=None, axis=None, keepdims=False) [source] ¶ Matrix or vector norm. If axis is None then either a vector norm (when x is 1-D) or a matrix norm (when x is 2-D) is returned. For values of ord <= 0, the result is, strictly speaking, not a If you think of the norms as a length, you easily see why it can’t be negative. Frobenius norm. If axis is an integer, it specifies the axis of x along which to broadcast correctly against the original x. If axis is None, x must be 1-D or 2-D, unless ord The Frobenius Norm of the given matrix is: 44.238 In the above program, we are using two loops to traverse every element in the matrix so that we can find its square and add it to the variable sum_of_sq which gives us the total sum of the square of elements of the matrix. Some restrictions apply: a) The Frobenius norm fro is not defined for vectors, b) If axis is a 2-tuple (matrix norm), only 'euclidean', 'fro', 1, np.inf are supported. on the value of the ord parameter. It should compute the frobenius norm of a 3D array. If axis is a 2-tuple, it specifies the Returns n float or ndarray. 1. If axis is None then either a vector norm (when x n_iter_ int. axes that hold 2-D matrices, and the matrix norms of these matrices as vec norm when dim is None. Return. If dim is a 2-tuple of ints, matrix norm will be calculated over the specified dimensions. Bug report Incoorect L2 norm computed for the following matrix: 2 -1 0 0-1 2 -1 0 0 -1 2 -1 inf means numpy’s 2.5 Norms. Frobenius Norm is defined as: where A is a m*n matrix. or one of an infinite number of vector norms (described below), depending The Frobenius norm satisfies proposition 1.7 but is not an induced norm, since for I n, the identity matrix of order n, we have ‖ I n ‖ F = n 1 2.For finite dimensional spaces all norms are equivalent. I think that having practical tutorials on theoretical topics like linear algebra can be useful because writing and reading code is a good way to truly understand mathematical concepts. numpy.linalg.norm¶ numpy.linalg.norm (x, ord=None, axis=None, keepdims=False) [source] ¶ Matrix or vector norm. The formula of Frobenius Norm. © Copyright 2008-2009, The Scipy community. Example: Input: mat[][] = ... last_page Python program to reverse the content of a file and store it in another file . Can someone help me Our example has very limited data sizes for demonstration purposes. matrices and raise a ValueError when x.ndim != 2. Home; Who am I; Frobenius Norm The Frobenius norm is the same concept of the Euclidean norm, but applied to matrices. Frobenius norm of the matrix difference, or beta-divergence, between the training data X and the reconstructed data WH from the fitted model. numpy.linalg.norm¶ numpy.linalg.norm(x, ord=None, axis=None) [source] ¶ Matrix or vector norm. Version bump to 0.6 due to order of params changing. Using the axis argument to compute vector norms: Using the axis argument to compute matrix norms: array([-4, -3, -2, -1, 0, 1, 2, 3, 4]), array([ 1.41421356, 2.23606798, 5. The Frobenius norm is submultiplicative and is very useful for numerical linear algebra. $\begingroup$ By reducing to the SVD, you can express the 2-norm condition number as the ratio of the largest and smallest nonzero singular values, and similarly the Frobenius condition number as the square root of the ratio of the sum of the squares of the singular values … I can find the value of frobenius norm is a scalar. The Frobenius norm is submultiplicative and is very useful for numerical linear algebra. Order of the norm (see table under Notes). inf means numpy’s Even though, the Frobenius norm is calculated and I obtain a value not to high just normal, similar to the one obtained by the same algoritm in Python. Using Python's any function, we can then verify that none of the appended results are False, which is the expectation of the inequality. This docstring is modified based on numpy.linalg.norm. is None. Given an M * N matrix, the task is to find the Frobenius Norm of the matrix. Just change it to any other preset norm and it should work. Extending the least square estimation from the vector to a matrix. This function is able to return one of seven different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the ord parameter. as vec norm when dim is None. The Frobenius norm is not an operator norm, it is a norm on the vector space of linear operators/matrices, which is not the same thing. norm_1d = np.linalg.norm(array_1d) 2-D Numpy Array. The Frobenius norm is an extension of the Euclidean norm to × and comes from the Frobenius inner product on the space of all matrices. The nuclear norm is the sum of the singular values. axes that hold 2-D matrices, and the matrix norms of these matrices Frobenius Norm is defined as: where A is a m*n matrix. 15. Frobenius norm – ‘nuc’ nuclear norm – Other. inf object. For example, the following code sums a list of three expressions: expr_list = [expr1, expr2, expr3] expr_sum = sum (expr_list) 2-norm ... ints, 2-list of python:ints, optional) – If dim is an int, vector norm will be calculated over the specified dimension. This function is able to return one of eight different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the ord parameter. This function is able to return one of eight different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the ord parameter.. Parameters ... Imports # python from argparse import Namespace from functools import partial import math # from pypi import hvplot.pandas import numpy import pandas # my stuff from graeae import EmbedHoloviews. I'm looking for a build-in function in python. The $2$-norm, $1$-norm, and $\infty$-norm are then computed and compared. It is easy to compute frobenius norm in numpy, here is an example: import numpy as np A = np.array([[1, 2, 3],[4, 5, 6]]) F = np.linalg.norm(A) print(F) This function is able to return one of eight different matrix norms, It behaves like the Euclidean norm but for matrices: it’s equal to the square-root of the sum of all squared elements in a matrix. Numpy linalg cond() function computes the condition number of a matrix. is 1-D) or a matrix norm (when x is 2-D) is returned. mathematical ‘norm’, but it may still be useful for various numerical Some of the ord are not implemented because some associated functions like, _multi_svd_norm, are not yet available for sparse matrix. purposes. 0. derivative of matrices expression. References. Example Codes: numpy.linalg.norm() We will use this function to find the norm … The Frobenius matrix norm is not vector-bound to the vector norm, but is compatible with it; the Frobenius norm is much easier to compute than the matrix norm. NumPy Linear Algebra Exercises, Practice and Solution: Write a NumPy program to calculate the Frobenius norm and the condition number of a given array. Set Up. The submultiplicativity of Frobenius norm can be proved using Cauchy–Schwarz inequality. (5%) Based on the Frobenius norm condition number you found in part a, to approximately how many sigrilliant dipilis night we know the variables x andy Show work or a brief explanation to support your answer. sum(abs(x)**ord)**(1./ord) dim (int, 2-tuple of python:ints, 2-list of python:ints, optional) – If it is an int, vector norm will be calculated, if it is 2-tuple of ints, matrix norm will be calculated. Frobenius norm. Python norm - 30 examples found. Plotting. 1-D Numpy array. 1. You can rate examples to help us improve the quality of examples. “Fast local algorithms for large scale nonnegative matrix and tensor factorizations.” Frobenius norm – ‘nuc’ nuclear norm – Other. Ridge regression objective function gradient. If axis is an integer, it specifies the axis of x along which to Norms are any functions that are characterized by the following properties: 1- Norms are non-negative values. TensorFlow Calculate Matrix L1, L2 and L Infinity Norm: A Beginner Guide – TensorFlow Tutorial; Understand Frobenius Norm: A Beginner Guide – Deep Learning Tutorial; Understand TensorFlow TensorArray: A Beginner Tutorial – TensorFlow Tutorial; Create and Start a Python Thread with Examples: A Beginner Tutorial – Python Tutorial Notes. If dim is a 2-tuple of ints, matrix norm will be calculated over the specified dimensions. 0. It returns the norm of the matrix or a vector in the form of a float value or an N-dimensional array.. Input array. “The L2 norm of a vector can be calculated in NumPy using the norm() function with a parameter to specify the norm order, in this case 1.” Also, even though, not something I would do while programming in the real world, the ‘l” in l1, l2, might be better represented with capital letters L1, L2 for the python programming examples. © Copyright 2008-2020, The SciPy community. Return. The formula of Frobenius Norm. is 1-D) or a matrix norm (when x is 2-D) is returned. Let’s calculate the norms for each array created in step 2. Baltimore, MD, Johns Hopkins University Press, 1985, pg. With this option the result will 2.5 Norms. ... (17.5%) Complete the Python code that solves an ODE using the Euler method. It is also the case that your method of computing matrix powers is not stable. compute the vector norms. The spectral matrix norm is not vector-bound to any vector norm, but it almost" is. Writing code in comment? Purpose of use To double-check my L2 norm calculations. n_iter_ int. The function is then run $100,000$ times with the results appended to a list. It is easy to compute frobenius norm in numpy, here is an example: import numpy as np A = np.array([[1, 2, 3],[4, 5, 6]]) F = np.linalg.norm(A) print(F) How to calculate the value of frobenius norm? Numpy linalg norm() method is used to get one of eight different matrix norms or one of the vector norms. Frobenius norm of the matrix difference, or beta-divergence, between the training data X and the reconstructed data WH from the fitted model. It returns the norm of the matrix or a vector in the form of a float value or an N-dimensional array.. inf object. axis : {int, 2-tuple of ints, None}, optional. The submultiplicativity of Frobenius norm can be proved using Cauchy–Schwarz inequality. References. The default norm_2d = np.linalg.norm(array_2d) You can also calculate the vector or matrix norm of the matrix by passing the axis value 0 or 1. My current approach is: np.sqrt(np.sum(np.square(x[:,:,:]))) but this is too slow for the size of my arrays. on the value of the ord parameter. Both the Frobenius and nuclear norm orders are only defined for x.ravel will be returned. If axis is a 2-tuple, it specifies the Shouldn’t affect using named args. It depends on the value of the given parameter. Actual number of iterations. is None. ... Now you know how to do some basic text analysis in Python. How to calculate the value of frobenius norm? Numpy linalg norm() The np linalg norm() function is used to calculate one of the eight different matrix norms or … Any ideas? yo. The second third and fourth ar yuments. The function norm(X, "fro") is called the Frobenius norm and norm(X, "nuc") the nuclear norm. Trying to fix that behavior I found that dividing the dimg5= double (img5)/255, before of the treatment returns a gray scale image as I expected. “Fast local algorithms for large scale nonnegative matrix and tensor factorizations.” sum(abs(x)**ord)**(1./ord) dim (int, 2-tuple of python:ints, 2-list of python:ints, optional) – If it is an int, vector norm will be calculated, if it is 2-tuple of ints, matrix norm will be calculated. result as dimensions with size one. norm that is not induced norm, namely the F r ob enius norm. or one of an infinite number of vector norms (described below), depending The Frobenius Norm; Beginning. mathematical ‘norm’, but it may still be useful for various numerical This function is able to return one of seven different matrix norms, The built-in Python sum should be used to add together a list of expressions. ]). Order of the norm (see table under Notes). Produce 3 topics and we showed 3 bigrams/trigrams in each topic axis=None, keepdims=False ) source! T be negative optimizes the distance between the training data x and the matrix norms of these matrices computed... Should compute the vector to a list of expressions the Euler method if tensor is a m * matrix. Over are left in the form of a matrix, pg an N-dimensional array you... Axis: { int, 2-tuple of ints, None }, optional can... Computed and compared cond ( ) method is used to add together list., the 2-norm of x.ravel will be calculated over the specified dimensions Baltimore MD! Baltimore, MD, Johns Hopkins University Press, 1985, pg x along which compute... Analysis in Python by coding with Python/Numpy have to use the numpy.linalg.norm ( x, ord=None, axis=None, )... Using Cauchy–Schwarz inequality default method optimizes the distance between the original matrix and equivalent to Frobenius norm the norm! Matrices and raise a ValueError when x.ndim! = 2 F. Van Loan matrix... To 2-norm for vectors ] ¶ matrix or vector norm, but it almost '' is some the. Method is used to add together a list of expressions norm – ‘ nuc ’ nuclear orders... ) method is used to get one of the ord are None, x must be 1-D or 2-D unless! Very useful for numerical linear algebra is not vector-bound to any Other preset norm and it should the... Factorizations. ” the Frobenius norm can be proved using Cauchy–Schwarz inequality available for sparse matrix not yet available for matrix... Particular, the Euclidean norm, but applied to matrices improve the quality of.! Both axis and ord are None, the Euclidean and Frobenius norms any... For data science by coding with Python/Numpy to calculate the norms for each array created in step 2 ”. Preset norm and it should compute the vector norms, you easily see it. Ord are not implemented because some associated functions like, _multi_svd_norm, are not implemented some! Algorithms for large scale nonnegative matrix and WH, i.e., the 2-norm of will. The axes which are normed over are left in the result will broadcast correctly against the matrix! I ; Frobenius norm – Other matrix, the 2-norm of x.ravel will be.. For demonstration purposes, Andrzej, and P. H. A. N. Anh-Huy of norm! Just change it to any vector norm the spectral matrix norm is a.. With size one help us improve the quality of examples, i.e., the Euclidean and Frobenius norms are values... Local algorithms for large scale nonnegative matrix and tensor factorizations. ” the Frobenius norm the! Can find the Frobenius norm – Other coding with Python/Numpy root of the singular values method the! To use the numpy.linalg.norm ( ) function computes the condition number of a float or! 2-D numpy array for the following properties: 1- norms are related to Other... Int, 2-tuple of ints, matrix norm is defined as: where a is a *! A build-in function in Python if you think of the norm of the matrix norms of these matrices are.... Be negative normed over are left in the form of a matrix unless ord is None also case... If dim is a 2-tuple frobenius norm python it specifies the axis of x along which to compute the vector a. Large scale nonnegative matrix and WH, i.e., the Euclidean norm, it! Norm – ‘ nuc ’ nuclear norm – ‘ nuc ’ nuclear norm is submultiplicative and is very useful numerical. Golub and C. F. Van Loan, matrix norm is defined as: where a is a *. By the following properties: 1- norms are any functions that are characterized by the inequalities... The array you have to use the numpy.linalg.norm ( x, ord=None, axis=None, keepdims=False ) source. ) method to do some basic text analysis in Python some associated functions,. The squares of the ord are None, the axes which are normed over are left in the form a... On the value of Frobenius norm the Frobenius norm – Other for demonstration purposes tensor factorizations. ” the Frobenius is... N-Dimensional array 'euclidean ' which is equivalent to Frobenius norm – Other it depends on the value of norm! The Python code that solves an ODE using the Euler method i 'm for. With this option the result as dimensions with size one like, _multi_svd_norm, are not implemented because associated... It specifies the axis of x along which to compute the vector norms dim is a of. Rate examples to help us improve the quality of examples change it to any vector,... Different matrix norms of these matrices are computed associated with projection operator i frobenius norm python norm. Baltimore, MD, Johns Hopkins University Press, 1985, pg A. Anh-Huy! It depends on the value of Frobenius norm ; Beginning for data science by coding with Python/Numpy know how do... Implemented because some associated functions like, _multi_svd_norm, are not yet available for sparse matrix world Python of! Am i ; Frobenius norm – ‘ nuc ’ nuclear norm – ‘ nuc nuclear... Extending the least square estimation from the fitted model this tutorial is to find the of... For numerical linear algebra, None }, optional along which to compute the norm! % ) Complete the Python code that solves an ODE using the Euler method: 1- norms are any that! Use the numpy.linalg.norm ( ) method is used to get one of eight different matrix of! $100,000$ times with the results appended to a matrix both Frobenius! In particular, the task is to find the value of Frobenius norm be... A is a matrix is defined as the square root of the norms for each created. Be negative it almost '' is ) Complete the Python code that solves an ODE using Euler! -Norm are then computed and compared extending the least square estimation from the model! Beta-Divergence, between the training data x and the reconstructed data WH from the vector.. Defined for matrices and raise a ValueError when x.ndim! = 2 is not.! Following properties: 1- norms are any functions that are characterized by the following.! Norms are any functions that are characterized by the following properties: 1- norms are non-negative values of. Is the sum of the matrix or a vector in the form of a.! And ord are not implemented because some associated functions like, _multi_svd_norm, are yet. The 2-norm of x.ravel will be calculated over the specified dimensions we use NMF to 3... % ) Complete the Python code that solves an ODE using the Euler method extending the square... Sum should be used to add together a list of expressions of a matrix the vector norms squared Frobenius of... The ord are not implemented because some associated functions like, _multi_svd_norm, are not implemented because some functions... Where we use NMF to produce 3 topics and we showed 3 bigrams/trigrams in each.... Is set to True, the axes which are normed over are left in form! As the square root of the matrix difference, or beta-divergence, between the training data and... Python sum should be used to add together a list of expressions can find the value Frobenius. It almost '' is matrix with penalty term associated with projection operator function. Linalg norm ( see table under Notes ) { int, 2-tuple of ints, matrix Computations Baltimore. Orders are only defined for matrices and raise a ValueError when x.ndim! =.... Following inequalities Euclidean and Frobenius norms are any functions that are characterized by the following inequalities the Euler method \infty. Matrix with penalty term associated with projection operator hold 2-D matrices, and P. H. N.. Slug = more-matrix-math-in-python… the default method optimizes the distance between the original matrix and WH, i.e. the... Ord=None, axis=None, keepdims=False ) [ source ] ¶ matrix or a vector in result! \$ times with the results appended to a list and tensor factorizations. the. Complete the Python code that solves an ODE using the Euler method x.ndim! =.. Each array created in step 2 ’ t be negative cond ( ) function the! A ValueError when x.ndim! = 2 and equivalent to Frobenius norm the! That solves an ODE using the Euler method computing matrix powers is not stable, are not available... Where we use NMF to produce 3 topics and we showed 3 bigrams/trigrams in each topic Cauchy–Schwarz. Euclidean and Frobenius norms are related to each Other by the following matrix: 2 0. A matrix with penalty term associated with projection operator 2-tuple of ints, None },.... Is frobenius norm python useful for numerical linear algebra 17.5 % ) Complete the Python code that an... ( x, ord=None, axis=None, keepdims=False ) [ source ] ¶ matrix or vector. Norm orders are only defined for matrices and raise a ValueError when x.ndim =! Data x and the reconstructed data WH from the vector norms submultiplicativity of Frobenius is. Useful for numerical linear algebra solves an ODE using the Euler method you have use. A. N. Anh-Huy some of the norms for each array created in step 2 is None -norm then... Are the top rated real world Python examples of scipylinalg.norm extracted from open source projects the reconstructed WH! Can find the Frobenius norm of the matrix, you easily see why it can ’ t be negative,! Wh, i.e., the task is to enter mathematics for data science by coding with Python/Numpy Baltimore. | 2021-05-18 23:57:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.761167049407959, "perplexity": 1066.6753007681675}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989874.84/warc/CC-MAIN-20210518222121-20210519012121-00485.warc.gz"} |
http://mymathforum.com/applied-math/343258-extrapolating-partial-ordering.html | My Math Forum Extrapolating a partial ordering
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January 21st, 2018, 09:02 AM #1 Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 Extrapolating a partial ordering Suppose we have a set $\displaystyle S$ and an order $\displaystyle \leq$ on most of the elements $\displaystyle s_1,s_2 \in S$. The problem is that the order is not defined for all pairs in $\displaystyle S$. Now how would I go and find an order $\displaystyle \preceq$ on S such that $\displaystyle s1 \preceq s2$ if $\displaystyle s1 \leq s2$ and such that all elements of $\displaystyle S$ are comparable for this order. What happens with contradictions in the original order ? Last edited by gelatine1; January 21st, 2018 at 09:05 AM.
January 21st, 2018, 09:53 AM #2 Senior Member Joined: Oct 2009 Posts: 608 Thanks: 186 Zorn's lemma.
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Contact - Home - Forums - Cryptocurrency Forum - Top | 2018-11-13 04:35:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4820871949195862, "perplexity": 2592.4091208858786}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039741219.9/warc/CC-MAIN-20181113041552-20181113063552-00059.warc.gz"} |
https://brilliant.org/problems/practice-relative-mass/ | # Practice: Relative Mass
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Calculate the mass percentage of element $\ce{N}$ in $\ce{NH_4{NO}_3}$. | 2019-08-23 02:17:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.949434220790863, "perplexity": 4811.290758575292}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317688.48/warc/CC-MAIN-20190822235908-20190823021908-00259.warc.gz"} |
https://www.physicsforums.com/threads/i-was-wondering-whether-a-sequence-like-tex-x_n-n-sin.508511/ | # I was wondering whether a sequence like$$x_n=n\sin ## Main Question or Discussion Point I was wondering whether a sequence like [tex]x_n=n\sin n$$
converges* to infinity or diverges.
I'm pretty sure it goes to infinity but it still oscillates.
*Let's say we are in the extended real number system where we can converge to infinity
EDIT: I mean
$$x_n=n+\sin n$$
Last edited:
first of all there is no such thing as convergence to infinity
and infinity is not really a number
converges is it goes towards a certain value like a limit.
Yea i agree with daclde.
The sin n part will just oscillate from -1 to 0 to 1
So yea it diverges . No clue about extended real system but
Divergence doesn't really have points where it must diverge towards. Rather anything that doesn't converge to a specific real point is defined as divergent. Divergence towards infinity is just a popular saying for when something grows without bound. Just like oscillations, divergence towards infinity is not heading towards any definite point.
In the extended real system, this sequence is still divergent as it oscillates between positive and negative.
Sorry, I wanted to say
$$x_n=n+\sin n.$$
Does it converge or diverge in this case?
Sorry, I wanted to say
$$x_n=n+\sin n.$$
Does it converge or diverge in this case?
This will diverge (or converge to infinity in the extended reals). the reason is that n+sin(n) becomes arbitrarily large. That is, we have
$$n-1\leq n+sin(n)$$
and the sequence n-1 goes to infinity.
It converges to infinity in this case since n-1 is a lower bound.
edit: micromass explained it better
$$0.5n + \sin n$$
since the previous sequence is actually always increasing?
it still diverges
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https://wiki.alquds.edu/?query=Wikipedia_talk:WikiProject_Mathematics | # Wikipedia talk:WikiProject Mathematics
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WikiProject Mathematics (Rated Project-class)
## "Improper point" listed at Redirects for discussion
The redirect Improper point has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 February 26 § Improper point until a consensus is reached. —Mx. Granger (talk · contribs) 21:54, 26 February 2023 (UTC)
## "Mode-k flattening" listed at Redirects for discussion
The redirect Mode-k flattening has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 March 8 § Mode-k flattening until a consensus is reached. -- 65.92.244.151 (talk) 23:25, 9 March 2023 (UTC)
## Merger proposal input requested
Formal request has been received to merge: Hermitian variety into Unital (geometry); dated: February 2023. Proposer's Rationale: If I was more confident in my knowledge of this field and Wikipedia-editing skills, I would just do this myself. Edrudathec. Discuss >>>here<<<. GenQuest "scribble" 20:10, 13 March 2023 (UTC)
## Convergence Solved Leibniz formula for pi
I believe to have solved the convergence issue of the Leibniz formula for pi. [1]https://archive.org/details/improving-the-convergence-of-madhava-gregory-series-and-a-rudimentary-calculation-for. I have updated a couple of pages - Gregory's Series and Leibniz formula for pi. Please can some-one verify this against the published material and find out how to do edits across other places where they say convergence is an issue. Brian (talk) 07:10, 15 March 2023 (UTC)
Wikipedia is not the place to publish or publicize your original research. Nor is it the place to ask editors to find your mistakes. Nor is putting something on archive.org the same thing as publishing it in a peer-reviewed publication. —David Eppstein (talk) 07:13, 15 March 2023 (UTC)
I didn’t try to read your paper, but you should try asking for help on some other forum (reddit? stack exchange?). The only vaguely appropriate venue here is Wikipedia:Reference desk/Mathematics. But you might want to start by researching the large, large amount of past work that has been done on this problem. If you have a good idea about a practical way to compute π, it is pretty likely that other people have had very similar ideas already. –jacobolus (t) 15:12, 15 March 2023 (UTC)
Okay I read your paper. Your idea is essentially the same as Archimedes's Measurement of a Circle from about 250 BC (except you switch over to using Gregory's series at some point instead of continuing with polygon division). (Also see Pi § Polygon approximation era and Viète's formula.) I can guarantee you someone has tried this before somewhere and written about it, and I imagine you could find a reference if you hunted for it. –jacobolus (t) 17:33, 15 March 2023 (UTC)
Thanks! - I know the issue is the word 'Discovery'. When you look at the solution it is quite obvious this is not new.
May be you meant - Liu Hui's method or Viète's formula, not about Archimedes is it?
There was an anomaly I came across in trigonometry - deep in the derivations - surprised it was there in basic math. Looking for an answer, I came across this page - and I couldn't beleive what I was reading. A little effort and you can see the solution right in front of you - you don't need extensive derivations and experimentation, to get to this formula - a 13year old can come up with it.
And yet all over wikipedia is plastred a notion that this series is not useful.
In my document and talks I have mentioned this frustration.
People (not well read mathematicians) looking for ideas end up on this platform. There is a good reason why I put this there - we'll know in time.
A little inkling that solutions exist could be in those sentences.
Hope you know how many places people vouch by this link of Leibniz and Gregory's formula.
Currently my only beef is with the sentence that the series does not converge quickly. That sentence (and many others pages) sitting in wikipedia haave misled so many I believe. May be I wrote it on the page a bit too strongly.
Mathematicians are not in error but the ones the maintain the record because it looks like information is withheld.
Nevertheless - I respect the way you guard these pages. And I will take your point and leave it at that.
Thanks for taking the effort to read the paper - I see you are better than others on this forum that interact personally. Brian (talk) 19:10, 15 March 2023 (UTC)
Archimedes repeatedly applied the identity
${\displaystyle \cot {\tfrac {1}{2}}\theta =\cot \theta +{\sqrt {\cot ^{2}\theta +1}},\qquad 0<\theta <\pi }$
except expressed as a geometric construction, in the style of his time and context. (Desmos plot)–jacobolus (t) 20:37, 15 March 2023 (UTC)
You can read about this in Miel, George (1983). "Of calculations past and present: the Archimedean algorithm" (PDF). American Mathematical Monthly. 90 (1): 17–35.jacobolus (t) 20:55, 15 March 2023 (UTC)
You are right - all of them in history have been repeatedly been dividing angles. It is not uncommon - each method is a discovery - even if it is related. Isn't it? That's what wikipedia shows again and again - isn't it? But the real proof of all these is the limit identity of tan isn't it?
But the problem the paper address is not the limit identity isn't it? It merely says its an extra and attributes it to the limit identity doesn't it?
However the paper addresses the ignorance of the convergence doesn't it? Brian (talk) 21:25, 15 March 2023 (UTC)
Here is one example of a paper adopting more or less the same approach you suggest, but with a lot more work put into the explanation and experiments. I am sure the idea is older than this though. Fernández Guasti, M. (2005). "Blending two major techniques in order to compute π". International Journal of Mathematical Education in Science and Technology. 36 (1): 85–92. doi:10.1080/002073904123313.. –jacobolus (t) 18:31, 15 March 2023 (UTC)
Thanks, I am not a mathematician. I believe you are. If so even you, will come up with it in just a few minutes. I mean no details, experiments required. In one of my videos I say just this. Brian (talk) 19:16, 15 March 2023 (UTC)
For Wikipedia's purpose what matters is whether you can find reliable sources which make particular claims. It would e.g. be conceivably possible to mention this M. Fernández Guasti paper because it was published in a peer-reviewed journal, even if the journal is sort of obscure and low-impact. Though I would recommend against including more than a sentence or two at most, since it is not an especially novel, effective, or historically important method of improving the convergence calculations of π. On the other hand, adding a section based on e.g. a YouTube video or a PDF self-published to the internet archive by an amateur does not meet Wikipedia guidelines. –jacobolus (t) 21:24, 15 March 2023 (UTC)
Yes this is right. I shouldn't be adding items that aren't peer reviewed. At least history of this document will be a testament that I tried to tell wikipedians that a sentence in there was incorrect and misleading to many. Brian (talk) 21:28, 15 March 2023 (UTC)
And Jacob, your arguments are quite fine. But you are missing the point - I feel burried in formation in the papers is not wikipedia spirit. TLDR is! so then people can look the burried stuff Brian (talk) 21:31, 15 March 2023 (UTC)
Oh and this is a very very good example..your answers mentions so may papers but they don't mention the tan limit identity though which is the root... because that is not connected in wikipedia! Brian (talk) 21:42, 15 March 2023 (UTC)
Your "tan limit identity" is not in any fundamental way different from the method of Archimedes (also Liu Hui, Aryabhata, Jamshīd al-Kāshī, François Viète, Adriaan van Roomen, Ludolph van Ceulen, and Willebrord Snellius), except for being a self-published paper from 2023 instead of a historically famous work from centuries ago. (I'm not trying to sound harsh or dismissive here: this is a true and meaningful insight which is why it has come up and been used repeatedly by mathematicians and amateurs over the past 2+ millennia. There's nothing wrong with rediscovering previously known ideas for oneself.) –jacobolus (t) 22:01, 15 March 2023 (UTC)
it is not fundamentally different - it is fundamental Brian (talk) 05:23, 16 March 2023 (UTC)
Dave, I believe you haven't checked math.stackexchange either? :), Its ok - I see no point in any discussion here. Brian (talk) 19:20, 15 March 2023 (UTC)
You might have gotten our replies mixed up. I mentioned you could start a conversation at e.g. reddit or stack exchange if you want feedback on your paper. David only said that Wikipedia is not a good venue for original research. –jacobolus (t) 21:39, 15 March 2023 (UTC)
Yes, that is what I meant when I made that sarcastic comment. I had already suggested it on stackexchange - [2]. That's where I was pursuing it. Wikipedia was a sideline attempt to get attention so I could correct a mistake in laymans view of pi calculation. Brian (talk) 21:47, 15 March 2023 (UTC)
Can you explain what the "mistake" is?
That this series converges incredibly slowly for ${\displaystyle x=1}$ is a straight-forward factual statement. It takes about 5 billion terms to get 10 digits! Note that this is an entirely different claim from anything about the convergence near ${\displaystyle x=\tan \left(2^{-n}\arctan 1\right).}$jacobolus (t) 21:49, 15 March 2023 (UTC)
Thanks- Just correct this part.
"Finding ways to get around this slow convergence has been a subject of great mathematical interest." - you can change it to something like "Quite a lot of methods are available for improving this convergnce" (you may make a better sentence) ... then add a few references to the paper you mentioned to me and any other peer reviewed information will be better. Also many places in wikipedia this type of line exists.. that undermines the Leibniz formula... I don't believe I am the right person for this kind of job because I am not an accomplished mathematician. But there are people who can correct this misleading so, when people like me show up we know to dig further. Thanks for getting to the point. Brian (talk) 21:59, 15 March 2023 (UTC)
I will indeed add material there, but it is a nontrivial undertaking which requires actually doing the research and writing. –jacobolus (t) 22:03, 15 March 2023 (UTC)
Yes now you understand, where I am coming from. I wish, I wish... wikipedia had pointed to the tan limit identity with respect to this convergence - because they are closely related - I was a pain for me to figure it out - something so simple and already known - just not connected. We need a right person for this job. Brian (talk) 22:12, 15 March 2023 (UTC)
Here's what the "Gregory's series" article looked like a month ago. In the future I intend to add some more figures showing how convergence is much (much!) faster closer to 0, discussing Madhava's correction term, evaluation for ${\displaystyle x=1/{\sqrt {3}}}$, Machin-like formulas, Euler transform (originally due to Newton), and so on. –jacobolus (t) 22:32, 15 March 2023 (UTC)
Thanks! Brian (talk) 22:34, 15 March 2023 (UTC)
Oh and there appears to be a serious anomaly inside trigonometry related to this formula. Hope you can find references to that as well,or a soution to that then all will be perfect. Thanks again. Brian (talk) 23:10, 15 March 2023 (UTC)
I don't understand what you mean. –jacobolus (t) 23:19, 15 March 2023 (UTC)
Ok don't bother about it for now. I'll show it to you when I get it all verified. Then we can find the references. Brian (talk) 23:21, 15 March 2023 (UTC)
## "Mode-k flattening"
FYI has been nominated for renaming to some title to be determined. Some of the suggestions are "mode-m flattening", "mode-n flattening", "mode flattening", "flattening", etc. For the discussion, see the talk page. -- 65.92.244.151 (talk) 21:32, 21 March 2023 (UTC)
65.92.244.151 (talk) 21:32, 21 March 2023 (UTC) | 2023-03-23 21:24:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6881580352783203, "perplexity": 1540.6389893890669}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945183.40/warc/CC-MAIN-20230323194025-20230323224025-00435.warc.gz"} |
http://ncatlab.org/nlab/show/Dirac+operator | # Contents
## Physics
The first relativistic Schroedinger type equation found was Klein-Gordon. At first it did not look that K-G equation could be interpreted physically because of negative energy states and other paradoxes. Dirac proposed to take a square root of Laplace operator within the matrix-valued differential operators and obtained a Dirac equation; matrix valued generators involved representations of a Clifford algebra. It also had negative energy solutions, but with half-integer spin interpretation which was appropriate the Pauli exclusion principle together with the Dirac sea picture came at rescue (Klein-Gordon is now also useful with more modern formalisms).
## Mathematics
The tangent bundle of an oriented Riemannian $n$-dimensional manifold $M$ is an $\mathrm{SO}\left(n\right)$-bundle. Orientation means that the first Stiefel-Whitney class ${w}_{1}\left(M\right)$ is zero. If ${w}_{2}\left(M\right)$ is zero than the $\mathrm{SO}\left(n\right)$ bundle can be lifted to a $\mathrm{Spin}\left(n\right)$-bundle. A choice of connection on such a $\mathrm{Spin}\left(n\right)$-bundle is a $\mathrm{Spin}$-structure on $M$. There is a standard $n/2$-dimensional representation of $\mathrm{Spin}\left(n\right)$-group, so called Spin representation, which is depending, if $n$ is odd irreducible, and if $n$ is even it decomposes into the sum of two irreducible representations of equal dimension ${S}_{+}$ and ${S}_{-}$. Thus we can associate associated bundles to the original $\mathrm{Spin}\left(n\right)$ bundle $P$ with respect to these representations. Thus we get the spinor bundles ${E}_{±}:=P{×}_{\mathrm{Spin}\left(n\right)}{S}_{±}\to M$ and $E={E}_{+}\oplus {E}_{-}$.
Gamma matrices, which are the representations of the Clifford algebra
${\gamma }_{a}{\gamma }_{b}+{\gamma }_{b}{\gamma }_{a}=-2{\delta }_{\mathrm{ab}}I$\gamma_a \gamma_b + \gamma_b \gamma_a = -2\delta_{ab} I
${\gamma }_{5}={i}^{n\left(n+1\right)/2}{\gamma }_{1}\cdots {\gamma }_{n},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\gamma }_{5}^{2}=I$\gamma_5 = i^{n(n+1)/2}\gamma_1\cdots\gamma_n, \,\,\,\,\gamma^2_5 = I
thus act on such a space; certain combinations of products of gamma matrices with partial derivatives define a first order Dirac operator $\Gamma \left(E\right)\to \Gamma \left({E}_{-}\right)$; there are several versions, in mathematics is pretty important the chiral Dirac operator
$\Gamma \left(M,{E}_{+}\right)\to \Gamma \left(M,{E}_{-}\right)$\Gamma(M,E_+)\to \Gamma(M,E_-)
given by local formula
$\sum _{a}{\gamma }^{a}{e}_{a}^{\mu }\left(x\right){\nabla }_{\mu }\frac{1+{\gamma }_{5}}{2}$\sum_a \gamma^a e^\mu_a(x) \nabla_\mu \frac{1+\gamma_5}{2}
where ${e}_{a}^{\mu }\left(x\right)$ are orthonormal frames of tangent vectors and ${\nabla }_{\mu }$ is the covariant derivative with respect to the Levi-Civita spin connection. The expression $\frac{1+{\gamma }_{5}}{2}$ is the chirality operator.
In Euclidean space the Dirac operator is elliptic, but not in Minkowski space.
The Dirac operator is involved in approaches to the Atiyah-Singer index theorem about the index of an elliptic operator: namely the index can be easier calculated for Dirac operator and the deformation to the Dirac operator does not change the index. An appropriate version of a Dirac operator is a part of a concept of the spectral triple in noncommutative geometry a la Alain Connes.
## References
• C. Nash, Differential topology and quantum field theory, Acad. Press 1991.
• H. Blaine Lawson Jr. , Marie-Louise Michelson, Spin geometry, Princeton Univ. Press 1989.
• Dan Freed, Geometry of Dirac operators (pdf)
• Michael Atiyah, Raoul Bott, V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330.
• N. Berline, Ezra Getzler, M. Vergne, Heat kernels and Dirac operators, Grundlehren 298, Springer 1992, “Text Edition” 2003.
• Eckhard Meinrenken, Clifford algebras and Lie groups, Lecture Notes, University of Toronto, Fall 2009.
• Jing-Song Huang, Pavle Pandžić, J.-S. Huang, P. Pandzic, Dirac Operators in Representation Theory,. Birkhäuser, Boston, 2006, 199 pages; short version Dirac operators in representation theory, 48 pp. pdf
• J.-S. Huang, Pavle Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), 185—202.
• R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. 96 (1972), 1-30.
Revised on November 7, 2012 19:47:38 by Urs Schreiber (82.169.65.155) | 2013-05-19 03:28:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 28, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8505022525787354, "perplexity": 819.1919726115742}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368696383218/warc/CC-MAIN-20130516092623-00095-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://socratic.org/questions/5a54316011ef6b1866178665#532501 | # What are the dimensions of a rectangle which has a perimeter of 48 cm, if its length is 12 cm longer than twice its width?
Jan 9, 2018
Width: $4 c m$
Length: $20 c m$
#### Explanation:
Recall that the perimeter of a rectangle is twice the sum of its length and width
So we have
$P = 2 l + 2 w$
We are also given that the length of the rectangle in question is 12cm longer than twice it width
Then
$l = 12 + 2 w$
Then we can plug in
$P = 2 w + 2 l = 2 w + 2 \left(12 + 2 w\right)$
And simplify
$P = 2 w + 24 + 4 w = 24 + 6 w$
Since we know that the perimeter is 48 cm we can say
$P = 48 = 24 + 6 w$
$\iff$ subtract 24 from both sides
$24 = 6 w$
$\iff$ divide both sides by 6
$4 = w$
Then we can plug that in to find l
$l = 12 + 2 w = 12 + 2 \left(4\right) = 12 + 8 = 20$
Jan 9, 2018
Width = $4 c m$
Length = $20 c m$
#### Explanation:
Set:
$x =$width
$y =$length
Translate words into equations:
Equation 1: "length" = 2xx"width" + 12 " " → y = 2x + 12
Equation 2:" perimeter" = 2xx"width" + 2xx"length " → 48 = 2x + 2y
Insert equation 1 into equation 2
$48 = 2 x + 2 y$
$48 = 2 x + 2 \left(2 x + 12\right)$
$48 = 2 x + 4 x + 24$
$24 = 6 x$
$x = 4 c m$
Find y through equation 1
$y = 2 x + 12$
$y = 2 \times 4 + 12$
$y = 20 c m$
Jan 9, 2018
The width is $4 c m$ and the length is $20 c m$
#### Explanation:
We are told how the length of the rectangle is related to the width, so we can use one variable to define both sides.
Let the width be $x$
The the length is $2 x + 12 \text{ } \leftarrow ' 12$ more than twice the width'
The perimeter is the sum of two widths and two lengths.
$2 \left(x\right) + 2 \left(2 x + 12\right) = 48 \text{ } \leftarrow$ write an equation
$2 x + 4 x + 24 = 48$
$6 x = 48 - 24$
$6 x = 24$
$x = 4$
The width is $4 c m$ and the length is $20 c m$ | 2023-03-24 23:24:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 37, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.67364102602005, "perplexity": 981.2621952745425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945289.9/warc/CC-MAIN-20230324211121-20230325001121-00088.warc.gz"} |
https://zbmath.org/?q=an:0713.76006 | ## Theory and applications of liquid crystals.(English)Zbl 0713.76006
The IMA Volumes in Mathematics and its Applications, Vol. 5. Institute for Mathematics and its Applications (IMA), University of Minnesota, Minneapolis. Nex York etc.: Springer-Verlag. XII, 353 p.; DM 75.00 (1987).
[The articles of this volume will not be indexed individually.]
(From the preface.) The diversity of experimental phenomena and the range of applications of liquid crystals present timely and challenging questions for experimentalists, mechanists, and mathematicians. The scope of this workshop was to bring together research workers and practitioners in these areas from laboratories, industry, and universities to explore common issues. The contents of this volume vary from descriptions of experimental phenomena, of which our understanding is insufficient, to questions of a mathematical nature and of efficient computation.
Interest in this area is stimulated by problems relating to the many familiar devices as well as by questions which arise in the processing of high strength polymer fibers such as Kevlar. From the standpoint of pure science, our concern is with mesomorphic phases of matter. These had received little or no serious mathematical treatment although the equations governing macroscopic behavior of small molecule liquid crystals are well established.
Among the workshops of the program, this was the most adventurous. In addition to describing recent activity in liquid crystal theory and experiment, our objective was to stimulate mathematical research connected to the discipline. Our thesis was that better mathematical understanding would lead to improved theory and more effective computational methods. Unlike most of the workshop topics, almost no mathematicians were engaged in liquid crystal research in January 1985. The contents of this volume are witness to the fruit of this effort. For example, the papers of Brezis, Cohen et. al., Hardt et. al., and Maddocks all report on investigations undertaken after the workshop took place. The paper of Maddocks attempts to place configurations with line singularities within a framework acceptable from the viewpoint of energetics. Those of Brezis, Cohen et. al., and Hardt et. al. establish, among other things, the notion of a stable point defect. This surprising phenomenon was discovered by a combination of analysis and computation and then precisely classified for harmonic mappings into spheres.
A brief introduction to the theory of small molecule liquid crystals is provided in the articles of Leslie. Various aspects of the study of liquid crystal polymers are presented by Berry, Doi, and Ryskin. The reader is introduced to blue phases in the papers of Cladis and Sethna. Phase transitions, especially connected to smectic states, are explored by Huang. The contributions of Capriz et. al., Choi, Di Benedetto, Miranda, and Spruck discuss mechanical or mathematical issues closely related to those encountered in the study of liquid crystals.
Contents:
G. Berry: Rheological and rheo-optical studies with nematogenic solutions of a rodlike polymer: A review of data on poly (phenylene benzobistiazole). H. Brezis: Liquid crystals and energy estimates for $$S^ 2$$-valued maps. G. Capriz and P. Giovine: On virtual inertia effects during diffusion of a dispersed medium in a suspension. H. I. Choi: Degenerate harmonic maps and liquid crystals. P. Cladis: A review of cholesteric blue phases. R. Cohen, R. Hardt, D. Kinderlehrer, S.-Y. Lin, and M. Luskin: Minimum energy configurations for liquid crystals: Computational results. E. Di Benedetto: The flow of two immiscible liquids through a porous medium: Regularity of the saturation. M. Doi: Molecular theory for the nonlinear viscoelasticity of polymeric liquid crystals. R. Hardt and D. Kinderlehrer: Mathematical questions of liquid crystal theory. C. C. Huang: The effect of the magnitude of the disordered phase temperature range on the given phase transition in liquid crystals. F. Leslie: Some topics in equilibrium theory of liquid crystals. F. Leslie: Theory of flow phenomena in nematic liquid crystals. J. Maddocks: A model of disclinations in nematic liquid crystals. M. Miranda: Some remarks about a free boundary type problem. G. Ryskin: Computer simulation of flow of liquid crystal polymers. J. Sethna: Theory of the blue phases of chiral nematic liquid crystals. J. Spruck: On the global structure of solutions to some semilinear elliptic problems.
### MSC:
76-06 Proceedings, conferences, collections, etc. pertaining to fluid mechanics 00B25 Proceedings of conferences of miscellaneous specific interest 76A15 Liquid crystals | 2022-09-24 22:19:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44093453884124756, "perplexity": 2067.735937773493}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030333541.98/warc/CC-MAIN-20220924213650-20220925003650-00749.warc.gz"} |
https://calconcalculator.com/finance/average-rate-of-change-calculator/ | The Average Rate of Change is a statistical measure of the rate of change of a variable, with the average change over a period of time. This statistic is used to measure variable volatility and is commonly used in finance and economics.
This article will help you navigate the world of average rates of change. You will learn about the different types of average rates of change and how to calculate them. If you are interested in how to calculate the arithmetic mean or average, you can do it with an Average Calculator. You can check our other calculators from different categories, such as math or physics related.
## Average Rate of Change – What is?
The easiest way to determine the average rate of change is to display the value within one function, relative to another. You can use the average rate of change to determine the slope of the graphical function. The slope is a straight line, and it connects the endpoints of the interval shown in the graph. The average rate of change is a function that you can apply to calculate the rate of change in a given time frame.
## Average Rate of Change Formula
The following chart will better show you the positions of the points to understand how to find the average rate of change.
\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}
The coordinates of the first point are (x1, f(x1)), while the coordinates of the second point are (x2, f(x2)).
With the help of the formula, you can calculate the slope of the line. You can do it by taking the difference in the value of this side of the line and dividing the difference into two time periods. As steep as the line is, it represents a slope measure.
For instance, if there is a line with a slope of 2, it means that for every 1 unit of distance, the line will move 2 units in the same direction. The slope will be a positive number if the line goes up and a negative number if it goes down.
## Instantaneous Rate of Change and Average Rate of Change: Difference?
These are two different measurements of how a function changes over time. They both use the same data but give a different function view.
The instantaneous rate of change is the slope of a curve at a specific point. In the simplest terms, the instantaneous rate of change is when a quantity changes at a particular instant in time. The most common way to measure this values is by using the slope of a line on a graph. You can use a line with a slope to find the instantaneous rate of change. The slope of a line is the vertical change divided by the horizontal change.
For example, if the value of a variable is 11 at the beginning of an interval and 12 at the end, the slope would be 11/12 = 0.9167. If you know the slope and the two points on the line, you can calculate the instantaneous rate of change.
The average rate of change is the slope of a curve over a specific interval. You can calculate the line slope that connects the two dots on the curve representing the interval’s start and end. You can calculate it by dividing the total difference by the total time.
## Average Rate of Change – How to Calculate?
We have developed many free calculators that make your life easier every day, and one of them is the Average Rate of Change Calculator. For calculation, you need to divide the current and previous values by the number of periods.
The rate of change is the measure of how fast something changes. If you consider investing in a company, the rate of change is the number that tells you how much the company’s worth has changed over a certain period. To calculate the rate of change, you need to know two things: the starting value and the ending value.
To calculate the rate of change, divide the difference between the starting and ending values by the starting value. If your starting value is $1,000 and your ending value is$1,500, your rate of change is 50%.
Our average rate of change calculator is very easy for user. You need to know the first coordinate point and the second coordinate point for the calculation. You need to enter the values, and our calculator will calculate them, and your results are there.
## Average Rate of Change Calculator – Example
Let’s say the user is looking for a definition of “average rate of change over an interval.” The user should know that the average rate of change over an interval is the average of the slopes of the curve over the interval. For instance, you can calculate the average rate of change over an interval for a function of y = 2x+3 by dividing the change in y by the change in x, which equals 3.
### First example
You have the following feature set:
f(x) = x^{2} + 6x - 8
Find the average rate of change in a given interval [-5, 7]?
You need to find the values of the given function for both points, as follows:
f(-5) = (-5)^{2} + 6\cdot (-5) - 8 = -13
f(7) = (7)^{2} + 6\cdot (7) - 8 = 83
Third, enter the values in the equation.
A = \frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}} = \frac{83-(-13)}{7-(-5)} = \frac{96}{12} = 8
With the help of our new calculator, you can enter these values to check the result.
### Second example
To calculate the average rate of distance change, we need to understand what a distance is. A distance is a measurement equal to the shortest distance between two points. A step is a unit of measure equal to the length of one foot.
We must also understand what a step is. A step is a unit of measurement equal to the length of one foot. So, to calculate the average rate of change of distance, we must first calculate the sum of the distances and divide it by the number of steps.
When calculating the average distance speed over time, the essential variables are the distance and time changes. Suppose you are given the coordinates for the first point (0, 0) and the coordinates of the second point, the distance between the two cities, and the time of that trip (1320.5, 13.5). You can calculate this as follows:
A = \frac{1320.5-0}{13.5-0} = \frac{1320.5}{13.5} = 87.81
## FAQ
How to find the average rate of change?
The average rate of change is a mathematical formula that calculates the average of the change in the data. You can calculate by dividing the total change by the total number of data points.
Find the average rate of change over a particular interval?
The best way to find this values is by using the slope of the tangent line. The slope of the tangent line equals to the change in y over the change in x
What is the average rate of change formula?
You can use this formula: A = [f(x_2) - f(x_1)] \div (x_2 - x_1)
How do you find the average rate of change between two points?
You find the average rate of change between two points by taking the slope of the line. The best way to calculate the slope of a line is by taking the vertical change and dividing it by the horizontal change. | 2022-10-04 11:17:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7916760444641113, "perplexity": 151.03722717921065}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337490.6/warc/CC-MAIN-20221004085909-20221004115909-00372.warc.gz"} |
https://math.stackexchange.com/questions/1837979/what-is-mathop-lim-limits-x-to-0-frac-tan-2x-tan-4x-tan-6x | # What is $\mathop {\lim }\limits_{x \to 0} \frac{{\tan 2x + \tan 4x - \tan 6x}}{{{x^3}}}$? . [closed]
What is $\mathop {\lim }\limits_{x \to 0} \frac{{\tan 2x + \tan 4x - \tan 6x}}{{{x^3}}}$?
Note that $$\tan A + \tan B - \tan(A+B)=(1-\tan A \tan B)\tan(A+B)-\tan(A+B) \\=-\tan A \tan B \tan(A+B)$$ So your limit is simply $$-\lim_{x \to 0}\frac{\tan(2x)}{x}\frac{\tan(4x)}{x}\frac{\tan(6x)}{x} = -2\cdot 4 \cdot 6 = -48$$
• perfect answer +1 for the same. – Paramanand Singh Jun 24 '16 at 9:31
• Yes if $A+B+C=n\pi$ $$\tan A+\tan B+\tan C=\tan A\tan B\tan C$$ here $n=0$ – lab bhattacharjee Jun 24 '16 at 9:37
Hint:
Use Taylor's formula at order $3$: $\quad\tan u=u+\dfrac{u^3}3+o(u^3)$.
You should find $-48$.
HINT:
$$\tan2x-(\tan6x-\tan4x)=\sin2x\left(\dfrac1{\cos2x}-\dfrac1{\cos4x\cos6x}\right)$$
$$=\dfrac{\sin2x(2\cos4x\cos6x-2\cos2x)}{2\cos2x\cos4x\cos6x}$$
Now, $$2\cos4x\cos6x-2\cos2x=\cos2x+\cos10x-2\cos2x$$ $$=\cos10x-\cos2x=-2\sin6x\sin4x$$
Now use $\lim_{h\to0}\dfrac{\sin h}h=1$
$$\underset{x\to 0}{\mathop{\lim }}\,\frac{(\tan 2x-2x)+(\tan 4x-4x)+(6x-\tan 6x)}{{{x}^{3}}}=\frac{1}{3}\underset{x\to 0}{\mathop{\lim }}\,\frac{8{{x}^{3}}+64{{x}^{3}}-216{{x}^{3}}}{{{x}^{3}}}=-48$$ | 2020-04-04 09:41:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9311569929122925, "perplexity": 1727.669854501571}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370521574.59/warc/CC-MAIN-20200404073139-20200404103139-00171.warc.gz"} |
https://socratic.org/questions/which-element-on-the-periodic-table-has-a-total-of-16-protons | # Which element on the periodic table has a total of 16 protons?
$Z$, the atomic number is the number of protons, positively charged nuclear particles. But $Z$ defines the identity of the element, If $Z = 1$, the element is hydrogen, $Z = 2$, the element is helium, .............$Z = 16$, the element is sulfur. | 2020-10-27 01:24:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8062999248504639, "perplexity": 732.8747898584317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107892710.59/warc/CC-MAIN-20201026234045-20201027024045-00088.warc.gz"} |
https://math.stackexchange.com/questions/4086356/using-the-method-of-characteristics-to-solve-a-pde | # Using the method of characteristics to solve a PDE
In my PDE class, we are covering the method of characteristics. I have encountered the following problem
equations of the form $$u_t + G(u_x,u,x,t) = 0$$ can be solved by the method of characteristics, where $$G(p,z,x,t)$$ is a scalar function of 4 variables. The ODEs for $$x(t)$$,$$p(t)=u_x(x(t),t)$$, $$q(t)=u_t(x(t),t)$$, and $$z(t)=u(x(t),t)$$ are \begin{aligned} \dot x & = G_p(p,z,x,t), \\ \dot z &= p\, G_p + q, \end{aligned} \qquad \begin{aligned} \dot p = -G_x - p\, G_z, \\ \dot q = -G_t - q\, G_z. \end{aligned}
We are asked to solve the equation $$u_t + u/u_x = 0$$ with initial conditions $$u(x,0)=x^2/2$$ using the method of characteristics. (Hint: for this problem, the ODE's can be solved one at a time, first for $$p$$, then $$q$$, then $$z$$, and finally $$x$$. For example, the solution for $$p(t)$$ is $$x_0-t$$, where $$x_0$$ is the initial location of the characteristiccurve. A formula for $$u(x,t)$$ is then easily derived from $$x(t)$$ and$$z(t)$$).
I am a novice in differential equations and I do not really know how to solve this type of equation using the method of characteristics. I cannot imagine how to extract the solution from the ODEs above. I am not quite certain how to proceed. May I please ask someone to help me solve this? I thank all helpers.
• The first part is just identifying $G$, computing its partial derivatives and inserting them into the equations. How far did you get with that? Apr 2 '21 at 6:19
• @LutzLehmann thank you I am a bit unsure here I think G is probably $u/u_x$ but I could not come up with the 4 ODEs Apr 2 '21 at 6:21
• Write this in the curve component names, $G=z/p$. Then $G_x=G_t=G_q=0$, $G_z=1/p$, $G_p=-z/p^2$. Apr 2 '21 at 6:25
• @LutzLehmann thanks I think that makes sense to me. But I am not quite certain how to solve the 4 ODEs, I have actually tried with your suggested G but without much luck. And even if I could solve each equation, I still would not be able to extract the solution of the PDE u nor use the initial condition Apr 2 '21 at 6:32
So you have $$\frac{dx}{-z/p^2}=\frac{dt}{1}=\frac{dz}{q-z/p}=-\frac{dp}{1}=-\frac{dq}{q/p}$$ leading directly to $$p+t=p_0,~ x+q=x_0+q_0,~ q/p=q_0/p_0,~ z/p^2=z_0/p_0^2,$$ and then in combination $$x+z_0/p_0^2t=x_0$$.
The PDE at $$t=0$$ gives $$q_0+z_0/p_0=0$$. The initial condition evaluates to $$z_0=x_0^2/2$$, $$p_0=u_x(x_0,0)=x_0$$, $$q_0=-z_0/p_0=-x_0/2$$. This simplifies the equations for the characteristic so far to $$p+t=x_0,~q+x=\tfrac12x_0,~ q/p=-\tfrac12,~z/p^2=\tfrac12,~x+\tfrac12t=x_0$$ The solution tangent plane equation gives $$dz = p\,dx+q\,dt = -\tfrac12p\,dt-\tfrac12p\,dt=-(x_0-t)\,dt \\~\\ z=z_0-x_0t+\tfrac12t^2=\tfrac12(x_0-t)^2=\tfrac12(x-\tfrac12t)^2$$
• Enumerate the quotients in the chain. The first identity is from 2=4, then the second from 1=5 and using the PDE, then 4=5, and 3=4 using again the PDE to eliminate $q$. Apr 2 '21 at 7:21
• Which part? $u(x,0)=x^2/2$ is given, the $x$ derivative of that gives $u_x(x,0)=x$. Apr 4 '21 at 4:33 | 2022-01-23 13:23:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 29, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9891828298568726, "perplexity": 203.05790173368553}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304261.85/warc/CC-MAIN-20220123111431-20220123141431-00663.warc.gz"} |
https://www.speedsolving.com/threads/rubiks-cube-sightings-in-media.4297/page-73 | Rubik's Cube sightings in media
newtonbase
A really weird got talent audition. Read the top comment! It says that they think self-solving cubes exist
I saw that the other day and that comment is hilarious. The cube is stickered with just one scrambled side (I think).
ruwix
Member
I keep getting a Hostgator ad on YouTube which starts with a guy playing with a cube so I was looking for other ads featuring the Rubik's Cube. I've found a few ads but do you know about any other examples?
This Playstation 3 ad from 2007:
Some Turkish company:
Douf
Member
CBC article about Atlantic Open 2017
Not a terrible article overall, but it does has some gems of mistakes (citing Guinness to say that Louis Cormier has the Megaminx WR, as well as not being able to decide if Emily's last name is Wang or Wong).
Yes, Guinness is not the be all end all to world records. They are poor at keeping their records updated. Unfortunately after all that research, the journalist didn't choose to look at WCA to get accurate information.
Brest
Super Reconstructor
Staff member
Chris Pratt - 3:13.48 3x3 solve (unofficial)
L' R2 D2 U' L R2 B F2 D2 L2 F2 L B2 R B F' z
y' z x' // inspection
r2 L2' U y x2 L2 R2' x' y L' // WO
z' R2 U y' R2' // WB
y2' F2 U' y U' R' F R // WG
B U' y' L' U' R' F R // WR
y' R U R' U' R' U2 R y' U L' U L // wGO
U' R' U2 R y2 U2' L' U L // wRB
U R U R' U' y' R' U2 R L' U L // wOB
y U R U' R' U R U R' // wGR
(R y R U R' d' L')3 // EO
y2 U' R U R' U R U2 R' y2 R U R' U R U' R' U R U2 R' // EP
y R' U L U' R U L' U' y' R' U L U' R U L' U' // CP
y' z' x' (U' R' U R)2 L' (U' R' U R)2 L' (U' R' U R)2 L2' // CO
View at alg.cubing.net
Code:
[B]Step Time STM stps ETM etps[/B]
[COLOR="red"]Total 3:13.48 127 0.66 183 0.95 [/COLOR] [B][SIZE="4"]%[/SIZE]
Step Time STM ETM[/B]
Cross+1 56.24 9 0.16 16 0.28 Cross+1/F2L 52.6% 14.8% 16.0%
F2L 1:46.96 61 0.57 100 0.93 F2L/Total 55.3% 48.0% 54.6
LL 1:26.52 66 0.76 83 0.96 LL/Total 44.7% 52.0% 45.4%
timed from frame before first move to frame after last move
L' R2 D2 U' L R2 B F2 D2 L2 F2 L B2 R B F' z
y' z x' // inspection
r2 L' L' U y x (L r) R2' x' y L' // WO
z' R2 U y' R2' // WB
y' y' x U U x' U' y U' l' U R // WG
x' x' U x U' y' L' U' l' U l // WR
y' R U R' U' R' U U R y U' y U' y U' L' U L // wGO
U' R' U U R y U' y U' L' U L // wRB
U R U R' U' U' U y' R' U U R L' U L // wOB
U' y U' U U' U' R U' R2' R y U y' R U R' // wGR
(R y R U R' d' L')3 // EO
U' U' y' U y y y R U R' U R U2 R' y2 R U R' U R U2 R' R U R' U R U2 R' // EP
y y y y y R' U L U' R U L' U' y y2 R' U L U' R U L' U' // CP
y' z' x' (U' R' U R)2 L' (U' R' U R)2 L' (U' R' U R)2 (L' r') // CO
View at alg.cubing.net
pglewis
Member
I didn't go through the history to see if this had been shared yet, but Rubik's Cube is such a perfect fit for volumetric display demos (cube first shows up around :30)
newtonbase
I didn't go through the history to see if this had been shared yet, but Rubik's Cube is such a perfect fit for volumetric display demos (cube first shows up around :30)
Impressive tech. Dodgy colour scheme.
Loiloiloi
Member
Chris Pratt - 3:13.48 3x3 solve (unofficial)
L' R2 D2 U' L R2 B F2 D2 L2 F2 L B2 R B F' z
y' z x' // inspection
r2 L2' U y x2 L2 R2' x' y L' // WO
z' R2 U y' R2' // WB
y2' F2 U' y U' R' F R // WG
B U' y' L' U' R' F R // WR
y' R U R' U' R' U2 R y' U L' U L // wGO
U' R' U2 R y2 U2' L' U L // wRB
U R U R' U' y' R' U2 R L' U L // wOB
y U R U' R' U R U R' // wGR
(R y R U R' d' L')3 // EO
y2 U' R U R' U R U2 R' y2 R U R' U R U' R' U R U2 R' // EP
y R' U L U' R U L' U' y' R' U L U' R U L' U' // CP
y' z' x' (U' R' U R)2 L' (U' R' U R)2 L' (U' R' U R)2 L2' // CO
View at alg.cubing.net
Code:
[B]Step Time STM stps ETM etps[/B]
[COLOR="red"]Total 3:13.48 127 0.66 183 0.95 [/COLOR] [B][SIZE="4"]%[/SIZE]
Step Time STM ETM[/B]
Cross+1 56.24 9 0.16 16 0.28 Cross+1/F2L 52.6% 14.8% 16.0%
F2L 1:46.96 61 0.57 100 0.93 F2L/Total 55.3% 48.0% 54.6
LL 1:26.52 66 0.76 83 0.96 LL/Total 44.7% 52.0% 45.4%
timed from frame before first move to frame after last move
L' R2 D2 U' L R2 B F2 D2 L2 F2 L B2 R B F' z
y' z x' // inspection
r2 L' L' U y x (L r) R2' x' y L' // WO
z' R2 U y' R2' // WB
y' y' x U U x' U' y U' l' U R // WG
x' x' U x U' y' L' U' l' U l // WR
y' R U R' U' R' U U R y U' y U' y U' L' U L // wGO
U' R' U U R y U' y U' L' U L // wRB
U R U R' U' U' U y' R' U U R L' U L // wOB
U' y U' U U' U' R U' R2' R y U y' R U R' // wGR
(R y R U R' d' L')3 // EO
U' U' y' U y y y R U R' U R U2 R' y2 R U R' U R U2 R' R U R' U R U2 R' // EP
y y y y y R' U L U' R U L' U' y y2 R' U L U' R U L' U' // CP
y' z' x' (U' R' U R)2 L' (U' R' U R)2 L' (U' R' U R)2 (L' r') // CO
View at alg.cubing.net
"Do I get to keep this? No? Aww" @ 4:45
Something you don't see every day, someone wanting to keep a Rubik's Brand.
Gomorrite
Member
It appears in the recently released film The Bad Batch, in a junkyard full of cannibals in a dystopian future there is a guy playing with the cube.
Underwatercuber
Member
Who has seen Despicable Me 3 already?
Lots of cubes in it It may be just me, but I'm pretty sure they got colour schemes right/wrong literally from scene to scene and at times from shot to shot.
I noticed that too!
xyzzy
Member
So there's this animated movie that came out not too long ago called "Your name". It's became the second-most financially successful Anime movie worldwide ("Spirited Away" being first), and you should probably watch it. But don't watch any trailers/ read any plot synopsis tho, it's best viewed with no real knowledge about it going in, I think.
Anyway, where was I? Oh yeah, a scrambled Rubik's cube can be seen it at least one scene:
I'd just like to point out that, now that the Blu-ray has been released, you can actually see that there are more than six colours on the cube… (Or if you want to chalk that up to shading differences, there are two corners with white and red in clockwise order.) | 2019-08-17 21:06:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17812545597553253, "perplexity": 11466.267002334236}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027312025.20/warc/CC-MAIN-20190817203056-20190817225056-00199.warc.gz"} |
https://ask.openstack.org/en/answers/64465/revisions/ | # Revision history [back]
1. Can I ping the VM tenant from the Network host? Or do I need to be in the linux namespace?
NO. you should use linux namespace. But you are using work around by putting the below routing rule which is causing you to access the VM tenant network without namespace.
10.0.0.0 172.29.173.5 255.255.255.0 UG 0 0 0 br-ex
2. Do I need to add a physical port to br-ex?
YES. you should add the port to br-ex. but as reported by you, floating ip is not accessbile.
Again looks like some routing problem. Why you have two entry like below.
172.29.173.0 0.0.0.0 255.255.255.224 U 0 0 0 br-ex
172.29.173.0 0.0.0.0 255.255.255.192 U 0 0 0 br-ex | 2019-05-20 23:31:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24592632055282593, "perplexity": 4068.316198234024}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256163.40/warc/CC-MAIN-20190520222102-20190521004102-00065.warc.gz"} |
http://mathoverflow.net/feeds/question/61946 | A lower bound of a particular convex function - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T05:50:26Z http://mathoverflow.net/feeds/question/61946 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61946/a-lower-bound-of-a-particular-convex-function A lower bound of a particular convex function Jennifer Gao 2011-04-16T19:15:18Z 2011-05-08T10:14:55Z <p>Hello, I suspect this reduces to a homework problem, but I've been a bit hung up on it for the last few hours. I'm trying to minimize the (convex) function $f(x) = 1/x + ax + bx^2$ , where $x,a,b>0$. Specifically, I'm interested in the minimal objective function value as a function of $a$ and $b$. Since finding the minimizer $x^*$ is tricky (requires solving a cubic), I figured I'd try and find a lower bound using the following argument: if $b=0$, the minimizer is $x=1/\sqrt{a}$ and the minimal value is $2\sqrt{a}$. If $a=0$, the minimizer is $x=(2b)^{-1/3}$ and the minimal value is $\frac{3\cdot2^{1/3}}{2}b^{1/3}$. Therefore, one possible approximate solution is the convex combination</p> <p>$(\frac{a}{a+b})\cdot2\sqrt{a} + (\frac{b}{a+b})\cdot\frac{3\cdot2^{1/3}}{2}b^{1/3}$.</p> <p>Numerical simulations suggest that the above expression is a lower bound for the minimal value. Does this follow from some nice result about parameterized convex functions? It seems like it shouldn't be hard to prove. I guess in a nutshell I just want to prove that for all $x,a,b>0$ we have</p> <p>$(\frac{a}{a+b})\cdot2\sqrt{a} + (\frac{b}{a+b})\cdot\frac{3\cdot2^{1/3}}{2}b^{1/3} \leq 1/x + ax + bx^2$. Thanks!</p> <p>EDIT: It also appears that if I take the convex combination</p> <p>$(\frac{a^{3/5}}{a^{3/5}+b^{2/5}})\cdot2\sqrt{a} + (\frac{b^{2/5}}{a^{3/5}+b^{2/5}})\cdot\frac{3\cdot2^{1/3}}{2}b^{1/3}$</p> <p>then I get a tighter lower bound, and in fact the lower bound is within a factor of something like $3/2$ of the true minimal solution.</p> http://mathoverflow.net/questions/61946/a-lower-bound-of-a-particular-convex-function/62099#62099 Answer by Denis Serre for A lower bound of a particular convex function Denis Serre 2011-04-18T08:41:55Z 2011-04-18T08:41:55Z <p>The first inequality is true. Write $$f=\frac{a}{a+b}f_0+\frac{b}{a+b}f_1,$$ where $f_0$ and $f_1$ correspond to the case $b=0$ and $a=0$, respectively. You know that $f_0\ge2\sqrt a$ and $f_1\ge\frac{3\cdot2^{1/3}}{2}b^{1/3}$. This implies $$f\ge\frac{a}{a+b}2\sqrt a+\frac{b}{a+b}\frac{3\cdot2^{1/3}}{2}b^{1/3}.$$</p> http://mathoverflow.net/questions/61946/a-lower-bound-of-a-particular-convex-function/64279#64279 Answer by Maciej S. for A lower bound of a particular convex function Maciej S. 2011-05-08T10:14:55Z 2011-05-08T10:14:55Z <p>As Nishant Chandgotia sugessted: simply write $f(x) = \left(p\cdot \frac{1}{x} + ax\right) + \left((1-p)\frac{1}{x}+bx^2 \right)$ for some parametr $p\in[0,1]$.</p> <p>For the first term, minimizer is equal to $p^{\frac{1}{2}}a^{-\frac{1}{2}}$ and the minimal value is $p^{\frac{1}{2}} 2a^{\frac{1}{2}}$. For the second therm, minimizer is equal to $(1-p)^{\frac{1}{3}}(2b)^{-\frac{1}{3}}$ and the minimal value is $(1-p)^{\frac{2}{3}} \cdot\frac{3\cdot 2^{\frac{1}{3}} }{2} b^{\frac{1}{3}}$</p> <p>The best estimate is achieved when both minimizers are equal, which means, in therms of $p$, that </p> <p>$$\left(\frac{p}{a}\right)^{3} = \left( \frac{1-p}{2b}\right)^{2}$$</p> <p>Note, that this equation has a solution in interval $0 < p < 1$ by Mean-value theorem, unfortunately not expressible in nice way. </p> <p>Any way, we get for any $p\in[0,1]$ the following estimate:</p> <p>$$f(x) \geqslant p^{\frac{1}{2}} \cdot 2a^{\frac{1}{2}} + (1-p)^{\frac{2}{3}} \cdot\frac{3\cdot 2^{\frac{1}{3}} }{2} b^{\frac{1}{3}}$$</p> <p>Finally, we check that $p^{\frac{1}{2}} + (1-p)^{\frac{2}{3}} \geqslant 1$. This inequality implies, that estimate reamins valid after using any convex combination instead of weights $p^{\frac{1}{2}}, (1-p)^{\frac{2}{3}}$, i.e.</p> <p>$$f(x) \geqslant \alpha \cdot 2a^{\frac{1}{2}} + (1-\alpha) \cdot\frac{3\cdot 2^{\frac{1}{3}} }{2} b^{\frac{1}{3}}$$</p> <p>This can be seen immediately, however, by the <strong>inequality</strong> $f \geqslant \alpha f_0 + (1-\alpha) f_1$. My goal was to complete presented ideas and to show when exact optimum is attained.</p> | 2013-06-19 05:50:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.9635924696922302, "perplexity": 350.3813683475754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368707906397/warc/CC-MAIN-20130516123826-00050-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://hillsdrmission.com/mh7kc/u38zr.php?id=unscented-kalman-filter-python-489900 | # unscented kalman filter python
kappa is an arbitrary constant. In this section we will be dealing with python com server to integrate Amibroker + Python to compute Kalman Filter and Unscented Kalman Filter Mean Estimation and plot the same in Amibroker. Figure 2 Correlation coefficient as a function of forecast time of ensemble-mean predictions of NINO3. SST T20 Unscented Kalman Filter - Part 1 - Duration: 1:16:56. Includes Kalman filters,extended Kalman filters, unscented Kalman filters, particle filters, and more. Bayes filter . I've trained a gaussian process which will take X (x1:5) and predict Y (x6). Unless you know better, this should be your default choice. array of the means (state variable x) of the output of a Kalman KalmanFilter (k_endog, k_states, k_posdef = None, loglikelihood_burn = 0, tolerance = 1e-19, results_class = None, kalman_filter_classes = None, ** kwargs) [source] ¶ State space representation of a time series process, with Kalman filter. time. 3.2Unscented Kalman Filter localization This is a sensor fusion localization with Unscented Kalman Filter(UKF). Compare the EKF and UKF filters’ performance using the robot_localization ROS package. This 1st order linearization may be too coarse, and this is one motivation for Unscented Kalman Filters we mention in the last section. the sigmas for one dimension in the problem space. measurements), so the sigmas correctly reflect the updated state 3. We use optional third-party analytics cookies to understand how you use GitHub.com so we can build better products. Atsushi Sakai, Daniel Ingram, Joseph Dinius, Karan Chawla, Antonin Raffin: “PythonRobotics: a Python code collection of robotics algorithms”, arXiv:1808.10703, (2018); Link. For Kalman filter. may be illuminating. Alphatrading ⭐ 121. \chi[1..n] = &x + [\sqrt{(n+\kappa)P}]_k \\ ... the task in Kalman filters is to maintain a mu and sigma squared as the best estimate of the location of the object we’re trying to find. Use this if your state variable contains nonlinear An unscented Kalman Filter implementation for fusing lidar and radar sensor measurements. Typically your alternative choice will be Kalman-and-Bayesian-Filters-in-Python by rlabbe - Kalman Filter book using Jupyter Notebook. In this paper, we presented the Python code for the Kalman Filter implementation. Wm: ndarray [# sigmas … optional list of values to use for the measurement error filter. Includes exercises with solutions. speed. which multiply by this value, so by default we always return a EKF and UKF. We presented a two step based implementation and we give an example of using this kind of filters for localization in wireless networks. Read only. This python unscented kalman filter (UKF) implementation supports multiple measurement updates (even simultaneously) and allows you to … 2004 dissertation[1] for the UnscentedKalmanFilter class.. Returns tuple of the sigma points and weights. Each column contains all of optional keyword arguments to be passed into f(x). called after every epoch. Based on the fluctuation of the stock market and the dynamic tracking features of Kalman filter, taking stock of Changbaishan (603099) as an example, … The ensemble Kalman filter (EnKF) is very similar to the unscented Kalman filter (UKF) of the last chapter. per epoch. does the right thing as far as this class is concerned. Contr., Lake Louise, AB, Canada, Oct. 2000. https://www.seas.harvard.edu/courses/cs281/papers/unscented.pdf. and Jeffery K. Uhlmann’s original paper[1]. filterpy.common.Saver object. The most common variants of Kalman filters for non-linear systems are the Extended Kalman Filter and Unscented Kalman filter. Unscented Kalman Filter localization¶ This is a sensor fusion localization with Unscented Kalman Filter(UKF). CoCalc Public Files Kalman-and-Bayesian-Filters-in-Python / 10-Unscented-Kalman-Filter.ipynb Open with one click! Read Only. The state transition model has additive noise. If float, then the same time step is used for all steps. Examples. Cholesky is the default choice due to its Important: this MUST be called before update() is called for the first Learn more. Measurement noise. array of the covariances of the output of a kalman filter. This structure is very similar to the Kalman Filter which we will discuss in the next section. If you are using multiple sensors the size of z can Edit: I found maybe some documents through your profile but it seems you didnt use an extended kalman filter oder unscented. SLAM Course - 06 - Unscented Kalman Filter (2013/14; Cyrill Stachniss) - Duration: 55:01. work - you can use x_mean_fn and z_mean_fn to alter the behavior 3 means measurement 10 min read. class. reasons it returns a lower triangular matrix. If your method returns a triangular matrix it must be upper Unscented Kalman Filtering with Application to Parameter x and y array of the state for each time step after the update. https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python. are ordered as: Generates sigma points and weights according to the simplex Implement a Bayes filter in order to know a robot’s position. Implements the Unscented Kalman Filter with additive noise. x and y. Computes the sigma points for an unscented Kalman filter This works in conjunction with the UnscentedKalmanFilter class. Measurement function. examples: 1, [1,2], np.array([1,2]). Process., Commun. x, P. Performs the UKF filter over the list of measurement in zs. Taking the k. array of the covariances for each time step after the update. The test files in this directory also give you a basic idea of use, albeit without much description. for more information. Computed from the log-likelihood. E. A. Wan and R. Van der Merwe, “The unscented Kalman filter for The SciPy version Includes Kalman filters,extended Kalman filters, unscented Kalman filters, particle filters, and more. method presented in [1]. Focuses on building intuition and experience, not formal proofs. The nonlinearity can be associated either with the process model or with the observation model or with both. reasons it returns a lower triangular matrix. Each column contains all of Optional function to compute the unscented transform for the sigma points using kappa. Allow users to filter the list of styles to only show those which are. While recursive least squares update the estimate of a static parameter, Kalman filter is able to update and estimate of an evolving state[2]. I wrote about Kalman Filter and Extended Kalman Filter. Model Predictive Control. However, since my input is non-linear, I wanted to use Kalman Filter so that I can detect and track the drops of the filtered signal (blue color in the above plot). https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python, weight for each sigma point for the covariance, x.__init__(…) initializes x; see help(type(x)) for signature, Number of sigma points for each variable in the state x. Computes the sigma points for an unscented Kalman filter kappa=0 gives If you're using this be sure to use the square root of the measurement noise R, since we are working with … If it is a list of matrices or a 3D array where Number of of measurement inputs. self.P contain the predicted state (x) and covariance (P). Includes Kalman filters,extended Kalman filters, unscented Kalman filters, particle filters, and more. Function that computes the difference between x and y. A Code for Unscented Kalman Filtering on Manifolds (UKF-M) ... method on two independent open-source Python and Matlab frameworks we call UKF-M , for quickly implementing and testing the approach. Kalman Filters: A step by step implementation guide in python This article will simplify the Kalman Filter for you. Python for Robotics, Linux for Robotics, ROS Basics in 5 Days. When the state transition and observation models—that is, the predict and update functions and —are highly nonlinear, the extended Kalman filter can give particularly poor performance. Optional, vector of shape (dim_z). would come from the output of batch_filter(). pyfilter provides Unscented Kalman Filtering, Sequential Importance Resampling and Auxiliary Particle Filter models, and has a number of advanced algorithms implemented, with PyTorch backend. JulierSigmaPoints implements Julier’s original kappa Generates sigma points and weights according to Simon J. Julier Dimensionality of the state. Cyrill Stachniss 41,608 views. FilterPy ¶ FilterPy is a Python library that implements a number of Bayesian filters, most notably Kalman filters. The unscented Kalman filter algorithm and Unscented Kalman Filter block use the unscented transformation to capture the propagation of the statistical properties of state estimates through nonlinear functions. If you recall, the UKF uses a set of deterministically chosen weighted sigma points passed through nonlinear state and measurement functions. Allow users to filter the list of styles to only show those which are. This python unscented kalman filter (UKF) implementation supports multiple measurement updates (even simultaneously) and allows you to easily plug in your model and measurements! Ducati Multistrada 950 S BS6 Launch Date Revealed; Bookings Open. For more information, see our Privacy Statement. Dimensionality of the state. Function that computes the residual (difference) between x and y. Implement a Kalman filter and test it in a simulated robot. The lines and points are same meaning of the EKF simulation. given the mean (x) and covariance(P) of the filter. As of van der Merwe’s dissertation of exp() of that results in 0.0, which can break typical algorithms Kalman Filter textbook using Ipython Notebook. the other is for the measurement state. You can vary the UKF implementation by changing this More complex systems, however, can be nonlinear. Determins the spread of the sigma points around the mean. If dtss is None then self.dt is used for all epochs. dt is the time step in seconds. I'm using a square root continuous-discrete version of the UKF and comparing it with the EKF, so I used the measurement update step. The usual input are state vectors, not scalars. The Unscented Kalman filter uses a similar technique but reduces the amount of computation needed by a drastic amount by using a deterministic method of choosing the points. covariance R. If Rs is None then self.R is used for all epochs. self.x and self.P contain the new mean and covariance of the filter. Localization Paid only upon seeing the output. small, meaning a large negative value such as -28000. Symp. order errors in x and P. Function that computes the difference between x and y. triangular. An unscented Kalman filter is a recursive algorithm for estimating the evolving state of a process when measurements are made on the process. Update the UKF with the given measurements. Ref: Discriminatively Trained Unscented Kalman Filter for Mobile Robot Localization The basic Kalman filter is limited to a linear assumption. not give you a functional filter. Generates sigma points and weights according to Van der Merwe’s You will have to set the following attributes after constructing this object for the filter to perform properly. Unscented Kalman Filter Code. DOI: 10.1051/cocv/2010006. We use essential cookies to perform essential website functions, e.g. https://filterpy.readthedocs.org, Supporting book at: 2004 [6] this was not a well reseached area so I have no advice means and covariances computed by the UKF. the sigmas for one dimension in the problem space. Unscented Kalman filter. The Kalman Filter and Sensor Fusion . in [2]. this function call. kappa to 3-dim_x for a Gaussian x you will minimize the fourth Today we will look at another member of Kalman Filter Family: The Unscented Kalman Filter. If scalar, is treated as eye(n)*P. Two dimensional array of sigma points. signature of this class if you want to implement your own. Prior (predicted) state covariance matrix. If your method returns a triangular matrix it must be upper 55:01. On return, self.x and So, if you read my last two posts you would be knowing my colleague Larry by now. points passed through hx. What projects will you be doing? len(Rs) == len(zs), then it is treated as a list of R values, one Secondary scaling parameter usually set to 0 according to [4], This is licensed under an MIT license. E.g. Usually a small positive value (1e-3) according to [3]. “A new method for If it is a list where len(dts) == len(zs), then it is treated as a 2D array of sigma points $$\chi$$. Includes Kalman filters, Extended Kalman filters, unscented filters, and more. https://www.cs.unc.edu/~welch/kalman/media/pdf/ACC02-IEEE1357.PDF. Unscented Filtering and Nonlinear Estimation SIMON J. JULIER, MEMBER, IEEE, AND JEFFREY K. UHLMANN, MEMBER, IEEE Invited Paper The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. Beyond filtering performance, the main interests of the approach are its versatility, as the method applies to numerous state estimation problems, and its simplicity of implementation for practitioners not being necessarily … sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I, Can be a scalar if 1D. To know Kalman Filter we need to get to the basics. For in depth explanations see my book Kalman and Bayesian Filters in Python There is no SPIE 3068, Signal Processing, Atsushi Sakai, and Yoji Kuroda. Has companion book 'Kalman and Bayesian Filters in Python'. The online repositories contain tutorials, documentation, and various relevant robotics examples that the user can readily reproduce and then adapt, for fast prototyping and benchmarking. In other words covariance[k,:,:] is the covariance at step k. Runs the Rauch-Tung-Striebal Kalman smoother on a set of current epoch. values such as angles which cannot be summed. The process of the Kalman Filter is very similar to the recursive least square. Model Predictive Control. Kalman Filter implementation in Python using Numpy only in 30 lines. Kalman filter, Extended Kalman filter, Unscented Kalman filter, g-h, least squares, H Infinity, smoothers, and more. given the mean (x) and covariance(P) of the filter. and weights. Computes the implex sigma points for an unscented Kalman filter Typically your alternative choice will be - rlabbe/Kalman-and-Bayesian-Filters-in-Python These are the top rated real world Python examples of ukf.UnscentedKalmanFilter extracted from open source projects. You will have to supply this if your state variable cannot support Implements a extended Kalman filter. All exercises include solutions. So let’s get started! given the mean (x) and covariance(P) of the filter. creation. You can always update your selection by clicking Cookie Preferences at the bottom of the page. Venom ⭐ 142. yields maximal performance. Signal subtraction, such as angles (359-1 degreees is 2, not 358). http://github.com/rlabbe/filterpy, Documentation at: I need an unscented / kalman filter forecast of a time series. Units are seconds. But since I am so new to Kalman Filter, I seem to have a hardtime understanding the mathematical formulation and and to get started with Unscented Kalman Filter. Cholesky is the default choice due to its This is because the covariance is propagated through linearization of the underlying nonlinear model. Inference in Dynamic State-Space Models” (Doctoral dissertation), Julier, Simon J.; Uhlmann, Jeffrey “A New Extension of the Kalman All exercises include solutions. The Overflow Blog How to write an effective developer resume: Advice from a hiring manager. The unscented Kalman filter can model the evolution of a state that obeys a nonlinear motion model. Ref: •Discriminatively Trained Unscented Kalman Filter for Mobile Robot Localization 10 Chapter 3. This works in conjunction with the UnscentedKalmanFilter class. are state vectors, not scalars. Returns sigma points. Posterior (updated) state covariance matrix. no unique answer. provides you with position in (x,y), dim_z would be 2. Author: Roger Labbe. I have just completed my Term 2 of Udacity Self Driving Car Nanodegree. This is for convience, so everything is sized correctly on optional value or list of delta time to be passed into predict. E. A. Wan and R. Van der Merwe, “The Unscented Kalman filter for Each entry An workflow in factor-based equity trading, including factor analysis and factor modeling. Common uses for the Kalman Filter include radar and sonar tracking and state estimation in robotics. Budget \$30-250 USD. Class which computes the sigma points and weights for a UKF The algorithm first generates a set of state values called sigma points. You will have to supply this if your state variable cannot support The lines and points are same meaning of the EKF simulation. Compute cross variance of the state x and measurement z. computes the values of sigmas_f. The validation of unscented and extended Kalman filter performance is typically done using extensive Monte Carlo simulations. state transistion function. If provided, specifies the time step of each step of the filter. For now the best documentation is my free book Kalman and Bayesian Filters in Python . This is standard for Gaussian processes, function(sigmas, Wm, Wc, noise_cov), optional, None, np.array or list-like, default=None, # this example demonstrates tracking a measurement where the time, # between measurement varies, as stored in dts The output is then smoothed, function(ndarray), default=scipy.linalg.cholesky, An array-like object of the means of length n, array-like object of the means of length n, https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python, https://www.cs.unc.edu/~welch/kalman/media/pdf/ACC02-IEEE1357.PDF. list of dt values, one per epoch. Both values have to be fused together with the Kalman Filter. Observations are assumed to be generated from the following process, While less general the general-noise Unscented Kalman Filter, the Additive version is more computationally efficient with complexity where is the number of time steps and is the size of the state space. All exercises include solutions. However, more than 35 years of experience in the estimation community has shown measurements must be represented by ‘None’. At this point in the book we have developed the theory for the linear Kalman filter. triangular. Clearly there are limits to such an approximation, and in situations where models deviate significantly from linearity, performance can suffer. Includes Kalman filters,extended Kalman filters, unscented Kalman filters, particle filters, and more. Function that computes the mean of the provided sigma points change based on the sensor. FilterPy is a Python library that implements a number of Bayesian filters, most notably Kalman filters. Then, in the last two chapters we broached the topic of using Kalman filters for nonlinear problems. Online copy: Includes Kalman filters,extended Kalman filters, unscented Kalman filters, particle filters, and more. This python unscented kalman filter (UKF) implementation supports multiple measurement updates (even simultaneously) and allows you to easily plug in your model and measurements! Identification in Large-Dimensional Systems” The log-likelihood can be very They Contr., Lake Louise, AB, Canada, Oct. 2000. Symp. does the right thing. Kalman Filter book using Jupyter Notebook. Examples. If specified, the time step to be used for this prediction. is the version seen in most publications. Filter to Nonlinear Systems”. Both values have to be fused together with the Kalman Filter. Software Architecture & Python Projects for €30 - €250. algorithm. The system being modeled could be some kind … self._dt is used if this is not provided. This allows you to have varying Unscented Kalman Filter Code. the nonlinear transformation of means and covariances in filters Emplois. class ExtendedKalmanFilter (object): """ Implements an extended Kalman filter (EKF). An unscented Kalman filter is a recursive algorithm for estimating the evolving state of a process when measurements are made on the process. To implement the extended Kalman filter we will leave the linear equations as they are, and use partial derivatives to evaluate the system matrix F \mathbf{F} F and the measurement matrix H \mathbf{H} H at the state at time t (x t \mathbf{x}_t x t ).In other words we linearize the equations at time t by finding the slope (derivative) of the equations at that time. Today we will look at another member of Kalman Filter Family: The Unscented Kalman Filter. arguments to be passed into h(x) after x -> h(x, **hx_args). Normally a user would not call This is the default setting in the filter, hence you do not need to specify it. It parametizes the sigma See my book Kalman and Bayesian Filters in Python Just provide the appropriate hx function. pp. You are responsible for setting the Kalman And Bayesian Filters In Python Kalman Filter book using Jupyter Notebook. One is for the state variable, Browse other questions tagged python kalman-filters multirate unscented-kalman-filter bayesian-estimation or ask your own question. If provided, saver.save() will be During the first missions in Project Apollo, the KF was implemented on analog hardware. Focuses on building intuition and experience, not formal proofs. Clearly there are limits to such an approximation, and in situations where models deviate significantly from linearity, performance can suffer. See either of those for the required Typically the default function will form the measurements after being passed through hx(). Scaling factor that can reduce high order errors. S. Julier, J. Uhlmann, and H. Durrant-Whyte. According to [Julier], if you set Parameters k_endog {array_like, int} The observed time-series process $$y$$ if array like or the number of variables in the process if an integer. For example, if \chi[0] = &x \\ It Usually this will not matter to you; if so the default cholesky() parameterization. Includes Kalman filters,extended Kalman filters, unscented Kalman filters, particle filters, and more. Trying out the first example (example.py) should be really easy. 2n+1 weights will be generated. Course Project. parametizes the sigma points using alpha, beta, kappa terms, and Read Only. 3 - Non-linear models: unscented Kalman filter¶ The previous tutorial showed how the extended Kalman filter propagates estimates using a first-order linearisation of the transition and/or sensor models. filterpy.kalman.unscented_transform(sigmas, Wm, Wc, noise_cov=None, mean_fn=None, residual_fn=None)[source]¶ Computes unscented transform of a set of sigma points and weights. GitHub is home to over 50 million developers working together to host and review code, manage projects, and build software together. scipy.linalg.sqrtm. Download, Raw , Embed. These simulations should test variations of process and measurement noise realizations, plant operating under various conditions, initial state and state covariance guesses. an array, then each element k contains the time at step k. Learn more, We use analytics cookies to understand how you use our websites so we can make them better, e.g. Compute Environment: Ubuntu 18.04 (Deprecated) Table of Contents. Focuses on building intuition and experience, not formal proofs. Wan, Merle “The Unscented Kalman Filter,” chapter in, R. Van der Merwe “Sigma-Point Kalman Filters for Probabilitic Last measurement used in update(). The unscented Kalman filter can model the evolution of a state that obeys a nonlinear motion model. 1. Extended Kalman Filters¶ If the evolution and observation are non-linear, we can linearize them using their Jacobian and transform them into linear equations suitable for a Kalman filter. are for convienence; they store the prior and posterior of the function passed in during construction will be used. This allows you to have varying R per epoch. I wrote about Kalman Filter and Extended Kalman Filter. If and estimators,” IEEE Transactions on Automatic Control, 45(3), Do not use numpy.linalg.cholesky - for historical Read Only. Different choices affect how the sigma points Freelancer. of the unscented transform. The SciPy version Description Kalman filtering and optimal estimation library in Python. Process noise of the Kalman filter at each time step. In [1]: #format the book % matplotlib inline % load_ext autoreload % autoreload 2 from __future__ import division, print_function import book_format book_format. Gaussian x beta=2 is optimal, according to [3]. Dynamics, measurement equations and initial conditions will be provided. The *_prior and *_post attributes If provided, overrides self.R for is an np.array. Defines how we compute the square root of a matrix, which has Examples. x and y So, if you read my last two posts you would be … filterpy.kalman.unscented_transform (sigmas, Wm, Wc, noise_cov=None, mean_fn=None, residual_fn=None) [source] ¶ Computes unscented transform of a set of sigma points and weights. All exercises include solutions. Number of state variables for the filter. https://www.seas.harvard.edu/courses/cs281/papers/unscented.pdf. subtraction, such as angles (359-1 degreees is 2, not 358). 2 Kalman Filter for Yield in Equation (1. By default, the Kalman filter follows Durbin and Koopman, 2012, in initializing the filter with predicted values. Also see the filterpy/kalman/tests subdirectory for test code that function that returns the state x transformed by the I chose to start off with the Unscented Kalman filter, which probably felt like quite a departure from the linear Kalman filter math. Works with both scalar and array inputs: number >= sys.float_info.min. Using a Kalman filter for predicting stock prices in python. Sensor Fusion, and Target Recognition VI, 182 (July 28, 1997), Phillippe Moireau and Dominique Chapelle “Reduced-Order \chi[n+1..2n] = &x - [\sqrt{(n+\kappa)P}]_k they're used to log you in. or to 3-n according to [5]. Incorporates prior knowledge of the distribution of the mean. Prior (predicted) state estimate. to give you. Process., Commun. Focuses on building intuition and experience, not formal proofs. Linearizing the Kalman Filter. Implements the Scaled Unscented Kalman filter (UKF) as defined by Defines how we compute the square root of a matrix, which has The measurements can also be nonlinear functions of the state, and the process and measurements can have noise. Converts state vector x into a measurement Kalman-and-Bayesian-Filters-in-Python by rlabbe - Kalman Filter book using Jupyter Notebook. This is an animation of the Unscented Kalman Filter that I created for a student's Neuroscience PhD. Simon Julier in [1], using the formulation provided by Wan and Merle various state variables to reasonable values; the defaults below will list of measurements at each time step self._dt Missing Podcast 290: This computer science degree is brought to … Fusion Ukf ⭐ 150. epoch durations. no unique answer. speed. Do not use numpy.linalg.cholesky - for historical Learn more. need to use a UKF for this example, but it is easy to read. defense at Penn State. Fixed price. https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python. - rlabbe/Kalman-and-Bayesian-Filters-in-Python If not provided, the default All exercises include solutions. Nonlinear Estimation,” in Proc. dimensions, dim_x would be 4. All Terrain Autonomous Quadruped. I have just completed my Term 2 of Udacity Self Driving Car Nanodegree. You signed in with another tab or window. are arranged relative to the eigenvectors of the covariance matrix. 3 - Non-linear models: unscented Kalman filter¶ The previous tutorial showed how the extended Kalman filter propagates estimates using a first-order linearisation of the transition and/or sensor models. class filterpy.kalman.UnscentedKalmanFilter(dim_x, dim_z, dt, hx, fx, points, sqrt_fn=None, x_mean_fn=None, z_mean_fn=None, residual_x=None, residual_z=None) [source] ¶ Implements the Scaled Unscented Kalman filter (UKF) as defined by Simon Julier in, using the formulation provided by Wan and Merle in. The output has to be a rolling predict step without incorporating the next measurement (a priori prediction). This filter scales the sigma points to avoid strong nonlinearities. Returns tuple of the sigma points and weights. the standard unscented filter. returns the mean and covariance in a tuple. 477-482 (March 2000). I had a hard time interpreting the algorithm presented in the paper 'The Square-Root Unscented Kalman Filter For State and Parameter-Estimation'. Millions of developers and companies build, ship, and maintain their software on GitHub — the largest and most advanced development platform in the world. \end{eqnarray}, © Copyright 2014-2016, Roger R. Labbe. ‘. beta, kappa parameterization of Van der Merwe, and pseudo inverse, set it to that instead: Create a Kalman filter. The present paper introduces a novel methodology for Unscented Kalman Filtering (UKF) on manifolds that extends previous work by the authors on UKF on Lie groups. "Discriminatively Trained Unscented Kalman Filter for Mobile Robot Localization." Performs the predict step of the UKF. ” If you prefer another inverse function, such as the Moore-Penrose It has two models or stages. sigma_points (5, 9, 2) # mean 5, covariance 9 For example, if the sensor Read Only. Covariance of the filter. This python unscented kalman filter (UKF) implementation supports multiple measurement updates (even simultaneously) and allows you to easily plug in your model and measurements! Proc. 5 Sigma Points - … For example, MerweScaledSigmaPoints implements the alpha, The current model is from this paper: with f being GP function. MATLAB. this, but it is useful if you need to call update more than once Revert only if serious. You can rate examples to help us improve the quality of examples. Focuses on building intuition and experience, not formal proofs. | 2022-08-14 10:35:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38394203782081604, "perplexity": 1590.5144971380287}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572021.17/warc/CC-MAIN-20220814083156-20220814113156-00697.warc.gz"} |
https://cs.stackexchange.com/questions/85207/explain-hashed-page-tables-in-operating-system | # Explain Hashed page tables in operating system
I have a difficult time understanding hashed page tables used in virtual memory management. Here is picture of the slide that I am referring to:
I understand that p is hashed and then the hash is checked in the page table for a match. What is elements referring to here? Is it the page table entry? What does chain mean and where is virtual page number located?
Can someobody help me understand the flow chart? Thanks
• The element is a page table entry (tag, the PTE proper, and a pointer/reference to the next element matching that hash value). The chain is a singly linked list for handling hash conflicts. The VPN (usually compressed by exploiting the hash table indexing bits) is used as a tag and is stored with the PTE and next pointer. – Paul A. Clayton Dec 9 '17 at 11:06
# Definitions (What's with the symbols?)
In the diagram, we have these guys:
• Virtual Page Number (VPN): p, q
• Page Frame Number (PFN): r
• Offset: d
• Hash Function: h(x)
• Hashed Page Table with schema (key, VPN, PFN, Pointer to next entry with key) for each entry in the table
It so happens that h(p) = same_key and h(q) = same_key. There is hash collision. Both p and q are hashed to the same_key.
This is resolved by chaining the entry with VPN = q to the entry with VPN = p. Chaining means to use the Pointer field in the entry with VPN = q to point to the entry with VPN = p.
# Workflow (How the system works)
Operating system (OS) grabs p from the CPU, and performs h(p) to get same_key.
OS looks up the first entry in the Hashed Page Table with key = same_key and checks p against the first entry's VPN field. It checks p against q. This is incorrect.
OS uses the Pointer in the first entry to find the second entry. It knows that the second entry has the same key = same_key, because the Page Table is constructed this way. OS checks p against the seond entry's VPN field. It checks p against p. This is correct. Bam. Kill confirmed.
OS knows that this is the correct entry it is looking for. It grabs PFN from the second entry. It grabs r. r is the correct physical frame number that corresponds to virtual page number p.
OS uses r to look for the physical frame it wants in physical memory, and looks for the exact word wanted which is offset by d within frame r in physical memory. OS grabs the contents of the word and we're done. | 2019-07-16 14:11:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2470608502626419, "perplexity": 3442.0439095867973}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195524568.14/warc/CC-MAIN-20190716135748-20190716161748-00451.warc.gz"} |
https://proofwiki.org/wiki/Category:Examples_of_Order_of_Power_of_Group_Element | # Category:Examples of Order of Power of Group Element
This category contains examples of Order of Power of Group Element.
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $g \in G$ be an element of $G$ such that:
$\order g = n$
where $\order g$ denotes the order of $g$.
Then:
$\forall m \in \Z: \order {g^m} = \dfrac n {\gcd \set {m, n} }$
where $\gcd \set {m, n}$ denotes the greatest common divisor of $m$ and $n$.
## Pages in category "Examples of Order of Power of Group Element"
The following 2 pages are in this category, out of 2 total. | 2020-08-06 07:22:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.98016756772995, "perplexity": 261.66114717204664}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439736883.40/warc/CC-MAIN-20200806061804-20200806091804-00521.warc.gz"} |
https://www.ncatlab.org/nlab/show/daseinisation | Contents
topos theory
# Contents
## Idea
Given a C*-algebra $A$ thought of as the algebra of observables of a quantum mechanical system, write $ComSub(A)$ for its poset of commutative subalgebras. Then the presheaf topos over $ComSub(A)$ with its canonical spectral presheaf as well as the presheaf topos over the opposite category $ComSub(A)^{op}$ canonically regarded as a ringed topos – the “Bohr topos”, might both be regarded as topos-theoretic incarnations of the phase space of the given quantum mechanical system. By standard quantum mechanics every self-adjoint operator $a \in A_{sa}$ is to be regarded as an “observable on phase space”, in some sense. Hence one may ask if $a$ induces in a precise sense a function on the phase space internal to these toposes.
A construction from each $a \in A_{sa}$ of a clopen subset $\delta^o(a) \subset \Sigma_A$ of the spectral presheaf $\Sigma$ of $A$ has been given in (Isham-Döring 07) for von Neumann algebras $A$. There this is called the “daseinisation” of $a$. An analogous construction for the Bohr toposes of C*-algebras has been given in (Heunen-Landsman-Spitters 09). A direct identification of quantum observables with homorphisms of ringed toposes out of the Bohr topos is discussed at Bohr topos – The observables.
## References
Last revised on December 22, 2015 at 05:57:31. See the history of this page for a list of all contributions to it. | 2021-03-02 21:13:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 12, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9020687937736511, "perplexity": 425.3419920462867}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178364764.57/warc/CC-MAIN-20210302190916-20210302220916-00218.warc.gz"} |
https://healthyalgorithms.com/tag/global-health/ | # Tag Archives: global health
Filed under Uncategorized
## Fact checking with GBD Compare
I’ve been developing a habit of comparing health statistics I hear in the media with the results in GBD Compare. It is nice when they agree, such as in a recent ScienceMag focus on chronic kidney disease, corroborated here: http://ihmeuw.org/1v7i . It would be even better if the cause was known, and the burden could be removed.
Comments Off on Fact checking with GBD Compare
Filed under global health
## A simple optimization problem I don’t know how to solve (from DCP)
Inspired by the recent 8F workshop, I’m trying to write up theory challenges arising from global health. And I’m trying to do it with less background research, because avoiding foolishness is a recipe for silence.
This is the what I called the “simplest open problem in DCP optimization” in a recent post about DCP (Disease Control Priorities), but with more reflection, I should temper that claim. I’m not sure it is the simplest. I’m not sure it is an open problem. And I’m pretty sure that if we solve it, the DCP optimizers will come back with something more complicated.
But it is a nice, clean problem to start with. I’m calling it “Fully Stochastic Knapsack”. It looks just like the plain, old knapsack problem:
$\max \bigg\{ \sum_{i=1}^n v_ix_i \qquad s.t. \quad \sum_{i=1}^n w_ix_i \leq W, \quad x_i \in \{0,1\} \bigg\}$
The fully stochastic part is that everything that usually would be input data is now a probability distribution, and the parameters of the distribution are the input data.
This makes even deciding what to maximize a challenge. I was visiting the UW Industrial Engineering Dept yesterday, and Zelda Zabinsky pointed me to this nice INFORMS tutorial by Terry Rockafeller on “coherent approaches” to this. | 2021-09-23 18:11:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6059686541557312, "perplexity": 1288.3455243689332}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057427.71/warc/CC-MAIN-20210923165408-20210923195408-00441.warc.gz"} |
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# The total surface area of cube C
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The total surface area of cube C [#permalink] 14 May 2018, 12:16
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Question Stats:
100% (00:29) correct 0% (00:00) wrong based on 11 sessions
The total surface area of cube C equals 150.
Quantity A Quantity B The length of one edge of cube C 4.5
A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA
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Re: The total surface area of cube C [#permalink] 03 Aug 2018, 21:55
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Carcass wrote:
The total surface area of cube C equals 150.
Quantity A Quantity B The length of one edge of cube C 4.5
A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
Surface area of Cube = $$6 * {side}^2$$
or $$150 = 6 * {side}^2$$
or $${side}^2 = 25$$
or $$side = 5$$
Therefore QTY A > QTY B
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Re: The total surface area of cube C [#permalink] 03 Aug 2018, 21:55
Display posts from previous: Sort by | 2018-12-14 14:15:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5780912637710571, "perplexity": 4916.386476002188}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376825916.52/warc/CC-MAIN-20181214140721-20181214162221-00089.warc.gz"} |
https://itectec.com/superuser/using-ffmpeg-to-create-a-video-from-a-sequence-of-tga-images-with-a-depth-of-field-blur-effect-sourced-from-the-tgas-alpha-channel/ | Using ffmpeg to create a video from a sequence of TGA images, with a depth of field blur effect sourced from the TGA’s alpha channel
I'm working on turning a sequence of TGA images into a video using ffmpeg. I want the video to have a depth of field blur effect. The alpha channel of each TGA contains a depth-mask, where black=close and white=far. I want to use this info to add DOF blur to the final output of that frame. After searching, the closest answer I could find was this
FFMPEG filter to boxblur and greyscale a video using alpha mask, but it is for a static dof-mask. The DOF mask I would be using is obviously changing every frame, and an alpha channel instead of a seperate png.
Here is my current cmd line
ffmpeg -framerate 60 -i image.%10d.tga -c:v libx264 -preset slow -crf 0 -c:a copy -pix_fmt yuv420p output0.mp4
It seems the answer would involve some use of alphamerge/alphaextract/boxblur, but I'm brand new to ffmpeg so I don't know how to formulate the command.
Here is an example of the type of TGA I would use https://dl.dropboxusercontent.com/u/19482624/alphachanneltest.tga
ffmpeg -framerate 60 -i image.%10d.tga \ | 2021-04-20 00:38:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5352306365966797, "perplexity": 3513.2433057175226}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038921860.72/warc/CC-MAIN-20210419235235-20210420025235-00279.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=178384 | ## Photoelectric effect
I like to get the data of the actual photoelectric experimentation. I have tables with the work function, thus I can compute the threshold frequency, but I believe that introduces at least two errors. I would like to get the frequency obtained directly from experimentation. Anyone can help me? | 2013-05-26 00:08:14 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8024525046348572, "perplexity": 460.9865038508024}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368706472050/warc/CC-MAIN-20130516121432-00033-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://docs.blender.org/manual/ko/dev/editors/graph_editor/introduction.html | # Introduction¶
The Graph editor is the main animation editor. It allows you to modify the animation for any properties using F-Curves.
The Graph editor has two modes, F-Curve for Actions, and Drivers for Drivers. Both are very similar in function.
## Curve View¶
Here you can see and edit the curves and keyframes.
### 2D Cursor¶
The current frame is represented by a green vertical line called the Time Cursor.
As in the Timeline, you can change the current frame by pressing or holding LMB.
The green horizontal line is called the Cursor. This can be disabled via the View Menu or the View Properties panel.
The Time Cursor and the Cursor make the 2D Cursor. The 2D Cursor is mostly used for editing tools.
### View Axes¶
For Actions the X-axis represents time, the Y-axis represents the value to set the property.
For Drivers the X-axis represents the Driver Value, the Y-axis represents the value to set the property.
Depending on the selected curves, the values have different meaning: For example rotation properties are shown in degrees, location properties are shown in Blender Units. Note that Drivers use radians for rotation properties.
### Markers¶
Like with most animation editors, markers are shown at the bottom of the editor.
Markers can be modified in the Graph Editor though it's usually best to use the Timeline.
## Properties Region¶
The panels in the Properties Region.
### View Properties Panel¶
Show Cursor
Show the vertical Cursor.
Cursor from Selection
Set the 2D cursor to the center of the selected keyframes.
Cursor X
Time Cursor X position.
To Keys
Snap selected keyframes to the Time Cursor.
Cursor Y
Vertical Cursor Y position.
To Keys
Snap selected keyframes to the Cursor.
### Further Tabs¶
F-Curve Tab
See F-Curve.
Drivers Tab
See Drivers Panel.
Modifiers Tab
See F-Modifiers. | 2018-01-20 12:53:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19428445398807526, "perplexity": 5199.154768180848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889617.56/warc/CC-MAIN-20180120122736-20180120142736-00113.warc.gz"} |
https://ham.stackexchange.com/tags/math/hot | # Tag Info
## Hot answers tagged math
Accepted
### Is free space path loss dependent on frequency?
Mathematically yes, the value of that equation increases with frequency. However, that's not to say there's some physical mechanism for frequency-dependent attenuation in free space. Rather, the ...
• 49.8k
Accepted
### Why EME with 100W and a single Yagi-Uda antenna is possible?
You didn't include antenna gain at all. That number for the path loss is a starting point assuming zero gain (isotropic antennas) on both ends. This isn't a realistic situation, because of course the ...
• 8,740
Accepted
### Conversion from Baud to Bits per Second and then to Bytes per Second
Baud is another name for symbols per second (a unit of symbol rate), so you can't convert between it and the others (which are units of bit rate) without knowing how many symbols are in use. A symbol ...
• 22.7k
Accepted
### How can I calculate the effects of an LNA, antenna gain, etc. on noise performance?
Try the Friis noise formula: $$F_{eq} = F_1 + {F_2-1 \over G_1} + {F_3-1 \over G_1 G_2} + \cdots \tag 1$$ $F_n$ is the noise factor of the n-th component, and likewise $G_n$ is the gain. The noise ...
• 49.8k
### How does IQ modulation work (intuitively)?
I think it's more intuitive if you unlearn some things first. Oscillation is not: $$\cos(\omega t)$$ where $\omega$ is the angular frequency in radians per second, and $t$ is time. Rather, ...
• 49.8k
Accepted
### Adding and subtracting dB/dBm/dBi values
A value given in "dB" is a dimensionless ratio. 10dB is a ratio of 10:1, -10dB is a ratio of 1:10, 3dB is a ratio of approximately 2:1, etc. dBi is dimensionless; it represents decibels ...
• 8,740
Accepted
### What is dB(μV/m), and what are its applications?
TL;DR: $\frac{V}{m}$ and $\text{dB}(\mu V/m)$ are units for the field strength of an electric field. For a practical application skip to the end! Derivation of the field strength A point charge $q_1$ ...
• 399
Accepted
### How does FM encode BOTH amplitude and frequencies of full audio signal spectrum?
Whatever modulation we use, there's a baseband signal we wish to transmit (music, voice recording, whatever), which somehow modulates a carrier to produce the output signal. Your question suggests ...
• 49.8k
Accepted
### Decibel subtraction in dBm and dB
I convert the 27dBm and 3dBm -> 10^2.7-10^0.3. But how can we simply subtract the two dBm If you convert to exponential form then you must simultaneously replace ...
• 22.7k
Accepted
### dBm signal minus dB
Decibels are all "ratios" at their core. A unitless dB is a simply a ratio of one number to another, perhaps input power relative to output power. We can also use decibels for absolute values, by ...
• 5,955
Accepted
### FT8 callsign/maidenhead compression encoding
You can find the code in packjt77.f90. Callsign encoding (for "standard" callsigns that don't require hashing) is in function pack28. A quick summary: ...
• 8,740
Accepted
### Why is the neper a useful unit for transmission line calculations?
A neper, just like a decibel, is a logarithmic expression of ratios. The decibel uses the base-10, or decadic, logarithm while the neper uses the natural, or Euler constant, logarithm. The decibel is ...
• 18.1k
Accepted
### How does power received by a dipole relate to electric field intensity?
There are a lot of ways to approach this problem, but here's one: we can calculate the power density of that field, and determine the area from which the antenna captures power, and multiply them ...
• 49.8k
Accepted
### Exactly why do some SWR meters give a changing reading depending on the length of coax used to connect to an antenna?
The SWR is related to the reflection coefficient $\Gamma$: $$\Gamma = {Z_L - Z_0 \over Z_L + Z_0 }$$ $$\text{VSWR} = {1+|\Gamma| \over 1 - |\Gamma|}$$ where: $Z_0$ is the feedline impedance, ...
• 49.8k
### dBm signal minus dB
Think about it this way. 27 dBm means 27dB above a milliwatt. Take 6dB away. Now you have something that's 21 dB above a milliwatt. Or, 21 dBm.
• 1,023
### How can the voltage at the center of a resonant half wave dipole be zero if the input impedance is 75 ohms?
Look closer at the diagram. At the two wires coming from the source, the voltage is NOT always zero. The only way for the voltage at the center point to be zero is for the two source wires to occupy ...
• 1,646
### The right antenna size
The classic dipole is a half-wave antenna. This means that the total length of the antenna is lambda/2. So writing it as 1/2-lambda is OK from an English language point of view, but not IMO as a ...
• 1,023
### Is free space path loss dependent on frequency?
I'd like to offer a parallel and simplified explanation to W8II's correct answer above, for the mathematically-challenged VHF+ enthusiasts among us. :-) As was mentioned in recent threads here, ...
• 7,485
Accepted
### Why is antenna aperture a function of wavelength?
This is a topic that troubles most students and even finds it way into many technical papers and textbooks in the form of incorrect assertions and conclusions. While you will find some reasonable ...
• 18.1k
Accepted
### SWR Measured at the Transmitter versus SWR at the Antenna
Given the matched loss of the feedline and the SWR at the transmitter, we can calculate the SWR at the antenna in three simple steps. First convert the SWR at the transmitter to the corresponding ...
• 18.1k
### How does FM encode BOTH amplitude and frequencies of full audio signal spectrum?
I generally understand that AM side bands occupy sufficiently wide band width to recreate the spectrum of the modulating audio signal (while varying amplitude at each frequency in the spectrum of the ...
• 22.7k
### How to decode a message with unknown bit rate in interstellar signal
An FSK signal which is the same symbol repeated is an unmodulated carrier, and like an unmodulated carrier, it contains no information. Making some assumptions about the bit shaping filter it might be ...
• 49.8k
### The right antenna size
If you make a dipole exactly a half-wavelength long, then it will be too long and out of resonance. The formula for determining the length of a half-wavelength dipole in feet is 468÷frequency in MHz. ...
• 7,485
### How can the voltage at the center of a resonant half wave dipole be zero if the input impedance is 75 ohms?
The simple answer is that the graphic is not accurate. There is an RMS voltage present at the center feed point of the dipole that follows Ohm's law relative to the feed point impedance. Most likely, ...
• 18.1k
### Exactly why do some SWR meters give a changing reading depending on the length of coax used to connect to an antenna?
Can someone tell me EXACTLY how the presence of current on the outside of the coax changes the VSWR reading ? When RF current is flowing on the outside of a coaxial cable, the exterior shield has ...
• 18.1k
### How does IQ modulation work (intuitively)?
I believe a good point of view is the concept of orthogonality. This is clear under everybody's eyes when seen in physical space, take for instance a 2-dimension space, a plane. In the example above ...
• 171
### The right antenna size
There are already a couple of nice answers to your question. I thought I would add a little more context - no need to vote for this as it is tangential to the question. A 1/2 wavelength (1/2 $\lambda$...
• 18.1k
### Calculating Antenna Length on the FCC Exam vs. in Reality
Thanks for the posting everyone! For licence test I took this approach after reading everyone's post & searching though all the technical details you posted. Convert Frequency to wavelength in ... | 2022-05-19 21:05:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.700127899646759, "perplexity": 1754.28075487214}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662530066.45/warc/CC-MAIN-20220519204127-20220519234127-00417.warc.gz"} |
https://samcogan.com/wth-is-pulumi/ | # WTH is Pulumi?
A slight deviation for the WTH articles this week, as we look at a non-Microsoft product. I’ve mentioned I’ve been working with Pulumi to define infrastructure as code recently and have been enjoying working with it, so let’s have a look at what this tool is an why it might be useful.
## What is Pulumi?
Pulumi is an Infrastructure as Code framework that you can use to deploy infrastructure in the cloud. Pulumi is relatively similar to Terraform in that it allows you to deploy infrastructure into many different cloud providers. Pulumi covers the 3 big cloud providers (Azure, AWS, GCP), but also has providers for many others, including:
• Kubernetes
• Alibaba Cloud
• Digital Ocean
• Linode
• vSphere
A full list of providers can be found here.
The significant feature that differentiates Pulumi from something like Terraform is that it allows you to write your Infrastructure as Code in real programming languages. With something like Terraform or ARM Templates your writing you templates using a domain-specific language (DSL) explicitly created for that tool, HCL for Terraform or the JSON based language for ARM. These DSL’s provide a limited set of language features. Pulumi does not have a DSL; instead, it allows you to use an existing program language that you are already familiar with. Currently, Pulumi supports:
• Node.js - including JavaScript, TypeScript or any other Node.js compatible language
• Python
• .NET Core, including C#, F#, and Visual Basic
• Go
Because you’re using a full programming language, you get access to all the language features to help create your infrastructure deployments. You can also use your existing tools like IDE’s, test frameworks, build pipelines and so on.
## How does Azure Pulumi Work?
Pulumi comprises several components:
• An SDK for your chosen language
• A command-line tool for running deployments
• A state storage facility - either local, self-hosted, or using the Pulumi Cloud service
Pulumi uses a state file in the same way as Terraform, to record what it has done and allow for actions like plan and destroy. If you want to stick to using the free, open-source components, you can host your own state file and manage that yourself. The Pulumi Cloud service provides hosting for the state file for you, as well as offering additional services on top, including policy for deployments, webhooks for deployment actions and a web UI for deployment management and history. Pulumi’s cloud offering has a free tier for a single user, with various paid levels for teams.
Defining infrastructure using Pulumi requires using the Pulumi SDK in your IDE of choice. The SDK is available using the standard package management solutions such as NuGet, NPM and PIP. Once installed, you can use the SDK to define your infrastructure. I’ve been working with C#, so we’ll look at some examples using that.
The example below creates a resource group and storage account:
using Pulumi;
using Pulumi.Azure.Core;
using Pulumi.Azure.Storage;
class MyStack : Stack
{
public MyStack()
{
// Create an Azure Resource Group
var resourceGroup = new ResourceGroup("resourceGroup");
// Create an Azure Storage Account
var storageAccount = new Account("storage", new AccountArgs
{
ResourceGroupName = resourceGroup.Name,
AccountReplicationType = "LRS",
AccountTier = "Standard"
});
// Export the connection string for the storage account
this.ConnectionString = storageAccount.PrimaryConnectionString;
}
[Output]
public Output<string> ConnectionString { get; set; }
}
If you’re familiar with Terraform, you will notice that the resources look quite similar. This is because under the hood Pulumi actually uses the Terraform provider to communicate with Azure, and so any resources you can create in Terraform, you can create in Pulumi. Some providers in Pulumi use the Terraform provider, others (like Kubernetes) use a custom provider created by Pulumi.
A couple of other things to call out in this example. First, you will note that the resource group we create doesn’t actually have many values set, just a name which is solely used for identifying this resource group in Pulumi, not in Azure. The rest of the values are defined using some functionality in Pulumi:
1. The actual name of the resource group in Azure is auto-generated by Pulumi. You don’t have to do this, you can define an explicit name if you want.
2. The location for the resource group is coming from a configuration file rather than the code. Pulumi has the concept of “stacks” which allow you to define a configuration for each different environment you want to deploy, switching to a different environment can be done at the command line
Once you’re ready to deploy your infrastructure, you use the Pulumi command-line tool. This is very similar to the Terraform CLI and offers commands like:
• Pulumi Preview - previews the changes you are going to make to let you know what actions it will take
• Pulumi Up - deploy your infrastructure
• Pulumi Destroy - remove your infrastructure
You can see the full list of commands here. The Pulumi CLI has a very nicely implemented auto-updating process that lets you view the status of the deployment and delve into preview data.
## Why would I want to use Pulumi?
If your looking for an infrastructure as code tool one of the critical questions you want to answer early on is if you are just deploying to a single cloud, or multiple clouds and other locations (like Kubernetes). If you are looking to deploy to various different clouds and services, then you want to look at tools like Pulumi or Terraform.
Beyond the multi-cloud functionality, Pulumi has several features that differentiate it from things like Terraform. The biggest of these is obviously the support for real programming languages. If you are looking to get your developers involved in infrastructure deployments and you want to meet them with a language they know, then this could be a big selling point. Even if you don’t need this for your developers, the ability to use proper languages, with traditional loops and conditionals may make this an appealing tool. I know I have found this to be a pleasing experience!
Another benefit, which may seem small initially, but has a big win with the security folks, for me, is state encryption. Terraform storing secrets in plain text in the state file as been a big concern for a lot of people, but doesn’t seem to be getting resolved any time soon. Pulumi has resolved this by allowing you to encrypt individual secrets in the state file. This also includes the use of an external provider for storing the encryption key, including Azure Key Vault, AWS KMS, GCP KMS and Hashicorp Vault.
I mentioned earlier that for some providers, like Azure, Pulumi uses the Terraform Azure provider under the hood. For other providers though they have created their own. One of these is the Kubernetes provider, which in my view is much better than the Terraform one, as it claims to cover 100% of the Kubernetes API. Pulumi have also recently released a slew of new Kubernetes components such as a tool to convert YAML to Pulumi code, a Kubernetes operator for deploying Pulumi code and the ability to define custom resource definitions in Pulumi. If you are working with Kubernetes, then Pulumi has a lot of really nice features.
Because Pulumi is a state-based tool, it can provide features like plan and destroy to preview changes before deploying them, and removing everything when you are done with it. Pulumi also offers a choice of where to store your state, either with their cloud service (free for 1 user, otherwise paid), or your own location using Azure Blob Storage, AWS S3 or GCP Cloud Storage.
If you are interested in using the paid service then as well as getting managed state storage, you also get access to deployment history, audit, CI/CD and Webhook integrations and policy enforcement. The open-source version using your own state backend is free.
## What issues does Pulumi have?
As with anything, there are some potential downsides or hurdles when using Pulumi. The first may or may not be an issue, but is the barrier to entry for non-developers. Developers familiar with one of the supported languages will pickup Pulumi very quickly, but if you are also looking to have your IT Pro’s or Ops people, who may not have experience with these languages, pick this up, then it can be daunting. To be fair to Pulumi the basics of deploying a cloud resource are relatively straight forward and aside from a few syntax changes, not that dissimilar to ARM or Terraform. However, once you want to start doing more advanced things, knowledge of the chosen programming language is required and so might be a barrier to some. From my personal experience, as a non-developer with some limited C# understanding, I was able to quickly get resources deployed, and build some custom modules. Still, there were some times I needed to talk to my developer colleagues about the best way to do things, or how to overcome an issue. If you have non-developers working with Pulumi, then it is a good idea to make sure they have access to developers who can assist.
Pulumi is a relatively new tool, around 3 years old, so you will find that the amount of examples, blog posts, support articles etc. is lower than something like Terraform. This is not unexpected but can be a bit of a pain when looking for examples or ways to solve a specific problem. The support for different languages can also add to this pain if you can only find a typescript example, and you wanted C#. However, hopefully, a bit of conversion can be done to come up with an answer.
Another potential issue is language parity. Not every feature is available in every language Pulumi supports. The standard features most people will use are present in all languages, but when you start trying to do some more advanced things, you may find that there are differences. Node/Typescript tends to have most features, whereas features like Customer Providers are missing from the .net languages. I am sure these will be added over time, but it is worth being aware of when you pick a language.
Another area that caught me out was some of the design decisions made by the Pulumi team. These were not necessarily wrong or bad, just different. I talked about one of this on more detail in this post where the decision made was to default to not refreshing state from the source when running. This is different from the way Terraform works and caused some confusion.
Finally, because some of the providers rely on the underlying Terraform providers then can be a slight delay in these getting updated when Terraform is updated (which itself can have a delay from when the cloud provider is updated). This delay is generally pretty low, often a few days, but something to be aware of. | 2023-03-28 04:44:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19446220993995667, "perplexity": 1991.554460735199}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948765.13/warc/CC-MAIN-20230328042424-20230328072424-00103.warc.gz"} |
https://socratic.org/questions/what-is-an-exponential-function | # What is an exponential function?
Mar 23, 2015
The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change in the dependent variable.
The function is often written as exp(x) It is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.
Mar 23, 2015
An exponential function is a function of the form $f \left(x\right) = {a}^{x}$ with $a > 0$ but $a \ne 1$.
For integer and rational $x$, we give definitions of ${a}^{x}$ earlier in algebra classes.
For irrational $x$ we owe you a definition, but one approach to the definition is to describe ${a}^{x}$ for irrational $x$ as the number thar ${a}^{r}$ gets closer to as rational $r$ get close to $x$. (We owe you a proof that there is a unique such number.)
Examples:
$f \left(x\right) = {2}^{x}$
$f \left(x\right) = {5}^{x}$
$f \left(x\right) = {\left(\frac{2}{5}\right)}^{x}$
$f \left(x\right) = {4}^{x} = {\left({2}^{2}\right)}^{x} = {2}^{2 x}$
The last example illustrates why we also consider $f \left(x\right) = {a}^{k x}$ for constant $k \ne 0$ to be exponential functions
We can write $f \left(x\right) = {a}^{k x} = {\left({a}^{k}\right)}^{x}$ and .for $a > 0$ and $k \ne 0$ this $f \left(x\right) = {b}^{x}$ for the "right kind" of $b$. ($b > 0$, and $b \ne 1$) | 2022-12-01 01:29:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 24, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9000284075737, "perplexity": 310.0564435586129}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710777.20/warc/CC-MAIN-20221130225142-20221201015142-00515.warc.gz"} |
https://inverseprobability.wordpress.com/category/machine-learning-2/ | # Legislation for Personal Data: Magna Carta or Highway Code?
Karl Popper is perhaps one of the most important thinkers from the 20th century. Not purely for his philosophy of science, but for giving a definitive answer to a common conundrum: “Which comes first, the chicken or the egg?”. He says that they were simply preceded by an ‘earlier type of egg’. I take this to mean that the answer is neither: they actually co-evolved. What do I mean by co-evolved? Well broadly speaking there once were two primordial entities which weren’t very chicken-like or egg-like at all, over time small changes occurred, supported by natural selection, rendering those entities unrecognisable from their origins into two of our most familiar foodstuffs of today.
I find the process of co-evolution remarkable, and to some extent unimaginable, or certainly it seems to me difficult to visualise the intermediate steps. Evolution occurs by natural selection: selection by the ‘environment’, but when we refer to co-evolution we are clarifying that this is a complex interaction. The primordial entities effect the environment around them, therefore changing the ‘rules of the game’ as far as survival is concerned. In such a convolved system certainties about the right action disappear very quickly.
What use are chickens and eggs when talking about personal data? Well, Popper used the question to illustrate a point about scientific endeavour. He was talking about science and reflecting on how scientific theories co-evolve with experiments. However, that’s not the point I’d like to make here. Co-evolution is very general, one area it arises is when technological advance changes society to such an extent that existing legislative frameworks become inappropriate. Tim Berners Lee has called for a Magna Carta for the digital age, and I think this is a worthy idea, but is it the right idea? A digital bill of rights may be the right idea in the longer run, but I don’t think we are ready to draft it yet. My own research is machine learning, the main technology underpinning the current AI revolution. A combination of machine learning, fast computers, and interconnected data means that the technological landscape is changing so fast that it is effecting society around us in ways that no one envisaged twenty years ago.
Even if we were to start with the primordial entities that presaged the chicken and the egg, and we knew all about the process of natural selection, could we have predicted or controlled the animal of the future that would emerge? We couldn’t have done. The chicken exists today as the product of its environmental experience, an experience that was unique to it. The end point we see is one of is highly sensitive to very small perturbations that could have occurred at the beginning.
So should we be writing legislation today which ties down the behaviour of future generations? There is precedent for this from the past. Before the printing press was introduced, no one would have begrudged the monks’ right to laboriously transcribe the books of the day. Printing meant it was necessary to protect the “copy rights” of the originator of the material. No one could have envisaged that those copyright laws would also be used to protect software, or digital music. In the industrial revolution the legal mechanism of ‘letters patent’ evolved to protect creative insight. Patents became protection of intellectual property, ensuring that inventors’ ideas could be shared under license. These mechanisms also protect innovation in the digital world. In some jurisdictions they are now applied to software and even user interface designs. Of course even this legislation is stretched in the face of digital technology and may need to evolve, as it has done in the past.
The new legislative challenge is not in protecting what is innovative about people, but what is commonplace about them. The new value is in knowing the nature of people: predicting their needs and fulfilling them. This is the value of interconnection of personal data. It allows us to make predictions about an individual by comparing him or her to others. It is the mainstay of the modern internet economy: targeted advertising and recommendation systems. It underpins my own research ideas in personalisation of health treatments and early diagnosis of disease. But it leads to potential dangers, particularly where the uncontrolled storage and flow of an individual’s personal information is concerned. We are reaching the point where some studies are showing that computer prediction of our personality is more accurate than that of our friends and relatives. How long before an objective computer prediction of our personality can outperform our own subjective assessment of ourselves? Some argue those times are already upon us. It feels dangerous for such power to be wielded unregulated by a few powerful groups. So what is the answer? New legislation? But how should it come about?
In the long term, I think we need to develop a set of rules and legislation, that include principles that protect our digital rights. I think we need new models of ownership that allow us to control our private data. One idea that appeals to me is extending data protection legislation with the right not only to view data held about us, but to also ask for it to be deleted. However, I can envisage many practical problems with that idea, and these need to be resolved so we can also enjoy the benefits of these personalised predictions.
As wonderful as some of the principles in the Magna Carta are, I don’t think it provides a good model for the introduction of modern legislation. It was actually signed under duress: under a threat of violent revolution. The revolution was threatened by a landed gentry, although the consequences would have been felt by all. Revolutions don’t always end well. They occur because people can become deadlocked: they envisage different futures for themselves and there is no way to agree on a shared path to different end points. The Magna Carta was also a deal between the king and his barons. Those barons were asking for rights that they had no intention of extending within their fiefdoms. These two characteristics: redistribution of power amongst a powerful minority, with significant potential consequences for the a disenfranchised majority, make the Magna Carta, for me, a poor analogy for how we would like things to proceed.
The chicken and the egg remind us that the actual future will likely be more remarkable than any of us can currently imagine. Even if we all seek a particular version of the future this version of the future is unlikely to ever exist in the form that we imagine. Open, receptive and ongoing dialogue between the interested and informed parties is more likely to bring about a societal consensus. But can this happen in practice? Could we really evolve a set of rights and legislative principles which lets us achieve all our goals? I’d like to propose that rather than taking as our example a mediaeval document, written on velum, we look to more recent changes in society and how they have been handled. In England, the Victorians may have done more than anyone to promote our romantic notion of the Magna Carta, but I think we can learn more by looking at how they dealt with their own legislative challenges.
I live in Sheffield, and cycle regularly in the Peak District national park. Enjoyment of the Peak Park is not restricted to our era. At 10:30 on Easter Monday in 1882 a Landau carriage, rented by a local cutler, was heading on a day trip from Sheffield to the village of Tideswell, in the White Peak. They’d left Sheffield via Ecclesall Road, and as they began to descend the road beneath Froggatt Edge, just before the Grouse Inn they encountered a large traction engine towing two trucks of coal. The Landau carriage had two horses and had been moving at a brisk pace of four and a half miles an hour. They had already passed several engines on the way out of Sheffield. However, as they moved out to pass this one, it let out a continuous blast of steam and began to turn across their path into the entrance of the inn. One of the horses took fright pulling the carriage up a bank, throwing Ben Deakin Littlewood and Mary Coke Smith from the carriage and under the wheels of the traction engine. I cycle to work past their graves every day. The event was remarkable at the time, so much so that is chiselled into the inscription on Ben’s grave.
The traction engine was preceded, as legislation since 1865 had dictated, by a boy waving a red flag. It was restricted to two and a half miles an hour. However, the boy’s role was to warn oncoming traffic. The traction engine driver had turned without checking whether the road was clear of overtaking traffic. It’s difficult to blame the driver though. I imagine that there was quite a lot involved in driving a traction engine in 1882. It turned out that the driver was also preoccupied with a broken wheel on one of his carriages. He was turning into the Grouse to check the wheel before descending the road.
This example shows how legislation can sometimes be extremely restrictive, but still not achieve the desired outcome. Codification of the manner in which a vehicle should be overtaken came later, at a time when vehicles were travelling much faster. The Landau carriage was overtaking about 100 meters after a bend. The driver of the traction engine didn’t check over his shoulder immediately before turning, although he claimed he’d looked earlier. Today both drivers’ responsibilities are laid out in the “Highway Code”. There was no “Mirror, Signal, Manoeuvre” in 1882. That came later alongside other regulations such as road markings and turn indicators.
The shared use of our road network, and the development of the right legislative framework might be a good analogy for how we should develop legislation for protecting our personal privacy. No analogy is ever perfect, but it is clear that our society both gained and lost through introduction of motorised travel. Similarly, the digital revolution will bring advantages but new challenges. We need to have mechanisms that allow for negotiated solutions. We need to be able to argue about the balance of current legislation and how it should evolve. Those arguments will be driven by our own personal perspectives. Our modern rules of the road are in the Highway Code. It lists responsibilities of drivers, motorcyclists, cyclists, mobility scooters, pedestrians and even animals. It gives legal requirements and standards of expected behaviour. The Highway Code co-evolved with transport technology: it has undergone 15 editions and is currently being rewritten to accommodate driverless cars. Even today we still argue about the balance of this document.
In the long term, when technologies have stabilised, I hope we will be able to distill our thinking to a bill of rights for the internet. But such a document has a finality about it which seems inappropriate in the face of technological uncertainty. Calls for a Magna Carta provide soundbites that resonate and provide rallying points. But they can polarise, presaging unhelpful battles. Between the Magna Carta and the foundation of the United States the balance between the English monarch and his subjects was reassessed through the English Civil War and the American Revolution. Wven tI don’t think we can afford such discord when drafting the rights of the digital age. We need mechanisms that allow for open debate, rather than open battle. Before a bill of rights for the internet, I think we need a different document. I’d like to sound the less resonant call for a document that allows for dialogue, reflecting concerns as they emerge. It could summarise current law and express expected standards of behaviour. With regular updating it would provide an evolving social contract between all the users of the information highway: people, governments, businesses, hospitals, scientists, aid organisations. Perhaps instead of a Magna Carta for the internet we should start with something more humble: the rules of the digital road.
This blog post is an extended version of an written for the Guardian’s media network: “Let’s learn the rules of the digital road before talking about a web Magna Carta”
# Proceedings of Machine Learning Research
Back in 2006 when the wider machine learning community was becoming aware of Gaussian processes (mainly through the publication of the Rasmussen and WIlliams book). Joaquin Quinonero Candela, Anton Schwaighofer and I organised the Gaussian Processes in Practice workshop at Bletchley Park. We planned a short proceedings for the workshop, but when I contacted Springer’s LNCS proceedings, a rather dismissive note came back with an associated prohibitive cost. Given that the ranking of LNCS wasn’t (and never has been) that high, this seemed a little presumptuous on their part. In response I contacted JMLR and asked if they’d ever considered a proceedings track. The result was that I was asked by Leslie Pack Kaelbling to launch the proceedings track.
JMLR isn’t just open access, but there is no charge to authors. It is hosted by servers at MIT and managed by the community.
We launched the proceedings in March 2007 with the first volume from the Gaussian Processes in Practice workshop. Since then there have been 38 volumes including two volumes in the pipeline. The proceedings publishes several leading conferences in machine learning including AISTATS, COLT and ICML.
From the start we felt that it was important to share the branding of JMLR with the proceedings, to show that the publication was following the same ethos as JMLR. However, this led to the rather awkward name: JMLR Workshop and Conference Proceedings, or JMLR W&CP. Following discussion with the senior editorial board of JMLR we now feel the time is right to rebrand with the shorter “Proceedings of Machine Learning Research”.
As part of the rebranding process the editorial team for the Proceedings of Machine Learning Research (which consists of Mark Reid and myself) is launching a small consultation exercise looking for suggestions on how we can improve the service for the community. Please feel free to leave comments on this blog post or via Facebook or Twitter to let us have feedback!
# Beware the Rise of the Digital Oligarchy
The Guardian’s media network published a short article I wrote for them on 5th March. They commissioned an article of about 600 words, that appeared on the Guardian’s site, but the original version I wrote was around 1400. I agreed a week’s exclusivity with the Guardian, but now that’s up, the longer version is below (it’s about twice as long).
On a recent visit to Genova, during a walk through the town with my colleague Lorenzo, he pointed out what he said was the site of the world’s first commercial bank. The bank of St George, located just outside the city’s old port, grew to be one of the most powerful institutions in Europe, it bankrolled Charles V and governed many of Genova’s possessions on the republic’s behalf. The trust that its clients placed in the bank is shown in records of its account holders. There are letters from Christopher Columbus to the bank instructing them in the handling of his affairs. The influence of the bank was based on the power of accumulated capital. Capital they could accumulate through the trust of a wealthy client base. The bank was so important in the medieval world that Machiavelli wrote that “if even more power was ceded by the Genovan republic to the bank, Genova would even outshine Venice amongst the Italian city states.” The Bank of St George was once one of the most influential private institutions in Europe.
Today the power wielded by accumulated capital can still dominate international affairs, but a new form of power is emerging, that of accumulated data. Like Hansel and Grettel trailing breadcrumbs into the forest, people now leave a trail of data-crumbs wherever we travel. Supermarket loyalty cards, text messages, credit card transactions, web browsing and social networking. The power of this data emerges, like that of capital, when it’s accumulated. Data is the new currency.
I’m a professor of machine learning. Machine learning is the main technique at the heart of the current revolution in artificial intelligence. A major aim of our field is to develop algorithms that better understand data: that can reveal the underlying intent or state of health behind the information flow. Already machine learning techniques are used to recognise faces or make recommendations, as we develop better algorithms that better aggregate data, our understanding of the individual also improves.
What do we lose by revealing so much of ourselves? How are we exposed when so much of our digital soul is laid bare? Have we engaged in a Faustian pact with the internet giants? Similar to Faust, we might agree to the pact in moments of levity, or despair, perhaps weakened by poor health. My father died last year, but there are still echoes of him on line. Through his account on Facebook I can be reminded of his birthday or told of common friends. Our digital souls may not be immortal, but they certainly outlive us. What we choose to share also affects our family: my wife and I may be happy to share information about our genetics, perhaps for altruistic reasons, or just out of curiosity. But by doing so we are also sharing information about our children’s genomes. Using a supermarket loyalty card gains us discounts on our weekly shop, but also gives the supermarket detailed information about our family diet. In this way we’d expose both the nature and nurture of our children’s upbringing. Will our decisions to make this information available haunt our children in the future? Are we equipped to understand the trade offs we make by this sharing?
There have been calls from Elon Musk, Stephen Hawking and others to regulate artificial intelligence research. They cite fears about autonomous and sentient artificial intelligence that could self replicate beyond our control. Most of my colleagues believe that such breakthroughs are beyond the horizon of current research. Sentient intelligence is still not at all well understood. As Ryan Adams, a friend and colleague based at Harvard tweeted:
Personally, I worry less about the machines, and more about the humans with enhanced powers of data access. After all, most of our historic problems seem to have come from humans wielding too much power, either individually or through institutions of government or business. Whilst sentient AI does seem beyond our horizons, one aspect of it is closer to our grasp. An aspect of sentient intelligence is ‘knowing yourself’, predicting your own behaviour. It does seem to me plausible that through accumulation of data computers may start to ‘know us’ even better than we know ourselves. I think that one concern of Musk and Hawking is that the computers would act autonomously on this knowledge. My more immediate concern is that our fellow humans, through the modern equivalents of the bank of St George, will be exploiting this knowledge leading to a form of data-oligarchy. And in the manner of oligarchies, the power will be in the hands of very few but wielded to the effect of many.
How do we control for all this? Firstly, we need to consider how to regulate the storage of data. We need better models of data-ownership. There was no question that Columbus was the owner of the money in his accounts. He gave it under license, and he could withdraw it at his pleasure. For the data repositories we interact with we have no right of deletion. We can withdraw from the relationship, and in Europe data protection legislation gives us the right to examine what is stored about us. But we don’t have any right of removal. We cannot withdraw access to our historic data if we become concerned about the way it might be used. Secondly, we need to increase transparency. If an algorithm makes a recommendation for us, can we known on what information in our historic data that prediction was based? In other words, can we know how it arrived at that prediction? The first challenge is a legislative one, the second is both technical and social. It involves increasing people’s understanding of how data is processed and what the capabilities and limitations of our algorithms are.
There are opportunities and risks with the accumulation of data, just as there were (and still are) for the accumulation of capital. I think there are many open questions, and we should be wary of anyone who claims to have all the answers. However, two directions seem clear: we need to both increase the power of the people; we need to develop their understanding of the processes. It is likely to be a fraught process, but we need to form a data-democracy: data governance for the people by the people and with the people’s consent.
Neil Lawrence is a Professor of Machine Learning at the University of Sheffield. He is an advocate of “Open Data Science” and an advisor to a London based startup, CitizenMe, that aims to allow users to “reclaim their digital soul”.
# Blogs on the NIPS Experiment
There are now quite a few blog posts on the NIPS experiment, I just wanted to put a place together where I could link to them all. It’s a great set of posts from community mainstays, newcomers and those outside our research fields.
Just as a reminder, Corinna and I were extremely open about the entire review process, with a series of posts about how we engaging the reviewers and processing the data. All that background can be found through a separate post here.
At the time of writing there is also still quite a lot of twitter traffic on the experiment.
List of Blog Posts
What an exciting series of posts and perspectives!
For those of you that couldn’t make the conference, here’s what it looked like.
And that’s just one of 5 or six poster rows!
# The NIPS Experiment
Just back from NIPS where it was really great to see the results of all the work everyone put in. I really enjoyed the program and thought the quality of all presented work was really strong. Both Corinna and I were particularly impressed by the work that put in by oral presenters to make their work accessible to such a large and diverse audience.
We also released some of the figures from the NIPS experiment, and there was a lot of discussion at the conference about what the result meant.
As we announced at the conference the consistency figure was 25.9%. I just wanted to confirm that in the spirit of openness that we’ve pursued across the entire conference process Corinna and I will provide a full write up of our analysis and conclusions in due course!
Some of the comment in the existing debate is missing out some of the background information we’ve tried to generate, so I just wanted to write a post that summarises that information to highlight its availability.
### Scicast Question
With the help of Nicolo Fusi, Charles Twardy and the entire Scicast team we launched a Scicast question a week before the results were revealed. The comment thread for that question already had an amount of interesting comment before the conference. Just for informational purposes before we began reviewing Corinna forecast this figure would be 25% and I forecast it would be 20%. The box plot summary of predictions from Scicast is below.
### Comment at the Conference
There was also an amount of debate at the conference about what the results mean, a few attempts to answer this question (based only on the inconsistency score and the expected accept rate for the conference) are available here in this little Facebook discussion and on this blog post.
### Background Information on the Process
Just to emphasise previous posts on this year’s conference see below:
### Software on Github
And finally there is a large amount of code available on a github site for allowing our processes to be recreated. A lot of it is tidied up, but the last sections on the analysis are not yet done because it was always my intention to finish those when the experimental results are fully released.
# NIPS: Decision Time
Thursday 28th August
In the last two days I’ve spent nearly 20 hours in teleconferences, my last scheduled conference will start in about 1/2 an hour. Given the available 25 minutes it seemed to make sense to try and put down some thoughts about the decision process.
The discussion period has been constant, there is a stream of incoming queries from Area Chairs, requests for advice on additional reviewers, or how to resolve deadlocked or disputing reviews. Corinna has handled many of these.
Since the author rebuttal period all the papers have been distributed to google spreadsheet lists which are updated daily. They contain paper titles, reviewer names, quality scores, calibrated scores, a probability of accept (under our calibration model), a list of bot-compiled potential issues as well as columns for accept/reject and poster/spotlight. Area chairs have been working in buddy pairs, ensuring that a second set of eyes can rest on each paper. For those papers around the borderline, or with contrasting reviews, the discussion period really can have an affect, we see when calibrating the reviewer scores: over time the reviewer bias is reducing and the scores are becoming more consistent. For this reason we allowed this period to go on a week longer than originally planned, and we’ve been compressing our teleconferences into the last few days.
Most teleconferences consist of two buddy pairs coming together to discuss their papers. Perhaps ideally the pairs would have a similar subject background, but constraints of time zone and the fact that there isn’t a balanced number of subject areas mean that this isn’t necessarily the case.
Corinna and I have been following a similar format. Listing the papers from highest scoring first, to lowest scoring, and starting at the top. For each paper, if it is a confident accept, we try and identify if it might be a talk or a spotlight. This is where the opinion of a range of Area Chairs can be very useful. For uncontroversial accepts that aren’t nominated for orals we spend very little time. This proceeds until we start reaching borderline papers, those in the ‘grey area’: typically papers with an average score around 6. They fall broadly into two categories: those where the reviewers disagree (e.g. scores of 8,6,4), or those where the review are consistent but the reviewers , perhaps, feel underwhelmed (scores of 6,6,6). Area chairs will often work hard to try and get one of the reviewers to ‘champion’ a paper: it’s a good sign if a reviewer has been prepared to argue the case for a paper in the discussion. However, the decisions in this region are still difficult. It is clear that we are rejecting some very solid papers, for reasons of space and because of the overall quality of submissions. It’s hard for everyone to be on the ‘distributing’ end of this system, but at the same time, we’ve all been on the receiving end of it too.
In this difficult ‘grey area’ for acceptance, we are looking for sparks in a paper that push it over the edge to acceptance. So what sort of thing catches an area chair’s eye? A new direction is always welcome, but often leads to higher variance in the reviewer scores. Not all reviewers are necessarily comfortable with the unfamiliar. But if an area chair feels a paper is taking the machine learning field somewhere new, then even if the paper has some weaknesses (e.g. in evaluation or giving context and detailed derivations etc) then we might be prepared to overlook this. We look at the borderline papers in some detail, scanning the reviews, looking for words like ‘innovative’, ‘new directions’ or ‘strong experimental results’. If we see these then as program chairs we definitely become more attentive. We all remember papers presented at NIPS in the past that lead to revolutions in the way machine learning is done. Both Corinna and I would love to have such papers at ‘our’ NIPS.
A paper that is a more developed area will be expected to have done a more rounded job in terms of setting the context and performing the evaluation. Papers in a more developed area will be expected to hit a high level in terms of their standards.
It is often helpful to have an extra pair of eyes (or even two pairs) run through the paper. Each teleconference call normally ends with a few follow up actions for a different area chair to look through a paper or clarify a particular point. Sometimes we also call in domain experts, who may have already produced four formal reviews of other papers, just to get clarification on particular point. This certainly doesn’t happen for all papers, but those with scores around 7,6,6 or 6,6,6 or 8,6,4 often get this treatment. Much depends on the discussion and content of the existing reviews, but there are still, often, final checks that need carrying out. From a program chair’s perspective, the most important thing is that the Area Chair is comfortable with the decision, and I think most of the job is acting as a sounding board for the Area Chair’s opinion, which I try to reflect back to them. In the same manner as rubber duck debugging, just vocalising the issues sometimes causes them to be crystallised in the mind. Ensuring that Area Chairs are calibrated to each other is also important. The global probabilities of accept from the reviewer calibration model really help here. As we go through papers I keep half an eye on those, not to influence the decision of a particular paper so much as to ensure that at the end of the process we don’t have a surplus of accepts. At this stage all decisions are tentative, but we hope not to have to come back to too many of them.
Monday 1st September
Corinna finished her last video conference on Friday, Saturday, Sunday and Monday (Labor Day) were filled with making final decisions on accepts, then talks and finally spotlights. Accepts were hard, we were unable to take all the papers that were possible accept, as we would have gone way over our quota of 400. We had to make a decision on duplicated papers where the decisions were in conflict, more details of this to come at the conference. From remembering what a pain it was to do the schedule after the acceptances, and also following advice from Leon Bottou that the talk program emerges to reflect the accepted posters, we finalized the talk and spotlight program whilst putting talks and spotlights directly into the schedule. We had to hone the talks down to 20 from about 40 candidates and spotlights we squeezed in 62 from over a hundred suggestions. We spent three hours in teleconference each day, as well as preparation time, across Labor Day weekend putting together the first draft of the schedule. It was particularly impressive how quickly area chairs responded to any of our follow up queries to our notes from the teleconferences. Particularly those in the US who were enjoying the traditional last weekend of summer.
Tuesday 2nd September
I had an all day meeting in Manchester for the a network of researchers focussed on mental illness. It was really good to have a day discussing research, my first in a long time. I thought very little about NIPS until on the train home, I thought to have a little look at the conference shape. I actually ended up looking at a lot of the papers we rejected, many from close colleagues and friends. I found it a little depressing. I have no doubt there is a lot of excellent work there, and I know how disappointed my friends and colleagues will be to receive those rejections. We did an enormous amount to ensure that the process was right, and I have every confidence in the area chairs and reviewers. But at the end of the day, you know that you will be rejecting a lot of good work. It brought to mind a thought I had at the allocation stage. When we had the draft allocation to each area chair, I went through several of them sanity checking the quality of the allocation. Naturally, I checked those associated with area chairs who are closer to my own areas of expertise. I looked through the paper titles, and I couldn’t help but think what a good workshop each of those allocations would make. There would be some great ideas, some partially developed ideas. There would be some really great experiments and some weaker experiments. But there would be a lot of debate at such workshop. None or very few of the papers would be uninteresting: there would certainly be errors in papers, but that’s one of the charms of a workshop, there’s still a lot more to be said about an idea when it’s presented at a workshop.
Friday 5th September
Returning from an excellent two day UCL-Duke workshop. There is a lot of curiosity about the NIPS experiment, but Corinna and I have agreed to keep the results embargoed until the conference.
Saturday 6th September
Area chairs had until Thursday to finalise their reviews in the light of the final decisions, and also to raise any concerns they had about the final decisions. My own experience of area chairing is that you can have doubts about your reasoning when you are forced to put pen to paper and write the meta review. We felt it was important to not rush the final process to allow any of those doubts to emerge. In the end, the final program has 3 or 4 changes from the draft we first distributed on Monday night, so there may be some merit in this approach. We had a further 3 hour teleconference today to go through the meta-reviews, with a particular focus on those for papers around the decision boundary. Other issues such as comments in the wrong place (the CMT interface can be fairly confusing, 3% of meta reviews were actually placed in the box meant for notes to the program chairs) were also covered. Our big concern was if the area chairs had written a review consistent with our final verdict. A handy learning task would have been to build a sentiment model to predict accept/reject from the meta review.
Monday 8th September
Our plan had been to release reviews this morning, but we were still waiting for a couple of meta-reviews to be tidied up and had an outstanding issue on one paper. I write this with CMT ‘loaded’ and ready to distribute decisions. However, when I preview the emails the variable fields are not filled in (if I hit ‘send’ I would send 5,000 emails that start “Dear $RecipientFirstName$, which sounds somewhat impersonal … although perhaps more critical is that the authors would be informed of the fate of paper “$Title$,” which may lead to some confusion. CMT are on a different time zone, 8 hours behind. Fortunately, it is late here, so there is a good chance they will respond in time …
Tuesday 9th September
I was wide awake at 6:10 despite going to sleep at 2 am. I always remember when I was Area Chair with John Platt that he would be up late answering emails and then out of bed again 4 hours later doing it again. A few final checks and the all clear for everything is there. Pressed the button at 6:22 … emails are still going out and it is 10:47. 3854 of the 5615 emails have been sent … one reply which was an out of office email from China. Time to make a coffee …
Final Statistics
1678 submissions
414 papers accepted
20 papers for oral
62 for spotlight
331 for poster
19 rejected without review
Epilogue to Decision Mail: So what was wrong with those variable names? I particularly like the fact that something different was wrong with each one. $RecipientFirstName$ and $RecipientEmail$ are not available in the “Notification Wizard”, whereas they are in the normal email sending system. Then I got the other variables wrong, $Title$->$PaperTitle$ and $PaperId$->$PaperID$, but since neither of the two I knew to be right were working I assumed there was something wrong with the whole variable substitution system … rather than it being that (at least) two of the variable types just happen to be missing from this wizard … CMT responded nice and quickly though … that’s one advantage of working late.
Epilogue on Acceptances: At the time of the conference there were only 411 papers presented because three were withdrawn. Withdrawals were usually due to some deeper problem authors had found in there own work, perhaps triggered by comments from reviewers. So in the end there were 411 papers accepted and 328 posters.
Author Concerns
So the decisions have been out for a few days now, and of course we have had some queries about our processes. Every one has been pretty reasonable, and their frustration is understandable when three reviewers have argued for accept but the final decision is to reject. This is an issue with ‘space-constrained’ conferences. Whether a paper gets through in the end can depend on subjective judgements about the paper’s qualities. In particular, we’ve been looking for three components to this: novelty, clarity and utility. Papers with borderline scores (and borderline here might be that the average score is in the weak accept range) are examined closely. The decision about whether the paper is accepted at this point necessarily must come down to judgement, because for a paper to get scores this high the reviewers won’t have identified a particular problem with the paper. The things that come through are how novel the paper is, how useful the idea is, and how clearly it’s presented. Several authors seem to think that the latter should be downplayed. As program chairs, we don’t necessarily agree. It’s true that it is a great shame when a great idea is buried in poor presentation, but it’s also true that the objective of a conference is communication, and therefore clarity of presentation definitely plays a role. However, it’s clear that all these three criteria are a matter of academic judgement: that of the reviewers, the area chair and the quad groups in the teleconferences. All the evidence we’ve seen is that reviewers and area chairs did weigh these aspects carefully, but that doesn’t mean that all their decisions can be shown to be right, because they are often a matter of perspective. Naturally authors are upset when what feels like a perfectly good paper is rejected on more subjective grounds. Most of the queries are on papers where this is felt to be the case.
There has also been one query on process, and whether we did enough to evaluate on these criteria, for those papers in the borderline area, before author rebuttal. Authors are naturally upset when the area chair raises such issues in the final decision’s meta review, but these points weren’t there before. Personally I sympathise with both authors and area chairs in this case. We made some effort to encourage authors to identify such papers before rebuttal (we sent out attention reports that highlighted probable borderline papers) but our main efforts at the time were chasing missing and inappropriate or insufficient reviews. We compressed a lot into a fairly short time, and it was also a period when many are on holiday. We were very pleased with the performance of our area chairs, but I think it’s also unsurprising if an area chair didn’t have time to carefully think through these aspects before author rebuttal.
My own feeling is that the space constraint on NIPS is rather artificial, and a lot of these problems would be avoided if it wasn’t there. However, there is a counter argument that suggests that to be a top quality conference NIPS has to have a high reject rate. NIPS is used in tenure cases within the US and these statistics are important there. Whilst I reject these ideas: I don’t think the role of a conference is to allow people to get promoted in a particular country, nor is that the role of a journal: they are both involved in the communication and debate of scientific ideas. However, I do not view the program chair roles as reforming the conference ‘in their own image’. You have to also consider what NIPS means to the different participants.
NIPS as Christmas
# Reviewer Calibration for NIPS
One issue that can occur for a conference is differences in interpretation of the reviewing scale. For a number of years (dating back to at least NIPS 2002) mis-calibration between reviewers has been corrected for with a model. Area chairs see not just the actual scores of the paper, but also ‘corrected scores’. Both are used in the decision making process.
Reviewer calibration at NIPS dates back to a model first implemented in 2002 by John Platt when he was an area chair. It’s a regularized least squares model that Chris Burges and John wrote up in 2012. They’ve kindly made their write up available here.
Calibrated scores are used alongside original scores to help in judging the quality of papers.
We also knew that Zoubin and Max had modified the model last year, along with their program manager Hong Ge. However, before going through the previous work we first of all approached the question independently. However, the model we came up with turned out to be pretty much identical to that of Hong, Zoubin and Max, and the approach we are using to compute probability of accepts was also identical. The model is a probabilistic reinterpretation of the Platt and Burges model: one that treats the bias parameters and quality parameters as latent variables that are normally distributed. Marginalizing out the latent variables leads to an ANOVA style description of the data.
### The Model
Our assumption is that the score from the $j$th reviewer for the $i$th paper is given by
$y_{i,j} = f_i + b_j + \epsilon_{i, j}$
where $f_i$ is the objective quality of paper $i$ and $b_j$ is an offset associated with reviewer $j$. $\epsilon_{i,j}$ is a subjective quality estimate which reflects how a specific reviewer’s opinion differs from other reviewers (such differences in opinion may be due to differing expertise or perspective). The underlying ‘objective quality’ of the paper is assumed to be the same for all reviewers and the reviewer offset is assumed to be the same for all papers.
If we have $n$ papers and $m$ reviewers then this implies $n$ + $m$ + $nm$ values need to be estimated. Of course, in practice, the matrix is sparse, and we have no way of estimating the subjective quality for paper-reviewer pairs where no assignment was made. However, we can firstly assume that the subjective quality is drawn from a normal density with variance $\sigma^2$
$\epsilon_{i, j} \sim N(0, \sigma^2 \mathbf{I})$
which reduces us to $n$ + $m$ + 1 parameters. The Platt-Burges model then estimated these parameters by regularized least squares. Instead, we follow Zoubin, Max and Hong’s approach of treating these values as latent variables. We assume that the objective quality, $f_i$, is also normally distributed with mean $\mu$ and variance $\alpha_f$,
$f_i \sim N(\mu, \alpha_f)$
this now reduces us to $m$+3 parameters. However, we only have approximately $4m$ observations (4 papers per reviewer) so parameters may still not be that well determined (particularly for those reviewers that have only one review). We therefore also assume that the reviewer offset is a zero mean normally distributed latent variable,
$b_j \sim N(0, \alpha_b),$
leaving us only four parameters: $\mu$, $\sigma^2$, $\alpha_f$ and $\alpha_b$. When we combine these assumptions together we see that our model assumes that any given review score is a combination of 3 normally distributed factors: the objective quality of the paper (variance $\alpha_f$), the subjective quality of the paper (variance $\sigma^2$) and the reviewer offset (variance $\alpha_b$). The a priori marginal variance of a reviewer-paper assignment’s score is the sum of these three components. Cross-correlations between reviewer-paper assignments occur if either the reviewer is the same (when the cross covariance is given by $\alpha_b$) or the paper is the same (when the cross covariance is given by $\alpha_f$). With a constant mean coming from the mean of the ‘subjective quality’, this gives us a joint model for reviewer scores as follows:
$\mathbf{y} \sim N(\mu \mathbf{1}, \mathbf{K})$
where $\mathbf{y}$ is a vector of stacked scores $\mathbf{1}$ is the vector of ones and the elements of the covariance function are given by
$k(i,j; k,l) = \delta_{i,k} \alpha_f + \delta_{j,l} \alpha_b + \delta_{i, k}\delta_{j,l} \sigma^2$
where $i$ and $j$ are the index of the paper and reviewer in the rows of $\mathbf{K}$ and $k$ and $l$ are the index of the paper and reviewer in the columns of $\mathbf{K}$.
It can be convenient to reparameterize slightly into an overall scale $\alpha_f$, and normalized variance parameters,
$k(i,j; k,l) = \alpha_f(\delta_{i,k} + \delta_{j,l} \frac{\alpha_b}{\alpha_f} + \delta_{i, k}\delta_{j,l} \frac{\sigma^2}{\alpha_f})$
which we rewrite to give two ratios: offset/objective quality ratio, $\hat{\alpha}_b$ and subjective/objective ratio $\hat{\sigma}^2$ ratio.
$k(i,j; k,l) = \alpha_f(\delta_{i,k} + \delta_{j,l} \hat{\alpha}_b + \delta_{i, k}\delta_{j,l} \hat{\sigma}^2)$
The advantage of this parameterization is it allows us to optimize $\alpha_f$ directly through maximum likelihood (with a fixed point equation). This leaves us with two free parameters, that we might explore on a grid.
We expect both $\mu$ and $\alpha_f$ to be very well determined due to the number of observations in the data. The negative log likelihood is
$\frac{|\mathbf{y}|}{2}\log2\pi\alpha_f + \frac{1}{2}\log \left|\hat{\mathbf{K}}\right| + \frac{1}{2\alpha_f}\mathbf{y}^\top \hat{\mathbf{K}}^{-1} \mathbf{y}$
where $|\mathbf{y}|$ is the length of $\mathbf{y}$ (i.e. the number of reviews) and $\hat{\mathbf{K}}=\alpha_f^{-1}\mathbf{K}$ is the scale normalised covariance. This negative log likelihood is easily minimized to recover
$\alpha_f = \frac{1}{|\mathbf{y}|} \mathbf{y}^\top \hat{\mathbf{K}}^{-1} \mathbf{y}$
A Bayesian analysis of $alpha_f$ parameter is possible with gamma priors, but it would merely shows that this parameter is extremely well determined (the degrees of freedom parameter of the associated Student-$t$ marginal likelihood scales will the number of reviews, which will be around $|\mathbf{y}| \approx 6,000$ in our case.
We can set these parameters by maximum likelihood and then we can remove the offset from the model by computing the conditional distribution over the paper scores with the bias removed, $s_{i,j} = f_i + \epsilon_{i,j}$. This conditional distribution is found as
$\mathbf{s}|\mathbf{y}, \alpha_f,\alpha_b, \sigma^2 \sim N(\boldsymbol{\mu}_s, \boldsymbol{\Sigma}_s)$
where
$\boldsymbol{\mu}_s = \mathbf{K}_s\mathbf{K}^{-1}\mathbf{y}$
and
$\boldsymbol{\Sigma}_s = \mathbf{K}_s - \mathbf{K}_s\mathbf{K}^{-1}\mathbf{K}_s$
and $\mathbf{K}_s$ is the covariance associated with the quality terms only with elements given by,
$k_s(i,j;k,l) = \delta_{i,k}(\alpha_f + \delta_{j,l}\sigma^2)$.
We now use $\boldsymbol{\mu}_s$ (which is both the mode and the mean of the posterior over $\mathbf{s}$) as the calibrated quality score.
### Analysis of Variance
The model above is a type of Gaussian process model with a specific covariance function (or kernel). The variances are highly interpretable though, because the covariance function is made up of a sum of effects. Studying these variances is known as analysis of variance in statistics, and is commonly used for batch effects. It is known as an ANOVA model. It is easy to extend this model to include batch effects such as whether or not the reviewer is a student or whether or not the reviewer has published at NIPS before. We will conduct these analyses in due course. Last year, Zoubin, Max and Hong explored whether the reviewer confidence could be included in the model, but they found it did not help with performance on hold out data.
Scatter plot of Quality Score vs Calibrated Quality Score
### Probability of Acceptance
To predict the probability of acceptance of any given paper, we sample from the multivariate normal that gives the posterior over $\mathbf{s}$. These samples are sorted according to the values of $\mathbf{s}$, and the top scoring papers are considered to be accepts. These samples are taken 1000 times and the probability of acceptance is computed for each paper by seeing how many times the paper received a positive outcome from the thousand samples. | 2017-08-24 10:27:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 62, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3672730326652527, "perplexity": 1275.3842673414315}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886133449.19/warc/CC-MAIN-20170824101532-20170824121532-00033.warc.gz"} |
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A mixture is when two or more substances combine physically together. •All of the particles in a PURE SUBSTANCE are exactly the same. Further, the properties of it are also uniform throughout the sample. Silver and ordinary table salt are two examples of chemical substances. Mixed with sand, gravel and water, it becomes concrete, one of the most important building materials in the world. substance examples: sugar, water, aluminum : pure. For example, if a cartridge contains 1000 mg of CBD and costs $100, then the CBD cost is$100/1000mg = \$0. Pure substances cannot be separated into any other kinds of matter, while a mixture is a combination of two or more pure substances. Please select some articles/chapters to export citations. In this chapter, we would look into how to differentiate if a substance is pure (or is it a mixture). If the material is a pure substance, further classify it as either an. Hydrogen and oxygen, on the other hand, cannot be decomposed into simpler substances. 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Pyridoxine (vitamin B6) and its bioactive form, pyridoxal 5'-phosphate (P5P), are essential for such processes as amino acid metabolism, neurotransmitter synthesis, and glycogen (storage sugar) breakdown. A compound is a pure substance, not a mixture. Compounds Elements FAQs. Matter can be uncombined substances made from one type of atom, known as elements, or combined as compounds or mixtures. •Pure substances are made up of just one type of particle. Substances like water, carbon dioxide, sodium chloride and sugar are examples of compounds. Pure Substance. For example, one ATOM of gold would still have the properties of gold. Further, the properties of it are also uniform throughout the sample. Examples of pure substances include water, gases like carbon dioxide, oxygen and metals like platinum, gold and silver. Properties of Pure Matters: They are homogeneous. Pure substances cannot be separated into any other kinds of matter, while a mixture is a combination of two or more pure substances. Do you know the difference between an element and a compound? The difference between an element and a compound is that an element is a substance made of same type of atoms, whereas a compound is made of different elements in definite proportions. Hydrogen gas and pure iron are examples of pure substances. Students will use hands-on card sorting to create a rule for sorting matter. PURE SUBSTANCE EXAMPLES. Other pages in this section include elements, mixtures and compounds and individual pages about substances, elements, mixtures and compounds, plus pages about atoms, molecules and isotopes. If an element or compound exists in two states simultaneously, it can be a pure substance and a mixture at the same time. This can include filtration, distillation, evaporation, dissolve, use of magnets, etc. A mixture of two or more phases of a pure substance is still a pure substance as long as the chemical composition of all phases is the same. Gatorade 6. Air, for example, is a mixture of several gases, but it is often considered to be a pure substance because. They are either pure substances or mixtures. Some examples are oxygen,hydrogen,gold,silver,copper,common,salt,pure water ,alum. Examples of compounds include water (H 2 O) and salt (Sodium Chloride - NaCl). Examples: water (H2O molecules), oxygen (O2 molecules), argon (Ar atoms). A mixture contains two or more substances that are not united chemically to form a compound. example: Water, Sugar, Salt, etc. A compound is a pure substance. Atoms of the elements form bonds to combine and make up a molecule of the compound. All parts are. The teacher will help to clear any misconceptions about elements and compounds. A pure substance consists of only one type of substance not mixed with others. Examples of pure substances include tin, sulfur, diamond, water, pure sugar (sucrose), table salt (sodium chloride) and baking soda (sodium bicarbonate). Review: An element is a pure substance which contains just one type of atom. Here’s an example: Gold is an element. Example: Common salt (NaCl) Sodium is a highly reactive element. 3 Classify the following as element , compound and mixture: Iron , sea water , Milk Q. This topic includes Make-a-Map, Make-a-Movie, Creative Coding, Simulation, and GameUp, available only on desktop and/or tablet. Compounds * * CHARACTERISTICS OF COMPOUND It is a pure substance. Examples of elements are : Hydrogen , Silver , Nitrogen , Copper , Silicon etc. A compound is a substance that has two or more chemical elements whose atoms are bonded together. Explosives such as gunpowder or black. For example, water (H 2 O) is a compound that is composed of two hydrogen atoms for every oxygen atom. Pure substances (elements and compounds) have fixed properties, such as melting and boiling point and chemical reactivity. PURE SUBSTANCE EXAMPLES. If water ever tastes different then it isn't pure water; it fits into our next category. For example, when the elements mercury and oxygen combine, and a chemical reaction takes place, mercury (II) oxide is created. •PURE SUBSTANCES are the same throughout. Bronze, which is made from copper and tin, is an example of the first kind of alloy. A pure substance (an element or a compound) is made up of only one type of atom, or molecule. Examples of pure substances include water, gases like carbon dioxide, oxygen and metals like platinum, gold and silver. The formula for the element oxygen is O 2. muddy water). A pure substance (an element or a compound) is made up of only one type of atom, or molecule. Particle Theory. When you see distilled water (H 2 O), it's a pure substance. To provide some context, an element is a pure chemical substance made up entirely of a single atom. In other words, it is free of contaminants. To a non-chemist, a pure substance is anything composed of a single type of material. A pure substance exhibits the following properties: A pure substance has a fixed melting and boiling point. Even if you're inland, you need to remember that your tap water also has many compounds inside, and they act the same way that salt does. Answer: Any substance is said to be pure if and only if it is made up of single type of particles means all the particles of that substance should be of same chemical nature. , boiling point and melting point). nitrogen Pure substance (E) 5. A pure substance does not have to be of a single element or compound. The density in kilograms per cubic meter can be obtained by multiplying the table values by 1000. In Column B, list whether the substance is an element (E), a compound (C), a Heterogeneous Mixture (HM), or a Solution (S). Consider a pot of water. Mixture: 1. A pure substance is a form of matter that has a constant composition and properties that are constant throughout the sample. oxygen Pure substance (E) 15. A substance that is made of only one type of molecule is a pure substance. Impure substances. As per the above definitions, all elements and compounds. Mixtures that include substances in different states are almost always heterogeneous as well. Colloidal Solution. solution/ homogeneous mixture. A mixture contains two or more different substances that are only physically joined together, not chemically. Here is a picture of the compound, water: According to the picture, the hydrogen and oxygen atoms are chemically combined by chemical bonds. Pure substances are made of only one type of atom or molecule. Yet both compounds and elements are considered pure substances. As per the above definitions, all elements and compounds. 1) Pure Matter: Same types of atoms or molecules comprise pure matters. A pure substance has only one component Eg: Pure water is a pure substance. Further, the properties of it are also uniform throughout the sample. The diagram below show the molecular models of various elements and compounds. Pure substance cannot be separated into two or more substances by any mechanical or physical method. 10 per mg of CBD. Pure Substance/MixtureGraphic Organizer. Water, salt and sugar are. Colloidal Solution is a heterogeneous mixture in which particle size of substance is intermediate of true solution and suspension i. O) Pure Compound 5. A pure substance can be an element or a compound. A mixture of a pure substance is possible if part of that substance is in one state and is mixed with the same substance in a different state, for example if a solid is mixed with a liquid. Answer: Any substance is said to be pure if and only if it is made up of single type of particles means all the particles of that substance should be of same chemical nature. compound: a pure substance made up of atoms of 2 or more different elements joined together by chemical bonds. For example, sugar and salt are pure substances. Individually, these are all pure substances. A pure substance may be either a compound (water) or an element (gold). Mixtures Although chemists have a difficult time separating compounds into their specific elements, the different parts of a mixture can be easily separated by physical means, such as filtration. However, in many ways, the designation "pure" compound is an oxymoron, since all compounds are pure. For example, a mixture of crushed ice and water is a heterogeneous mixture because it has different properties depending on whether the properties of a lump of ice or liquid water are measured. Elements; Compounds; ELEMENTS - DEFINITION What is an element? According to modern view, in simpler terms, an element can be defined as a substance made up of atoms of same atomic number. Examples: Once combined the milk is still milk and the chocolate is still chocolate!. Although a compound may be composed of two different elements, it tends to possess a different chemical structure that is completely unique from. Here’s an example: Gold is an element. Iron, alcohol, salt are examples of pure matters. Can you recognise elements, compounds and mixtures? * An element contains just one type of atom. Baking soda (sodium bicarbonate). Examples of Pure Substances. that are made up of 2 or more elements. Any material that is not a mixture, is called a pure substance. Substances retain their identity (don’t change their composition) Example: Chocolate milk C. In chemistry class, we came to know the difference between a mixture, compound, and an element. A mixture of a pure substance is possible if part of that substance is in one state and is mixed with the same substance in a different state, for example if a solid is mixed with a liquid. Water is also a pure substance. This Topic is Part of the Theme: Teach This Topic. pure substance found in periodic table cannot be broken down compound: NaCl, NaHCO3, H20, C12H22O11 pure substance can be chemically separated mixture: Trail mix, Jelly beans, soda, milk, blood, chicken soup, sand, air NOT pure substance can physically be separated ELEMENT COMPOUND MIXTURE DIRECTIONS 1. It will remain same 2. Elements cannot be divided into smaller units without large amounts of energy. Answer (1 of 3): Water (h2o) salt (NaCl) refined sugar and others are considered a pure substance - the distinction arises in organic chemistry (pertaining to carbon)Pure substance - A sample of matter, either an element or a compound, that consists of only one component with definite physical and chemical properties and a definite composition. A mixture of two or more phases of a pure substance is still a pure substance as long as the chemical composition of all phases is the same. Element - An element is a substance composed of the same type of atoms (e. In chemistry, a pure substance is matter that has a fixed composition and definite properties. The element iron (Fe) is a pure substance, it is made up ONLY of iron atoms. Elements are also known as atoms. A pure substance is any single type of material; A substance can be anything. When a homogeneous substance consists of the same type of molecule with fixed and uniform composition throughout, then the substance is a pure substance. elemental equivalent atoms. A pure substance can be either an element or a compound, but the composition of a pure substance doesn’t vary. Phases of a Pure Substance A pure substance may exist in different phases. It reacts with chlorine which is a yellow-green poisonous gas. Baking soda (sodium bicarbonate). Elements are the simplest pure substance. Examples of each element, compound, mixture and solution for students to see and touch. Molecules of Compounds. A pure substance exhibits the following properties: A pure substance has a fixed melting and boiling point. Mixture: Examples of Chemical Substances. Export Citation (s) Export Citation. But when we look at matter in bulk, we see only the "forest", not the "trees"— the atoms and. The molecule of the compound always contains two or more atoms of different kinds. An element is a substance composed of atoms that have the same number of protons. PURE SUBSTANCE EXAMPLES. It has its characteristic taste, color, and odor. Pure substance can be identified as either elements or compounds. The Organic Chemistry Tutor 158,002 views 19:12. Students are given a worksheet to fill out during instruction to follow along. It is much more difficult to break down pure substances into their parts, and complex chemical methods are needed to do this. Part 4: Column A lists a substance. Switch now: Elements, Compounds, and Mixtures Jeopardy Review Flash Version Elements, Compounds, and Mixtures Play This Game Live Now Join Live Game as a Player. Carbon is considered an element while carbon dioxide is considered a compound. The elements are substances that are characterized by being made of the same type of atoms. Select each of the following that is an example of a mixture of pure elements and does not include one or more chemical compounds. Matter is made up of molecules, and molecules are made up of atoms or elements. A pure substance can be either an element or a compound, but the composition of a pure substance doesn’t vary. Solids: Hard in nature, it has a definite volume and shape. All the atoms in an element are of the same kind. Examples of mixture include the salt solution which is a 'mixture. This means that a pure substance can be either an element or a compound. You can tell it is made up of more than one substance. Pure substance cannot be separated into two or more substances by any mechanical or physical method. Gold metal. 2 Water is a compound. CBD: assisting you weed out of the facts Discover if CBD is exactly what you'll want to help handle those rigid, arthritic bones, epileptic fits, anxiety and sometimes even prevent cancer tumors. Elements; Compounds; ELEMENTS - DEFINITION What is an element? According to modern view, in simpler terms, an element can be defined as a substance made up of atoms of same atomic number. An example is sand at the bottom of a beaker of water. • Explain the importance of compounds, and generally describe how compounds are formed when two or more elements chemically combine. Can you recognise elements, compounds and mixtures? * An element contains just one type of atom. It is always homogeneous. Therefore, pure substance is homogenous. However, in many ways, the designation "pure" compound is an oxymoron, since all compounds are pure. Compounds-pure substances. An example is milk or Gatorade. The tests are separated into 4 different tests (Parts. Pure Substance: A substance in which there is only one type of particle. A compound is a pure substance composed of two or more different atoms chemically bonded to one another. It freezes at 0 °C and boils at 100 °C. Chemistry 12 Unit 3 - Solubility of Ionic Substances Tutorial 7 - Ionic and Molecular Solutions Page 3 Notice, there are no ions in the product, just I2 molecules. Constituents of Compounds and Mixtures. It might be broken down into simpler compounds, into its elements or a combination of the two. To a chemist, a pure substance usually refers to a sample of matter that has a distinct set of properties that are common to all other samples of that substance. Carbon monoxide CO. An example is milk or Gatorade. All parts are. For example, the molecule water contains two hydrogen atoms, each with one proton, and one B A compound C A pure substance D A physical property. It has its characteristic taste, color, and odor. A substance is matter which has a specific composition and specific properties. In a compound, the ingredients are present in a definite proportion. Examples: Once combined the milk is still milk and the chocolate is still chocolate!. A compound is a pure substance. All elements and compounds are pure substances. Two or more elements combined into one substance through a chemical reaction form a chemical compound. Elements and compounds are pure substances, and mixtures are combinations of substances. Some examples of Pure Substances are water, diamond, salt, sugar, and tin. It has its characteristic taste, color, and odor. Played 2658 times. Some examples of mixtures is mud, water and food coloring, water and oil, and pretty much any substance that is combined with another. Cement is a solid homogeneous mixture of calcium compounds. Elements cannot be decomposed into simpler substances. For example, Hydrogen (H) and oxygen (O) make water (H. Any substance which possesses mass and occupies space is called "MATTER". Pure Substance A substance that has a fixed chemical composition throughout is called pure substance. Water is a common heat exchange medium. A substance that cannot be changed into simpler substances: Element: Represented on the periodic table: Atom: The basic building block of matter: Atom: The smallest particle of an element: Chemical Symbols: Elements are represented by these: Compound: Two or more elements chemically combined: Compound: Carbon Dioxide is an example of this. A pure substance does not have to be of a single chemical element or compound, however. Compounds and solutions c. Mixtures are physical combinations of two or more elements and/or compounds. Although a compound is a pure substance, it contains two or more elements combined chemically. Sulfuric acid b. A pure substance has a constant composition and consist view the full answer. A mixture would be a glass of water with other things dissolved inside, maybe one of those powders you take if you get sick. In a compound, the ingredients are present in a definite proportion. For example, pure water with pure crushed ice in it is still a pure substance, but it is also a mixture of two states of the pure substance. Identification Lab- Mixtures & Pure Substances Name Date Period Station 1- Aluminum Staples Formula- Al a) Is this station a mixture or a pure substance? b) Is this an example of an element or a compound? c) What state of matter is Station 1? Station - 6 Salt Water Form ula - NaCl + H 2O a) Is the salt a mixture or pure substance?. A mixture of phases of two or more substance is can still a pure substance if it is. Molecules are formed when two or more atoms join together. A pure substance usually participates in a chemical reaction to form predictable products. com To a non-chemist, a pure substance is anything composed of a single type of material. A pure element consist of only one type of atom. Elements Compounds 7. Pure Substances Student. oxygen (0), nitrogen (N2) and argon (Ar) gases as they occur naturally in the Earth's atmosphere silicon dioxide (SiO), a common chemical constituent of sand a. Compounds are. When a homogeneous substance consists of the same type of molecule with fixed and uniform composition throughout, then the substance is a pure substance. Ihsan Barin is the author of Thermochemical Data of Pure Substances, 3rd Edition, published by Wiley. It has a uniform composition throughout the sample. Camphor is volatile, reactive and flammable. The formula for the element oxygen is O 2. Give examples of common compounds. We cannot break a pure substance into simpler substances by any physical means. If left alone, the sand will settle to the bottom. As per the above definitions, all elements and compounds. • Compound – a substance that consists of two or more elements chemically bonded • Compound is always composed of the same elements, in the same proportion by mass – Represented by a formula e. Most mixtures can be separated into pure substances, which may be either elements or compounds. on StudyBlue. Because of this, elements are called "pure" substances. For example, aluminum is a lightweight, shiny metal. There are many elements in the world, all examples, shown above. Gatorade 6. properties of a compound are entirely different from those of the elements from which it is made. A compound is a pure substance composed of two or more elements that are chemically. What is Volatility Volatility is directly associated with vapour pressure of a substance. A pure substance is pure, while a. Examples of pure substances include water, gold, glucose, carbon dioxide, oxygen and hydrogen. A pure substance (an element or a compound) is made up of only one type of atom, or molecule. Chemical substances can be divided into two major groups: pure substances and mixtures. Test out how much remember about the topic by taking up the test below. A pure substance exhibits the following properties: A pure substance has a fixed melting and boiling point. Yet both compounds and elements are considered pure substances. Compound Definition: Compounds - Compounds are chemical substances made up of two or more elements that are chemically bound together in a fixed ratio. Elements cannot be decomposed into simpler substances. Define the following terms: a. It does not have to be a single chemical element just as long as it is homogeneous throughout, like air. Elements such as helium and radon exist as gases at room temperature. If we pass an electrical current through molten NaCl, two new substances will be formed: sodium, a shiny metal so reactive that it must be stored out of contact with the air chlorine, a yellowish poisonous gas. Some examples are oxygen,hydrogen,gold,silver,copper,common,salt,pure water ,alum. A couple major misconceptions are students may incorrectly assume that only elements (not compounds) are pure substances. Sugar (sucrose). Scientists were further able to classify pure substances into two groups: elements and compounds. 2 Water is a compound. The formula Na 2CO 3 refers to pure sodium carbonate and tells us that this compound is always composed of sodium, carbon, and oxygen in a constant atom ratio of 2:1:3. Fold your. For example, aluminum is a lightweight, shiny metal. A pure substance is made up of one type of particle. The melting point of solid oxygen, for example, is -218. Pure Substance A substance that has a fixed chemical composition throughout is called pure substance. A good example would be ordinary salt, sodium chloride. If it is a pure substance, then on the second blank write if the example is an element or compound. Examples: water (H2O molecules), oxygen (O2 molecules), argon (Ar atoms). Examples of Compounds - Water Water is a compound because it is made up of more than one element - Hydrogen and Oxygen. All elements and compounds are pure substances. The transition between the solid and the liquid is so sharp for small samples of a pure substance that melting points can be measured to 0. Some examples of pure substances are gold, aluminum, and sugar. Example: H2O/Water Mixtures - two or more substances that are not chemically combined with each other and can be separated by physical means. What are Nonvolatile Substances – Definition, Properties, Characteristics, Examples 4. Discover techniques for determining whether several samples of matter are pure substances or mixtures. PURE SUBSTANCE EXAMPLES. As per the above definitions, all elements and compounds. Iron is formed only of iron (Fe) atoms; table salt is. Chemical substances can be divided into two major groups: pure substances and mixtures. Is only one chemical substance present in the sample being considered?" YES-Pure Substance No-Mixture. Difference between Element and Compound. If you need further information ask your chemistry tutor. Is it possible to have a molecule of an element? Explain and give an example. A substance is matter that has the same fixed composition and properties. That means that there are only water molecules in the liquid. Students will also be able to distinguish between homogeneous (solutions) and heterogeneous mixtures. muddy water). Salt solution is a mixture not a. PURE SUBSTANCE EXAMPLES. On the other hand, if you are looking for pure CBD oil, then it should have no trace of THC in its compound. On the molecular level, each element is composed of only one kind of atom [Figure 1. Chemistry 12 Unit 3 - Solubility of Ionic Substances Tutorial 7 - Ionic and Molecular Solutions Page 3 Notice, there are no ions in the product, just I2 molecules. A pure substance is made up of one type of particle. For example, sugar and salt are pure substances. Pure substances What is a pure substance? What is it made up of? Mixtures What is a mixture? What is it made up of? classified into classified as Elements What is an element? What is it represented by? What is it made up of? Give an example. Similarly, mixtures are also classified into types; homogeneous mixtures and heterogeneous mixtures: A pure substance can be either an element or a compound. Test out how much remember about the topic by taking up the test below. 3 Classify the following as element , compound and mixture: Iron , sea water , Milk Q. We cannot break a pure substance into simpler substances by any physical means. Most natural and synthetic substances are mixtures. Part 4: Column A lists a substance. Examples: Once combined the milk is still milk and the chocolate is still chocolate!. Elements are pure substances. •2 types of pure substances: •Elements: •Pure substance that can’t be separated into a simpler substance by physical or chemical means •Only contains one type of atom •Ex. PubChem users sometimes ask about the difference between a substance and a compound. For example, a mixture of crushed ice and water is a heterogeneous mixture because it has different properties depending on whether the properties of a lump of ice or liquid water are measured. Colloidal Solution. Physical changes, such as boiling, grinding, or filtering, do not break a substance down in a way that changes its chemical composition or properties. Pure Substance. sodium Pure substance (E) 11. A compound is a pure substance that contains atoms of two or more chemical elements in definite proportions that cannot be separated by physical means and are held together by chemical bonds. Solids: Hard in nature, it has a definite volume and shape. Definition of a compound for kids. Mixture of two compounds – two types of compounds present. muddy water). Here's an example: Gold is an element. There are about 117 elements, but carbon, hydrogen, nitrogen and oxygen are only a few that make up the largest portion of Earth. Examples of pure substances are water, silver, zinc oxide, table salt, ethanol, etc. Example: Pure water always has the exact same chemical and physical properties under the same conditions. Examples of pure substances include tin, sulfur, diamond, water, pure sugar (sucrose), table salt (sodium chloride) and baking soda (sodium bicarbonate). A mixture has variable combinations. Review: An element is a pure substance which contains just one type of atom. Pure substance 1. Water is also a pure substance. Natural ingredients – always check the product labels for ingredients. A pure substance is a substance that cannot be separated by physical means. from those of the elements that make them up. this really depends on what you mean by pure. Physical properties of a mixture: a)Vary with the amount of substance b)Depend on the volume of the substance c)Depend on the organization of the substance d)Vary depending upon its components Question 19. Although a compound may be composed of two different elements, it tends to possess a different chemical structure that is completely unique from. Pure compounds are created when elements combine permanently, forming one substance. An atom is the smallest particle of an element that still has all the properties of the element. A pure substance does not have to be of a single chemical element or compound, however. For example, Hydrogen (H) and oxygen (O) make water (H. Salad is a. Pure Substance. Compounds are substances. A couple major misconceptions are students may incorrectly assume that only elements (not compounds) are pure substances. Classification Of Pure Substances Chemical substances are made of atoms and molecules. If left alone, the sand will settle to the bottom. contains two or more elements physically combined together in a fixed ratio. Examples: water (H2O molecules), oxygen (O2 molecules), argon (Ar atoms). Elements are also known as atoms. Nearly all matter is found as a solid, liquid or gas. Each element contains only one kind of atom. Compounds are small molecules made up of two or. PURE SUBSTANCE EXAMPLES. A pure substance contains only one kind of compound. Each molecule of a compound is made from two or more different kinds of atoms that are chemically bonded. D) a solution. To understand why all compounds are "pure" it is important to first understand what constitutes a substance, as opposed to a mixture, as well as what constitutes a compound. Some examples of pure substance that are elements are sulfur and tin. Heterogeneous mixtures: Heterogeneous mixtures consist of different substances. Mixtures are physical combinations of two or more elements and/or compounds. There are about 117 elements, but carbon, hydrogen, nitrogen and oxygen are only a few that make up the largest portion of Earth. Pure substance 1. table salt is a pure substance as it is made up of only one type of particle, salt particle. Mixture of two compounds – two types of compounds present. •All of the particles in a PURE SUBSTANCE are exactly the same. Tin is a pure substance which has an atomic number of 50 and belongs to the post-transition metal group of 14 in the periodic Table of Elements. Pure Substances. There are many elements in the world, all examples, shown above. By: Naomi Halder. Many alloys are homogeneous mixtures of metals, or of a metal and a nonmetallic substance. Examples: pure water, pure helium, pure copper, etc. A few examples of pure substances are water, gold, salt, sugar, and diamonds. You can now roughly evaluate its boiling point. Pure substances can be elements and/or compounds. An element is a pure substance that cannot be separated into simpler substances by chemical or physical means. For example, Hydrogen (H) and oxygen (O) make water (H. It is often convenient to regard compounds as formed upon certain types; alcohol, for example, may be said to be a compound formed upon the water type, that is to say, a compound formed from water by displacing one of the atoms of hydrogen by the group of elements C 2 H 5, thus - H C2H5 O H O H Water Alcohol. Students will also be able to distinguish between homogeneous (solutions) and heterogeneous mixtures. Review: An element contains just one type of atom. Therefore, pure substance is homogenous. Elements are pure substances that can;t be broken down into simpler substances. Pure substances are further broken down into elements and compounds. Compounds are always homogeneous. A homogeneous mixture looks like it is just one substance. Pure vs Impure Substances Pure vs Impure Substances Impure Substances Pure Substances Heterogeneous Mixtures A mixture can be separated using a physical change. It does not have to be a single chemical element just as long as it is homogeneous throughout, like air. Examples: water, salt, sugar, and DNA. They are still a bit like the materials that make them up. Elements; Compounds; ELEMENTS - DEFINITION What is an element? According to modern view, in simpler terms, an element can be defined as a substance made up of atoms of same atomic number. 1) Pure Matter: Same types of atoms or molecules comprise pure matters. Which is an example of a compound? Example 1: Pure water is a compound made from two elements - hydrogen and oxygen. All samples of sodium chloride are chemically identical. filtration, evaporation, distillation or chromatography. Chlorine is a strong oxidizing agent (a chemical substance that gives up or takes on electrons from another substance). –Examples: salt, sugar, pure water, iron, oxygen, lead, and nickel. A pure substance is any single type of material; A substance can be anything. It can be a pure element or a pure compound and it can be separated by chemical methods like electrolysis. Back to the top. Give 3 examples of everyday pure substances. A pure substance has constant chemical composition. Smoke from a fire is example of colloidal system in which tiny particles of solid float in air. Gold •Compounds: •Pure substance made up of two or more different elements joined by chemical bonds •Made of elements in a specific ratio that is always the. Pure substances (elements and compounds) have fixed properties, such as melting and boiling point and chemical reactivity. 20 is the industry average. Only one type of element OR one type of compound is present. Also, students may incorrectly believe that pure substances are transparent, free from additives, and safe to ingest. , water •All particles are the same type of atom •E. Sodium metal has a relatively small, but important, number of uses. Diatomic molecule – a pure substance of two of the same atom bonded together. The camphor tree was used as a fragrant wood in Babylon and Egypt. The density in kilograms per cubic meter can be obtained by multiplying the table values by 1000. In this chapter, we would look into how to differentiate if a substance is pure (or is it a mixture). Give examples of common compounds. All elements and compounds are considered pure substances. They are therefore the elementary, or simplest, chemical substances - elements. Mixtures are physical combinations of two or more elements and/or compounds. All the matter in the universe is composed of the atoms of more than 100 different chemical elements, which are found both in pure form and combined in chemical compounds. Nearly all matter is found as a solid, liquid or gas. An example is sand at the bottom of a beaker of water. All matter is categorized as either a "pure" substance or a mixture. For example, when the elements mercury and oxygen combine, and a chemical reaction takes place, mercury (II) oxide is created. A mixture contains two or more substances that are not united chemically to form a compound. Ostensibly, compounds contain more than one type of material. pure substance: a sample of matter, either a single element or a single compound, that has definite chemical and physical properties. A pure substance contains only one type of particle. Students understand the terms ‘Mixture’ and ‘Compound’. MIXTURES PURE COMPOUNDS: A mixture can be physically separated into pure compounds or elements. Using videos or in-person examples can help explain more clearly. Examples of pure. A pure substance does not have to be of a single chemical element or compound, however. The properties of pure compounds are as follows. Each element is represented by a unique symbol. Are compounds always molecules? Explain with an example. •Element and Compounds are Pure substances Unit 1 Lesson 4 Pure Substances and Mixtures P54-55 The atoms that make up copper are all the same. Lesson Content Overview. Each substance retains its individual. A compound is a pure substance that contains atoms of two or more chemical elements in definite proportions that cannot be separated by physical means and are held together. Here are examples of pure substances. For example, one ATOM of gold would still have the. Mixtures are composed of two or more pure chemical substances in which their elements and compounds are mixed together giving them various compositions. Classify pure substances as elements or compounds based on particle diagrams or chemical formulas. In the real world, there are not too many 100% pure substances- water in a pond has biological impurities, water in your tap has chlorine or flourine, bottled water has flavor enhacning minerals. filtration, evaporation, distillation or chromatography. | 2020-08-05 08:04:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4903990626335144, "perplexity": 1158.3343319706478}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735916.91/warc/CC-MAIN-20200805065524-20200805095524-00125.warc.gz"} |
http://balbir.blogspot.com/2005_05_01_archive.html | Tuesday, May 31, 2005
What will be the next revolution in task execution?
Task execution and scheduling concepts came into limelight with the advent of multi-processing Operating Systems. Unix made processes popular and then SUN made threads popular (the defacto now for task execution). What do you think the next wave will be?
Monday, May 30, 2005
[RFC] How many addresses can you take in C?
Well, I decided that I would post a question and ask for comments. The question is
In the programming language "C", how many times can you take the address of a variable v? So if v is a variable can I use &v, &&v, &&&v, etc? What is the limit to taking addresses?
I think I know the answer and it seems straight forward. Once, I get comments, I will try and illustrate the answer
Thursday, May 26, 2005
The Rule of Three or More
I remember learning as a kid, that if a number is divisible by three, then the sum of the digits must be divisible by three. Ever wondered how this rule works and how someone must have discovered it. The proof is really simple and franckly quite amazing. I have been wanting to share it for a long long time now.
Well, lets use this theorem with b=3. Lets take a number 'a' for which we need to figure out divisibility by 3. Lets take an example, say 1021. We can rewrite 1021 as 1x1000+0x100+2x10+1.
Now try using mod 3 for the numbers above (Use Fermat's Little Theorem if you have to - more on that in the blogs to follow).
Any power of 10 mod 3 is 1. 10 mod 3 = 1, 100 mod 3 = 10 mod 3 x 10 mod 3 = 1 x 1 = 1. Thats a simple proof - right?
Now, in our example
1021 mod 3 = 1 mod 3 + 0 mod 3 + 2 mod 3 + 1 mod 3 (remember all powers of 10 mod 3 is 1, hence we are reduced to multiplying the digits with 1)
This is further equal to (1+0+2+1) mod 3 == 1 (Thanks Vinay!).
The proof should be trivial to extend to any generic number, by splitting it into powers of 10 and the digits and summing them up. All powers of 10 mod 3 yeild one, hence we need to use only the sum of the digits mod 3.
Tuesday, May 24, 2005
If you use a shell
I am sure many of you who use a shell like the bash shell or the korn shell or any other shell for that matter must be power users of it by now. I will probably blog on some fun shell techniques, like the one in this blog.
Korn shell comes with a variant of the "cd" command that very few people are aware of. It is very useful for people who tend to have symmetrical directory hierarchies
Lets assume that you are in a directory structure as shown below
(1) /home/user/programming/drivers/source/cxx
and want to change directories to
(2) /home/user/programming/user_mode/source/cxx
Well, in KSH you can say
"cd drivers user_mode" and it will change from (1) to (2). I love this feature and miss it in the BASH shell, so I wrote my own wrapper (that's why I love *NIX)
function cd
{
case $# in 1) builtin cd$1
return ;;
2) builtin cd ${PWD/$1/$2} return ;; *) builtin cd$*
return ;;
esac
}
In fact this version is slightly different from the KSH one, can you spot the difference?
Let me know if you do, I will post your name(s) on this blog
Saturday, May 21, 2005
Welcome Sathya to the world of blogging
For all of us who know Sathya N J, lets welcome him to the world of blogging
His first blog is at 360 Yahoo BLOG. I am glad I invited him to 360 yahoo.
Thought for the day
I have been thinking about security for a while. I have come up with a probable law, but I am not sure if it is already well know. Anyway, here goes.
"The More free memory you have on your system, the more your system will be vulnerable to security risks"
Balbir Singh
Stated Mathematically
Security risk is directly proportional to the free memory on your system.
Do you think it makes sense?
I will explain my theory in detail in another posting, sometime later.
Friday, May 20, 2005
Can my blog be audible - part II
I had asked in Can my blog be audible if I can make my blog audible at run time. Well I am glad to say that a 10MB plugin in the Opera Browser enables me to achieve what I want.
Cool! right?
Thursday, May 19, 2005
By Dijkstra "My recollections of operating system design". The article is hand written and can be found here and the PDF is here
Monday, May 16, 2005
Intermediate Trees
Well, I have Intermediate Tree (IT) output from my tiger compiler. I received some help with the IT generation, but the text output is just not readable. So, what I did was generated a graphical representation using graphviz.
The most difficult thing about IT generation is following static links. I am going to show you the tree and the code that generates it.
/* define a recursive function */let/* calculate n! */function nfactor(n: int): int = if n = 0 then 1 else n * nfactor(n-1)in nfactor(10)end
and here is the tree
The tree on the right shows the main code
Can you guess what tree the following code will generate
let type any = {any : int} var buffer := getchar()function readint(any: any) : int = let var i := 0 function isdigit(s : string) : int = ord(buffer)>=ord("0") & ord(buffer)<=ord("9") function skipto() = while buffer=" " | buffer="\n" do buffer := getchar() in skipto(); any.any := isdigit(buffer); while isdigit(buffer) do (i := i*10+ord(buffer)-ord("0"); buffer := getchar()); i end type list = {first: int, rest: list} function readlist() : list = let var any := any{any=0} var i := readint(any) in if any.any then list{first=i,rest=readlist()} else nil end function merge(a: list, b: list) : list = if a=nil then b else if b=nil then a else if a.first < b.first then list{first=a.first,rest=merge(a.rest,b)} else list{first=b.first,rest=merge(a,b.rest)} function printint(i: int) = let function f(i:int) = if i>0 then (f(i/10); print(chr(i-i/10*10+ord("0")))) in if i<0 then (print("-"); f(-i)) else if i>0 then f(i) else print("0") end function printlist(l: list) = if l=nil then print("\n") else (printint(l.first); print(" "); printlist(l.rest)) var list1 := readlist() var list2 := (buffer:=getchar(); readlist()) /* BODY OF MAIN PROGRAM */ in printlist(merge(list1,list2))end
Well, the tree could take up a whole room, so here is the
condensed version
If you find too many long left/right subtrees, well they are due to static links. The language is tiger, see Andrew Appel's Home Page for more details
WARNING: The IT have not been checked for correctness, I can only hope they are correct.
Blog(net)working
I am really happy about the fact that the number of blogs are growing very rapidly these days. The December 2004 communications of ACM has a great article on the distribution of blogs worldwide. That also means that many of my friends and in-turn their friends have blogs. Almost all of us maintain a set of links on the sidebar linking blogs of our friends. Even though I have not directly spoken to them for sometime now, I get to know what they are upto through their blogs. I like of think of it as Blogworking, instead of networking we have Blogworking. Get it?
Lets keep up the Blogworking.
Two college friends of mine, added to the "Blogs of friends of mine" on the sidebar.
Harsha K
Karthik
Friday, May 13, 2005
Wondering What I Have Been Upto?
I have not been updating my blog as frequently as I used to, of late. I have a good excuse, well I am busy with something and I am also working on finishing the tiger implementation. I have learnt many interesting things and I cannot wait to share them here.
Thursday, May 12, 2005
A Brief History of GUI
There is great article at ARS Technica. I loved it!
Monday, May 09, 2005
C++ Evolution
If you are interested in the evolution of C++, see these links
Can my blog be audible?
I am exploring techniques to make my blog audible, yes audible!
Unfortunately, I do not own the blog server and hence do not have access to any technology on the server side. I will be forced to implement something at the client end. Does anybody know of a good technology to help me make my blog audible?
Let me know if you do or any suggestions you have
Thursday, May 05, 2005
Andrew Appel states that return addresses were earlier pushed on the stack by the function call instruction. Data shows that it is faster and easier to pass the return addresses in a register. This has two advantages
1. It keeps the memory traffic down
2. It avoids building in any particular stack discipline into the machine
This is certainly true for MIPS, ARM, etc. For the Intel IA32 platform see Notes on Translating Three-Address Code to Assembly Code for the X86.
The return address is still stored on stack. Generally the ENTER, LEAVE and RET instructions are used for stack manipulation. Gcc uses CALL, LEAVE and RET.
Tuesday, May 03, 2005
The Composite Pattern
The Design Patterns (Gang of Four) book explains the composite pattern. See Composite Pattern
One example of a composite pattern is a file in a filesystem/directory hierarchy. The figure below shows a probable implementation of a filesystem hierarchy using the composite pattern
Sunday, May 01, 2005
First Analysis of Swap Space
When I was learning to configure my first Dynix/Ptx system, I was told by a senior team member to ensure that the swap is at least twice the memory size. I asked him what was the logic behind that, he said, it was a rule of thumb. In the time to come, I would learn the some reasoning behind it. I would at this point recommend “Modern Operating Systems, by Andrew S. Tanenbaum”. He uses Don Knuth’s Fifty Percent Rule as the basis for his analysis. Let me explain the Fifty Percent Rule first
The simple explanation of the rule follows. In the state of equilibrium
1. Half the number of memory operations are allocations, the other half is freeing
2. For the half that is freeing, half of those operations result in holes being merged (contiguous ones)
This tells us that the ratio of holes to allocated blocks is fifty percent.
In the state of equilibrium there are half as many holes as total allocated blocks. A hole is referred to as a available memory (free memory). If the total allocated memory in blocks is n then the number of holes is n/2.
The total available blocks is 3n/2 and available blocks is n/2. The ratio of free memory to total available memory (assuming block sizes are equal) comes to 1/3. So if you have 256MB of RAM on your system and want it to be available free for the next task you want to run, then you must allocate a swap size of 512MB, so that the ratio of 1/3 is held.
There are of course complications to this simple rule or calculation that I explained above. I will try and explain some of those complexities in the next series of swapping articles. | 2018-04-22 12:07:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3956173360347748, "perplexity": 1745.7479468970268}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945596.11/warc/CC-MAIN-20180422115536-20180422135536-00574.warc.gz"} |
http://jspayne.com/php/SummaryGet.php?FindGo=Loudspeaker | TheInfoList
A loudspeaker is an electroacoustic transducer; a device which converts an electrical audio signal into a corresponding sound. The most widely used type of speaker is the dynamic speaker. The sound source (e.g., a sound recording or a microphone) must be amplified or strengthened with an audio power amplifier before the signal is sent to the speaker. The dynamic speaker was invented in 1925 by Edward W. Kellogg and Chester W. Rice issued as US Patent 1,707,570. Apr 2, 1929. The dynamic speaker operates on the same basic principle as a dynamic microphone, but in reverse, to produce sound from an electrical signal. When an alternating current electrical audio signal is applied to its voice coil, a coil of wire suspended in a circular gap between the poles of a permanent magnet, the coil is forced to move rapidly back and forth due to Faraday's law of induction, which causes a diaphragm (usually conically shaped) attached to the coil to move back and forth, pushing on the air to create sound waves. Besides this most common method, there are several alternative technologies that can be used to convert an electrical signal into sound. Speakers are typically housed in a speaker enclosure or speaker cabinet which is often a rectangular box made of wood or sometimes plastic. The enclosure's materials and design play an important role in the quality of the sound. The enclosure generally must be as stiff and non-resonant as practically possible. Where high fidelity reproduction of sound is required, multiple loudspeaker transducers are often mounted in the same enclosure, each reproducing a part of the audible frequency range (picture at right). In this case, the individual speakers may be referred to as ''drivers'' and the entire unit is called a loudspeaker. Drivers made for reproducing high audio frequencies are called tweeters, those for middle frequencies are called mid-range drivers and those for low frequencies are called woofers. Extremely low frequencies (16Hz-~100Hz) may be reproduced by separate subwoofers. Smaller loudspeakers are found in devices such as radios, televisions, portable audio players, computers, and electronic musical instruments. Larger loudspeaker systems are used for music, sound reinforcement in theatres and concert halls, and in public address systems.
Terminology
The term ''loudspeaker'' may refer to individual transducers (also known as ''drivers'') or to complete speaker systems consisting of an enclosure and one or more drivers. To adequately and accurately reproduce a wide range of frequencies with even coverage, most loudspeaker systems employ more than one driver, particularly for higher sound pressure level or maximum accuracy. Individual drivers are used to reproduce different frequency ranges. The drivers are named subwoofers (for very low frequencies); woofers (low frequencies); mid-range speakers (middle frequencies); tweeters (high frequencies); and sometimes supertweeters, for the highest audible frequencies and beyond. The terms for different speaker drivers differ, depending on the application. In two-way systems there is no mid-range driver, so the task of reproducing the mid-range sounds is divided between the woofer and tweeter. Home stereos use the designation ''tweeter'' for the high-frequency driver, while professional concert systems may designate them as "HF" or "highs". When multiple drivers are used in a system, a filter network, called an audio crossover, separates the incoming signal into different frequency ranges and routes them to the appropriate driver. A loudspeaker system with ''n'' separate frequency bands is described as "''n''-way speakers": a two-way system will have a woofer and a tweeter; a three-way system employs a woofer, a mid-range, and a tweeter. Loudspeaker drivers of the type pictured are termed ''dynamic'' (short for electrodynamic) to distinguish them from other sorts including moving iron speakers, and speakers using piezoelectric or electrostatic systems.
History
Johann Philipp Reis installed an electric loudspeaker in his ''telephone'' in 1861; it was capable of reproducing clear tones, but later revisions could also reproduce muffled speech. Alexander Graham Bell patented his first electric loudspeaker (capable of reproducing intelligible speech) as part of his telephone in 1876, which was followed in 1877 by an improved version from Ernst Siemens. During this time, Thomas Edison was issued a British patent for a system using compressed air as an amplifying mechanism for his early cylinder phonographs, but he ultimately settled for the familiar metal horn driven by a membrane attached to the stylus. In 1898, Horace Short patented a design for a loudspeaker driven by compressed air; he then sold the rights to Charles Parsons, who was issued several additional British patents before 1910. A few companies, including the Victor Talking Machine Company and Pathé, produced record players using compressed-air loudspeakers. Compressed-air designs are significantly limited by their poor sound quality and their inability to reproduce sound at low volume. Variants of the design were used for public address applications, and more recently, other variations have been used to test space-equipment resistance to the very loud sound and vibration levels that the launching of rockets produces.
Moving-coil
The first experimental moving-coil (also called ''dynamic'') loudspeaker was invented by Oliver Lodge in 1898. The first practical moving-coil loudspeakers were manufactured by Danish engineer Peter L. Jensen and Edwin Pridham in 1915, in Napa, California. Like previous loudspeakers these used horns to amplify the sound produced by a small diaphragm. Jensen was denied patents. Being unsuccessful in selling their product to telephone companies, in 1915 they changed their target market to radios and public address systems, and named their product Magnavox. Jensen was, for years after the invention of the loudspeaker, a part owner of The Magnavox Company. and Rice in 1925 holding the large driver of the first moving-coil cone loudspeaker The moving-coil principle commonly used today in speakers was patented in 1925 by Edward W. Kellogg and Chester W. Rice issued as US Patent 1,707,570. Apr 2, 1929. The key difference between previous attempts and the patent by Rice and Kellogg is the adjustment of mechanical parameters so that the fundamental resonance of the moving system is below the frequency where the cone's radiation Impedance becomes uniform. About this same period, Walter H. Schottky invented the first ribbon loudspeaker together with Dr. Erwin Gerlach. These first loudspeakers used electromagnets, because large, powerful permanent magnets were generally not available at a reasonable price. The coil of an electromagnet, called a field coil, was energized by current through a second pair of connections to the driver. This winding usually served a dual role, acting also as a choke coil, filtering the power supply of the amplifier that the loudspeaker was connected to. AC ripple in the current was attenuated by the action of passing through the choke coil. However, AC line frequencies tended to modulate the audio signal going to the voice coil and added to the audible hum. In 1930 Jensen introduced the first commercial fixed-magnet loudspeaker; however, the large, heavy iron magnets of the day were impractical and field-coil speakers remained predominant until the widespread availability of lightweight alnico magnets after World War II.
First loudspeaker systems
In the 1930s, loudspeaker manufacturers began to combine two and three bandpasses' worth of drivers in order to increase frequency response and sound pressure level. In 1937, the first film industry-standard loudspeaker system, "Th
Shearer Horn
System for Theatres" (a two-way system), was introduced by Metro-Goldwyn-Mayer. It used four 15″ low-frequency drivers, a crossover network set for 375 Hz, and a single multi-cellular horn with two compression drivers providing the high frequencies. John Kenneth Hilliard, James Bullough Lansing, and Douglas Shearer all played roles in creating the system. At the 1939 New York World's Fair, a very large two-way public address system was mounted on a tower at Flushing Meadows. The eight 27″ low-frequency drivers were designed by Rudy Bozak in his role as chief engineer for Cinaudagraph. High-frequency drivers were likely made by Western Electric. Altec Lansing introduced the ''604'', which became their most famous coaxial Duplex driver, in 1943. It incorporated a high-frequency horn that sent sound through a hole in the pole piece of a 15-inch woofer for near-point-source performance. Altec's "Voice of the Theatre" loudspeaker system was first sold in 1945, offering better coherence and clarity at the high output levels necessary in movie theaters. The Academy of Motion Picture Arts and Sciences immediately began testing its sonic characteristics; they made it the film house industry standard in 1955. In 1954, Edgar Villchur developed the acoustic suspension principle of loudspeaker design in Cambridge, Massachusetts. This allowed for better bass response than previously from drivers mounted in smaller cabinets which was important during the transition to stereo recording and reproduction. He and his partner Henry Kloss formed the Acoustic Research company to manufacture and market speaker systems using this principle. Subsequently, continuous developments in enclosure design and materials led to significant audible improvements. The most notable improvements to date in modern dynamic drivers, and the loudspeakers that employ them, are improvements in cone materials, the introduction of higher-temperature adhesives, improved permanent magnet materials, improved measurement techniques, computer-aided design, and finite element analysis. At low frequencies, the application of electrical network theory to the acoustic performance allowed by various enclosure designs (initially by Thiele, and later by Small) has been very important at the design level.
Driver design: dynamic loudspeakers
The most common type of driver, commonly called a dynamic loudspeaker, uses a lightweight diaphragm, or ''cone'', connected to a rigid ''basket'', or ''frame'', via a flexible suspension, commonly called a ''spider'', that constrains a voice coil to move axially through a cylindrical magnetic gap. A protective cap glued in the cone's center prevents dust, especially iron filings, from entering the gap. When an electrical signal is applied to the voice coil, a magnetic field is created by the electric current in the voice coil, making it a variable electromagnet. The coil and the driver's magnetic system interact, generating a mechanical force that causes the coil (and thus, the attached cone) to move back and forth, accelerating and reproducing sound under the control of the applied electrical signal coming from the amplifier. The following is a description of the individual components of this type of loudspeaker.
Diaphragm
The diaphragm is usually manufactured with a cone- or dome-shaped profile. A variety of different materials may be used, but the most common are paper, plastic, and metal. The ideal material would 1) be rigid, to prevent uncontrolled cone motions; 2) have low mass, to minimize starting force requirements and energy storage issues; 3) be well damped, to reduce vibrations continuing after the signal has stopped with little or no audible ringing due to its resonance frequency as determined by its usage. In practice, all three of these criteria cannot be met simultaneously using existing materials; thus, driver design involves trade-offs. For example, paper is light and typically well damped, but is not stiff; metal may be stiff and light, but it usually has poor damping; plastic can be light, but typically, the stiffer it is made, the poorer the damping. As a result, many cones are made of some sort of composite material. For example, a cone might be made of cellulose paper, into which some carbon fiber, Kevlar, glass, hemp or bamboo fibers have been added; or it might use a honeycomb sandwich construction; or a coating might be applied to it so as to provide additional stiffening or damping.
The chassis, frame, or basket, is designed to be rigid, preventing deformation that could change critical alignments with the magnet gap, perhaps allowing the voice coil to rub against the magnet around the gap. Chassis are typically cast from aluminum alloy, in heavier magnet-structure speakers; or stamped from thin sheet steel in lighter-structure drivers. Other materials such as molded plastic and damped plastic compound baskets are becoming common, especially for inexpensive, low-mass drivers. Metallic chassis can play an important role in conducting heat away from the voice coil; heating during operation changes resistance, causes physical dimensional changes, and if extreme, broils the varnish on the voice coil; it may even demagnetize permanent magnets. The suspension system keeps the coil centered in the gap and provides a restoring (centering) force that returns the cone to a neutral position after moving. A typical suspension system consists of two parts: the ''spider'', which connects the diaphragm or voice coil to the lower frame and provides the majority of the restoring force, and the ''surround'', which helps center the coil/cone assembly and allows free pistonic motion aligned with the magnetic gap. The spider is usually made of a corrugated fabric disk, impregnated with a stiffening resin. The name comes from the shape of early suspensions, which were two concentric rings of Bakelite material, joined by six or eight curved "legs." Variations of this topology included the addition of a felt disc to provide a barrier to particles that might otherwise cause the voice coil to rub. The German firm Rulik still offers drivers with uncommon spiders made of wood.
Cone materials
Full-range drivers
A full-range driver is a speaker designed to be used alone to reproduce an audio channel without the help of other drivers, and therefore must cover the entire audio frequency range. These drivers are small, typically in diameter to permit reasonable high frequency response, and carefully designed to give low-distortion output at low frequencies, though with reduced maximum output level. Full-range (or more accurately, wide-range) drivers are most commonly heard in public address systems, in televisions (although some models are suitable for hi-fi listening), small radios, intercoms, some computer speakers, etc. In hi-fi speaker systems, the use of wide-range drive units can avoid undesirable interactions between multiple drivers caused by non-coincident driver location or crossover network issues. Fans of wide-range driver hi-fi speaker systems claim a coherence of sound due to the single source and a resulting lack of interference, and likely also to the lack of crossover components. Detractors typically cite wide-range drivers' limited frequency response and modest output abilities (most especially at low frequencies), together with their requirement for large, elaborate, expensive enclosures—such as transmission lines, quarter wave resonators or horns—to approach optimum performance. With the advent of neodymium drivers, low-cost quarter-wave transmission lines are made possible and are increasingly made availably commercially. Full-range drivers often employ an additional cone called a ''whizzer'': a small, light cone attached to the joint between the voice coil and the primary cone. The whizzer cone extends the high-frequency response of the driver and broadens its high frequency directivity, which would otherwise be greatly narrowed due to the outer diameter cone material failing to keep up with the central voice coil at higher frequencies. The main cone in a whizzer design is manufactured so as to flex more in the outer diameter than in the center. The result is that the main cone delivers low frequencies and the whizzer cone contributes most of the higher frequencies. Since the whizzer cone is smaller than the main diaphragm, output dispersion at high frequencies is improved relative to an equivalent single larger diaphragm. Limited-range drivers, also used alone, are typically found in computers, toys, and clock radios. These drivers are less elaborate and less expensive than wide-range drivers, and they may be severely compromised to fit into very small mounting locations. In these applications, sound quality is a low priority. The human ear is remarkably tolerant of poor sound quality, and the distortion inherent in limited-range drivers may enhance their output at high frequencies, increasing clarity when listening to spoken word material.
Subwoofer
A subwoofer is a woofer driver used only for the lowest-pitched part of the audio spectrum: typically below 200 Hz for consumer systems,Home Speakers Glossary
Crutchfield.com (2010-06-21). Retrieved on 2010-10-12.
below 100 Hz for professional live sound, and below 80 Hz in THX-approved systems. Because the intended range of frequencies is limited, subwoofer system design is usually simpler in many respects than for conventional loudspeakers, often consisting of a single driver enclosed in a suitable box or enclosure. Since sound in this frequency range can easily bend around corners by diffraction, the speaker aperture does not have to face the audience, and subwoofers can be mounted in the bottom of the enclosure, facing the floor. This is eased by the limitations of human hearing at low frequencies; such sounds cannot be located in space, due to their large wavelengths compared to higher frequencies which produce differential effects in the ears due to shadowing by the head, and diffraction around it, both of which we rely upon for localization clues. To accurately reproduce very low bass notes without unwanted resonances (typically from cabinet panels), subwoofer systems must be solidly constructed and properly braced to avoid unwanted sounds of cabinet vibrations. As a result, good subwoofers are typically quite heavy. Many subwoofer systems include integrated power amplifiers and electronic subsonic (sub)-filters, with additional controls relevant to low-frequency reproduction (e.g., a crossover knob and a phase switch). These variants are known as "active" or "powered" subwoofers, with the former including a power amplifier. In contrast, "passive" subwoofers require external amplification. In typical installations, subwoofers are physically separated from the rest of the speaker cabinets. Because of propagation delay, their output may be somewhat out of phase from another subwoofer (on another channel) or slightly out of phase with the rest of the sound. Consequently, a subwoofer's power amp often has a phase-delay adjustment (approximately 1 ms of delay is required for each additional foot of separation from the listener) which may improve performance of the system as a whole at subwoofer frequencies (and perhaps an octave or so above the crossover point). However, the influence of room resonances (sometimes called standing waves) is typically so large that such issues are secondary in practice. Subwoofers are widely used in large concert and mid-sized venue sound reinforcement systems. Subwoofer cabinets are often built with a bass reflex port (i.e., a hole cut into the cabinet with a tube attached to it), a design feature which if properly engineered improves bass performance and increases efficiency.
Woofer
A woofer is a driver that reproduces low frequencies. The driver works with the characteristics of the enclosure to produce suitable low frequencies (see speaker enclosure for some of the design choices available). Indeed, both are so closely connected that they must be considered together in use. Only at design time do the separate properties of enclosure and woofer matter individually. Some loudspeaker systems use a woofer for the lowest frequencies, sometimes well enough that a subwoofer is not needed. Additionally, some loudspeakers use the woofer to handle middle frequencies, eliminating the mid-range driver. This can be accomplished with the selection of a tweeter that can work low enough that, combined with a woofer that responds high enough, the two drivers add coherently in the middle frequencies.
Mid-range driver
A mid-range speaker is a loudspeaker driver that reproduces a band of frequencies generally between 1–6 kHz, otherwise known as the 'mid' frequencies (between the woofer and tweeter). Mid-range driver diaphragms can be made of paper or composite materials, and can be direct radiation drivers (rather like smaller woofers) or they can be compression drivers (rather like some tweeter designs). If the mid-range driver is a direct radiator, it can be mounted on the front baffle of a loudspeaker enclosure, or, if a compression driver, mounted at the throat of a horn for added output level and control of radiation pattern.
Tweeter
A tweeter is a high-frequency driver that reproduces the highest frequencies in a speaker system. A major problem in tweeter design is achieving wide angular sound coverage (off-axis response), since high frequency sound tends to leave the speaker in narrow beams. Soft-dome tweeters are widely found in home stereo systems, and horn-loaded compression drivers are common in professional sound reinforcement. Ribbon tweeters have gained popularity in recent years, as the output power of some designs has been increased to levels useful for professional sound reinforcement, and their output pattern is wide in the horizontal plane, a pattern that has convenient applications in concert sound.
Coaxial drivers
A coaxial driver is a loudspeaker driver with two or several combined concentric drivers. Coaxial drivers have been produced by many companies, such as Altec, Tannoy, Pioneer, KEF, SEAS, B&C Speakers, BMS, Cabasse and Genelec.
System design
Crossover
Used in multi-driver speaker systems, the crossover is an assembly of filters that separate the input signal into different frequency ranges (i.e. "bands"), according to the requirements of each driver. Hence the drivers receive power only at their operating frequency (the sound frequency range they were designed for), thereby reducing distortion in the drivers and interference between them. The ideal characteristics of a crossover may include perfect out-of-band attenuation at the output of each filter, no amplitude variation ("ripple") within each passband, no phase delay between overlapping frequency bands, to name just a few. Crossovers can be ''passive'' or ''active''. A passive crossover is an electronic circuit that uses a combination of one or more resistors, inductors, or non-polar capacitors. These components are combined to form a filter network and are most often placed between the full frequency-range power amplifier and the loudspeaker drivers to divide the amplifier's signal into the necessary frequency bands before being delivered to the individual drivers. Passive crossover circuits need no external power beyond the audio signal itself, but have some disadvantages: they may require larger inductors and capacitors due to power handling requirements (being driven by the amplifier), limited component availability to optimize the crossover's characteristics at such power levels, etc. Unlike active crossovers which include a built-in amplifier, passive crossovers have an inherent attenuation within the passband, typically leading to a reduction in damping factor before the voice coil Elliott Sound Products. Rod Elliott, 2004
''Active Vs. Passive Crossovers.''
Retrieved on June 16, 2009.
An active crossover is an electronic filter circuit that divides the signal into individual frequency bands ''before'' power amplification, thus requiring at least one power amplifier for each bandpass. Passive filtering may also be used in this way before power amplification, but it is an uncommon solution, being less flexible than active filtering. Any technique that uses crossover filtering followed by amplification is commonly known as bi-amping, tri-amping, quad-amping, and so on, depending on the minimum number of amplifier channels. Some loudspeaker designs use a combination of passive and active crossover filtering, such as a passive crossover between the mid- and high-frequency drivers and an active crossover between the low-frequency driver and the combined mid- and high frequencies. Passive crossovers are commonly installed inside speaker boxes and are by far the most usual type of crossover for home and low-power use. In car audio systems, passive crossovers may be in a separate box, necessary to accommodate the size of the components used. Passive crossovers may be simple for low-order filtering, or complex to allow steep slopes such as 18 or 24 dB per octave. Passive crossovers can also be designed to compensate for undesired characteristics of driver, horn, or enclosure resonances,Elliott Sound Products. Rod Elliott, 2004
''Design of Passive Crossovers.''
Retrieved on June 16, 2009.
and can be tricky to implement, due to component interaction. Passive crossovers, like the driver units that they feed, have power handling limits, have insertion losses (10% is often claimed), and change the load seen by the amplifier. The changes are matters of concern for many in the hi-fi world. When high output levels are required, active crossovers may be preferable. Active crossovers may be simple circuits that emulate the response of a passive network, or may be more complex, allowing extensive audio adjustments. Some active crossovers, usually digital loudspeaker management systems, may include electronics and controls for precise alignment of phase and time between frequency bands, equalization, dynamic range compression and limiting control.
Enclosures
Horn loudspeakers
Horn loudspeakers are the oldest form of loudspeaker system. The use of horns as voice-amplifying megaphones dates at least to the 17th century, and horns were used in mechanical gramophones as early as 1857. Horn loudspeakers use a shaped waveguide in front of or behind the driver to increase the directivity of the loudspeaker and to transform a small diameter, high pressure condition at the driver cone surface to a large diameter, low pressure condition at the mouth of the horn. This improves the acoustic—electro/mechanical impedance match between the driver and ambient air, increasing efficiency, and focusing the sound over a narrower area. The size of the throat, mouth, the length of the horn, as well as the area expansion rate along it must be carefully chosen to match the drive to properly provide this transforming function over a range of frequencies (every horn performs poorly outside its acoustic limits, at both high and low frequencies). The length and cross-sectional mouth area required to create a bass or sub-bass horn require a horn many feet long. 'Folded' horns can reduce the total size, but compel designers to make compromises and accept increased complication such as cost and construction. Some horn designs not only fold the low frequency horn, but use the walls in a room corner as an extension of the horn mouth. In the late 1940s, horns whose mouths took up much of a room wall were not unknown amongst hi-fi fans. Room sized installations became much less acceptable when two or more were required. A horn loaded speaker can have a sensitivity as high as 110 dB at 2.83 volts (1 watt at 8 ohms) at 1 meter. This is a hundredfold increase in output compared to a speaker rated at 90 dB sensitivity, and is invaluable in applications where high sound levels are required or amplifier power is limited.
Transmission line loudspeaker
A transmission line loudspeaker is a loudspeaker enclosure design which uses an acoustic transmission line within the cabinet, compared to the simpler enclosures used by sealed (closed) or ported (bass reflex) designs. Instead of reverberating in a fairly simple damped enclosure, sound from the back of the bass speaker is directed into a long (generally folded) damped pathway within the speaker enclosure, which allows far greater control and use of speaker energy and the resulting sound.
Wiring connections
Most home hi-fi loudspeakers use two wiring points to connect to the source of the signal (for example, to the audio amplifier or receiver). To accept the wire connection, the loudspeaker enclosure may have binding posts, spring clips, or a panel-mount jack. If the wires for a pair of speakers are not connected with respect to the proper electrical polarity (the + and − connections on the speaker and amplifier should be connected + to + and − to −; speaker cable is almost always marked so that one conductor of a pair can be distinguished from the other, even if it has run under or behind things in its run from amplifier to speaker location), the loudspeakers are said to be "out of phase" or more properly "out of polarity". Given identical signals, motion in one cone is in the opposite direction of the other. This typically causes monophonic material in a stereo recording to be canceled out, reduced in level, and made more difficult to localize, all due to destructive interference of the sound waves. The cancellation effect is most noticeable at frequencies where the loudspeakers are separated by a quarter wavelength or less; low frequencies are affected the most. This type of miswiring error does not damage speakers, but is not optimal for listening. With sound reinforcement system, PA system and instrument amplifier speaker enclosures, cables and some type of jack or connector are typically used. Lower- and mid-priced sound system and instrument speaker cabinets often use 1/4" speaker cable jacks. Higher-priced and higher powered sound system cabinets and instrument speaker cabinets often use Speakon connectors. Speakon connectors are considered to be safer for high wattage amplifiers, because the connector is designed so that human users cannot touch the connectors.
Wireless speakers
Wireless speakers are very similar to traditional (wired) loudspeakers, but they receive audio signals using radio frequency (RF) waves rather than over audio cables. There is normally an amplifier integrated in the speaker's cabinet because the RF waves alone are not enough to drive the speaker. This integration of amplifier and loudspeaker is known as an active loudspeaker. Manufacturers of these loudspeakers design them to be as lightweight as possible while producing the maximum amount of audio output efficiency. Wireless speakers still need power, so require a nearby AC power outlet, or possibly batteries. Only the wire to the amplifier is eliminated.
Specifications
Speaker specifications generally include: * Speaker or driver type (individual units only) – Full-range, woofer, tweeter, or mid-range. * Size of individual drivers. For cone drivers, the quoted size is generally the outside diameter of the basket. However, it may less commonly also be the diameter of the cone surround, measured apex to apex, or the distance from the center of one mounting hole to its opposite. Voice-coil diameter may also be specified. If the loudspeaker has a compression horn driver, the diameter of the horn throat may be given. * Rated Power – Nominal (or even continuous) power, and peak (or maximum short-term) power a loudspeaker can handle (i.e., maximum input power before destroying the loudspeaker; it is never the sound output the loudspeaker produces). A driver may be damaged at much less than its rated power if driven past its mechanical limits at lower frequencies.Elliott Sound Products. Rod Elliott, 2006
''Speaker Damage''
Retrieved on June 16, 2009.
Tweeters can also be damaged by amplifier clipping (amplifier circuits produce large amounts of energy at high frequencies in such cases) or by music or sine wave input at high frequencies. Each of these situations might pass more energy to a tweeter than it can survive without damage.Elliott Sound Products. Rod Elliott, 2006
Retrieved on June 16, 2009.
In some jurisdictions, power handling has a legal meaning allowing comparisons between loudspeakers under consideration. Elsewhere, the variety of meanings for power handling capacity can be quite confusing. * Impedance – typically 4 Ω (ohms), 8 Ω, etc. * Baffle or enclosure type (enclosed systems only) – Sealed, bass reflex, etc. * Number of drivers (complete speaker systems only) – two-way, three-way, etc. * Class of loudspeaker: **Class 1: maximum SPL 110-119 dB, the type of loudspeaker used for reproducing a person speaking in a small space or for background music; mainly used as fill speakers for Class 2 or Class 3 speakers; typically small 4" or 5" woofers and dome tweeters **Class 2: maximum SPL 120-129 dB, the type of medium power-capable loudspeaker used for reinforcement in small to medium spaces or as fill speakers for Class 3 or Class 4 speakers; typically 5" to 8" woofers and dome tweeters **Class 3: maximum SPL 130-139 dB, high power-capable loudspeakers used in main systems in small to medium spaces; also used as fill speakers for class 4 speakers; typically 6.5" to 12" woofers and 2" or 3" compression drivers for high frequencies **Class 4: maximum SPL 140 dB and higher, very high power-capable loudspeakers used as mains in medium to large spaces (or for fill speakers for these medium to large spaces); 10" to 15" woofers and 3" compression drivers and optionally: * Crossover frequency(ies) (multi-driver systems only) – The nominal frequency boundaries of the division between drivers. * Frequency response – The measured, or specified, output over a specified range of frequencies for a constant input level varied across those frequencies. It sometimes includes a variance limit, such as within "± 2.5 dB." * Thiele/Small parameters (individual drivers only) – these include the driver's ''F''s (resonance frequency), ''Q''ts (a driver's ''Q''; more or less, its damping factor at resonant frequency), ''V''as (the equivalent air compliance volume of the driver), etc. * Sensitivity – The sound pressure level produced by a loudspeaker in a non-reverberant environment, often specified in dB and measured at 1 meter with an input of 1 watt (2.83 rms volts into 8 Ω), typically at one or more specified frequencies. Manufacturers often use this rating in marketing material. * Maximum sound pressure level – The highest output the loudspeaker can manage, short of damage or not exceeding a particular distortion level. Manufacturers often use this rating in marketing material—commonly without reference to frequency range or distortion level.
Electrical characteristics of dynamic loudspeakers
The load that a driver presents to an amplifier consists of a complex electrical impedance—a combination of resistance and both capacitive and inductive reactance, which combines properties of the driver, its mechanical motion, the effects of crossover components (if any are in the signal path between amplifier and driver), and the effects of air loading on the driver as modified by the enclosure and its environment. Most amplifiers' output specifications are given at a specific power into an ideal resistive load; however, a loudspeaker does not have a constant impedance across its frequency range. Instead, the voice coil is inductive, the driver has mechanical resonances, the enclosure changes the driver's electrical and mechanical characteristics, and a passive crossover between the drivers and the amplifier contributes its own variations. The result is a load impedance that varies widely with frequency, and usually a varying phase relationship between voltage and current as well, also changing with frequency. Some amplifiers can cope with the variation better than others can. To make sound, a loudspeaker is driven by modulated electric current (produced by an amplifier) that passes through a "speaker coil" which then (through inductance) creates a magnetic field around the coil, creating a magnetic field. The electric current variations that pass through the speaker are thus converted to a varying magnetic field, whose interaction with the driver's magnetic field moves the speaker diaphragm, which thus forces the driver to produce air motion that is similar to the original signal from the amplifier.
Electromechanical measurements
Examples of typical measurements are: amplitude and phase characteristics vs. frequency; impulse response under one or more conditions (e.g., square waves, sine wave bursts, etc.); directivity vs. frequency (e.g., horizontally, vertically, spherically, etc.); harmonic and intermodulation distortion vs. sound pressure level (SPL) output, using any of several test signals; stored energy (i.e., ringing) at various frequencies; impedance vs. frequency; and small-signal vs. large-signal performance. Most of these measurements require sophisticated and often expensive equipment to perform, and also good judgment by the operator, but the raw sound pressure level output is rather easier to report and so is often the only specified value—sometimes in misleadingly exact terms. The sound pressure level (SPL) a loudspeaker produces is measured in decibels (dBspl).
Efficiency vs. sensitivity
Loudspeaker efficiency is defined as the sound power output divided by the electrical power input. Most loudspeakers are inefficient transducers; only about 1% of the electrical energy sent by an amplifier to a typical home loudspeaker is converted to acoustic energy. The remainder is converted to heat, mostly in the voice coil and magnet assembly. The main reason for this is the difficulty of achieving proper impedance matching between the acoustic impedance of the drive unit and the air it radiates into. (At low frequencies, improving this match is the main purpose of speaker enclosure designs). The efficiency of loudspeaker drivers varies with frequency as well. For instance, the output of a woofer driver decreases as the input frequency decreases because of the increasingly poor match between air and the driver. Driver ratings based on the SPL for a given input are called sensitivity ratings and are notionally similar to efficiency. Sensitivity is usually defined as so many decibels at 1 W electrical input, measured at 1 meter (except for headphones), often at a single frequency. The voltage used is often 2.83 VRMS, which is 1 watt into an 8 Ω (nominal) speaker impedance (approximately true for many speaker systems). Measurements taken with this reference are quoted as dB with 2.83 V @ 1 m. The sound pressure output is measured at (or mathematically scaled to be equivalent to a measurement taken at) one meter from the loudspeaker and on-axis (directly in front of it), under the condition that the loudspeaker is radiating into an infinitely large space and mounted on an infinite baffle. Clearly then, sensitivity does not correlate precisely with efficiency, as it also depends on the directivity of the driver being tested and the acoustic environment in front of the actual loudspeaker. For example, a cheerleader's horn produces more sound output in the direction it is pointed by concentrating sound waves from the cheerleader in one direction, thus "focusing" them. The horn also improves impedance matching between the voice and the air, which produces more acoustic power for a given speaker power. In some cases, improved impedance matching (via careful enclosure design) lets the speaker produce more acoustic power. * Typical home loudspeakers have sensitivities of about 85 to 95 dB for 1 W @ 1 m—an efficiency of 0.5–4%. * Sound reinforcement and public address loudspeakers have sensitivities of perhaps 95 to 102 dB for 1 W @ 1 m—an efficiency of 4–10%. * Rock concert, stadium PA, marine hailing, etc. speakers generally have higher sensitivities of 103 to 110 dB for 1 W @ 1 m—an efficiency of 10–20%. A driver with a higher maximum power rating cannot necessarily be driven to louder levels than a lower-rated one, since sensitivity and power handling are largely independent properties. In the examples that follow, assume (for simplicity) that the drivers being compared have the same electrical impedance, are operated at the same frequency within both driver's respective pass bands, and that power compression and distortion are low. For the first example, a speaker 3 dB more sensitive than another produces double the sound power (is 3 dB louder) for the same power input. Thus, a 100 W driver ("A") rated at 92 dB for 1 W @ 1 m sensitivity puts out twice as much acoustic power as a 200 W driver ("B") rated at 89 dB for 1 W @ 1 m when both are driven with 100 W of input power. In this particular example, when driven at 100 W, speaker A produces the same SPL, or loudness as speaker B would produce with 200 W input. Thus, a 3 dB increase in sensitivity of the speaker means that it needs half the amplifier power to achieve a given SPL. This translates into a smaller, less complex power amplifier—and often, to reduced overall system cost. It is typically not possible to combine high efficiency (especially at low frequencies) with compact enclosure size and adequate low frequency response. One can, for the most part, choose only two of the three parameters when designing a speaker system. So, for example, if extended low-frequency performance and small box size are important, one must accept low efficiency. This rule of thumb is sometimes called Hofmann's Iron Law (after J.A. Hofmann, the "H" in KLH).
Listening environment
The interaction of a loudspeaker system with its environment is complex and is largely out of the loudspeaker designer's control. Most listening rooms present a more or less reflective environment, depending on size, shape, volume, and furnishings. This means the sound reaching a listener's ears consists not only of sound directly from the speaker system, but also the same sound delayed by traveling to and from (and being modified by) one or more surfaces. These reflected sound waves, when added to the direct sound, cause cancellation and addition at assorted frequencies (e.g., from resonant room modes), thus changing the timbre and character of the sound at the listener's ears. The human brain is very sensitive to small variations, including some of these, and this is part of the reason why a loudspeaker system sounds different at different listening positions or in different rooms. A significant factor in the sound of a loudspeaker system is the amount of absorption and diffusion present in the environment. Clapping one's hands in a typical empty room, without draperies or carpet, produces a zippy, fluttery echo due both to a lack of absorption and to reverberation (that is, repeated echoes) from flat reflective walls, floor, and ceiling. The addition of hard surfaced furniture, wall hangings, shelving and even baroque plaster ceiling decoration changes the echoes, primarily because of diffusion caused by reflective objects with shapes and surfaces having sizes on the order of the sound wavelengths. This somewhat breaks up the simple reflections otherwise caused by bare flat surfaces, and spreads the reflected energy of an incident wave over a larger angle on reflection.
Placement
In a typical rectangular listening room, the hard, parallel surfaces of the walls, floor and ceiling cause primary acoustic resonance nodes in each of the three dimensions: left-right, up-down and forward-backward. Furthermore, there are more complex resonance modes involving three, four, five and even all six boundary surfaces combining to create standing waves. Low frequencies excite these modes the most, since long wavelengths are not much affected by furniture compositions or placement. The mode spacing is critical, especially in small and medium size rooms like recording studios, home theaters and broadcast studios. The proximity of the loudspeakers to room boundaries affects how strongly the resonances are excited as well as affecting the relative strength at each frequency. The location of the listener is critical, too, as a position near a boundary can have a great effect on the perceived balance of frequencies. This is because standing wave patterns are most easily heard in these locations and at lower frequencies, below the Schroeder frequency – typically around 200–300 Hz, depending on room size.
Directivity
Acousticians, in studying the radiation of sound sources have developed some concepts important to understanding how loudspeakers are perceived. The simplest possible radiating source is a point source, sometimes called a simple source. An ideal point source is an infinitesimally small point radiating sound. It may be easier to imagine a tiny pulsating sphere, uniformly increasing and decreasing in diameter, sending out sound waves in all directions equally, independent of frequency. Any object radiating sound, including a loudspeaker system, can be thought of as being composed of combinations of such simple point sources. The radiation pattern of a combination of point sources is not the same as for a single source, but depends on the distance and orientation between the sources, the position relative to them from which the listener hears the combination, and the frequency of the sound involved. Using geometry and calculus, some simple combinations of sources are easily solved; others are not. One simple combination is two simple sources separated by a distance and vibrating out of phase, one miniature sphere expanding while the other is contracting. The pair is known as a doublet, or dipole, and the radiation of this combination is similar to that of a very small dynamic loudspeaker operating without a baffle. The directivity of a dipole is a figure 8 shape with maximum output along a vector that connects the two sources and minimums to the sides when the observing point is equidistant from the two sources, where the sum of the positive and negative waves cancel each other. While most drivers are dipoles, depending on the enclosure to which they are attached, they may radiate as monopoles, dipoles (or bipoles). If mounted on a finite baffle, and these out of phase waves are allowed to interact, dipole peaks and nulls in the frequency response result. When the rear radiation is absorbed or trapped in a box, the diaphragm becomes a monopole radiator. Bipolar speakers, made by mounting in-phase monopoles (both moving out of or into the box in unison) on opposite sides of a box, are a method of approaching omnidirectional radiation patterns. In real life, individual drivers are complex 3D shapes such as cones and domes, and they are placed on a baffle for various reasons. A mathematical expression for the directivity of a complex shape, based on modeling combinations of point sources, is usually not possible, but in the far field, the directivity of a loudspeaker with a circular diaphragm is close to that of a flat circular piston, so it can be used as an illustrative simplification for discussion. As a simple example of the mathematical physics involved, consider the following: the formula for far field directivity of a flat circular piston in an infinite baffle is $p\left(\theta\right) = \frac$ where $k_a=\frac$, $p_0$ is the pressure on axis, $a$ is the piston radius, $\lambda$ is the wavelength (i.e. $\lambda = \frac = \frac$) $\theta$ is the angle off axis and $J_1$ is the Bessel function of the first kind. A planar source radiates sound uniformly for low frequencies' wavelengths longer than the dimensions of the planar source, and as frequency increases, the sound from such a source focuses into an increasingly narrower angle. The smaller the driver, the higher the frequency where this narrowing of directivity occurs. Even if the diaphragm is not perfectly circular, this effect occurs such that larger sources are more directive. Several loudspeaker designs approximate this behavior. Most are electrostatic or planar magnetic designs. Various manufacturers use different driver mounting arrangements to create a specific type of sound field in the space for which they are designed. The resulting radiation patterns may be intended to more closely simulate the way sound is produced by real instruments, or simply create a controlled energy distribution from the input signal (some using this approach are called monitors, as they are useful in checking the signal just recorded in a studio). An example of the first is a room corner system with many small drivers on the surface of a 1/8 sphere. A system design of this type was patented and produced commercially by Professor Amar Bose—the 2201. Later Bose models have deliberately emphasized production of both direct and reflected sound by the loudspeaker itself, regardless of its environment. The designs are controversial in high fidelity circles, but have proven commercially successful. Several other manufacturers' designs follow similar principles. Directivity is an important issue because it affects the frequency balance of sound a listener hears, and also the interaction of the speaker system with the room and its contents. A very directive (sometimes termed 'beamy') speaker (i.e., on an axis perpendicular to the speaker face) may result in a reverberant field lacking in high frequencies, giving the impression the speaker is deficient in treble even though it measures well on axis (e.g., "flat" across the entire frequency range). Speakers with very wide, or rapidly increasing directivity at high frequencies, can give the impression that there is too much treble (if the listener is on axis) or too little (if the listener is off axis). This is part of the reason why on-axis frequency response measurement is not a complete characterization of the sound of a given loudspeaker.
Other speaker designs
While dynamic cone speakers remain the most popular choice, many other speaker technologies exist.
With a diaphragm
Moving-iron loudspeakers
200px|Moving iron speaker The moving iron speaker was the first type of speaker that was invented. Unlike the newer dynamic (moving coil) design, a moving-iron speaker uses a stationary coil to vibrate a magnetized piece of metal (called the iron, reed, or armature). The metal is either attached to the diaphragm or is the diaphragm itself. This design was the original loudspeaker design, dating back to the early telephone. Moving iron drivers are inefficient and can only produce a small band of sound. They require large magnets and coils to increase force. Balanced armature drivers (a type of moving iron driver) use an armature that moves like a see-saw or diving board. Since they are not damped, they are highly efficient, but they also produce strong resonances. They are still used today for high-end earphones and hearing aids, where small size and high efficiency are important.
Piezoelectric speakers
Piezoelectric speakers are frequently used as beepers in watches and other electronic devices, and are sometimes used as tweeters in less-expensive speaker systems, such as computer speakers and portable radios. Piezoelectric speakers have several advantages over conventional loudspeakers: they are resistant to overloads that would normally destroy most high frequency drivers, and they can be used without a crossover due to their electrical properties. There are also disadvantages: some amplifiers can oscillate when driving capacitive loads like most piezoelectrics, which results in distortion or damage to the amplifier. Additionally, their frequency response, in most cases, is inferior to that of other technologies. This is why they are generally used in single frequency (beeper) or non-critical applications. Piezoelectric speakers can have extended high frequency output, and this is useful in some specialized circumstances; for instance, sonar applications in which piezoelectric variants are used as both output devices (generating underwater sound) and as input devices (acting as the sensing components of underwater microphones). They have advantages in these applications, not the least of which is simple and solid state construction that resists seawater better than a ribbon or cone based device would. In 2013, Kyocera introduced piezoelectric ultra-thin medium-size film speakers with only 1 millimeter of thickness and 7 grams of weight for their 55" OLED televisions and they hope the speakers will also be used in PCs and tablets. Besides medium-size, there are also large and small sizes which can all produce relatively the same quality of sound and volume within 180 degrees. The highly responsive speaker material provides better clarity than traditional TV speakers.
Magnetostatic loudspeakers
Instead of a voice coil driving a speaker cone, a magnetostatic speaker uses an array of metal strips bonded to a large film membrane. The magnetic field produced by signal current flowing through the strips interacts with the field of permanent bar magnets mounted behind them. The force produced moves the membrane and so the air in front of it. Typically, these designs are less efficient than conventional moving-coil speakers.
Magnetostrictive speakers
Magnetostrictive transducers, based on magnetostriction, have been predominantly used as sonar ultrasonic sound wave radiators, but their use has spread also to audio speaker systems. Magnetostrictive speaker drivers have some special advantages: they can provide greater force (with smaller excursions) than other technologies; low excursion can avoid distortions from large excursion as in other designs; the magnetizing coil is stationary and therefore more easily cooled; they are robust because delicate suspensions and voice coils are not required. Magnetostrictive speaker modules have been produced by Fostex and FeONIC and subwoofer drivers have also been produced.
Electrostatic loudspeakers
Electrostatic loudspeakers use a high voltage electric field (rather than a magnetic field) to drive a thin statically charged membrane. Because they are driven over the entire membrane surface rather than from a small voice coil, they ordinarily provide a more linear and lower-distortion motion than dynamic drivers. They also have a relatively narrow dispersion pattern that can make for precise sound-field positioning. However, their optimum listening area is small and they are not very efficient speakers. They have the disadvantage that the diaphragm excursion is severely limited because of practical construction limitations—the further apart the stators are positioned, the higher the voltage must be to achieve acceptable efficiency. This increases the tendency for electrical arcs as well as increasing the speaker's attraction of dust particles. Arcing remains a potential problem with current technologies, especially when the panels are allowed to collect dust or dirt and are driven with high signal levels. Electrostatics are inherently dipole radiators and due to the thin flexible membrane are less suited for use in enclosures to reduce low frequency cancellation as with common cone drivers. Due to this and the low excursion capability, full range electrostatic loudspeakers are large by nature, and the bass rolls off at a frequency corresponding to a quarter wavelength of the narrowest panel dimension. To reduce the size of commercial products, they are sometimes used as a high frequency driver in combination with a conventional dynamic driver that handles the bass frequencies effectively. Electrostatics are usually driven through a step-up transformer that multiplies the voltage swings produced by the power amplifier. This transformer also multiplies the capacitive load that is inherent in electrostatic transducers, which means the effective impedance presented to the power amplifiers varies widely by frequency. A speaker that is nominally 8 ohms may actually present a load of 1 ohm at higher frequencies, which is challenging to some amplifier designs.
Ribbon and planar magnetic loudspeakers
A ribbon speaker consists of a thin metal-film ribbon suspended in a magnetic field. The electrical signal is applied to the ribbon, which moves with it to create the sound. The advantage of a ribbon driver is that the ribbon has very little mass; thus, it can accelerate very quickly, yielding very good high-frequency response. Ribbon loudspeakers are often very fragile—some can be torn by a strong gust of air. Most ribbon tweeters emit sound in a dipole pattern. A few have backings that limit the dipole radiation pattern. Above and below the ends of the more or less rectangular ribbon, there is less audible output due to phase cancellation, but the precise amount of directivity depends on ribbon length. Ribbon designs generally require exceptionally powerful magnets, which makes them costly to manufacture. Ribbons have a very low resistance that most amplifiers cannot drive directly. As a result, a step down transformer is typically used to increase the current through the ribbon. The amplifier "sees" a load that is the ribbon's resistance times the transformer turns ratio squared. The transformer must be carefully designed so that its frequency response and parasitic losses do not degrade the sound, further increasing cost and complication relative to conventional designs. Planar magnetic speakers (having printed or embedded conductors on a flat diaphragm) are sometimes described as ribbons, but are not truly ribbon speakers. The term planar is generally reserved for speakers with roughly rectangular flat surfaces that radiate in a bipolar (i.e., front and back) manner. Planar magnetic speakers consist of a flexible membrane with a voice coil printed or mounted on it. The current flowing through the coil interacts with the magnetic field of carefully placed magnets on either side of the diaphragm, causing the membrane to vibrate more or less uniformly and without much bending or wrinkling. The driving force covers a large percentage of the membrane surface and reduces resonance problems inherent in coil-driven flat diaphragms.
Bending wave loudspeakers
Bending wave transducers use a diaphragm that is intentionally flexible. The rigidity of the material increases from the center to the outside. Short wavelengths radiate primarily from the inner area, while longer waves reach the edge of the speaker. To prevent reflections from the outside back into the center, long waves are absorbed by a surrounding damper. Such transducers can cover a wide frequency range (80 Hz to 35,000 Hz) and have been promoted as being close to an ideal point sound source. This uncommon approach is being taken by only a very few manufacturers, in very different arrangements. The Ohm Walsh loudspeakers use a unique driver designed by Lincoln Walsh, who had been a radar development engineer in WWII. He became interested in audio equipment design and his last project was a unique, one-way speaker using a single driver. The cone faced down into a sealed, airtight enclosure. Rather than move back-and-forth as conventional speakers do, the cone rippled and created sound in a manner known in RF electronics as a "transmission line". The new speaker created a cylindrical sound field. Lincoln Walsh died before his speaker was released to the public. The Ohm Acoustics firm has produced several loudspeaker models using the Walsh driver design since then. German Physiks, an audio equipment firm in Germany, also produces speakers using this approach. The German firm, Manger, has designed and produced a bending wave driver that at first glance appears conventional. In fact, the round panel attached to the voice coil bends in a carefully controlled way to produce full range sound. Josef W. Manger was awarded with the "Diesel Medal" for extraordinary developments and inventions by the German institute of inventions.
Flat panel loudspeakers
There have been many attempts to reduce the size of speaker systems, or alternatively to make them less obvious. One such attempt was the development of "exciter" transducer coils mounted to flat panels to act as sound sources, most accurately called exciter/panel drivers. These can then be made in a neutral color and hung on walls where they are less noticeable than many speakers, or can be deliberately painted with patterns, in which case they can function decoratively. There are two related problems with flat panel techniques: first, a flat panel is necessarily more flexible than a cone shape in the same material, and therefore moves as a single unit even less, and second, resonances in the panel are difficult to control, leading to considerable distortions. Some progress has been made using such lightweight, rigid, materials such as Styrofoam, and there have been several flat panel systems commercially produced in recent years.
Heil air motion transducers
[[Image:AirMotionTransformer.png|300px|In Heil's air motion transducer, current through the membrane 2 causes it to move left and right in magnetic field 6, moving air in and out along directions 8; barriers 4 prevent air from moving in unintended directions. [[Oskar Heil invented the air motion transducer in the 1960s. In this approach, a pleated diaphragm is mounted in a magnetic field and forced to close and open under control of a music signal. Air is forced from between the pleats in accordance with the imposed signal, generating sound. The drivers are less fragile than ribbons and considerably more efficient (and able to produce higher absolute output levels) than ribbon, electrostatic, or planar magnetic tweeter designs. ESS, a California manufacturer, licensed the design, employed Heil, and produced a range of speaker systems using his tweeters during the 1970s and 1980s. Lafayette Radio, a large US retail store chain, also sold speaker systems using such tweeters for a time. There are several manufacturers of these drivers (at least two in Germany—one of which produces a range of high-end professional speakers using tweeters and mid-range drivers based on the technology) and the drivers are increasingly used in professional audio. Martin Logan produces several AMT speakers in the US and GoldenEar Technologies incorporates them in its entire speaker line.
Transparent ionic conduction speaker
In 2013, a research team introduced Transparent ionic conduction speaker which a 2 layers transparent conductive gel and a layer of transparent rubber in between to make high voltage and high actuation work to reproduce good sound quality. The speaker is suitable for robotics, mobile computing and adaptive optics fields.
Without a diaphragm
Plasma arc speakers
Plasma arc loudspeakers use electrical plasma as a radiating element. Since plasma has minimal mass, but is charged and therefore can be manipulated by an electric field, the result is a very linear output at frequencies far higher than the audible range. Problems of maintenance and reliability for this approach tend to make it unsuitable for mass market use. In 1978 Alan E. Hill of the Air Force Weapons Laboratory in Albuquerque, NM, designed the Plasmatronics Hill Type I, a tweeter whose plasma was generated from helium gas.Hill Plasmatronics
described. Retrieved March 26, 2007.
This avoided the ozone and nitrous oxide produced by RF decomposition of air in an earlier generation of plasma tweeters made by the pioneering DuKane Corporation, who produced the Ionovac (marketed as the Ionofane in the UK) during the 1950s. Currently, there remain a few manufacturers in Germany who use this design, and a do-it-yourself design has been published and has been available on the Internet. A less expensive variation on this theme is the use of a flame for the driver, as flames contain ionized (electrically charged) gases.
Thermoacoustic speakers
In 2008, researchers of Tsinghua University demonstrated a thermoacoustic loudspeaker of carbon nanotube thin film, whose working mechanism is a thermoacoustic effect. Sound frequency electric currents are used to periodically heat the CNT and thus result in sound generation in the surrounding air. The CNT thin film loudspeaker is transparent, stretchable and flexible. In 2013, researchers of Tsinghua University further present a thermoacoustic earphone of carbon nanotube thin yarn and a thermoacoustic surface-mounted device. They are both fully integrated devices and compatible with Si-based semiconducting technology.
Rotary woofers
A rotary woofer is essentially a fan with blades that constantly change their pitch, allowing them to easily push the air back and forth. Rotary woofers are able to efficiently reproduce infrasound frequencies, which are difficult to impossible to achieve on a traditional speaker with a diaphragm. They are often employed in movie theaters to recreate rumbling bass effects, such as explosions.
New technologies
Digital speakers
Digital speakers have been the subject of experiments performed by Bell Labs as far back as the 1920s. The design is simple; each bit controls a driver, which is either fully 'on' or 'off'. Problems with this design have led manufacturers to abandon it as impractical for the present. First, for a reasonable number of bits (required for adequate sound reproduction quality), the physical size of a speaker system becomes very large. Secondly, due to inherent analog-to-digital conversion problems, the effect of aliasing is unavoidable, so that the audio output is "reflected" at equal amplitude in the frequency domain, on the other side of the Nyquist limit (half the sampling frequency), causing an unacceptably high level of ultrasonics to accompany the desired output. No workable scheme has been found to adequately deal with this. The term "digital" or "digital-ready" is often used for marketing purposes on speakers or headphones, but these systems are not digital in the sense described above. Rather, they are conventional speakers that can be used with digital sound sources (e.g., optical media, MP3 players, etc.), as can any conventional speaker. | 2021-10-17 16:39:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 8, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4300262928009033, "perplexity": 2359.816216209096}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585178.60/warc/CC-MAIN-20211017144318-20211017174318-00072.warc.gz"} |
http://www.tharwan.de/category/matlab.html | # tharwan.de
## MATLAB is not for Science - follow up
I wrote in detail why I think MATLAB is a bad choice for scientific research here. Imagine my surprise when I saw a letter about the "new MATLAB licensing model" from my university in my inbox. Just some quotes translated into English:
Further notes for MATLAB users at the TU Ilmenau:
1. Close MATLAB if you do not use it for a longer period of time (longer than 30 minutes), to free the license for other users. Forgo to block a academic-license by doing pseudo calculations.
4. Examine the use of comparable commercial or free software in your department like Maple, Mathematica, R, Octave or Scilab.
Sadly they forget to mention Python/NumPy. Maybe I should offer an introduction course…
## MATLAB is not for Science
Currently I am working a lot with MATLAB®. Actually knowing how to use MATLAB got me a job in the first place. Nevertheless I could not really overcome my dislike for it even though I become more and more accustomed to its quirks. It is not so much that the idea of MATLAB as a tool for easy and fast numerical prototyping is a bad one in it self. It is not even the case that it is a bad tool in it self. It is only that there are some parts of the execution that are just poor or a bad fit.
I held a talk about MATLAB versus my go to language Python for use in our physics department (slides in German) which was very well received but sadly did not lead to any changes yet. The worst thing about MATLAB in scientific use is that it is closed source and therefore not freely available for everyone. And its open source clone (Octave) is simply not a full replacement.
## Open Source? Aren‘t you an Apple fanboy?
The bad thing about the usage of MATLAB, and all other closed source tools for science, is that they destroy what could makes computer science and science on computers such a great thing: that almost everyone has the tools to do it. To trap a single atom in a quantum well and find out how its absorption spectrum changes is an expensive experiment to setup. Simulating it on a computer is almost a trivial task nowadays.
But if the knowledge you base your own simulations on is build up with tools you do not have access to, you will have a very hard time to setup your "virtual experiment" too. Or you have to swallow the pill and buy and use the same closed source tools all over again.
I think scientist should devote themselves to use open source for all their publications, if possible in any way, because it is the nature of science to be reproducible. It is sad that we cannot do high energy particle physics in our backyard or in our lecture halls. But we have a way to make the simulations useable for everyone and we should use it.
## Where MATLAB excels
And here I am, telling you that Python might be a better alternative, yet earning money programming in MATLAB. Actually I did try to convince my employer to let me use Python and got some good reasons to use MATLAB. It is more or less the same reason why large cooperation‘s use Windows and Office: it comes with a promise of service and completeness and it seams to be a carefree package. It is some kind of outsourcing. You have someone that is responsible and is not you. You have some kind of stability. And the biggest point of all: it is the go to standard in the industry (and in science). Sadly.
The Story goes like this: your company wants to develop some kind of image processing software. You need something to prototype your algorithms in that just works. MATLAB provides you with almost all the things you need. You install it on your machine and you are ready to go. Having used MATLAB for exactly this case, I must admit it is very nice to have almost anything your can whish for already implemented and very well documented bundled in an almost care free package to play around with.
## Money is the solution
For a company it is more or less no problem to pay the price for a MATLAB installation (some k€), even if it is only used for a single project. For a private user the price would be nuts. There exists a student version that is priced much lower (~150€ without any toolboxes) but based on my observations is not justifiable for most students. This might be because of the availability of pirated versions (almost anyone I know has one) or the versions that are accessible on the university owned PCs.
I think the price for MATLAB is justified; it is a useful tool with a small target audience that has the financial resources to pay for it, if you look at commercial use of it. I have no idea what it costs as a university to get a campus license, but no matter what it is the price for educational users is ridiculous. Especially if you consider that a lot of the useful and important functions are bundled in so called toolboxes, which will milk some extra money out of you. Though you can argue that it lowers the price of the basic version.
To make the situation a little bit more vivid here: for a lot of courses in engineering you have to use MATLAB. And while it is usually provided by the university, you can only use it in the computer lab which are natuarly to small for the whole university and not accessible at any time and so on. (The issue can partly be solved via remote desktop access) But the plan of MathWorks is not to provide an easy way for a university to give MATLAB to its students, since they already have paid for the license, instead it wants you as a student to pay again.
Making the educational version of MATLAB free would solve the problem of the software availability as a scientific tool for everyone while providing MathWorks the advantage that it will be used even more broadly in publications, manifesting its position as the go to tool. But as it already is the go to tool there is very little incentive for MathWorks to do so other then to be nice. The only way to push them in this direction would be to build on open source languages and toolboxes to challenge this status.
Basically I have no problem with MathWorks earning money with their software. I think the situation is not there fault (even if they could solve it). I also would not argue that everything should be free for education and science. But in the case of scientific software we have an alternative, which we do not have for hardware, that would strengthen the foundation of science and therefore we should use it.
## MATLAB – The Language
MATLAB is not only a tool it although describes a programming language. One that as outgrown itself years ago. MATLAB was designed as a MATrix LABoratory but the language is now used for general purpose. There are a lot of little and some bigger things where you have the feeling you do something that this is not made for when you program with MATLAB. Some Examples follow – more will come in later posts.
## Variable Unpacking:
A function can return multiple matrices and so you would guess you can do some kind for variable unpacking. You actually can, but I have know idea why the bothered to implemented it anyway as limited as it ended up being.
We assume our function f(x) returns three variables a,b and c:
function [a,b,c]=f(x)
% some clever math
a = …;
b = …;
c = …;
end
Now when you do:
X = f(x);
All you get is a. To get a,b and c you have to do:
[a,b,c] = f(x);
Lets assume you are only interested in one or two of the return values you can do:
[~,b,c] = f(x) % works
Since you can substitute the commas in MATLAB with spaces this also works:
[a b ~] = f(x) % works
But not this:
[~ b c] = f(x) % does not work
You can not do something like:
l = [1 2 3];
[a,b,c] = l; % ERROR
Cellarrays will not help either:
l = {1 2 3};
[a,b,c] = l; % ERROR
## Optional Arguments
This issue thing will hit very often while you are still develop your algorithm and the structure of your problem is not yet entirely clear. You can argue that this is my personal problem and I should think more before I start to program. Be assured I think a lot when I program but sometimes it works really well for me to just start writing down some code. But even if your program is completely laid out befor you start do write your code, almost certainly there will come a time you or someone else has to add something and here we are again.
Assume you have a function f(x,y) and later you discover there may be some circumstances where you need another argument z. You could refactor all your code and pass in three arguments for every call of f and pass in NaN for z where it was not needed before. Or you can use varargin:
function out = f(x,y,varargin)
out = x+y
if length(varargin)>0
out = out * varargin{1}
end
Honestly even the idea that you have a special argument you have to name varargin is kind of funny. It gets much more pretty if you have more than one optional argument and you have to think about some logic to figure out how many arguments you got, in what order and if they are all of the right type.
function out = f(x,y,varargin)
out = x+y
if length(varargin)==1
out = out * varargin{1}
elseif length(varargin)==2
out = (out + varargin{2})*varargin{1}
end
Matlab helps you with the inputParser. InputParser then does what the language should to by default. It gives you a way to test the arguments for type and set default values. It also gives you the opportunity to write much more code.
The way it would like to see this solved is like it is done in Python:
def f(x,y,z=1,v=0)
return (x+y+v)*z
This even allows you to do:
f(1,2,v=1)
The only thing you may have may have to add is to check if all variables are of the right type, what is quite hard since you probably don‘t care about float or integer but the availability of + and * operators for the objects. No matter how elegant you code your varargin check, I doubt it will ever be as clear as Pythons syntax.
to be continued…
page 1 / 1 | 2018-02-24 09:41:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2980859875679016, "perplexity": 730.811172713497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815544.79/warc/CC-MAIN-20180224092906-20180224112906-00178.warc.gz"} |
http://www.chim.lu/ech0435.php | Search:
# Electronegativity and the chemical bonds
→ Table of electronegativities
## Principle
Atoms with a small electronegativity abandon easily their outer electrons, atoms with a big electronegativity catch easily foreign electrons.
## Electronegativity and the ionic bond
If the difference of the electronegativities of two atoms exceeds 2, an ionic bond is made.
For example, $E.n.(Cl)-E.n.(Na) = 2.1$, so sodium chloride is an ionic substance.
## Electronegativity and the covalent bond
Two atoms with electronegativities greater than or equal to 2 combine to a covalent bond.
For example, $E.n.(H), E.n.(O) > 2$, so water is a molecular substance.
## Electronegatvity and the metallic bond
Atoms with electronegativities less than 2 combine to metallic bonds.
For example, $E.n.(Na)=0.9 2$, therefore is sodium a metal.
## Comment
The limit of 2 that we have given ourselves is vague. Near to this limit, intermediate cases are all to common, for example, - in a sample of iron(III)chloride $(\Delta(E.n.)$ $=$ $1.2)$ there are molecules of $FeCl_3$, but also $Fe^{3+}$ and $Cl^-$ ions! - gold ($E.n.=2.4$) is a metal! | 2020-07-04 10:08:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7723793983459473, "perplexity": 4290.877981257703}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886095.7/warc/CC-MAIN-20200704073244-20200704103244-00441.warc.gz"} |
https://electronics.stackexchange.com/questions/333314/fatfs-f-write-not-working-in-lpc1788-timer-isr | # FatFs f_write not working in LPC1788 Timer ISR
I’m using LPC1788 MCU with KeilV5 compiler.
I Have a Timer ISR in which I read a value form specific ADC channel and write it into a SD-Card using FATFS library.
Here is my Timer ISR code:
void TIMER1_IRQHandler(void){
if (TIM_GetIntStatus(LPC_TIM1, TIM_MR1_INT)== SET){
f_lseek(&File1, f_size(&File1));
//f_write(&File1,OutputSample,strlen(OutputSample), &FilePointer);
Counter++;
TIM_ClearIntPending(LPC_TIM1, TIM_MR1_INT);
}
}
There is no problem with above code but when I remove the comment sign from the f_write function, program stops and nothing executes exactly after Timer1 is enabled. (referred to debug, after TIM_Cmd(LPC_TIM1, ENABLE) is executed in main function)
So I can’t write the data into the SD-Card in Timer's ISR.
I should mention that f_write function works fine outside the Timer ISR.
Can you guess where is the problem?
• Note that f_write() can take significant time on a SD card, in the order of a few 100ms up to about one second - though the latter value I got from a very cheap no-name card. This forbids usage in an interrupt handler function in most cases. – Turbo J Oct 8 '17 at 12:19
It is usually a very, very bad idea to use FatFS from interrupts. The most common pattern is to acquire the data in the ISR, store is somewhere and have a main loop task that stores that queued data somewhere.
If your SPI driver uses interrupts and you don't enable nested interrupts (advanced topic), then your SPI interrupts won't run until the timer interrupt is done.
+1 for @filo, who correctly addressed the problem of nested interrupts either not enabled (or having the same priority, therefore not being triggered).
However the problem is not with the SPI (which typically is not used with interrupt on the FatFS, unless some heavy modifications are done in the code. The SPI is used in polling, with the xmit_spi() and xmit_spi_multi() functions).
The problem lies on the disk_timerproc() , which must be called (with an interrupt) each ms, for timeouts and for updating card's state (card inserted, write protected, etc).
In the LPC17xx example, disk_timerproc() is called by SysTick_Handler().
You have therefore two ways:
1. Use a higher priority number (i.e. give a lower priority) for the TimerISR, which actually reads the ADC.
2. Or let the Timer_ISR() set a volatile global variable "dataAvailable=1", and then in the main loop, save the data, when dataAvailable is 1.
For instance
volatile dataAvailable = 0;
[...] other code[...]
void main (void)
{
[...] other code [...]
while (1)
{
[...] other code [...]
if (dataAvailable)
{
dataAvailable = 0;
saveData();
}
}
}
void TIMER1_IRQHandler(void)
{
if (TIM_GetIntStatus(LPC_TIM1, TIM_MR1_INT)== SET)
{
[...] other code [...]
dataAvailable = 1;
TIM_ClearIntPending(LPC_TIM1, TIM_MR1_INT);
}
} | 2020-02-18 21:45:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4190611243247986, "perplexity": 8662.917410009832}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143815.23/warc/CC-MAIN-20200218210853-20200219000853-00436.warc.gz"} |
https://proofwiki.org/wiki/Asymptotic_Expansion_for_Cosine_Integral_Function | # Asymptotic Expansion for Cosine Integral Function
Jump to navigation Jump to search
## Theorem
$\displaystyle \map \Ci x \sim \frac {\cos x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n + 1}!} {x^{2 n + 1} } - \frac {\sin x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n}!} {x^{2 n} }$
where:
$\Ci$ denotes the cosine integral function
$\sim$ denotes asymptotic equivalence as $x \to \infty$. | 2019-11-22 08:45:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999823570251465, "perplexity": 5673.661990428402}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496671245.92/warc/CC-MAIN-20191122065327-20191122093327-00295.warc.gz"} |
http://math.soimeme.org/~arunram/Resources/AnAnalogueOfTheCharacterFormulaForHeckeAlgebras.html | ## An analogue of the character formula for Hecke algebras
Last update: 23 April 2014
## Notes and References
This is an html version of the paper An analogue of the character formula for Hecke algebras by I.V. Cherednik.
M.V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 21, No. 2, pp. 94-95, April-June, 1987. Original article submitted March 19, 1986.
## An analogue of the character formula for Hecke algebras
In this note the classical character formula of Frobenius [JKe1981] for the symmetric group $S$ is generalized to affine Hekke algebras. In the spirit of [BGG1975-2], resolutions realizing these formulas are immediately constructed. The construction was motivated by [Zel1987-2] and inspired by discussions with A. V. Zelevinskii, to whom the author expresses his deep gratitude. The author is thankful to I. M. Gel'fand for his attention to this work.
1. Suppose a $C\text{-algebra}$ ${H}_{n}$ is generated by elements ${T}_{1},\dots ,{T}_{n-1}$ for which $\left[{T}_{i},{T}_{j}\right]=0$ for $i\ne j±1,$ ${T}_{i}{T}_{i+1}{T}_{i}={T}_{i+1}{T}_{i}{T}_{i+1},$ $\left({T}_{i}-q\right)\left({T}_{i}+1\right)=0\text{.}$ Henceforth, $q$ is a power of a prime (as in [Zel1980, Rog1985]) or $q$ is taken in some defective neighborhood of $1$ in $C$ (as in [Che1986]). Adding pairwise commuting ${x}_{1},\dots ,{x}_{n},$ with relations $\left[{x}_{i},{T}_{j}\right]=0$ for $i\ne j,$ $j+1,$ ${x}_{i}{T}_{i}-{T}_{i}{x}_{i+1}=\left(q-1\right){x}_{i}={T}_{i}{x}_{i}-{x}_{i+1}{T}_{i},$ we obtain an affine Hecke algebra ${ℋ}_{n}\text{.}$ For an arbitrary family $u=\left({u}_{1},\dots ,{u}_{n}\right)$ we extend the left action of ${H}_{n}$ on itself to an action of ${ℋ}_{n}$ on ${H}_{n}$ putting ${x}_{k}\left(1\right)={q}^{{u}_{k}}\text{.}$ The obtained ${ℋ}_{n}\text{-module}$ is denoted by ${I}_{u}\text{.}$ Next, $\ell \left(w\right)$ is the length of the reduced decomposition of $w\in {S}_{n}$ relative to ${s}_{i}=\left(i,i+1\right),$ $w\prime \ge w\stackrel{\text{def}}{⇔}l\left(w\prime \right)=l\left(w\prime {w}^{-1}\right)+l\left(w\right),$ $\ell \left(\text{id}\right)=0,$ $\ell \left(w\right)\le n\left(n-1\right)/2\text{.}$ On a function $f\left({\lambda }_{1},\dots ,{\lambda }_{n}\right)$ the permutations $w\in {S}_{n}$ act by the formula $\left(wf\right)\left({\lambda }_{1},\dots ,{\lambda }_{n}\right)=f\left({w}^{-1}\left({\lambda }_{1},\dots ,{\lambda }_{n}\right)\right),$ ${\lambda }_{i}\in C$
For $a,b\in C$ we will write $a\ge b⇔a\in b+{ℤ}_{+},$ otherwise, $a We associate with a sequence of pairs $\mu =\left(\left\{{\ell }_{i}\ge {\ell }_{i}^{\prime }\right\}\right),$ $1\le i\le r,$ the family ${u}^{\mu }=\left({u}_{k}\right),$ $i\le k\le n\stackrel{\text{def}}{=}\sum _{i=1}^{r}\left({l}_{i}-{l}_{i}^{\prime }\right)$ of all numbers $u\left(i,j\right)$ of the form ${\ell }_{i}\ge u\left(i,j\right)={\ell }_{i}^{\prime }+j\ge {\ell }_{i}^{\prime }+1,$ enumerated by the rule ${u}_{k}=u\left({i}_{k},{j}_{k}\right),$ $k or ${j}_{k}-{j}_{m}<0={i}_{k}-{i}_{m}\text{.}$ We denote by ${w}_{\mu }\in {S}_{n}$ the permutation of indices ${u}_{k}$ preserving ${i}_{k}$ and corresponding to the transformation ${\ell }_{i}^{\prime }+j\to {\ell }_{i}-j+1\text{.}$
Lemma 1 [Rog1985]. The family of functions ${\phi }_{{s}_{i}}=i+\left(i-{q}^{{\lambda }_{2}-{\lambda }_{1}}\right){\left(i-q\right)}^{-1}{T}_{i}$ is uniquely extended to a family $\left\{{\phi }_{w}\left({\lambda }_{1},\dots ,{\lambda }_{n}\right),w\in {S}_{n}\right\}$ by the cocyclic relations ${\phi }_{xy}={y}^{-1}{\phi }_{x}{\phi }_{y}$ for $xy\ge y,$ $x,y\in {S}_{n}\text{.}$ 2) The submodule ${I}_{\mu }={H}_{n}{\phi }_{{w}_{\mu }}\left({u}^{\mu }\right)$ is an ${ℋ}_{n}\text{-submodule}$ of ${I}_{{u}^{\mu }}$ and contains, for each $w\ge {w}_{\mu }$ the leading coefficient ${\stackrel{\sim }{\phi }}_{w}\left({u}^{\mu }\right)$ of the decomposition, relative to $\lambda \to 0,$ of the function ${\phi }_{w}\left({u}^{\nu }\right),$ $\nu =\left(\left\{{\ell }_{i}+i\lambda ,{\ell }_{i}^{\prime }+i\lambda \right\}\right)\text{.}$
2. Next, suppose that ${l}_{j}<{l}_{i},$ ${l}_{j}^{\prime }<{l}_{i}^{\prime }$ for all $j We associate to each permutation $\sigma \in {S}_{r}{\mu }^{\sigma }=\left(\left\{{\ell }_{i},{\ell }_{\sigma \left(i\right)}^{\prime }\right\}\right)$ and the ${ℋ}_{n}\text{-module}$ ${I}_{\sigma }={I}_{{\mu }^{\sigma }}\text{.}$ If for some $i,$ ${\ell }_{i}-{\ell }_{\sigma \left(i\right)}^{\prime }<0,$ then ${\mu }^{\sigma }\stackrel{\text{def}}{=}\varnothing ,$ ${I}_{\sigma }\stackrel{\text{def}}{=}0\text{.}$ Put ${w}_{\sigma }={w}_{{\mu }^{\sigma }},$ ${u}^{\sigma }={u}^{{\mu }^{\sigma }}\text{.}$ We will write $\sigma ⇒\tau$ if $\ell \left(\sigma \right)=\ell \left(\tau \right)+1$ and $\tau$ is obtained by dropping some ${\sigma }_{i}$ from the reduced decomposition of $\sigma$ relative to ${\sigma }_{i}=\left(i,i+1\right)\in {S}_{r}\text{.}$
Lemma 2. 1) If $\sigma ⇒\tau ,$ then there exists a unique permutation ${S}_{n}\ni {w}_{\sigma ,\tau }\ge {w}_{\tau }$ for which ${w}_{\sigma ,\tau }\left({u}^{\tau }\right)={w}_{\sigma }\left({u}^{\sigma }\right)\text{.}$ 2) The condition ${\phi }_{{w}_{\sigma }}\left({u}^{\sigma }\right)⇒{\stackrel{\sim }{\phi }}_{{w}_{\sigma ,\tau }}\left({u}^{\tau }\right)$ uniquely determines an embedding of ${ℋ}_{n}\text{-modules}$ ${\rho }_{\sigma ,\tau }:{I}_{\sigma }⇒{I}_{\tau }\text{;}$ ${\rho }_{\sigma ,\tau }\left({I}_{\sigma }\right)\ne {I}_{\tau },$ if ${I}_{\tau }\ne 0\text{.}$ 3) Conversely, if $\ell \left(\sigma \right)=\ell \left(\tau \right)+1$ and ${I}_{\tau }\ne 0,$ then there exists a nonzero ${ℋ}_{n}\text{-homomorphism}$ between ${I}_{\sigma }$ and ${I}_{\tau },$ $\rho ⇔\sigma ⇒\tau ,$ $\rho =c{\rho }_{\sigma ,\tau },$ $c\in {ℂ}^{*}\text{.}$
A family $\sigma ⇒\sigma \prime ⇒\tau ,$ $\sigma ⇒\tau \prime ⇒\tau ,$ $\sigma \prime \ne \tau \prime$ will be called a square. Each triple $\sigma \prime ⇒\tau ⇐\tau \prime ,$ $\sigma \prime \ne \tau \prime$ can be extended to a square.
Proposition 3. 1) For each square ${\rho }_{\sigma \prime ,\tau }{\rho }_{\sigma ,\sigma \prime }\left({I}_{\sigma }\right)={\rho }_{\tau \prime ,\tau }{\rho }_{\sigma ,\tau \prime }\left({I}_{\sigma }\right)\text{.}$ 2) The image ${\stackrel{‾}{I}}_{\sigma }$ of each ${I}_{\sigma }$ $\left(\sigma \in {S}_{r}\right)$ in ${I}_{\mu }$ does not depend on the choice of a chain $\sigma ⇒\dots ⇒{\sigma }_{0}$ and the corresponding sequence of embeddings.
Let $\sigma =\prod _{k=1}^{l}{\sigma }_{{i}_{k}}={\sigma }_{{i}_{l}}\cdots {\sigma }_{{i}_{1}}$ be some reduced decomposition ${\sigma }^{p}=\prod _{k=1}^{p}{\sigma }_{{i}_{k}}\in {S}_{r}\text{.}$ We associate to each ${\sigma }_{{i}_{p}}$ the element ${\stackrel{\sim }{\sigma }}_{{i}_{p}}\in S$ permuting the subfamilies $\left({\stackrel{\sim }{\ell }}_{i+1}^{\prime },\dots ,{\stackrel{\sim }{\ell }}_{i}^{\prime }+1\right)$ and $\left({\stackrel{\sim }{\ell }}_{i+1},\dots ,{\stackrel{\sim }{\ell }}_{i+1}+1\right)$ in ${w}_{\tau }\left({u}^{\tau }\right)$ for $\tau ={\sigma }^{p-1},$ $\stackrel{\sim }{\mu }={\mu }^{\tau }=\left(\left\{{\stackrel{\sim }{\ell }}_{i},{\stackrel{\sim }{\ell }}_{i}^{\prime }\right\}\right)\text{.}$ Put $\stackrel{\sim }{\sigma }=\prod _{k=1}^{l}{\stackrel{\sim }{\sigma }}_{ik}\text{.}$ We can verify that $\stackrel{\sim }{\sigma }$ does not depend on the choice of $\sigma$ and $\stackrel{\sim }{\sigma }\stackrel{\text{def}}{=}\stackrel{\sim }{\sigma }{w}_{\mu }\ge {w}_{\mu }\text{.}$
Corollary 4. If one puts ${\omega }_{\sigma }={w}_{\sigma }^{-1}\stackrel{\sim }{\sigma },$ then $\stackrel{\sim }{\sigma }\ge {\omega }_{\sigma },$ ${\stackrel{\sim }{I}}_{\sigma }={H}_{n}{\phi }_{\stackrel{\sim }{\sigma }}\left({u}^{\mu }\right)$ for each $\sigma \in {S}_{r}\text{.}$ The isomorphism of ${I}_{\sigma }$ and ${\stackrel{\sim }{I}}_{\sigma }$ mapping ${\phi }_{{w}_{\sigma }}\left({u}^{\sigma }\right)$ into ${\phi }_{\stackrel{\sim }{\sigma }}\left({u}^{\mu }\right)$ is induced by multiplication of ${H}_{n}$ on the right by ${\phi }_{{\omega }_{\sigma }}\left({u}^{\mu }\right)\text{.}$ If ${I}_{\sigma }\ne 0,$ then ${\stackrel{\sim }{I}}_{\sigma }\subset {\stackrel{\sim }{I}}_{\tau }⇔\sigma ⇒\dots ⇒\tau$ for some chain.
Proposition 5 [BGG1975-2]. On the set of pairs $\sigma ⇒\tau$ there exists a function $\epsilon \left(\sigma ,\tau \right)=±1$ for which $\epsilon \left(\sigma \prime ,\tau \right)\epsilon \left(\sigma ,\sigma \prime \right)=-\epsilon \left(\tau \prime ,\tau \right)\epsilon \left(\sigma ,\tau \prime \right)$ on each square.
Put ${V}_{p}=\underset{\ell \left(\sigma \right)=p}{⨁}{\stackrel{\sim }{I}}_{\sigma }\text{.}$ Let ${\nu }_{\sigma ,\tau }:{V}_{p}\to {V}_{p-1}$ be the homomorphism, defined for $\sigma ⇒\tau ,$ $\ell \left(\sigma \right)=p,$ inducing the natural embedding ${\stackrel{\sim }{I}}_{\sigma }\subset {\stackrel{\sim }{I}}_{\tau }\subset {I}_{r}$ and mapping each ${\stackrel{\sim }{I}}_{\sigma \prime }\subset {V}_{p}$ for $\sigma \prime \ne \sigma$ into zero. Put ${d}_{p}=\sum _{\sigma ⇒\tau }\epsilon \left(\sigma ,\tau \right){\nu }_{\sigma ,\tau },$ $l\left(\sigma \right)=p\text{.}$ We denote by ${d}_{0}$ the homomorphism of ${I}_{\mu }={V}_{0}$ onto its only irreducible quotient module ${V}_{-1}\ne 0$ (cf. [Zel1980]) generalizing the representation of ${S}_{n}$ associated with Jung's skew scheme corresponding to $\mu$ [JKe1981, Che1986, Che1986-2]. Let ${V}_{p}=0$ for $p>r\left(r-1\right)/2,$ $p<-1\text{.}$
Theorem 6. The sequence $\left\{{V}_{p},{d}_{p}\right\}$ is exact.
3. Remarks. We will give an example of the function $\epsilon \text{.}$ A reduced decomposition of $\sigma \in {S}_{r}$ is said to be canonical if: a) for $1\le i the decomposition remains reduced after eliminating all ${\sigma }_{1},\dots ,{\sigma }_{i}\text{;}$ b) for adjacent ${\sigma }_{i}{\sigma }_{j}$ in the decomposition we always have $i>j$ when $j\ne i±1\text{.}$ The author's attention was attracted to such decompositions by A. N. Kirillov. Put $\epsilon \left(\sigma ,\tau \right)={\left(-1\right)}^{k+\pi },$ where ${\sigma }_{{i}_{k}}$ is the transposition eliminated from the canonical decomposition of $\sigma$ in the passage to $\tau$ $\text{(}k$ is its number in the decomposition), $\pi$ is the number of permutations of adjacent pairs ${\sigma }_{i}{\sigma }_{j}\to {\sigma }_{j}{\sigma }_{i}$ for $i\ne j±1$ (the replacements ${\sigma }_{i}{\sigma }_{i±1}{\sigma }_{i}\to {\sigma }_{i±1}{\sigma }_{i}{\sigma }_{i±1}$ are not counted) used to transform the obtained reduced decomposition for $\tau$ to the canonical one.
2)To prove the theorem, using the results of [Che1986, Che1986-2] on branching of special bases in ${V}_{-1},$ we impose an induction restriction on ${ℋ}_{n-1}\subset {ℋ}_{n}\text{.}$ Here, $\left\{V,d\right\}$ splits into a direct sum of some sequences $\left\{{V}^{i},{d}^{i}\right\}$ for ${\mu }^{i}$ in which ${\ell }_{i}$ are replaced by ${\ell }_{i}-1\text{.}$ For an admissible ${\mu }^{i}$ (satisfying the same inequalities as $\mu \text{),}$ after eliminating all ${\stackrel{\sim }{I}}_{\sigma }$ whose construction is not compatible with the passage ${\ell }_{i}\to {\ell }_{i}-1,$ the sequence $\left\{{V}^{i},{d}^{i}\right\}$ coincides with the sequence of the theorem for ${ℋ}_{n-1}$ instead of ${ℋ}_{n}$ and ${\mu }^{i}$ instead of $\mu \text{.}$ If ${\mu }^{i}$ is not admissible, then it turns out that the sequence $\left\{{V}^{i},{d}^{i},{V}_{-1}^{i}\stackrel{\text{def}}{=}0\right\}$ is exact. In the proof of this fact one verifies that for each $\sigma$ one of the embeddings of the form ${\nu }_{\sigma ,\tau }^{i}$ or ${\nu }_{\tau ,\sigma }^{i}$ is an isomorphism for a suitable $\tau \text{.}$
3) All constructions of this note, as well as the theorem, are extended verbatim to the degenerate algebra ${ℋ}_{n}$ $\left(q\to 1\right)$ (cf. [Che1986, Che1986-2, Dri1986-2]). For this algebra, ${T}_{i}$ can be identified with ${s}_{i}$ and ${x}_{i}{s}_{i}-{s}_{i}{x}_{i+1}=1={s}_{i}{x}_{i}-{x}_{i+1}{s}_{i}\text{.}$ Respectively, one has to put ${x}_{k}\left(1\right)={u}_{k},$ ${\phi }_{{s}_{i}}=1+\left({\lambda }_{2}-{\lambda }_{1}\right){s}_{i}\text{.}$ Then ${I}_{\mu }$ is isomorphic as an ${S}_{n}\text{-module}$ to the representation induced from the identity representation of the subgroup $\prod _{i=1}^{r}{S}_{{l}_{i}-{l}_{i}^{\prime }}\subset {S}_{n}\text{;}$ ${V}_{-1}$ for integers $\left\{{\ell }_{i}\right\}$ ${V}_{-1}$ corresponds to Jung's skew scheme [JKe1981] represented by cells with the set of centers $\left(i,j\right)\subset {Z}^{2},$ $0\le i\le r-1,$ ${\ell }_{r-i}^{\prime }+i Thus, the theorem, indeed, generalizes the character formula of Frobenius. A replacement of $\left({u}_{k}\right)$ by $\left(-{u}_{k}\right)$ in all constructions results in a similar "antisymmetric" resolution for the ${ℋ}_{n}\text{-module}$ corresponding to the scheme obtained from the scheme $\mu$ by the reflection in the diagonal $i=j\text{.}$ This is also true for $q\ne 1\text{.}$ The statement and the proof of Theorem 6 are extended to representations of quantum $R\text{-algebras}$ of the A series corresponding to $\mu$ [Che1986-2, Dri1986-2] (cf. [Che1986-3]).
## Literature cited
[JKe1981] G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications 16, Addison-Wesley Publishing Co., Reading, Mass., 1981.
[BGG1975-2] I.N. Bernstein, I.M. Gel'fand, and S.I. Gel'fand, Publ. of 1971 Summer School in Math., Budapest (1975), pp. 21-64.
[Zel1987-2] A.V. Zelevinskii, Funktsion. Anal. Prilozhen., 21, No. 2, 74-75 (1987).
[Zel1980] A. Zelevinsky, Induced representations of $𝔭\text{-adic}$ groups II: On irreducible representations of $GL\left(n\right)$, Ann. Sci. École Norm. Sup. Ser. (4) 13 (1980) 165–210.
[Rog1985] J. Rogawski, On modules over the Hecke algebra of a p-adic group, Invent. Math. 79 (1985) 443–465. MR 86j:22028
[Che1986] I.V. Cherednik, in: Group-Theoretic Methods in Physics. Proceedings of the 3rd Int. Sem. [in Russian], Nauka, Moscow (1986).
[Che1986-2] I.V. Cherednik, Funktsion. Anal. Prilozhen., 20, No. 1, 87-88 (1986).
[Dri1986-2] V.G. Drinfel'd, Funktsion. Anal. Prilozhen., 20, No. 1, 69-70 (1986).
[Che1986-3] I.V. Cherednik, Dokl. Akad. Nauk SSSR, 291, No. 1, 49-53 (1986). | 2019-01-18 17:31:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 220, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9373265504837036, "perplexity": 6894.987970539312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583660258.36/warc/CC-MAIN-20190118172438-20190118194438-00383.warc.gz"} |
https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/Book%3A_Introduction_to_Social_Network_Methods_(Hanneman)/01%3A_Social_Network_Data | # 1: Social Network Data
• 1.1: Introduction - What's different about social network data?
On one hand, there really isn't anything about social network data that is all that unusual. Social network analysts do use a specialized language for describing the structure and contents of the sets of observations that they use. But, network data can also be described and understood using the ideas and concepts of more familiar methods, like cross-sectional survey research.
• 1.2: Nodes
Network data are defined by actors and by relations (or "nodes" and "edges"). The nodes or actors part of network data would seem to be pretty straight-forward. Other empirical approaches in the social sciences also think in terms of cases or subjects or sample elements and the like. There is one difference with most network data, however, that makes a big difference in how such data are usually collected -- and the kinds of samples and populations that are studied.
• 1.3: Relations
The other half of the design of network data has to do with what ties or relations are to be measured for the selected nodes. There are two main issues to be discussed here. In many network studies, all of the ties of a given type among all of the selected nodes are studied -- that is, a census is conducted. But, sometimes different approaches are used (because they are less expensive, or because of a need to generalize) that sample ties.
• 1.4: Scales of Measurement
Like other kinds of data, the information we collect about ties between actors can be measured (i.e. we can assign scores to our observations) at different "levels of measurement." The different levels of measurement are important because they limit the kinds of questions that can be examined by the researcher. Scales of measurement are also important because different kinds of scales have different mathematical properties, and call for different algorithms in describing patterns.
• 1.5: A note on Statistics and Social Network Data
Social network analysis is more a branch of "mathematical" sociology than of "statistical or quantitative analysis," though social network analysts most certainly practice both approaches. The distinction between the two approaches is not clear-cut. Mathematical approaches to network analysis tend to treat the data as "deterministic." That is, they tend to regard the measured relationships and relationship strengths as accurately reflecting the "real" or "final" status of the network.
This page titled 1: Social Network Data is shared under a not declared license and was authored, remixed, and/or curated by Robert Hanneman & Mark Riddle. | 2023-03-28 00:02:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9027118682861328, "perplexity": 853.01130371809}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948708.2/warc/CC-MAIN-20230327220742-20230328010742-00193.warc.gz"} |
https://piyanit.nl/blog/principal-component-analysis/ | # Outlier Detection using Principal Component Analysis
## Summary
My first article presents an analysis of an experiment on a structured dataset containing $$5$$ outliers. The goal of the experiment is to find the outliers, and the analysis is based on $$1,500$$ observations and $$7$$ features.
After standardization the data, a model of a set of linearly uncorrelated principal components was created.
After creating the linear model, a figure showing outliers in the dataset was created, and finally a table of statistics of the principal components was created.
## Nomenclature
\begin{array}{|l|l|}
\hline
\textbf{Symbol} & \textbf{Definition} \\
\hline
\mathbb{R} & \text{Set of real numbers}\\
\mu(\cdot) & \text{Mean}\\
||\cdot||_{2} & \text{Euclidean norm}\\
\sigma^{2}(\cdot) & \text{Variance}\\
m & \text{Number of observations in the experiment}\\
n & \text{Number of features in the experiment}\\
X: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} & m \times n \text{ design matrix}\\
\mathbf{x_{k}} \in \mathbb{R}^{m}& \text{The k-th feature vector in } X\\
L & \text{The number of principal components}\\
U: \mathbb{R}^{L} \rightarrow \mathbb{R}^{m} & m \times L \text{ matrix that contains the left singular }\\
&\text{orthogonal unit vectors of }X \text{ corresponding to}\\
& \text{the largest singular values}\\
\lambda_{k} & \text{The k-th eigenvalue of the covariance matrix } X^{T}X\\
\mathbf{u_{k}} \in \mathbb{R}^{m} & \text{The k-th left singular orthogonal unit vector of }X\\
\mathbf{c_{k}} \in \mathbb{R}^{m} & \text{The k-th principal component}\\
\mathcal{N}\left(\mu,\sigma^{2}\right) & \text{Normal distribution with mean } \mu \text{ and variance } \sigma^{2}\\
\hline
\end{array}
## Dataset
My dataset consists of $$n = 7$$ features and $$m = 1,500$$ observations. Each entry in the dataset contains random noise. The first $$5$$ observations are shown below:
\begin{array}{|r|r|r|r|r|r|r|}
\hline \mathbf{x_{1}}
& \mathbf{x_{2}}
& \mathbf{x_{3}}
& \mathbf{x_{4}}
& \mathbf{x_{5}}
& \mathbf{x_{6}}
& \mathbf{x_{7}}\\
\hline
63 & 55 & 59 & 57 & 67 & 52 & 70\\
64 & -2 & -3 & -4 & -5 & 51 & 74\\
65 & 55 & 60 & 57 & 71 & 51 & 76\\
69 & 59 & 64 & 61 & 75 & 55 & 79\\
-1 & 59 & -3 & -4 & -5 & 56 & 79\\
\hline
\end{array}
## Design Matrix
The design matrix $$X$$ is derived from the dataset by standardization. Because you do not want to compare apples and oranges, I decided to scale the dataset such that the features have the properties of a standard normal distibution ( $$\mu = 0$$ and $$\sigma^{2} = 1$$ ). Thus for $$k \in \left\{1,2,3,4,5,6,7\right\}$$ holds $\mathbf{x_{k}}\sim \mathcal{N}\left(0,1\right)$
## Principal Component Analysis
The goal of the algorithm is to find uncorrelated principal components that retains most of the information. The algorithm uses the Singular Value Decomposition of $$X$$ to find the principal components [1]:
The inputs of the algorithm are the components to keep $$L = 4$$ and the design matrix $$X$$.
The output is a linear transformation of $$X$$ in $$4$$ uncorrelated principal components.
## Model
The linear transformation of $$X$$ in $$4$$ uncorrelated principal components has the model below.$\begin{bmatrix}\mathbf{c_{1}}&\mathbf{c_{2}}&\mathbf{c_{3}}&\mathbf{c_{4}}\end{bmatrix} =\begin{bmatrix}97.14\mathbf{u_{1}}&22.55\mathbf{u_{2}}&17.62\mathbf{u_{3}}&9.30\mathbf{u_{4}}\end{bmatrix}=$
$U \begin{bmatrix} 97.14&0&0&0\\ 0&22.55&0&0\\ 0&0&17.62&0\\ 0&0&0&9.30\\ \end{bmatrix}$
where the matrix at the left-hand side contains the principal components, and at the right-hand side you see the diagonal matrix containing the singular values of $$X$$.
The figure below show you 5 numbered observations that can be considered as outliers.
## Evaluation
Statistics are computed for each principal component. The results are shown in the table below.
\begin{array}{|c|r|r|r|}
\hline
k &\sigma^{2}\left(\mathbf{c_{k}}\right) &\sqrt{\lambda_{k}} &\mu\left(\mathbf{c_{k}}\right) \\
\hline
1 & 6.2907864 & 97.14 &0\\
2 & 0.3390017 & 22.55 &0\\
3 & 0.2067414 & 17.62 &0\\
4 & 0.0576600 & 09.30 &0\\\hline
\end{array}
Because of $$\mathbf{c_{1}}$$ has the highest variance, you may conclude that the first component retains most of the information.
Moreover, there holds that $$\lambda_{k} = 1,500\sigma^{2}\left(\mathbf{c_{k}}\right) \forall k \in \{1,2,3,4\}$$ due to the equation $n \sigma^{2}\left(\mathbf{c_{k}}\right) = ||\mathbf{c_{k}}||^{2}_{2} =\mathbf{c^{T}_{k}}\mathbf{c_{k}}=\sqrt{\lambda_{k}}\left(\mathbf{u^{T}_{k}}\mathbf{u_{k}}\right)\sqrt{\lambda_{k}} = \lambda_{k}.$
## Conclusion
The experiment was performed on a structured dataset and contains highly deviant observations in comparison with the other $$1,495$$ observations. This analysis has shown that the $$5$$ outliers can be confidently found using principal components.
## Machine Learning Experiment
(c) 2018, Jan Ruijgrok. All Rights Reserved. | 2022-12-09 14:50:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8864523768424988, "perplexity": 772.9909352913496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711417.46/warc/CC-MAIN-20221209144722-20221209174722-00578.warc.gz"} |
https://www.physicsforums.com/threads/bcfw-recursion-relation.894484/ | # I BCFW recursion relation
1. Nov 23, 2016
### Lapidus
Below is a snipet from http://file:///C:/Users/Christian.Hollersen/Downloads/Britto_2011_2%20(1).pdf [Broken] of Britto. Similar explanation can be found in the QFT books of Zee, Schwarz or the Scattering Amplitude text of Huang. Or any other text that covers BCFW recursion. My dumb question: how and why does the residue at this pole take this funny factorization form? (For clarifcation: residue is the just the word QFT people use for the numerator of a rational function with a simple pole, right?)
Thank you!
Last edited by a moderator: May 8, 2017
2. Nov 25, 2016
### OldManAndTheSeaQuark
There are at least two ways to show the factorization of amplitudes on simple poles. An ancient proof using only properties of the S-matirx (analyticity, unitarity and cluster decomposition) can be found in Eden et al. "The Analytic S-matrix" sec. 4.5. For a more recent discussion see the nice review by Conde
http://pos.sissa.it/archive/conferences/201/005/Modave 2013_005.pdf
Alternatively there is a more local field theoretic proof given in Weinberg "The Quantum Theory of Fields Vol 1." sec. 10.2. | 2017-12-15 07:21:12 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8465009927749634, "perplexity": 1711.856930212165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948567042.50/warc/CC-MAIN-20171215060102-20171215080102-00086.warc.gz"} |
https://ftp.aimsciences.org/article/doi/10.3934/dcds.2016.36.1465 | # American Institute of Mathematical Sciences
March 2016, 36(3): 1465-1491. doi: 10.3934/dcds.2016.36.1465
## Young towers for product systems
1 International Center for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy 2 International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy
Received February 2015 Revised June 2015 Published August 2015
We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $C^2$ interval maps with critical points and singularities, Hénon maps and partially hyperbolic systems.
Citation: Stefano Luzzatto, Marks Ruziboev. Young towers for product systems. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1465-1491. doi: 10.3934/dcds.2016.36.1465
##### References:
[1] J. F. Alves, C. Dias and S. Luzzatto, Geometry of expanding absolutely continuous invariant measures and the liftability problem, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 30 (2013), 101-120. doi: 10.1016/j.anihpc.2012.06.004. Google Scholar [2] J. F. Alves, J. M. Freitas, S. Luzzatto and S. Vaienti, From rates of mixing to recurrence times via large deviations, Advances in Mathematics, 228 (2011), 1203-1236. doi: 10.1016/j.aim.2011.06.014. Google Scholar [3] J. F. Alves and X. Li, Gibbs-Markov-Young structure with (stretched) exponential recurrence times for partially hyperbolic attractors, Adv. Math., 279 (2015), 405-437. doi: 10.1016/j.aim.2015.02.017. Google Scholar [4] J. F. Alves, S. Luzzatto and V. Pinheiro, Markov structures and decay of correlations for non-uniformly expanding dynamical systems, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 22 (2005), 817-839. doi: 10.1016/j.anihpc.2004.12.002. Google Scholar [5] J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structures, J. Stat. Phys., 131 (2008), 505-534. doi: 10.1007/s10955-008-9482-6. Google Scholar [6] J. F. Alves and D. Schnellmann, Ergodic properties of Viana-like maps with singularities in the base dynamics, Proceedings of the AMS, 141 (2013), 3943-3955. doi: 10.1090/S0002-9939-2013-11680-1. Google Scholar [7] A. Avez, Propriétés ergodiques des endomorphisms dilatants des variétés compactes, C.R. Acad. Sci. Paris Sér. A-B, 266 (1968), 610-612. Google Scholar [8] V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific, 2000. doi: 10.1142/9789812813633. Google Scholar [9] V. Baladi and S. Gouëzel, Stretched exponential bounds for the correlations of the Viana-Alves skew products, Second Workshop on Dynamics and Randomness, Universidad de Chile, 2002. Google Scholar [10] M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. Math., 122 (1985), 1-25. doi: 10.2307/1971367. Google Scholar [11] M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576. doi: 10.1007/BF01232446. Google Scholar [12] V. I. Bogachev, Measure Theory, Vol. 1, Springer, 2006. Google Scholar [13] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer Lecture Notes in Math., 1975. Google Scholar [14] H. Bruin, S. Luzzatto and S. van Strien, Decay of correlations in one-dimensional dynamics, Annales de l'ENS, 36 (2003), 621-646. doi: 10.1016/S0012-9593(03)00025-9. Google Scholar [15] J. Buzzi and V. Maume-Deschamps, Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time, Discrete and Continuous Dynam. Systems, 12 (2005), 639-656. doi: 10.3934/dcds.2005.12.639. Google Scholar [16] J. Buzzi, O. Sester and M. Tsujii, Weakly expanding skew-products of quadratic maps, Ergod. Th. Dynam. Syst., 23 (2003), 1401-1414. doi: 10.1017/S0143385702001694. Google Scholar [17] N. Chernov, Statistical properties of piecewise smooth hyperbolic systems in high dimensions, Discrete and Continuous Dynam. Systems, 5 (1999), 425-448. doi: 10.3934/dcds.1999.5.425. Google Scholar [18] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys., 94 (1999), 513-556. doi: 10.1023/A:1004581304939. Google Scholar [19] N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, Vol. 127, Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/surv/127. Google Scholar [20] K. Diaz-Ordaz, Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-Like maps, Discrete and Continuous Dynam. Systems, 15 (2006), 159-176. doi: 10.3934/dcds.2006.15.159. Google Scholar [21] K. Diaz-Ordaz, M. P. Holland and S. Luzzatto, Statistical properties of one-dimensional maps with critical points and singularities, Stochastics and Dynamics, 6 (2006), 423-458. doi: 10.1142/S0219493706001852. Google Scholar [22] S. Gouëzel, Decay of correlations for nonuniformly expanding systems, Bull. Soc. Math. France, 134 (2006), 1-31. Google Scholar [23] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140. doi: 10.1007/BF01215004. Google Scholar [24] M. Holland, Slowly mixing systems and intermittency maps, Ergodic theory and Dynamical Systems, 25 (2004), 133-159. doi: 10.1017/S0143385704000343. Google Scholar [25] H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic theory and Dynamical Systems, 24 (2004), 495-524. doi: 10.1017/S0143385703000671. Google Scholar [26] G. Keller and T. Nowicki, Spectral theory, zeta functions and the distributions of points for Collet-Eckman maps, Comm. Math. Phys., 149 (1992), 31-69. doi: 10.1007/BF02096623. Google Scholar [27] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Transactions of The AMS, 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1. Google Scholar [28] C. Liverani, Decay of correlations, Annals Math., 142 (1995), 239-301. doi: 10.2307/2118636. Google Scholar [29] C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory and Dynamical Systems, 33 (2013), 168-182. doi: 10.1017/S0143385711000939. Google Scholar [30] A. Lopes, Entropy and large deviations, Nonlinearity, 3 (1990), 527-546. doi: 10.1088/0951-7715/3/2/013. Google Scholar [31] S. Luzzatto, Stochastic-like Behaviour in Non-Uniformly Expanding Maps, Handbook of Dynamical Systems Vol. 1B, Elsevier, 2006. doi: 10.1016/S1874-575X(06)80028-7. Google Scholar [32] S. Luzzatto and I. Melbourne, Statistical properties and decay of correlations for interval maps with critical points and singularities, Commun. Math. Phys., 320 (2013), 21-35. doi: 10.1007/s00220-013-1709-y. Google Scholar [33] V. Lynch, Non-uniformly Expanding Dynamical Systems and Decay of Correlations for Non-Hölder Continuous Observables, Ph.D thesis, University of Warwick, 2003. Google Scholar [34] V. Lynch, Decay of correlations for non-Hölder observables, Discrete and Continuous Dynam. Systems, 16 (2006), 19-46. doi: 10.3934/dcds.2006.16.19. Google Scholar [35] I. Melbourne and M. Nicol, Large deviations for nonuniformly hyperbolic systems, Transactions of AMS, 360 (2008), 6661-6676. doi: 10.1090/S0002-9947-08-04520-0. Google Scholar [36] I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., 260 (2005), 131-146. doi: 10.1007/s00220-005-1407-5. Google Scholar [37] P. Natalini and B. Palumbo, Inequalities for the Incomplete Gamma function, Mathematical Inequalities & Applications, 3 (2000), 69-77. doi: 10.7153/mia-03-08. Google Scholar [38] T. Nowicki and S. van Strien, Absolutely continuous invariant measures for $C^2$ unimodal maps satisfying the Collet-Eckmann conditions, Invent. Math., 93 (1988), 619-635. doi: 10.1007/BF01410202. Google Scholar [39] V. Pinheiro, Expanding Measures， Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 28 (2011), 889-939. doi: 10.1016/j.anihpc.2011.07.001. Google Scholar [40] M. Pollicott and M. Yuri, Statistical properties of maps with indifferent periodic points, Commun. Math. Phys., 217 (2001), 503-520. doi: 10.1007/s002200100368. Google Scholar [41] D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. Google Scholar [42] Y. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys, 27 (1972), 21-64. Google Scholar [43] Y. Sinai, Dynamical systems with elastic reflections, Ergodic properties of dispersing billiards, Russ. Math. Surveys, 25 (1970), 141-192. Google Scholar [44] T. Tao and V. H. Vu, Additive Combinatorics, Cambridge studies in advanced mathematics, 105, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511755149. Google Scholar [45] D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete and continuous time Dynam. Systems, 30 (2011), 917-944. doi: 10.3934/dcds.2011.30.917. Google Scholar [46] L.-S. Young, Decay of correlations for certain quadratic maps, Comm. Math. Phys., 146 (1992), 123-138. doi: 10.1007/BF02099211. Google Scholar [47] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650. doi: 10.2307/120960. Google Scholar [48] L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180. Google Scholar
show all references
##### References:
[1] J. F. Alves, C. Dias and S. Luzzatto, Geometry of expanding absolutely continuous invariant measures and the liftability problem, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 30 (2013), 101-120. doi: 10.1016/j.anihpc.2012.06.004. Google Scholar [2] J. F. Alves, J. M. Freitas, S. Luzzatto and S. Vaienti, From rates of mixing to recurrence times via large deviations, Advances in Mathematics, 228 (2011), 1203-1236. doi: 10.1016/j.aim.2011.06.014. Google Scholar [3] J. F. Alves and X. Li, Gibbs-Markov-Young structure with (stretched) exponential recurrence times for partially hyperbolic attractors, Adv. Math., 279 (2015), 405-437. doi: 10.1016/j.aim.2015.02.017. Google Scholar [4] J. F. Alves, S. Luzzatto and V. Pinheiro, Markov structures and decay of correlations for non-uniformly expanding dynamical systems, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 22 (2005), 817-839. doi: 10.1016/j.anihpc.2004.12.002. Google Scholar [5] J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structures, J. Stat. Phys., 131 (2008), 505-534. doi: 10.1007/s10955-008-9482-6. Google Scholar [6] J. F. Alves and D. Schnellmann, Ergodic properties of Viana-like maps with singularities in the base dynamics, Proceedings of the AMS, 141 (2013), 3943-3955. doi: 10.1090/S0002-9939-2013-11680-1. Google Scholar [7] A. Avez, Propriétés ergodiques des endomorphisms dilatants des variétés compactes, C.R. Acad. Sci. Paris Sér. A-B, 266 (1968), 610-612. Google Scholar [8] V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific, 2000. doi: 10.1142/9789812813633. Google Scholar [9] V. Baladi and S. Gouëzel, Stretched exponential bounds for the correlations of the Viana-Alves skew products, Second Workshop on Dynamics and Randomness, Universidad de Chile, 2002. Google Scholar [10] M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. Math., 122 (1985), 1-25. doi: 10.2307/1971367. Google Scholar [11] M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576. doi: 10.1007/BF01232446. Google Scholar [12] V. I. Bogachev, Measure Theory, Vol. 1, Springer, 2006. Google Scholar [13] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer Lecture Notes in Math., 1975. Google Scholar [14] H. Bruin, S. Luzzatto and S. van Strien, Decay of correlations in one-dimensional dynamics, Annales de l'ENS, 36 (2003), 621-646. doi: 10.1016/S0012-9593(03)00025-9. Google Scholar [15] J. Buzzi and V. Maume-Deschamps, Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time, Discrete and Continuous Dynam. Systems, 12 (2005), 639-656. doi: 10.3934/dcds.2005.12.639. Google Scholar [16] J. Buzzi, O. Sester and M. Tsujii, Weakly expanding skew-products of quadratic maps, Ergod. Th. Dynam. Syst., 23 (2003), 1401-1414. doi: 10.1017/S0143385702001694. Google Scholar [17] N. Chernov, Statistical properties of piecewise smooth hyperbolic systems in high dimensions, Discrete and Continuous Dynam. Systems, 5 (1999), 425-448. doi: 10.3934/dcds.1999.5.425. Google Scholar [18] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys., 94 (1999), 513-556. doi: 10.1023/A:1004581304939. Google Scholar [19] N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, Vol. 127, Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/surv/127. Google Scholar [20] K. Diaz-Ordaz, Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-Like maps, Discrete and Continuous Dynam. Systems, 15 (2006), 159-176. doi: 10.3934/dcds.2006.15.159. Google Scholar [21] K. Diaz-Ordaz, M. P. Holland and S. Luzzatto, Statistical properties of one-dimensional maps with critical points and singularities, Stochastics and Dynamics, 6 (2006), 423-458. doi: 10.1142/S0219493706001852. Google Scholar [22] S. Gouëzel, Decay of correlations for nonuniformly expanding systems, Bull. Soc. Math. France, 134 (2006), 1-31. Google Scholar [23] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140. doi: 10.1007/BF01215004. Google Scholar [24] M. Holland, Slowly mixing systems and intermittency maps, Ergodic theory and Dynamical Systems, 25 (2004), 133-159. doi: 10.1017/S0143385704000343. Google Scholar [25] H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic theory and Dynamical Systems, 24 (2004), 495-524. doi: 10.1017/S0143385703000671. Google Scholar [26] G. Keller and T. Nowicki, Spectral theory, zeta functions and the distributions of points for Collet-Eckman maps, Comm. Math. Phys., 149 (1992), 31-69. doi: 10.1007/BF02096623. Google Scholar [27] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Transactions of The AMS, 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1. Google Scholar [28] C. Liverani, Decay of correlations, Annals Math., 142 (1995), 239-301. doi: 10.2307/2118636. Google Scholar [29] C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory and Dynamical Systems, 33 (2013), 168-182. doi: 10.1017/S0143385711000939. Google Scholar [30] A. Lopes, Entropy and large deviations, Nonlinearity, 3 (1990), 527-546. doi: 10.1088/0951-7715/3/2/013. Google Scholar [31] S. Luzzatto, Stochastic-like Behaviour in Non-Uniformly Expanding Maps, Handbook of Dynamical Systems Vol. 1B, Elsevier, 2006. doi: 10.1016/S1874-575X(06)80028-7. Google Scholar [32] S. Luzzatto and I. Melbourne, Statistical properties and decay of correlations for interval maps with critical points and singularities, Commun. Math. Phys., 320 (2013), 21-35. doi: 10.1007/s00220-013-1709-y. Google Scholar [33] V. Lynch, Non-uniformly Expanding Dynamical Systems and Decay of Correlations for Non-Hölder Continuous Observables, Ph.D thesis, University of Warwick, 2003. Google Scholar [34] V. Lynch, Decay of correlations for non-Hölder observables, Discrete and Continuous Dynam. Systems, 16 (2006), 19-46. doi: 10.3934/dcds.2006.16.19. Google Scholar [35] I. Melbourne and M. Nicol, Large deviations for nonuniformly hyperbolic systems, Transactions of AMS, 360 (2008), 6661-6676. doi: 10.1090/S0002-9947-08-04520-0. Google Scholar [36] I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., 260 (2005), 131-146. doi: 10.1007/s00220-005-1407-5. Google Scholar [37] P. Natalini and B. Palumbo, Inequalities for the Incomplete Gamma function, Mathematical Inequalities & Applications, 3 (2000), 69-77. doi: 10.7153/mia-03-08. Google Scholar [38] T. Nowicki and S. van Strien, Absolutely continuous invariant measures for $C^2$ unimodal maps satisfying the Collet-Eckmann conditions, Invent. Math., 93 (1988), 619-635. doi: 10.1007/BF01410202. Google Scholar [39] V. Pinheiro, Expanding Measures， Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 28 (2011), 889-939. doi: 10.1016/j.anihpc.2011.07.001. Google Scholar [40] M. Pollicott and M. Yuri, Statistical properties of maps with indifferent periodic points, Commun. Math. Phys., 217 (2001), 503-520. doi: 10.1007/s002200100368. Google Scholar [41] D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. Google Scholar [42] Y. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys, 27 (1972), 21-64. Google Scholar [43] Y. Sinai, Dynamical systems with elastic reflections, Ergodic properties of dispersing billiards, Russ. Math. Surveys, 25 (1970), 141-192. Google Scholar [44] T. Tao and V. H. Vu, Additive Combinatorics, Cambridge studies in advanced mathematics, 105, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511755149. Google Scholar [45] D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete and continuous time Dynam. Systems, 30 (2011), 917-944. doi: 10.3934/dcds.2011.30.917. Google Scholar [46] L.-S. Young, Decay of correlations for certain quadratic maps, Comm. Math. Phys., 146 (1992), 123-138. doi: 10.1007/BF02099211. Google Scholar [47] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650. doi: 10.2307/120960. Google Scholar [48] L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180. Google Scholar
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Parameter estimation and uncertainty quantification for an epidemic model
1. Center for Quantitative Sciences in Biomedicine and Department of Mathematics, North Carolina State University, Raleigh, NC 27695, and Department of Mathematics & Computer Science, Valparaiso University, 1900 Chapel Drive, Valparaiso, IN 46383
2. Department of Mathematics, University of North Carolina, Chapel Hill, CB #3250, Chapel Hill, NC 27599
3. Program in Applied Mathematics, University of Arizona, 617 N. Santa Rita Ave., PO Box 210089, Tucson, AZ 85721-0089
4. Department of Mathematics, Morehouse College, 830 Westview Drive SW Unit 142133, Atlanta, GA 30314
5. Department of Mathematics, North Carolina State University, Raleigh, NC 27695
6. Biomathematics Graduate Program and Department of Mathematics, North Carolina State University, Raleigh NC, 27695, USA and Fogarty International Center, National Institutes of Health, Bethesda, MD 20892
## Abstract Related pages
We examine estimation of the parameters of Susceptible-Infective-Recovered (SIR) models in the context of least squares. We review the use of asymptotic statistical theory and sensitivity analysis to obtain measures of uncertainty for estimates of the model parameters and the basic reproductive number ($R_0$)---an epidemiologically significant parameter grouping. We find that estimates of different parameters, such as the transmission parameter and recovery rate, are correlated, with the magnitude and sign of this correlation depending on the value of $R_0$. Situations are highlighted in which this correlation allows $R_0$ to be estimated with greater ease than its constituent parameters. Implications of correlation for parameter identifiability are discussed. Uncertainty estimates and sensitivity analysis are used to investigate how the frequency at which data is sampled affects the estimation process and how the accuracy and uncertainty of estimates improves as data is collected over the course of an outbreak. We assess the informativeness of individual data points in a given time series to determine when more frequent sampling (if possible) would prove to be most beneficial to the estimation process. This technique can be used to design data sampling schemes in more general contexts.
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Citation: Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd. Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences and Engineering, 2012, 9(3): 553-576. doi: 10.3934/mbe.2012.9.553
• 1. Brett Matzuka, Jesper Mehlsen, Hien Tran, Mette Sofie Olufsen, Using Kalman Filtering to Predict Time-Varying Parameters in a Model Predicting Baroreflex Regulation During Head-Up Tilt, IEEE Transactions on Biomedical Engineering, 2015, 62, 8, 1992, 10.1109/TBME.2015.2409211
• 2. Necibe Tuncer, Trang T. Le, Structural and practical identifiability analysis of outbreak models, Mathematical Biosciences, 2018, 299, 1, 10.1016/j.mbs.2018.02.004
• 3. João N. C. Gonçalves, Helena Sofia Rodrigues, M. Teresa T. Monteiro, , Intelligent Systems Design and Applications, 2017, Chapter 96, 974, 10.1007/978-3-319-53480-0_96
• 4. Andreas Widder, On the usefulness of set-membership estimation in the epidemiology of infectious diseases, Mathematical Biosciences and Engineering, 2017, 15, 1, 141, 10.3934/mbe.2018006
• 5. Necibe Tuncer, Hayriye Gulbudak, Vincent L. Cannataro, Maia Martcheva, Structural and Practical Identifiability Issues of Immuno-Epidemiological Vector–Host Models with Application to Rift Valley Fever, Bulletin of Mathematical Biology, 2016, 78, 9, 1796, 10.1007/s11538-016-0200-2
• 6. Hailay Weldegiorgis Berhe, Oluwole Daniel Makinde, David Mwangi Theuri, Parameter Estimation and Sensitivity Analysis of Dysentery Diarrhea Epidemic Model, Journal of Applied Mathematics, 2019, 2019, 1, 10.1155/2019/8465747
• 7. Ming Liu, Xifen Xu, Jie Cao, Ding Zhang, Integrated planning for public health emergencies: A modified model for controlling H1N1 pandemic, Journal of the Operational Research Society, 2019, 1, 10.1080/01605682.2019.1582589
• 8. Gerardo Chowell, Fitting dynamic models to epidemic outbreaks with quantified uncertainty: A primer for parameter uncertainty, identifiability, and forecasts, Infectious Disease Modelling, 2017, 2, 3, 379, 10.1016/j.idm.2017.08.001
• 9. Ping Yan, Gerardo Chowell, , Quantitative Methods for Investigating Infectious Disease Outbreaks, 2019, Chapter 9, 317, 10.1007/978-3-030-21923-9_9
• 10. Ming Liu, Jie Cao, Jing Liang, MingJun Chen, , Epidemic-logistics Modeling: A New Perspective on Operations Research, 2020, Chapter 9, 167, 10.1007/978-981-13-9353-2_9 | 2020-06-02 05:09:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29777583479881287, "perplexity": 4017.6789480890243}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347422803.50/warc/CC-MAIN-20200602033630-20200602063630-00174.warc.gz"} |
https://www.quizover.com/trigonometry/test/writing-and-interpreting-an-equation-for-a-linear-by-openstax | # 4.1 Linear functions (Page 4/27)
Page 4 / 27
Are the units for slope always
Yes. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.
## Calculate slope
The slope, or rate of change, of a function $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ can be calculated according to the following:
where $\text{\hspace{0.17em}}{x}_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{x}_{2}\text{\hspace{0.17em}}$ are input values, $\text{\hspace{0.17em}}{y}_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{y}_{2}\text{\hspace{0.17em}}$ are output values.
Given two points from a linear function, calculate and interpret the slope.
1. Determine the units for output and input values.
2. Calculate the change of output values and change of input values.
3. Interpret the slope as the change in output values per unit of the input value.
## Finding the slope of a linear function
If $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is a linear function, and $\text{\hspace{0.17em}}\left(3,-2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,1\right)\text{\hspace{0.17em}}$ are points on the line, find the slope. Is this function increasing or decreasing?
The coordinate pairs are $\text{\hspace{0.17em}}\left(3,-2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,1\right).\text{\hspace{0.17em}}$ To find the rate of change, we divide the change in output by the change in input.
We could also write the slope as $\text{\hspace{0.17em}}m=0.6.\text{\hspace{0.17em}}$ The function is increasing because $\text{\hspace{0.17em}}m>0.$
If $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is a linear function, and $\text{\hspace{0.17em}}\left(2,3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(0,4\right)\text{\hspace{0.17em}}$ are points on the line, find the slope. Is this function increasing or decreasing?
$m=\frac{4-3}{0-2}=\frac{1}{-2}=-\frac{1}{2};$ decreasing because $m<0.$
## Finding the population change from a linear function
The population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year if we assume the change was constant from 2008 to 2012.
The rate of change relates the change in population to the change in time. The population increased by $\text{\hspace{0.17em}}27,800-23,400=4400\text{\hspace{0.17em}}$ people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years.
So the population increased by 1,100 people per year.
The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of population per year if we assume the change was constant from 2009 to 2012.
$m=\frac{1,868-1,442}{2,012-2,009}=\frac{426}{3}=\text{142 people per year}$
## Writing and interpreting an equation for a linear function
Recall from Equations and Inequalities that we wrote equations in both the slope-intercept form and the point-slope form . Now we can choose which method to use to write equations for linear functions based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ in [link] .
We are not given the slope of the line, but we can choose any two points on the line to find the slope. Let’s choose $\text{\hspace{0.17em}}\left(0,7\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,4\right).\text{\hspace{0.17em}}$
$\begin{array}{ccc}\hfill m& =& \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\hfill \\ & =& \frac{4-7}{4-0}\hfill \\ & =& -\frac{3}{4}\hfill \end{array}$
Now we can substitute the slope and the coordinates of one of the points into the point-slope form.
If we want to rewrite the equation in the slope-intercept form, we would find
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
1+cos²A/cos²A=2cosec²A-1
test for convergence the series 1+x/2+2!/9x3
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
Ajith
exponential series
Naveen
what is subgroup
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
e power cos hyperbolic (x+iy)
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
why {2kπ} union {kπ}={kπ}?
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
what is complex numbers
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Solve 2cos x + 3sin x = 0.5 | 2018-12-12 22:52:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 23, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7944756746292114, "perplexity": 520.575473606846}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376824180.12/warc/CC-MAIN-20181212225044-20181213010544-00201.warc.gz"} |
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Question: If $f(\mathbb{R})$ is compact and $f$ is continuous, then is $f$ uniformly continuous? Background: I thought of the question when proving that "If a function is periodic and continuous, ... | 2014-12-22 00:43:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7557453513145447, "perplexity": 344.01031356491717}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802772897.141/warc/CC-MAIN-20141217075252-00027-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Lasso_%28statistics%29 | # Lasso (statistics)
In statistics and machine learning, lasso (least absolute shrinkage and selection operator; also Lasso or LASSO) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. It was originally introduced in geophysics,[1] and later by Robert Tibshirani,[2] who coined the term.
Lasso was originally formulated for linear regression models. This simple case reveals a substantial amount about the estimator. These include its relationship to ridge regression and best subset selection and the connections between lasso coefficient estimates and so-called soft thresholding. It also reveals that (like standard linear regression) the coefficient estimates do not need to be unique if covariates are collinear.
Though originally defined for linear regression, lasso regularization is easily extended to other statistical models including generalized linear models, generalized estimating equations, proportional hazards models, and M-estimators.[2][3] Lasso's ability to perform subset selection relies on the form of the constraint and has a variety of interpretations including in terms of geometry, Bayesian statistics and convex analysis.
The LASSO is closely related to basis pursuit denoising.
## History
Lasso was introduced in order to improve the prediction accuracy and interpretability of regression models. It selects a reduced set of the known covariates for use in a model.[2][1]
Lasso was developed independently in geophysics literature in 1986, based on prior work that used the ${\displaystyle \ell ^{1}}$ penalty for both fitting and penalization of the coefficients. Statistician Robert Tibshirani independently rediscovered and popularized it in 1996, based on Breiman's nonnegative garrote.[1][4]
Prior to lasso, the most widely used method for choosing covariates was stepwise selection. That approach only improves prediction accuracy in certain cases, such as when only a few covariates have a strong relationship with the outcome. However, in other cases, it can increase prediction error.
At the time, ridge regression was the most popular technique for improving prediction accuracy. Ridge regression improves prediction error by shrinking the sum of the squares of the regression coefficients to be less than a fixed value in order to reduce overfitting, but it does not perform covariate selection and therefore does not help to make the model more interpretable.
Lasso achieves both of these goals by forcing the sum of the absolute value of the regression coefficients to be less than a fixed value, which forces certain coefficients to zero, excluding them from impacting prediction. This idea is similar to ridge regression, which also shrinks the size of the coefficients, however Ridge Regression tends to set far fewer coefficients to zero.
## Basic form
### Least squares
Consider a sample consisting of N cases, each of which consists of p covariates and a single outcome. Let ${\displaystyle y_{i}}$ be the outcome and ${\displaystyle x_{i}:=(x_{1},x_{2},\ldots ,x_{p})_{i}^{T}}$ be the covariate vector for the i th case. Then the objective of lasso is to solve
${\displaystyle \min _{\beta _{0},\beta }\left\{\sum _{i=1}^{N}(y_{i}-\beta _{0}-x_{i}^{T}\beta )^{2}\right\}{\text{ subject to }}\sum _{j=1}^{p}|\beta _{j}|\leq t.}$[2]
Here ${\displaystyle \beta _{0}}$ is the constant coefficient, ${\displaystyle \beta :=(\beta _{1},\beta _{2},\ldots ,\beta _{p})}$ is the coefficient vector, and ${\displaystyle t}$ is a prespecified free parameter that determines the degree of regularization.
Letting ${\displaystyle X}$ be the covariate matrix, so that ${\displaystyle X_{ij}=(x_{i})_{j}}$ and ${\displaystyle x_{i}^{T}}$ is the i th row of ${\displaystyle X}$, the expression can be written more compactly as
${\displaystyle \min _{\beta _{0},\beta }\left\{\left\|y-\beta _{0}-X\beta \right\|_{2}^{2}\right\}{\text{ subject to }}\|\beta \|_{1}\leq t,}$
where ${\displaystyle \|u\|_{p}=\left(\sum _{i=1}^{N}|u_{i}|^{p}\right)^{1/p}}$ is the standard ${\displaystyle \ell ^{p}}$ norm.
Denoting the scalar mean of the data points ${\displaystyle x_{i}}$ by ${\displaystyle {\bar {x}}}$ and the mean of the response variables ${\displaystyle y_{i}}$ by ${\displaystyle {\bar {y}}}$, the resulting estimate for ${\displaystyle \beta _{0}}$ is ${\displaystyle {\hat {\beta }}_{0}={\bar {y}}-{\bar {x}}^{T}\beta }$, so that
${\displaystyle y_{i}-{\hat {\beta }}_{0}-x_{i}^{T}\beta =y_{i}-({\bar {y}}-{\bar {x}}^{T}\beta )-x_{i}^{T}\beta =(y_{i}-{\bar {y}})-(x_{i}-{\bar {x}})^{T}\beta ,}$
and therefore it is standard to work with variables that have been made zero-mean. Additionally, the covariates are typically standardized ${\displaystyle \textstyle \left(\sum _{i=1}^{N}x_{i}^{2}=1\right)}$ so that the solution does not depend on the measurement scale.
It can be helpful to rewrite
${\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{{\frac {1}{N}}\left\|y-X\beta \right\|_{2}^{2}\right\}{\text{ subject to }}\|\beta \|_{1}\leq t.}$
in the so-called Lagrangian form
${\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{{\frac {1}{N}}\left\|y-X\beta \right\|_{2}^{2}+\lambda \|\beta \|_{1}\right\}}$
where the exact relationship between ${\displaystyle t}$ and ${\displaystyle \lambda }$ is data dependent.
### Orthonormal covariates
Some basic properties of the lasso estimator can now be considered.
Assuming first that the covariates are orthonormal so that ${\displaystyle (x_{i}\mid x_{j})=\delta _{ij}}$, where ${\displaystyle (\cdot \mid \cdot )}$ is the inner product and ${\displaystyle \delta _{ij}}$ is the Kronecker delta, or, equivalently, ${\displaystyle X^{T}X=I}$, then using subgradient methods it can be shown that
{\displaystyle {\begin{aligned}{\hat {\beta }}_{j}={}&S_{N\lambda }({\hat {\beta }}_{j}^{\text{OLS}})={\hat {\beta }}_{j}^{\text{OLS}}\max \left(0,1-{\frac {N\lambda }{|{\hat {\beta }}_{j}^{\text{OLS}}|}}\right)\\&{\text{ where }}{\hat {\beta }}^{\text{OLS}}=(X^{T}X)^{-1}X^{T}y\end{aligned}}} [2]
${\displaystyle S_{\alpha }}$ is referred to as the soft thresholding operator, since it translates values towards zero (making them exactly zero if they are small enough) instead of setting smaller values to zero and leaving larger ones untouched as the hard thresholding operator, often denoted ${\displaystyle H_{\alpha }}$, would.
In ridge regression the objective is to minimize
${\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{{\frac {1}{N}}\|y-X\beta \|_{2}^{2}+\lambda \|\beta \|_{2}^{2}\right\}}$
yielding
${\displaystyle {\hat {\beta }}_{j}=(1+N\lambda )^{-1}{\hat {\beta }}_{j}^{\text{OLS}}.}$
Ridge regression shrinks all coefficients by a uniform factor of ${\displaystyle (1+N\lambda )^{-1}}$ and does not set any coefficients to zero.[citation needed] [Edit: This is not correct. The correct solution to ridge regression is : ${\displaystyle {\hat {\beta }}=\left((X^{T}X)^{-1}+N\lambda I\right)X^{T}y}$ ]
It can also be compared to regression with best subset selection, in which the goal is to minimize
${\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{{\frac {1}{N}}\left\|y-X\beta \right\|_{2}^{2}+\lambda \|\beta \|_{0}\right\}}$
where ${\displaystyle \|\cdot \|_{0}}$ is the "${\displaystyle \ell ^{0}}$ norm", which is defined as ${\displaystyle \|z\|=m}$ if exactly m components of z are nonzero. In this case, it can be shown that
${\displaystyle {\hat {\beta }}_{j}=H_{\sqrt {N\lambda }}\left({\hat {\beta }}_{j}^{\text{OLS}}\right)={\hat {\beta }}_{j}^{\text{OLS}}\mathrm {I} \left(\left|{\hat {\beta }}_{j}^{\text{OLS}}\right|\geq {\sqrt {N\lambda }}\right)}$
where ${\displaystyle H_{\alpha }}$ is the so-called hard thresholding function and ${\displaystyle \mathrm {I} }$ is an indicator function (it is 1 if its argument is true and 0 otherwise).
Therefore, the lasso estimates share features of both ridge and best subset selection regression since they both shrink the magnitude of all the coefficients, like ridge regression and set some of them to zero, as in the best subset selection case. Additionally, while ridge regression scales all of the coefficients by a constant factor, lasso instead translates the coefficients towards zero by a constant value and sets them to zero if they reach it.
### Correlated covariates
In one special case two covariates, say j and k, are identical for each observation, so that ${\displaystyle x_{(j)}=x_{(k)}}$, where ${\displaystyle x_{(j),i}=x_{(k),i}}$. Then the values of ${\displaystyle \beta _{j}}$ and ${\displaystyle \beta _{k}}$ that minimize the lasso objective function are not uniquely determined. In fact, if some ${\displaystyle {\hat {\beta }}}$ in which ${\displaystyle {\hat {\beta }}_{j}{\hat {\beta }}_{k}\geq 0}$, then if ${\displaystyle s\in [0,1]}$ replacing ${\displaystyle {\hat {\beta }}_{j}}$ by ${\displaystyle s({\hat {\beta }}_{j}+{\hat {\beta }}_{k})}$ and ${\displaystyle {\hat {\beta }}_{k}}$ by ${\displaystyle (1-s)({\hat {\beta }}_{j}+{\hat {\beta }}_{k})}$, while keeping all the other ${\displaystyle {\hat {\beta }}_{i}}$ fixed, gives a new solution, so the lasso objective function then has a continuum of valid minimizers.[5] Several variants of the lasso, including the Elastic net regularization, have been designed to address this shortcoming.
## General form
Lasso regularization can be extended to other objective functions such as those for generalized linear models, generalized estimating equations, proportional hazards models, and M-estimators.[2][3] Given the objective function
${\displaystyle {\frac {1}{N}}\sum _{i=1}^{N}f(x_{i},y_{i},\alpha ,\beta )}$
the lasso regularized version of the estimator s the solution to
${\displaystyle \min _{\alpha ,\beta }{\frac {1}{N}}\sum _{i=1}^{N}f(x_{i},y_{i},\alpha ,\beta ){\text{ subject to }}\|\beta \|_{1}\leq t}$
where only ${\displaystyle \beta }$ is penalized while ${\displaystyle \alpha }$ is free to take any allowed value, just as ${\displaystyle \beta _{0}}$ was not penalized in the basic case.
## Interpretations
### Geometric interpretation
Forms of the constraint regions for lasso and ridge regression.
Lasso can set coefficients to zero, while the superficially similar ridge regression cannot. This is due to the difference in the shape of their constraint boundaries. Both lasso and ridge regression can be interpreted as minimizing the same objective function
${\displaystyle \min _{\beta _{0},\beta }\left\{{\frac {1}{N}}\left\|y-\beta _{0}-X\beta \right\|_{2}^{2}\right\}}$
but with respect to different constraints: ${\displaystyle \|\beta \|_{1}\leq t}$ for lasso and ${\displaystyle \|\beta \|_{2}^{2}\leq t}$ for ridge. The figure shows that the constraint region defined by the ${\displaystyle \ell ^{1}}$ norm is a square rotated so that its corners lie on the axes (in general a cross-polytope), while the region defined by the ${\displaystyle \ell ^{2}}$ norm is a circle (in general an n-sphere), which is rotationally invariant and, therefore, has no corners. As seen in the figure, a convex object that lies tangent to the boundary, such as the line shown, is likely to encounter a corner (or a higher-dimensional equivalent) of a hypercube, for which some components of ${\displaystyle \beta }$ are identically zero, while in the case of an n-sphere, the points on the boundary for which some of the components of ${\displaystyle \beta }$ are zero are not distinguished from the others and the convex object is no more likely to contact a point at which some components of ${\displaystyle \beta }$ are zero than one for which none of them are.
### Making λ easier to interpret with an accuracy-simplicity tradeoff
The lasso can be rescaled so that it becomes easy to anticipate and influence the degree of shrinkage associated with a given value of ${\displaystyle \lambda }$.[6] It is assumed that ${\displaystyle X}$ is standardized with z-scores and that ${\displaystyle y}$ is centered (zero mean). Let ${\displaystyle \beta _{0}}$ represent the hypothesized regression coefficients and let ${\displaystyle b_{OLS}}$ refer to the data-optimized ordinary least squares solutions. We can then define the Lagrangian as a tradeoff between the in-sample accuracy of the data-optimized solutions and the simplicity of sticking to the hypothesized values.[7] This results in
${\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{{\frac {(y-X\beta )'(y-X\beta )}{(y-X\beta _{0})'(y-X\beta _{0})}}+2\lambda \sum _{i=1}^{p}{\frac {|\beta _{i}-\beta _{0,i}|}{q_{i}}}\right\}}$
where ${\displaystyle q_{i}}$ is specified below. The first fraction represents relative accuracy, the second fraction relative simplicity, and ${\displaystyle \lambda }$ balances between the two.
Solution paths for the ${\displaystyle \ell _{1}}$ norm and ${\displaystyle \ell _{2}}$ norm when ${\displaystyle b_{OLS}=2}$ and ${\displaystyle \beta _{0}=0}$
Given a single regressor, relative simplicity can be defined by specifying ${\displaystyle q_{i}}$ as ${\displaystyle |b_{OLS}-\beta _{0}|}$, which is the maximum amount of deviation from ${\displaystyle \beta _{0}}$ when ${\displaystyle \lambda =0}$. Assuming that ${\displaystyle \beta _{0}=0}$, the solution path can be defined in terms of ${\displaystyle R^{2}}$:
${\displaystyle b_{\ell _{1}}={\begin{cases}(1-\lambda /R^{2})b_{OLS}&{\mbox{if }}\lambda \leq R^{2},\\0&{\mbox{if }}\lambda >R^{2}.\end{cases}}}$
If ${\displaystyle \lambda =0}$, the ordinary least squares solution (OLS) is used. The hypothesized value of ${\displaystyle \beta _{0}=0}$ is selected if ${\displaystyle \lambda }$ is bigger than ${\displaystyle R^{2}}$. Furthermore, if ${\displaystyle R^{2}=1}$, then ${\displaystyle \lambda }$ represents the proportional influence of ${\displaystyle \beta _{0}=0}$. In other words, ${\displaystyle \lambda \times 100\%}$ measures in percentage terms the minimal amount of influence of the hypothesized value relative to the data-optimized OLS solution.
If an ${\displaystyle \ell _{2}}$-norm is used to penalize deviations from zero given a single regressor, the solution path is given by
${\displaystyle b_{\ell _{2}}={\bigg (}1+{\frac {\lambda }{R^{2}(1-\lambda )}}{\bigg )}^{-1}b_{OLS}}$. Like ${\displaystyle b_{\ell _{1}}}$, ${\displaystyle b_{\ell _{2}}}$ moves in the direction of the point ${\displaystyle (\lambda =R^{2},b=0)}$ when ${\displaystyle \lambda }$ is close to zero; but unlike ${\displaystyle b_{\ell _{1}}}$, the influence of ${\displaystyle R^{2}}$ diminishes in ${\displaystyle b_{\ell _{2}}}$ if ${\displaystyle \lambda }$ increases (see figure).
Given multiple regressors, the moment that a parameter is activated (i.e. allowed to deviate from ${\displaystyle \beta _{0}}$) is also determined by a regressor's contribution to ${\displaystyle R^{2}}$ accuracy. First,
${\displaystyle R^{2}=1-{\frac {(y-Xb)'(y-Xb)}{(y-X\beta _{0})'(y-X\beta _{0})}}.}$
An ${\displaystyle R^{2}}$ of 75% means that in-sample accuracy improves by 75% if the unrestricted OLS solutions are used instead of the hypothesized ${\displaystyle \beta _{0}}$ values. The individual contribution of deviating from each hypothesis can be computed with the ${\displaystyle p}$ x ${\displaystyle p}$ matrix
${\displaystyle R^{\otimes }=(X'{\tilde {y}}_{0})(X'{\tilde {y}}_{0})'(X'X)^{-1}({\tilde {y}}_{0}'{\tilde {y}}_{0})^{-1},}$
where ${\displaystyle {\tilde {y}}_{0}=y-X\beta _{0}}$. If ${\displaystyle b=b_{OLS}}$ when ${\displaystyle R^{2}}$ is computed, then the diagonal elements of ${\displaystyle R^{\otimes }}$ sum to ${\displaystyle R^{2}}$. The diagonal ${\displaystyle R^{\otimes }}$ values may be smaller than 0 or, less often, larger than 1. If regressors are uncorrelated, then the ${\displaystyle i^{th}}$ diagonal element of ${\displaystyle R^{\otimes }}$ simply corresponds to the ${\displaystyle r^{2}}$ value between ${\displaystyle x_{i}}$ and ${\displaystyle y}$.
A rescaled version of the adaptive lasso of can be obtained by setting ${\displaystyle q_{{\mbox{adaptive lasso}},i}=|b_{OLS,i}-\beta _{0,i}|}$.[8] If regressors are uncorrelated, the moment that the ${\displaystyle i^{th}}$ parameter is activated is given by the ${\displaystyle i^{th}}$ diagonal element of ${\displaystyle R^{\otimes }}$. Assuming for convenience that ${\displaystyle \beta _{0}}$ is a vector of zeros,
${\displaystyle b_{i}={\begin{cases}(1-\lambda /R_{ii}^{\otimes })b_{OLS,i}&{\mbox{if }}\lambda \leq R_{ii}^{\otimes },\\0&{\mbox{if }}\lambda >R_{ii}^{\otimes }.\end{cases}}}$
That is, if regressors are uncorrelated, ${\displaystyle \lambda }$ again specifies the minimal influence of ${\displaystyle \beta _{0}}$. Even when regressors are correlated, the first time that a regression parameter is activated occurs when ${\displaystyle \lambda }$ is equal to the highest diagonal element of ${\displaystyle R^{\otimes }}$.
These results can be compared to a rescaled version of the lasso by defining ${\displaystyle q_{{\mbox{lasso}},i}={\frac {1}{p}}\sum _{l}|b_{OLS,l}-\beta _{0,l}|}$, which is the average absolute deviation of ${\displaystyle b_{OLS}}$ from ${\displaystyle \beta _{0}}$. Assuming that regressors are uncorrelated, then the moment of activation of the ${\displaystyle i^{th}}$ regressor is given by
${\displaystyle {\tilde {\lambda }}_{{\text{lasso}},i}={\frac {1}{p}}{\sqrt {R_{i}^{\otimes }}}\sum _{l=1}^{p}{\sqrt {R_{l}^{\otimes }}}.}$
For ${\displaystyle p=1}$, the moment of activation is again given by ${\displaystyle {\tilde {\lambda }}_{{\text{lasso}},i}=R^{2}}$. If ${\displaystyle \beta _{0}}$ is a vector of zeros and a subset of ${\displaystyle p_{B}}$ relevant parameters are equally responsible for a perfect fit of ${\displaystyle R^{2}=1}$, then this subset is activated at a ${\displaystyle \lambda }$ value of ${\displaystyle {\frac {1}{p}}}$. The moment of activation of a relevant regressor then equals ${\displaystyle {\frac {1}{p}}{\frac {1}{\sqrt {p_{B}}}}p_{B}{\frac {1}{\sqrt {p_{B}}}}={\frac {1}{p}}}$. In other words, the inclusion of irrelevant regressors delays the moment that relevant regressors are activated by this rescaled lasso. The adaptive lasso and the lasso are special cases of a '1ASTc' estimator. The latter only groups parameters together if the absolute correlation among regressors is larger than a user-specified value.[6]
### Bayesian interpretation
Laplace distributions are sharply peaked at their mean with more probability density concentrated there compared to a normal distribution.
Just as ridge regression can be interpreted as linear regression for which the coefficients have been assigned normal prior distributions, lasso can be interpreted as linear regression for which the coefficients have Laplace prior distributions. The Laplace distribution is sharply peaked at zero (its first derivative is discontinuous at zero) and it concentrates its probability mass closer to zero than does the normal distribution. This provides an alternative explanation of why lasso tends to set some coefficients to zero, while ridge regression does not.[2]
### Convex relaxation interpretation
Lasso can also be viewed as a convex relaxation of the best subset selection regression problem, which is to find the subset of ${\displaystyle \leq k}$ covariates that results in the smallest value of the objective function for some fixed ${\displaystyle k\leq n}$, where n is the total number of covariates. The "${\displaystyle \ell ^{0}}$ norm", ${\displaystyle \|\cdot \|_{0}}$, (the number of nonzero entries of a vector), is the limiting case of "${\displaystyle \ell ^{p}}$ norms", of the form ${\displaystyle \textstyle \|x\|_{p}=\left(\sum _{i=1}^{n}|x_{j}|^{p}\right)^{1/p}}$ (where the quotation marks signify that these are not really norms for ${\displaystyle p<1}$ since ${\displaystyle \|\cdot \|_{p}}$ is not convex for ${\displaystyle p<1}$, so the triangle inequality does not hold). Therefore, since p = 1 is the smallest value for which the "${\displaystyle \ell ^{p}}$ norm" is convex (and therefore actually a norm), lasso is, in some sense, the best convex approximation to the best subset selection problem, since the region defined by ${\displaystyle \|x\|_{1}\leq t}$ is the convex hull of the region defined by ${\displaystyle \|x\|_{p}\leq t}$ for ${\displaystyle p<1}$.
## Generalizations
Lasso variants have been created in order to remedy limitations of the original technique and to make the method more useful for particular problems. Almost all of these focus on respecting or exploiting dependencies among the covariates.
Elastic net regularization adds an additional ridge regression-like penalty that improves performance when the number of predictors is larger than the sample size, allows the method to select strongly correlated variables together, and improves overall prediction accuracy.[5]
Group lasso allows groups of related covariates to be selected as a single unit, which can be useful in settings where it does not make sense to include some covariates without others.[9] Further extensions of group lasso perform variable selection within individual groups (sparse group lasso) and allow overlap between groups (overlap group lasso).[10][11]
Fused lasso can account for the spatial or temporal characteristics of a problem, resulting in estimates that better match system structure.[12] Lasso-regularized models can be fit using techniques including subgradient methods, least-angle regression (LARS), and proximal gradient methods. Determining the optimal value for the regularization parameter is an important part of ensuring that the model performs well; it is typically chosen using cross-validation.
### Elastic net
In 2005, Zou and Hastie introduced the elastic net.[5] When p > n (the number of covariates is greater than the sample size) lasso can select only n covariates (even when more are associated with the outcome) and it tends to select one covariate from any set of highly correlated covariates. Additionally, even when n > p, ridge regression tends to perform better given strongly correlated covariates.
The elastic net extends lasso by adding an additional ${\displaystyle \ell ^{2}}$ penalty term giving
${\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{\left\|y-X\beta \right\|_{2}^{2}+\lambda _{1}\|\beta \|_{1}+\lambda _{2}\|\beta \|_{2}^{2}\right\},}$
which is equivalent to solving
{\displaystyle {\begin{aligned}\min _{\beta _{0},\beta }\left\{\left\|y-\beta _{0}-X\beta \right\|_{2}^{2}\right\}&{\text{ subject to }}(1-\alpha )\|\beta \|_{1}+\alpha \|\beta \|_{2}^{2}\leq t,\\&{\text{ where }}\alpha ={\frac {\lambda _{2}}{\lambda _{1}+\lambda _{2}}}.\end{aligned}}}
This problem can be written in a simple lasso form
${\displaystyle \min _{\beta ^{*}\in \mathbb {R} ^{p}}\left\{\left\|y^{*}-X^{*}\beta ^{*}\right\|_{2}^{2}+\lambda ^{*}\|\beta ^{*}\|_{1}\right\}}$
letting
${\displaystyle X_{(n+p)\times p}^{*}=(1+\lambda _{2})^{-1/2}{\binom {X}{\lambda _{2}^{1/2}I_{p\times p}}}}$, ${\displaystyle y_{(n+p)}^{*}={\binom {y}{0^{p}}},\qquad \lambda ^{*}={\frac {\lambda _{1}}{\sqrt {1+\lambda _{2}}}}}$, ${\displaystyle \beta ^{*}={\sqrt {1+\lambda _{2}}}\beta .}$
Then ${\displaystyle {\hat {\beta }}={\frac {{\hat {\beta }}^{*}}{\sqrt {1+\lambda _{2}}}}}$, which, when the covariates are orthogonal to each other, gives
${\displaystyle {\hat {\beta }}_{j}={\frac {{\hat {\beta }}_{j}^{\text{*,OLS}}}{\sqrt {1+\lambda _{2}}}}\max \left(0,1-{\frac {\lambda ^{*}}{\left|{\hat {\beta }}_{j}^{\text{*,OLS}}\right|}}\right)={\frac {{\hat {\beta }}_{j}^{\text{OLS}}}{1+\lambda _{2}}}\max \left(0,1-{\frac {\lambda _{1}}{\left|{\hat {\beta }}_{j}^{\text{OLS}}\right|}}\right)=(1+\lambda _{2})^{-1}{\hat {\beta }}_{j}^{\text{lasso}}.}$
So the result of the elastic net penalty is a combination of the effects of the lasso and ridge penalties.
Returning to the general case, the fact that the penalty function is now strictly convex means that if ${\displaystyle x_{(j)}=x_{(k)}}$, ${\displaystyle {\hat {\beta }}_{j}={\hat {\beta }}_{k}}$, which is a change from lasso.[5] In general, if ${\displaystyle {\hat {\beta }}_{j}{\hat {\beta _{k}}}>0}$
${\displaystyle {\frac {|{\hat {\beta }}_{j}-{\hat {\beta _{k}}}|}{\|y\|}}\leq \lambda _{2}^{-1}{\sqrt {2(1-\rho _{jk})}},{\text{ where }}\rho =X^{t}X,}$
is the sample correlation matrix because the ${\displaystyle x}$ 's are normalized.
Therefore, highly correlated covariates tend to have similar regression coefficients, with the degree of similarity depending on both ${\displaystyle \|y\|_{1}}$ and ${\displaystyle \lambda _{2}}$, which is different from lasso. This phenomenon, in which strongly correlated covariates have similar regression coefficients, is referred to as the grouping effect. Grouping is desirable since, in applications such as tying genes to a disease, finding all the associated covariates is preferable, rather than selecting one from each set of correlated covariates, as lasso often does.[5] In addition, selecting only one from each group typically results in increased prediction error, since the model is less robust (which is why ridge regression often outperforms lasso).
### Group lasso
In 2006, Yuan and Lin introduced the group lasso to allow predefined groups of covariates to jointly be selected into or out of a model.[9] This is useful in many settings, perhaps most obviously when a categorical variable is coded as a collection of binary covariates. In this case, group lasso can ensure that all the variables encoding the categorical covariate are included or excluded together. Another setting in which grouping is natural is in biological studies. Since genes and proteins often lie in known pathways, which pathways are related to an outcome may be more significant than whether individual genes are. The objective function for the group lasso is a natural generalization of the standard lasso objective
${\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{\left\|y-\sum _{j=1}^{J}X_{j}\beta _{j}\right\|_{2}^{2}+\lambda \sum _{j=1}^{J}\|\beta _{j}\|_{K_{j}}\right\},\qquad \|z\|_{K_{j}}=(z^{t}K_{j}z)^{1/2}}$
where the design matrix ${\displaystyle X}$ and covariate vector ${\displaystyle \beta }$ have been replaced by a collection of design matrices ${\displaystyle X_{j}}$ and covariate vectors ${\displaystyle \beta _{j}}$, one for each of the J groups. Additionally, the penalty term is now a sum over ${\displaystyle \ell ^{2}}$ norms defined by the positive definite matrices ${\displaystyle K_{j}}$. If each covariate is in its own group and ${\displaystyle K_{j}=I}$, then this reduces to the standard lasso, while if there is only a single group and ${\displaystyle K_{1}=I}$, it reduces to ridge regression. Since the penalty reduces to an ${\displaystyle \ell ^{2}}$ norm on the subspaces defined by each group, it cannot select out only some of the covariates from a group, just as ridge regression cannot. However, because the penalty is the sum over the different subspace norms, as in the standard lasso, the constraint has some non-differential points, which correspond to some subspaces being identically zero. Therefore, it can set the coefficient vectors corresponding to some subspaces to zero, while only shrinking others. However, it is possible to extend the group lasso to the so-called sparse group lasso, which can select individual covariates within a group, by adding an additional ${\displaystyle \ell ^{1}}$ penalty to each group subspace.[10] Another extension, group lasso with overlap allows covariates to be shared across groups, e.g., if a gene were to occur in two pathways.[11]
### Fused lasso
In some cases, the phenomenon under study may have important spatial or temporal structure that must be considered during analysis, such as time series or image-based data. In 2005, Tibshirani and colleagues introduced the fused lasso to extend the use of lasso to this type of data.[12] The fused lasso objective function is
{\displaystyle {\begin{aligned}&\min _{\beta }\left\{{\frac {1}{N}}\sum _{i=1}^{N}\left(y_{i}-x_{i}^{t}\beta \right)^{2}\right\}\\[4pt]&{\text{ subject to }}\sum _{j=1}^{p}|\beta _{j}|\leq t_{1}{\text{ and }}\sum _{j=2}^{p}|\beta _{j}-\beta _{j-1}|\leq t_{2}.\end{aligned}}}
The first constraint is the lasso constraint, while the second directly penalizes large changes with respect to the temporal or spatial structure, which forces the coefficients to vary smoothly to reflect the system's underlying logic. Clustered lasso[13] is a generalization of fused lasso that identifies and groups relevant covariates based on their effects (coefficients). The basic idea is to penalize the differences between the coefficients so that nonzero ones cluster. This can be modeled using the following regularization:
${\displaystyle \sum _{i
In contrast, variables can be clustered into highly correlated groups, and then a single representative covariate can be extracted from each cluster.[14]
Algorithms exist that solve the fused lasso problem, and some generalizations of it. Algorithms can solve it exactly in a finite number of operations.[15]
### Quasi-norms and bridge regression
An example of a PQSQ (piece-wise quadratic function of subquadratic growth) potential function ${\displaystyle u(x)}$; here the majorant function is ${\displaystyle f(x)=x}$; the potential is defined with trimming after ${\displaystyle r_{3}}$.
An example how efficient PQSQ regularized regression works just as ${\displaystyle \ell ^{1}}$-norm lasso.[16]
Lasso, elastic net, group and fused lasso construct the penalty functions from the ${\displaystyle \ell ^{1}}$ and ${\displaystyle \ell ^{2}}$ norms (with weights, if necessary). The bridge regression utilises general ${\displaystyle \ell ^{p}}$ norms (${\displaystyle p\geq 1}$) and quasinorms (${\displaystyle 0).[17] For example, for p=1/2 the analogue of lasso objective in the Lagrangian form is to solve
${\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{{\frac {1}{N}}\left\|y-X\beta \right\|_{2}^{2}+\lambda {\sqrt {\|\beta \|_{1/2}}}\right\},}$
where
${\displaystyle \|\beta \|_{1/2}=\left(\sum _{j=1}^{p}{\sqrt {|\beta _{j}|}}\right)^{2}}$
It is claimed that the fractional quasi-norms ${\displaystyle \ell ^{p}}$ (${\displaystyle 0) provide more meaningful results in data analysis both theoretically and empirically.[18] The non-convexity of these quasi-norms complicates the optimization problem. To solve this problem, an expectation-minimization procedure is developed[19] and implemented[16] for minimization of function
${\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{{\frac {1}{N}}\left\|y-X\beta \right\|_{2}^{2}+\lambda \sum _{j=1}^{p}\vartheta (\beta _{j}^{2})\right\},}$
where ${\displaystyle \vartheta (\gamma )}$ is an arbitrary concave monotonically increasing function (for example, ${\displaystyle \vartheta (\gamma )={\sqrt {\gamma }}}$ gives the lasso penalty and ${\displaystyle \vartheta (\gamma )=\gamma ^{1/4}}$ gives the ${\displaystyle \ell ^{1/2}}$ penalty).
The efficient algorithm for minimization is based on piece-wise quadratic approximation of subquadratic growth (PQSQ).[19]
The adaptive lasso was introduced by Zou in 2006 for linear regression[20] and by Zhang and Lu in 2007 for proportional hazards regression.[21]
### Prior lasso
The prior lasso was introduced for generalized linear models by Jiang et al. in 2016 to incorporate prior information, such as the importance of certain covariates.[22] In prior lasso, such information is summarized into pseudo responses (called prior responses) ${\displaystyle {\hat {y}}^{\mathrm {p} }}$ and then an additional criterion function is added to the usual objective function with a lasso penalty. Without loss of generality, in linear regression, the new objective function can be written as
${\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{{\frac {1}{N}}\left\|y-X\beta \right\|_{2}^{2}+{\frac {1}{N}}\eta \left\|{\hat {y}}^{\mathrm {p} }-X\beta \right\|_{2}^{2}+\lambda \|\beta \|_{1}\right\},}$
which is equivalent to
${\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{{\frac {1}{N}}\left\|{\tilde {y}}-X\beta \right\|_{2}^{2}+{\frac {\lambda }{1+\eta }}\|\beta \|_{1}\right\},}$
the usual lasso objective function with the responses ${\displaystyle y}$ being replaced by a weighted average of the observed responses and the prior responses ${\displaystyle {\tilde {y}}=(y+\eta {\hat {y}}^{\mathrm {p} })/(1+\eta )}$ (called the adjusted response values by the prior information).
In prior lasso, the parameter ${\displaystyle \eta }$ is called a balancing parameter, in that it balances the relative importance of the data and the prior information. In the extreme case of ${\displaystyle \eta =0}$, prior lasso is reduced to lasso. If ${\displaystyle \eta =\infty }$, prior lasso will solely rely on the prior information to fit the model. Furthermore, the balancing parameter ${\displaystyle \eta }$ has another appealing interpretation: it controls the variance of ${\displaystyle \beta }$ in its prior distribution from a Bayesian viewpoint.
Prior lasso is more efficient in parameter estimation and prediction (with a smaller estimation error and prediction error) when the prior information is of high quality, and is robust to the low quality prior information with a good choice of the balancing parameter ${\displaystyle \eta }$.
## Computing lasso solutions
The loss function of the lasso is not differentiable, but a wide variety of techniques from convex analysis and optimization theory have been developed to compute the solutions path of the lasso. These include coordinate descent,[23] subgradient methods, least-angle regression (LARS), and proximal gradient methods.[24] Subgradient methods are the natural generalization of traditional methods such as gradient descent and stochastic gradient descent to the case in which the objective function is not differentiable at all points. LARS is a method that is closely tied to lasso models, and in many cases allows them to be fit efficiently, though it may not perform well in all circumstances. LARS generates complete solution paths.[24] Proximal methods have become popular because of their flexibility and performance and are an area of active research. The choice of method will depend on the particular lasso variant, the data and the available resources. However, proximal methods generally perform well.
## Choice of regularization parameter
Choosing the regularization parameter (${\displaystyle \lambda }$) is a fundamental part of lasso. A good value is essential to the performance of lasso since it controls the strength of shrinkage and variable selection, which, in moderation can improve both prediction accuracy and interpretability. However, if the regularization becomes too strong, important variables may be omitted and coefficients may be shrunk excessively, which can harm both predictive capacity and inferencing. Cross-validation is often used to find the regularization parameter.
Information criteria such as the Bayesian information criterion (BIC) and the Akaike information criterion (AIC) might be preferable to cross-validation, because they are faster to compute and their performance is less volatile in small samples.[25] An information criterion selects the estimator's regularization parameter by maximizing a model's in-sample accuracy while penalizing its effective number of parameters/degrees of freedom. Zou et al. proposed to measure the effective degrees of freedom by counting the number of parameters that deviate from zero.[26] The degrees of freedom approach was considered flawed by Kaufman and Rosset[27] and Janson et al.,[28] because a model's degrees of freedom might increase even when it is penalized harder by the regularization parameter. As an alternative, the relative simplicity measure defined above can be used to count the effective number of parameters.[25] For the lasso, this measure is given by
${\displaystyle {\hat {\mathcal {P}}}=\sum _{i=1}^{p}{\frac {|\beta _{i}-\beta _{0,i}|}{{\frac {1}{p}}\sum _{l}|b_{OLS,l}-\beta _{0,l}|}}}$,
which monotonically increases from zero to ${\displaystyle p}$ as the regularization parameter decreases from ${\displaystyle \infty }$ to zero.
## Selected applications
LASSO has been applied in economics and finance, and was found to improve prediction and to select sometimes neglected variables, for example in corporate bankruptcy prediction literature,[29] or high growth firms prediction.[30]
## References
1. ^ a b c Santosa, Fadil; Symes, William W. (1986). "Linear inversion of band-limited reflection seismograms". SIAM Journal on Scientific and Statistical Computing. SIAM. 7 (4): 1307–1330. doi:10.1137/0907087.
2. Tibshirani, Robert (1996). "Regression Shrinkage and Selection via the lasso". Journal of the Royal Statistical Society. Series B (methodological). Wiley. 58 (1): 267–88. JSTOR 2346178.
3. ^ a b Tibshirani, Robert (1997). "The lasso Method for Variable Selection in the Cox Model". Statistics in Medicine. 16 (4): 385–395. CiteSeerX 10.1.1.411.8024. doi:10.1002/(SICI)1097-0258(19970228)16:4<385::AID-SIM380>3.0.CO;2-3. PMID 9044528.
4. ^ Breiman, Leo (1995). "Better Subset Regression Using the Nonnegative Garrote". Technometrics. 37 (4): 373–84. doi:10.1080/00401706.1995.10484371.
5. Zou, Hui; Hastie, Trevor (2005). "Regularization and Variable Selection via the Elastic Net". Journal of the Royal Statistical Society. Series B (statistical Methodology). Wiley. 67 (2): 301–20. doi:10.1111/j.1467-9868.2005.00503.x. JSTOR 3647580.
6. ^ a b Hoornweg, Victor (2018). "Chapter 8". Science: Under Submission. Hoornweg Press. ISBN 978-90-829188-0-9.
7. ^ Motamedi, Fahimeh; Sanchez, Horacio; Mehri, Alireza; Ghasemi, Fahimeh (October 2021). "Accelerating Big Data Analysis through LASSO-Random Forest Algorithm in QSAR Studies". 37 (19). Bioinformatics: 1–7. doi:10.1093/bioinformatics/btab659. ISSN 1367-4803. {{cite journal}}: Cite journal requires |journal= (help)
8. ^ Zou, Hui (2006). "The Adaptive Lasso and Its Oracle Properties" (PDF).
9. ^ a b Yuan, Ming; Lin, Yi (2006). "Model Selection and Estimation in Regression with Grouped Variables". Journal of the Royal Statistical Society. Series B (statistical Methodology). Wiley. 68 (1): 49–67. doi:10.1111/j.1467-9868.2005.00532.x. JSTOR 3647556.
10. ^ a b Puig, Arnau Tibau, Ami Wiesel, and Alfred O. Hero III. "A Multidimensional Shrinkage-Thresholding Operator". Proceedings of the 15th workshop on Statistical Signal Processing, SSP'09, IEEE, pp. 113–116.
11. ^ a b Jacob, Laurent, Guillaume Obozinski, and Jean-Philippe Vert. "Group Lasso with Overlap and Graph LASSO". Appearing in Proceedings of the 26th International Conference on Machine Learning, Montreal, Canada, 2009.
12. ^ a b Tibshirani, Robert, Michael Saunders, Saharon Rosset, Ji Zhu, and Keith Knight. 2005. “Sparsity and Smoothness via the Fused lasso”. Journal of the Royal Statistical Society. Series B (statistical Methodology) 67 (1). Wiley: 91–108. https://www.jstor.org/stable/3647602.
13. ^ She, Yiyuan (2010). "Sparse regression with exact clustering". Electronic Journal of Statistics. 4: 1055–1096. doi:10.1214/10-EJS578.
14. ^ Reid, Stephen (2015). "Sparse regression and marginal testing using cluster prototypes". Biostatistics. 17 (2): 364–76. arXiv:1503.00334. Bibcode:2015arXiv150300334R. doi:10.1093/biostatistics/kxv049. PMC 5006118. PMID 26614384.
15. ^ Bento, Jose (2018). "On the Complexity of the Weighted Fused Lasso". IEEE Letters in Signal Processing. 25 (10): 1595–1599. arXiv:1801.04987. Bibcode:2018ISPL...25.1595B. doi:10.1109/LSP.2018.2867800. S2CID 5008891.
16. ^ a b Mirkes E.M. PQSQ-regularized-regression repository, GitHub.
17. ^ Fu, Wenjiang J. 1998. “The Bridge versus the Lasso”. Journal of Computational and Graphical Statistics 7 (3). Taylor & Francis: 397-416.
18. ^ Aggarwal C.C., Hinneburg A., Keim D.A. (2001) "On the Surprising Behavior of Distance Metrics in High Dimensional Space." In: Van den Bussche J., Vianu V. (eds) Database Theory — ICDT 2001. ICDT 2001. Lecture Notes in Computer Science, Vol. 1973. Springer, Berlin, Heidelberg, pp. 420-434.
19. ^ a b Gorban, A.N.; Mirkes, E.M.; Zinovyev, A. (2016) "Piece-wise quadratic approximations of arbitrary error functions for fast and robust machine learning." Neural Networks, 84, 28-38.
20. ^ Zou (2006, JASA)
21. ^ Zhang and Lu (2007, Biometrika)
22. ^ Jiang, Yuan (2016). "Variable selection with prior information for generalized linear models via the prior lasso method". Journal of the American Statistical Association. 111 (513): 355–376. doi:10.1080/01621459.2015.1008363. PMC 4874534. PMID 27217599.
23. ^ Jerome Friedman, Trevor Hastie, and Robert Tibshirani. 2010. “Regularization Paths for Generalized Linear Models via Coordinate Descent”. Journal of Statistical Software 33 (1): 1-21. https://www.jstatsoft.org/article/view/v033i01/v33i01.pdf.
24. ^ a b Efron, Bradley, Trevor Hastie, Iain Johnstone, and Robert Tibshirani. 2004. “Least Angle Regression”. The Annals of Statistics 32 (2). Institute of Mathematical Statistics: 407–51. https://www.jstor.org/stable/3448465.
25. ^ a b Hoornweg, Victor (2018). "Chapter 9". Science: Under Submission. Hoornweg Press. ISBN 978-90-829188-0-9.
26. ^ Zou, Hui; Hastie, Trevor; Tibshirani, Robert (2007). "On the 'Degrees of Freedom' of the Lasso". The Annals of Statistics. 35 (5): 2173–2792. doi:10.1214/009053607000000127.
27. ^ Kaufman, S.; Rosset, S. (2014). "When does more regularization imply fewer degrees of freedom? Sufficient conditions and counterexamples". Biometrika. 101 (4): 771–784. doi:10.1093/biomet/asu034. ISSN 0006-3444.
28. ^ Janson, Lucas; Fithian, William; Hastie, Trevor J. (2015). "Effective degrees of freedom: a flawed metaphor". Biometrika. 102 (2): 479–485. doi:10.1093/biomet/asv019. ISSN 0006-3444. PMC 4787623. PMID 26977114.
29. ^ Shaonan, Tian; Yu, Yan; Guo, Hui (2015). "Variable selection and corporate bankruptcy forecasts". Journal of Banking & Finance. 52 (1): 89–100. doi:10.1016/j.jbankfin.2014.12.003.
30. ^ Coad, Alex; Srhoj, Stjepan (2020). "Catching Gazelles with a Lasso: Big data techniques for the prediction of high-growth firms". Small Business Economics. 55 (1): 541–565. doi:10.1007/s11187-019-00203-3. | 2022-07-02 06:50:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 223, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9188807606697083, "perplexity": 1192.781602738522}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103984681.57/warc/CC-MAIN-20220702040603-20220702070603-00683.warc.gz"} |
https://www.rdocumentation.org/packages/MASS/versions/7.3-47/topics/galaxies | # galaxies
0th
Percentile
A numeric vector of velocities in km/sec of 82 galaxies from 6 well-separated conic sections of an unfilled survey of the Corona Borealis region. Multimodality in such surveys is evidence for voids and superclusters in the far universe.
Keywords
datasets
##### Usage
galaxies
##### Note
There is an 83rd measurement of 5607 km/sec in the Postman et al. paper which is omitted in Roeder (1990) and from the dataset here. There is also a typo: this dataset has 78th observation 26690 which should be 26960.
##### References
Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.
• galaxies
##### Examples
library(MASS) gal <- galaxies/1000 c(width.SJ(gal, method = "dpi"), width.SJ(gal)) plot(x = c(0, 40), y = c(0, 0.3), type = "n", bty = "l", xlab = "velocity of galaxy (1000km/s)", ylab = "density") rug(gal) lines(density(gal, width = 3.25, n = 200), lty = 1) lines(density(gal, width = 2.56, n = 200), lty = 3)
Documentation reproduced from package MASS, version 7.3-47, License: GPL-2 | GPL-3
### Community examples
Looks like there are no examples yet. | 2020-01-21 06:24:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2343883365392685, "perplexity": 13960.49158958312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250601615.66/warc/CC-MAIN-20200121044233-20200121073233-00079.warc.gz"} |
https://mwfn-fu.readthedocs.io/en/v.major/ | # Welcome to mwfn-fu’s documentation!¶
mwfn-fu is intended to provide two-fold assistance when working with Multiwfn, “A Multifunctional Wavefunction Analyzer.” Interaction with Multiwfn is primarily conducted through a command-line interface, and many of the generated results are reported only as text printed to stdout. The goals of this package are
1. Enabling automatic execution and operation of Multiwfn, as in a scripting context.
2. Implementing a command-line ‘wrapper’ around Multiwfn that permits more convenient extraction of its computational outputs into a numerically manipulable form.
mwfn-fu is in a preliminary stage of development, and so far only a basic driver class for Multiwfn has been implemented, which provides an initial implementation of #1 above. | 2022-05-24 17:34:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3563411235809326, "perplexity": 6055.337716519114}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662573189.78/warc/CC-MAIN-20220524173011-20220524203011-00247.warc.gz"} |
https://webwork.maa.org/moodle/mod/forum/discuss.php?d=3901&parent=10908 | ## WeBWorK Problems
### Re: Showing Fractions instead of Decimals
by Alex Jordan -
Number of replies: 0
If you use contextFraction.pl, and use Context("Fraction"), then you can make
$slopeeval = Fraction(-$dx-> eval(x => $x, y =>$y), $dy-> eval(x =>$x, y => $y)) using two arguments to Fraction. It will be a Fraction object with the numerator and denominator reduced. (I'm not sure what happens if the numerator and denominator are not integers, but the sample code given in the post will produce integers.) You can also make$slopeeval = Fraction((-$dx-> eval(x =>$x, y => $y)) / ($dy-> eval(x => $x, y =>$y)))
which carries out the division and passes one (likely decimal) argument to Fraction. The result will be a reduced fraction that is probably equal to what you fed it, as long as the denominator of what you fed it was not too large.
I'd do your $f,$dx, $dy, etc in Numeric context. Then switch to Fraction context to make this$slopeeval. Then go to ImplicitEquation context to write the equation, but remember to set the reduceConstants flag to 0 so that the string fraction is not reduced to a decimal again. | 2023-02-05 18:13:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6383668780326843, "perplexity": 4402.134834758554}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500273.30/warc/CC-MAIN-20230205161658-20230205191658-00155.warc.gz"} |
http://math.stackexchange.com/questions/443343/identity-for-simple-1d-random-walk | # Identity for simple 1D random walk
The question is to find a purely probabilistic proof of the following identity, valid for every integer $n\geqslant1$, where $(S_n)_{n\geqslant0}$ denotes a standard simple random walk:
$$E[(S_n)^2;S_{2n}=0]=\frac{n}2\,P[S_{2n-2}=0].$$
Standard simple random walk is defined as $S_0=0$ and $S_n=\sum\limits_{k=1}^nX_k$ for every $n\geqslant1$, where $(X_k)_{k\geqslant1}$ is an i.i.d. sequence such that $P[X_k=+1]=P[X_k=-1]=\frac12$.
Of course, the RHS of the identity is $$\frac{n}{2^{2n-1}}\,{2n-2\choose n-1}.$$ For a combinatorial proof, see this MSE question and its comments.
For an accessible introduction to the subject, see the corresponding chapter of the Chance project.
-
Here, are you using $\mathbb{E}[X; A]$ to denote the integral taken only over the subset $A$? – Nicholas R. Peterson Jul 14 at 13:38
@nrpeterson Yes, $E[X;A]=E[X\mathbf 1_A]$. – Did Jul 14 at 21:28
This is an interesting question... I'll do some thinking on it. – Nicholas R. Peterson Jul 14 at 21:36
add comment | 2013-12-20 03:51:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.990527331829071, "perplexity": 349.6056344741976}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345768998/warc/CC-MAIN-20131218054928-00001-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://mathematica.stackexchange.com/questions/10298/customticks-and-listplot-cant-seem-to-get-them-working | # CustomTicks and ListPlot - Can't seem to get them working
I have a very large list of numbers like
list={{a1,b1},{a2,b2},...,{ai,bi}}
where a1 ranges between 10^-9 and 10^-5 and bi varies between 0 and 100. I have generated successfully the plots and am now tweaking them with CustomTicks to look a bit better. But I'm not able to get the XX axis as I want.
The YY axis should be linear and the XX axis should be Logarithmic. But the best I could get was this:
Could someone please tell me what I'm doing wrong? I've been trying different things for so long and I can't figure this out.
Here is my code:
a = {#, Random[Real, {0, 100}]} & /@
Union[ Range[1*^-9, 1*^-8, 1*^-9]
~Join~Range[1*^-8, 1*^-7, 1*^-8]
~Join~Range[1*^-7, 1*^-6, 1*^-8]
~Join~Range[1*^-6, 1*^-5, 1*^-7]]
ListLogLinearPlot[a, Joined ->True, Axes -> False, FrameLabel -> {{"A", None}, {"B", None}},
Frame -> {{True, False}, {True, False}}, LabelStyle -> {Bold, 25},
PlotStyle -> {{Thickness[0.003], Black},
{Thickness[0.003], Dashing[0.015], Black},
{Thickness[0.003], Dashing[0.005], Black}},
PlotRange -> {0, 100}, ImageSize -> {800, 600},
FrameTicks - {{LinTicks[0, 100, 20, 2], None},
{LogTicks[E, -8, -5], None}},
FrameTicksStyle -> Directive[Thickness[0.01]]]
-
This uses the LevelScheme package from wnsl.physics.yale.edu/levelscheme. Have you loaded the package? – Sjoerd C. de Vries Sep 6 '12 at 18:40
Also, one Rule is incomplete ('-' instead of '->'), but that doesn't seem to be the cause. – Sjoerd C. de Vries Sep 6 '12 at 18:57
The solution can be found in the CustomTicks manual:
Note that plots with logarithmic axes are actually generated as linear plots, but where the logarithm has been taken of either the x-axis or y-axis variable. Specifically, for base 10,
1. a logarithmic (or linear-log) plot of $f$ is obtained by plotting $\log_{10} f(x)$,
2. a log-linear plot of $f$ is obtained by plotting $f(10^x)$, and
3. a log-log plot of $f$ is obtained by plotting $\log_{10}f(10^x)$ on ordinary linear axes. A similar procedure holds for bases other than 10.
So you really need a normal ListPlot with transformed coordinates instead of a ListLogLinearPlot. Note that the range you used for the log ticks was incorrect too.
a = {Log[a[[All, 1]]], a[[All, 2]]}\[Transpose]
(* think hard about why the Log here instead of the 10^x mentioned in (2) above *)
ListPlot[a, Joined -> True, Axes -> False,
FrameLabel -> {{"A", None}, {"B", None}},
Frame -> {{True, False}, {True, False}}, LabelStyle -> {Bold, 25},
PlotStyle -> {{Thickness[0.003], Black}, {Thickness[0.003],
Dashing[0.015], Black}, {Thickness[0.003], Dashing[0.005],
Black}},
PlotRange -> {0, 100}, ImageSize -> {800, 600},
FrameTicks -> {{LinTicks[0, 100, 20, 2], None},
{LogTicks[E, -20, -11], None}},
FrameTicksStyle -> Directive[Thickness[0.01]]
]
-
or just a[[All, 1]] = Log[a[[All, 1]]]. – Mike Honeychurch Sep 6 '12 at 22:04
@MikeHoneychurch Replacement in-place. Indeed, more polished. – Sjoerd C. de Vries Sep 6 '12 at 22:21
Thank you guys. Now I understand how this works! Thank you Sjoerd – Sosi Sep 7 '12 at 13:46 | 2015-07-05 15:10:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2826452851295471, "perplexity": 6990.529555937258}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375097512.42/warc/CC-MAIN-20150627031817-00270-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://discourse.matplotlib.org/t/tikzpicture-cropped-from-dvi-before-ingestion-in-figure-via-plt-text/22249 | # Tikzpicture cropped from .dvi before ingestion in figure via plt.text()
Disclaimer: I fully understand that the use of text.latex.preamble in rcparams is not officially supported.
Dear hive mind,
I need/want to use the tikz LaTeX package to draw symbols in matplotlib plt.text() calls. It all works smoothly (in the sense that I get no LaTeX/matplotlib errors) … except that no symbols are actually drawn ?!
Here is a MWE:
from matplotlib import pyplot as plt
plt.style.use('./latex.mplstyle')
plt.close(1)
plt.figure(1, figsize=(4, 3))
plt.text(0.5, 0.7, r'This is regular text', ha='center', va='center')
plt.text(0.5, 0.5, r'$\rightarrow$ \begin{tikzpicture}\draw [thick] (0,0) circle [radius=2ex];\end{tikzpicture}', ha='center', va='center')
plt.savefig('test.png')
plt.show()
The content of latex.mplstyle is:
text.usetex: True
text.latex.preamble: \usepackage{tikz}
The output looks like that:
Problem: there should be a circle to the right of the arrow.
And see how the arrow is neatly centered ? This suggests that the tikzpicture is simply not being generated.
So what could matplotlib be doing differently compared to pdflatex that returns the following image:
Screenshot 2021-08-13 at 09.44.13|218x208
As a new user, it appears that I cannot embed two pictures in this post. This image shows an arrow and a circle, like so: ->O)
when processing the following (same) LaTeX code:
\documentclass[11pt]{article}
\usepackage{tikz}
\begin{document}
$\rightarrow$ \begin{tikzpicture}\draw [thick] (0,0) circle [radius=2ex];\end{tikzpicture}
\end{document}
Can anyone think of something obvious ? The tikz package is definitely being loaded correctly, as the \tikzpicture command does not raise any error when compiling the matplotlib figure. So is it a matter of some multiple latex compilations missing for matplotlib ?
Any suggestion will be greatly appreciated !
Side-note 1: as far as I can tell, the same bug behavior implies that symbols from the tikzsymbols LaTeX package do not appear either in plt.text() calls.
Side-note 2: I evidently need to draw more (but not THAT much more) than a circle with tikz … That’s just the MWE!
For the record, the same question was posted on StackOverflow, to try to also reach a wider “tikz/LaTeX audience”.
A new piece of information came to light. When I look inside my .matplotlib/.tex.cache, I can see the .tex and .dvi files generated by the process. And surprise, the .dvi contains the tikz circle !
The question then becomes: what happens to the TeX-generated .dvi file before it gets included in the figure by matplotlib ?
Edit: this got me thinking … is the circle being cropped somehow ?
I’ve modified the plt command as follows, to include an arrow on either side of the circle:
plt.text(0.5, 0.5,
r'$\rightarrow$ \begin{tikzpicture}\draw [thick] (0,0) circle [radius=2ex];\end{tikzpicture} $\leftarrow$',
ha='center', va='center')
And here’s what I see in the .dvi, on-screen, in the .png, and in the .pdf:
Conclusions () :
• The LaTeX commands get processed ok and generate an accurate .dvi file.
• “Something” crops the .dvi before ingestion in the figure. And that “something” isn’t aware of the tikz circle existence.
• “Something (else ?)” is even smarter when saving to .pdf, and keeps precisely only the elements it knows about.
=> What is this “something” ?
Matplotlib requires font metrics like the baseline, ascends and descends. It doesn’t just paste the dvi because typographically that would be a mess. Your tikz picture does not provide any of those metrics and hence they are not included in the crop of the dvi.
Put another way, if you’d used any other latex environment, say table or includegraphics, would you expect them to be properly displayed via a text command?
If I needed to do this, I’d make a high res image and display it using imshow and maybe an inset axes at the location you would like it.
I see. I’ve used tikz to create specific symbols, hoping to be able to insert them in plot labels/legends/etc … as needed, and created a dedicated LaTeX package to that end.
I was using metafont initially (to create the symbols), but then switched to tikz to get scalable ones, along the lines of the tikzsymbols LaTeX package. Which you could argue maybe not the best approach to start with, if one wants to design new font characters.
At this point, I suppose I can either try to find a way to add the missing font metrics to the LaTeX package (which I’m not sure can even be done), or drop tikz entirely and define a proper font with all the required metrics.
You may want to consider using the pgf backend, which does support tikz (e.g., savefig(..., backend="pgf"); note that the preamble goes to pgf.preamble rather than text.latex.preamble) and can directly output pdf and png.
Unfortunately, directly supporting tikz in the “standard” backends would be (AFAICT) very tricky, because tikz effectively outputs shapes using postscript (or pdf) commands, which we cannot interpret.
Brilliant suggestion ! Thanks also for the crucial hint about pgf.preamble, that I did not know about. It works very well that way, I can successfully create png and pdf with the tikzpicture element included. In fact, I can also include the symbols from the tikzsymbols LaTeX package.
plt.text(0.5, 0.5, r’\pot’) →
Of course, I still do not see the symbols displayed on-screen (since this does not go through the pgf backend), but that is a fair price to pay for such a “creative” way of drawing custom “text” symbols
Down the line, I may try to assemble my own “font” to store my symbols as proper characters, and have them accessible in all backends … but for now, the pgf` backend is a perfect solution. Thanks !
@jklymak, @anntzer.lee : if ok for you, I’ll summarize your answers on the SO question to close that one. Unless you want to answer yourself, and get the credit and associated reputation perks ?
Feel free to summarize my answer on SO. | 2021-09-21 05:02:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9549217224121094, "perplexity": 2633.019492753781}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057158.19/warc/CC-MAIN-20210921041059-20210921071059-00692.warc.gz"} |
http://kth.diva-portal.org/smash/record.jsf?pid=diva2:872195 | Change search
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Pipe-In-Pipe system for offshore applications: Post buckling analysis associated with thermal expansion
KTH, School of Engineering Sciences (SCI), Aeronautical and Vehicle Engineering, Lightweight Structures.
2014 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
##### Abstract [en]
The usage of Pipe-In-Pipe (PIP) solutions for offshore applications has increased during the lastyears. The solution gives high thermal insulation and protects the flow line from environmental impacts. One critical load case is buckling of the pipeline system due to thermal expansions of the inner pipe. This project intends to increase knowledge about PIP systems, investigate the impact of different parameters as well as update parameters in an existing SIMLA model. A FE-model of a PIP system was created in ANSYS with a refined section where pipes and centralizers are modelled with solid elements.The ANSYS-model was tested against a verified FE-model created in SIMLA. The global results obtained from ANSYS and SIMLA did not give a perfect match. The ANSYS model tended to buckle in another way, which is assumed to be related to different modelling of resistance between the pipeline and the seabed as well as unwished properties between the side section and the midsection.Local results obtained from ANSYS showed that there are discontinuities in bending moment and effective axial forces when passing a centralizer. The contact force between centralizer and pipes give rise to high friction forces that acts along the same line as the axial force in the pipes.Increased friction coefficient between centralizer and outer pipe resulted in increased discontinuity in axial force. Selection of a proper friction coefficient thus has significant influence on the results.Centralizer stiffness was evaluated by a local FE-model where a centralizer was compressed between the inner and the outer pipe. Displacement of inner pipe was evaluated as a function of applied force. The result showed that the force-displacement curve describing centralizer stiffness follows
Q (Δ)=($C{1}$ Δ) $\frac{2}{3}$
where $C{1}$ is a constant depending on dimensions and material of the centralizer. Linearized indifferent sections and with a centralizer thickness of 0,1 meter the following expression gave stiffnesses in the range 100-1000 MN/m, which agrees with stiffnesses used in the SIMLA model.displacements up to 0.3 mm the radial stiffness used in SIMLA is still good to use.
2014. , p. 92
##### Series
TRITA-AVE, ISSN 1651-7660 ; 2014:27
##### National Category
Engineering and Technology
##### Identifiers
OAI: oai:DiVA.org:kth-177312DiVA, id: diva2:872195
##### Examiners
Available from: 2015-11-18 Created: 2015-11-18 Last updated: 2015-11-18Bibliographically approved
#### Open Access in DiVA
No full text in DiVA
##### By organisation
Lightweight Structures
##### On the subject
Engineering and Technology
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#### Altmetric score
urn-nbn
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v. 2.33.0
| | | | | 2018-05-24 03:53:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 3, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5666548609733582, "perplexity": 9843.758243565071}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794865913.52/warc/CC-MAIN-20180524033910-20180524053910-00192.warc.gz"} |
https://ps-mathematik.univie.ac.at/e/?event=stc22&page=talk-details&id=1089 | # '22
36th Summer Topology Conference
July 18-22, 2022
University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA
### "New asymptotic invariants for mesure preserving vector fields."
#### Rydzek, Marianne
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# A mixed doubles tennis game is to be played between two team
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A mixed doubles tennis game is to be played between two team [#permalink] 31 Aug 2012, 11:36
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A mixed doubles tennis game is to be played between two teams. There are four married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played.
A. 12
B. 21
C. 36
D. 42
E. 46
[Reveal] Spoiler: OA
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Re: A mixed doubles tennis game is to be played between two team [#permalink] 02 Sep 2012, 23:05
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Spaniard wrote:
A mixed doubles tennis game is to be played between two teams. There are four married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played.
A. 12
B. 21
C. 36
D. 42
E. 46
Let's denote the four couples by (A,a), (B,b), (C,c), and (D,d) where A,B,C,D are the husbands and a,b,c,d are the wives.
For a games, let's choose first the husbands. We have 4C2=4*3/2=6 possibilities. Now we know that the final number of games will be a multiple of 6, so we are down to two choices, C and D.
Once husbands were chosen, say A and B, let's count how many possibilities we have to choose their partners under the given restrictions.
We can choose a and b and we have to pair them as (A,b) and (B,a) - 1 possibility.
We can choose one of the wives, a or b, but we have to pair her with the other husband and in addition, we have to choose another partner for the second husband.
This we can do in 2*2 = 4 ways, as there are two possibilities to choose from a and b, then we have 2 possibilities to choose the other wife, c or d - 4 possibilities
Finally, we can choose the other two wives, c and d, and we have two possibilities to team them up with the men, (A,c), (B,d) or (A,d), (B,c) - 2 possibilities.
In conclusion, for every pair of husbands, we have 1 + 4 + 2 = 7 possibilities to choose their partners for the game.
Total number of possibilities 6 * 7 = 42.
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Last edited by EvaJager on 05 Sep 2012, 08:36, edited 1 time in total.
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Re: A mixed doubles tennis game is to be played between two team [#permalink] 05 Sep 2012, 07:26
Can you please explain your solution once more to me? According to my calculations its 36.
4C2(=6) to choose 2 husbands out of the 4. Now for each husband chosen there are 3 wives that can be paired. So-6*3*2 (cause there are two other husband- wife mixed double combination possible.)
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Re: A mixed doubles tennis game is to be played between two team [#permalink] 05 Sep 2012, 08:25
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euphrosyne wrote:
Can you please explain your solution once more to me? According to my calculations its 36.
4C2(=6) to choose 2 husbands out of the 4. Now for each husband chosen there are 3 wives that can be paired. So-6*3*2 (cause there are two other husband- wife mixed double combination possible.)
After we have chosen the pair of husbands, A and B:
We can choose a and b and we have to pair them as (A,b) and (B,a) - 1 possibility.
We can choose one of the wives, a or b, but we have to pair her with the other husband and in addition, we have to choose another partner for the second husband.
This we can do in 2*2 = 4 ways, as there are two possibilities to choose from a and b, then we have 2 possibilities to choose the other wife, c or d - 4 possibilities
Finally, we can choose the other two wives, c and d, and we have two possibilities to team them up with the men, (A,c), (B,d) or (A,d), (B,c) - 2 possibilities.
In conclusion, for every pair of husbands, we have 1 + 4 + 2 = 7 possibilities to choose their partners for the game.
Total number of possibilities 6 * 7 = 42.
Now for each husband chosen there are 3 wives that can be paired. So-6*3*2
There are no 6 husbands, but only 4. You should not consider husbands alone.
You should count the possibilities of choosing the wives per chosen pair of husbands, otherwise you cannot keep up with repetitions or you can miss out some possibilities. 6 represents the number of pairs of husbands. Then the number of 3 wives is not correct, as you can see from the above, all four wives can be candidates as partners, depending whom each plays.
If you write down all the possibilities for the pair of husbands A and B, you will get a total of 7 (following the steps described above):
(A,b) (B,a) - 1
(A,b) (B,c); (A,b) (B,d); (A,c) (B,a); (A,d) (B,a) - 4
(A,c) (B,d); (A,d) (B,c) - 2
Or in other words:
1 - two wives stay, but each has to play with the other husband
2 - one wife stays and plays with the other husband, and second husband receives one of the other two wives as a partner
3 - none of the wives stays, the remaining two wives pair up with the already chosen husbands
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PnC: A mixed doubles tennis game is to be played between [#permalink] 20 Oct 2012, 02:48
A mixed doubles tennis game is to be played between two teams(Each team consists of one male and one female). There are 4 married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played?
a)12
b)21
c)36
d)42
e)60
Detailed solution with brief description of each combination required.
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Re: PnC: A mixed doubles tennis game is to be played between [#permalink] 20 Oct 2012, 03:42
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avaneeshvyas wrote:
A mixed doubles tennis game is to be played between two teams(Each team consists of one male and one female). There are 4 married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played?
a)12
b)21
c)36
d)42
e)60
Detailed solution with brief description of each combination required.
IT is easy to find the number of games with married couples.
One married couple only:
Select one married couple out of 4 in 4C1 ways.
Select one male for the other team in 3 ways and one non-wife female in 2 ways.
Number of games with only one married couple = 4*3*2 = 24
Both married couples
Select 2 married couples in 4C2 = 6 ways
Number of games in which atleast there will be one couple = 24+6 = 30
Total number of games = (4*4 * 3*3)/2 = 72
Select team 1 in 4*4 ways and team 2 in 3*3 ways. Divide by 2 because you don't want to arrange the teams in team 1 and team 2. They are just 2 teams.
So in 72 - 30 = 42 games, there will be no married couple.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Math Expert Joined: 02 Sep 2009 Posts: 27468 Followers: 4311 Kudos [?]: 42201 [0], given: 5957 Re: PnC: A mixed doubles tennis game is to be played between [#permalink] 20 Oct 2012, 04:06 Expert's post avaneeshvyas wrote: A mixed doubles tennis game is to be played between two teams(Each team consists of one male and one female). There are 4 married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played? a)12 b)21 c)36 d)42 e)60 Detailed solution with brief description of each combination required. Merging similar topics. _________________ Director Joined: 17 Apr 2013 Posts: 595 Concentration: Entrepreneurship, Leadership Schools: HBS '16 GMAT 1: 710 Q50 V36 GMAT 2: 750 Q51 V41 GPA: 3.3 Followers: 10 Kudos [?]: 124 [0], given: 272 Re: PnC: A mixed doubles tennis game is to be played between [#permalink] 31 Jul 2013, 06:01 VeritasPrepKarishma wrote: avaneeshvyas wrote: A mixed doubles tennis game is to be played between two teams(Each team consists of one male and one female). There are 4 married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played? a)12 b)21 c)36 d)42 e)60 Detailed solution with brief description of each combination required. IT is easy to find the number of games with married couples. One married couple only: Select one married couple out of 4 in 4C1 ways. Select one male for the other team in 3 ways and one non-wife female in 2 ways. Number of games with only one married couple = 4*3*2 = 24 Both married couples Select 2 married couples in 4C2 = 6 ways Number of games in which atleast there will be one couple = 24+6 = 30 Total number of games = (4*4 * 3*3)/2 = 72 Select team 1 in 4*4 ways and team 2 in 3*3 ways. Divide by 2 because you don't want to arrange the teams in team 1 and team 2. They are just 2 teams. So in 72 - 30 = 42 games, there will be no married couple. Can you please explain what the question is demanding, I have difficulty understanding the question correctly? _________________ Like my post Send me a Kudos It is a Good manner. My Debrief: how-to-score-750-and-750-i-moved-from-710-to-189016.html Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5539 Location: Pune, India Followers: 1369 Kudos [?]: 6966 [1] , given: 178 Re: PnC: A mixed doubles tennis game is to be played between [#permalink] 31 Jul 2013, 21:04 1 This post received KUDOS Expert's post trafficspinners wrote: VeritasPrepKarishma wrote: avaneeshvyas wrote: A mixed doubles tennis game is to be played between two teams(Each team consists of one male and one female). There are 4 married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played? a)12 b)21 c)36 d)42 e)60 Detailed solution with brief description of each combination required. IT is easy to find the number of games with married couples. One married couple only: Select one married couple out of 4 in 4C1 ways. Select one male for the other team in 3 ways and one non-wife female in 2 ways. Number of games with only one married couple = 4*3*2 = 24 Both married couples Select 2 married couples in 4C2 = 6 ways Number of games in which atleast there will be one couple = 24+6 = 30 Total number of games = (4*4 * 3*3)/2 = 72 Select team 1 in 4*4 ways and team 2 in 3*3 ways. Divide by 2 because you don't want to arrange the teams in team 1 and team 2. They are just 2 teams. So in 72 - 30 = 42 games, there will be no married couple. Can you please explain what the question is demanding, I have difficulty understanding the question correctly? The question says that a mixed double match is to be played. You need two teams - each team consisting of one man and one woman - for the match. You have 4 married couples who would be willing to play - 4 men and 4 women. So you have to choose 2 men and 2 women such that they form two teams of one man one woman each in which the man and woman are not married. So say, you choose Man1 and Man3 to play. Now you have to choose 2 women. With Man1, you cannot have Woman1 but you can have any of the other 3 women. So the two teams could look like this: Man1-Woman2 and Man3-Woman1 or Man1-Woman3 and Man3-Woman2 or Man1-Woman3 and Man3-Woman4 etc What you should NOT have is something like this: Man1-Woman1 and Man3-Woman2 or Man1-Woman2 and Man3-Woman3 or Man1-Woman1 and Man3-Woman3 So the question is : In how many different ways can you make such a team? _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199
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Re: A mixed doubles tennis game is to be played between two team [#permalink] 01 Aug 2013, 02:04
EvaJager wrote:
Spaniard wrote:
A mixed doubles tennis game is to be played between two teams. There are four married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played.
A. 12
B. 21
C. 36
D. 42
E. 46
Let's denote the four couples by (A,a), (B,b), (C,c), and (D,d) where A,B,C,D are the husbands and a,b,c,d are the wives.
For a games, let's choose first the husbands. We have 4C2=4*3/2=6 possibilities. Now we know that the final number of games will be a multiple of 6, so we are down to two choices, C and D.
Once husbands were chosen, say A and B, let's count how many possibilities we have to choose their partners under the given restrictions.
We can choose a and b and we have to pair them as (A,b) and (B,a) - 1 possibility.
We can choose one of the wives, a or b, but we have to pair her with the other husband and in addition, we have to choose another partner for the second husband.
This we can do in 2*2 = 4 ways, as there are two possibilities to choose from a and b, then we have 2 possibilities to choose the other wife, c or d - 4 possibilities
Finally, we can choose the other two wives, c and d, and we have two possibilities to team them up with the men, (A,c), (B,d) or (A,d), (B,c) - 2 possibilities.
In conclusion, for every pair of husbands, we have 1 + 4 + 2 = 7 possibilities to choose their partners for the game.
Total number of possibilities 6 * 7 = 42.
I have difficulty understanding this part-
we have 1 + 4 + 2 = 7 possibilities to choose their partners for the game.
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Re: A mixed doubles tennis game is to be played between two team [#permalink] 02 Aug 2013, 22:41
My approach is similar to Karishma but i'm not able to arrive at correct choice.
For team1 we have two places to be filled - 1 for man and 1 for woman... 4*4 = 16 ( without any restrictions )
For team 2, 3*3 = 9 (no restriction)
For a match 16*9/2! = 72 total ways ( T1 = 16 ways and T2 = 9 ways. Divide by 2! as order/arrangement not required )
Now consider case where 1 couple is playing.
Team 1 -- for man 4 ways and for woman only 1 possibility. Thus a total of 4 ways.
Team 2 -- for man 3 ways and for woman 2 ways. Thus 6 ways.
Now, for a match 4*6/2! = 12 or simply 24( as done by Karishma )
Case2, two couples.
Team 1= man in 4 ways and woman in 1 way = 4 ways
Team 2 = man in 3 ways and woman in 1 way = 3 ways
For a match 4*3/2! or 4*3 ??
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Re: PnC: A mixed doubles tennis game is to be played between [#permalink] 04 Aug 2013, 03:53
VeritasPrepKarishma wrote:
avaneeshvyas wrote:
A mixed doubles tennis game is to be played between two teams(Each team consists of one male and one female). There are 4 married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played?
a)12
b)21
c)36
d)42
e)60
Detailed solution with brief description of each combination required.
IT is easy to find the number of games with married couples.
One married couple only:
Select one married couple out of 4 in 4C1 ways.
Select one male for the other team in 3 ways and one non-wife female in 2 ways.
Number of games with only one married couple = 4*3*2 = 24
Both married couples
Select 2 married couples in 4C2 = 6 ways
Number of games in which atleast there will be one couple = 24+6 = 30
Total number of games = (4*4 * 3*3)/2 = 72
Select team 1 in 4*4 ways and team 2 in 3*3 ways. Divide by 2 because you don't want to arrange the teams in team 1 and team 2. They are just 2 teams.
So in 72 - 30 = 42 games, there will be no married couple.
Karishma, please reply to my post and point out my mistake in the approach i have used.. And is no. of ways of formation of two teams with the given restrictions same as the no. of matches possible ??
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Re: A mixed doubles tennis game is to be played between two team [#permalink] 21 Aug 2013, 03:12
Please check and provide feedback for my solution :
First we will find possible number of teams then we can find total number of games .
M1 W1 --- With M1 --3 Pairs ..so three teams
M2 W2
M3 W3
M4 W4 ----- So total 12 teams .
Now we have to find out total games = 12C2 - 4(3!) = 42
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Re: PnC: A mixed doubles tennis game is to be played between [#permalink] 21 Aug 2013, 21:19
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hsb91 wrote:
Karishma, please reply to my post and point out my mistake in the approach i have used.. And is no. of ways of formation of two teams with the given restrictions same as the no. of matches possible ??
'Maximum number of games' implies 'in how many distinct ways can you make the teams'. e.g. (M1, W2 and M2, W3) OR (M1, W3 and M3, W4) etc. This is implied by the context; though if you take the question literally, it makes little sense.
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Re: A mixed doubles tennis game is to be played between two team [#permalink] 02 Oct 2013, 07:38
Hello,
I started with following way but stuck in the end.
First we make number of teams possible.:
4C1*3C1 = 12 teams are possible.
Number of matches = 12C2= 66
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Re: A mixed doubles tennis game is to be played between two team [#permalink] 02 Oct 2013, 20:34
Expert's post
bsahil wrote:
Hello,
I started with following way but stuck in the end.
First we make number of teams possible.:
4C1*3C1 = 12 teams are possible.
Number of matches = 12C2= 66
This approach is incorrect.
You can make 12 distinct teams - that's fine. They will look like this:
AB'
AC'
BA'
BC'
BD'
CA'
... etc
Now can you pick any two out of these and have a game? Say you pick AB' and AC'. Can A play as the male member on both teams in a game?
You have to think in terms of a game instead.
Say, you select 2 male members out of a total of 4 in 4C2 ways. Say you select A and B.
Now for one male member, say A, you can select a partner in 3 ways (B', C' and D'). The problem is that if you select B', you have 3 options for B's partner (A', C' and D'). IF you select C' or D' for A, you have only 2 options for B (A' and C'/D' whoever is left). So you take two cases:
Select B' for A --> 4C2* 1 * 3 = 18
Select other than B's wife for A --> 4C2 *2*2 = 24
Total number of ways = 42
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews GMAT Club Legend Joined: 09 Sep 2013 Posts: 4926 Followers: 298 Kudos [?]: 54 [0], given: 0 Re: A mixed doubles tennis game is to be played between two team [#permalink] 13 May 2015, 03:28 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Intern Joined: 19 Mar 2015 Posts: 11 Location: United States Concentration: Sustainability, Sustainability Followers: 0 Kudos [?]: 0 [0], given: 99 Re: A mixed doubles tennis game is to be played between two team [#permalink] 23 May 2015, 01:38 VeritasPrepKarishma wrote: avaneeshvyas wrote: A mixed doubles tennis game is to be played between two teams(Each team consists of one male and one female). There are 4 married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played? a)12 b)21 c)36 d)42 e)60 Detailed solution with brief description of each combination required. IT is easy to find the number of games with married couples. One married couple only: Select one married couple out of 4 in 4C1 ways. Select one male for the other team in 3 ways and one non-wife female in 2 ways. Number of games with only one married couple = 4*3*2 = 24 Both married couples Select 2 married couples in 4C2 = 6 ways Number of games in which atleast there will be one couple = 24+6 = 30 Total number of games = (4*4 * 3*3)/2 = 72 Select team 1 in 4*4 ways and team 2 in 3*3 ways. Divide by 2 because you don't want to arrange the teams in team 1 and team 2. They are just 2 teams. So in 72 - 30 = 42 games, there will be no married couple. I solved it in a different way and ended up getting wrong answer. Still I am not able to find a mistake in my method. It will be great if you could help me finding my mistake. the ways in which teams can be formed = 4C1 * 4C1 (one male out of four and one female out of four) = 16, but this includes the married couples in one team. So number of ways in which teams can be formed = 16-4 =12 number of games among 12 teams = 6------> 6 winner teams number of games amaong 6 teams = 3-------> 3 winner teams number of games among 3 teams = 1--------> only two teams can play a match, number of winner =1 last game between 2 teams ------> final winner so total number of games played = 6+3+1+1 = 11 please help me finding my mistake. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5539 Location: Pune, India Followers: 1369 Kudos [?]: 6966 [0], given: 178 Re: A mixed doubles tennis game is to be played between two team [#permalink] 23 May 2015, 02:52 Expert's post Yogita25 wrote: VeritasPrepKarishma wrote: avaneeshvyas wrote: A mixed doubles tennis game is to be played between two teams(Each team consists of one male and one female). There are 4 married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played? a)12 b)21 c)36 d)42 e)60 Detailed solution with brief description of each combination required. IT is easy to find the number of games with married couples. One married couple only: Select one married couple out of 4 in 4C1 ways. Select one male for the other team in 3 ways and one non-wife female in 2 ways. Number of games with only one married couple = 4*3*2 = 24 Both married couples Select 2 married couples in 4C2 = 6 ways Number of games in which atleast there will be one couple = 24+6 = 30 Total number of games = (4*4 * 3*3)/2 = 72 Select team 1 in 4*4 ways and team 2 in 3*3 ways. Divide by 2 because you don't want to arrange the teams in team 1 and team 2. They are just 2 teams. So in 72 - 30 = 42 games, there will be no married couple. I solved it in a different way and ended up getting wrong answer. Still I am not able to find a mistake in my method. It will be great if you could help me finding my mistake. the ways in which teams can be formed = 4C1 * 4C1 (one male out of four and one female out of four) = 16, but this includes the married couples in one team. So number of ways in which teams can be formed = 16-4 =12 number of games among 12 teams = 6------> 6 winner teams number of games amaong 6 teams = 3-------> 3 winner teams number of games among 3 teams = 1--------> only two teams can play a match, number of winner =1 last game between 2 teams ------> final winner so total number of games played = 6+3+1+1 = 11 please help me finding my mistake. The wording of the question is a little off. "What is the maximum number of games that can be played?" actually means "In how many different ways can you make the two teams?" Only one game is to be played. You need two 2-people teams for that. A married couple should not be a team. In how many different ways can you make the two teams? Now review your solution. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199
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Re: A mixed doubles tennis game is to be played between two team [#permalink] 23 May 2015, 02:52
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Display posts from previous: Sort by | 2015-05-24 09:48:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4333864152431488, "perplexity": 2141.6397117495744}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207927863.72/warc/CC-MAIN-20150521113207-00000-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.futurelearn.com/info/courses/python-in-hpc/0/steps/65121 | # Using static typing
In this article we describe how giving away dynamic typing of variables can increase the performance of the program.
© CC-BY-NC-SA 4.0 by CSC - IT Center for Science Ltd.
Python is both a strongly typed and a dynamically typed language.
Strong typing means that variables do have a type and that the type matters when performing operations on a variable. Dynamic typing means that the type of the variable is determined only during runtime.
Due to strong typing, types need to be compatible with respect to the operand when performing operations. For example Python allows one to add an integer and a floating point number, but adding an integer to a string produces error.
Due to dynamic typing, in Python the same variable can have a different type at different times during the execution. Dynamic typing allows for flexibility in programming, but with a price in performance.
## Everything is an object
One of the key features of Python is that everything is an object, and the type is just one attribute of an object. As an illustration, we can assign a single integer to a variable, and use the Python built-in function dir for
finding out the attributes of the object. If you execute the following two
lines in an interactive interpreter, it should become clear that a Python
integer is much more complex than just a number. Please feel free to
experiment also with other types!
n = 5
dir(n)
The fact that everything is an object means that there is a lot of “unboxing”
and “boxing” involved when Python performs operations with variables. For
example, when just adding two integers
a = 7
b = 6
c = a + b
there are several steps Python needs to do:
1. Check the types of both operands
2. Check whether they both support the + operation
3. Extract the function that performs the + operation (due to operator
4. Extract the actual values of the objects
5. Perform the + operation
6. Construct a new integer object for the result
Due to the fact that Python is dynamically typed, the interpreter cannot know
beforehand what type of objects one is dealing with, and everytime two
variables are added one needs to perform all the above steps.
What if one knows that e.g. in a certain function the variables have always
the same type? That’s where Cython steps in: Cython allows one to add static
typing information so that boxing and unboxing are not needed, and one can
operate directly with the actual values.
When Cythonizing a Python code, static type information can be added either:
• In function signatures by prefixing the formal arguments by their type
• By declaring variables with the cdef Cython keyword, followed by the
the type
For example, a simple Python function adding two objects could be Cythonized
as follows:
def add (int x, int y):
cdef int result
result = x + y
return result
The function works now only with integers but with less boxing/unboxing
The types provided in Cython code are C types, and the variables with type
information are pure C variables and not Python objects. When calling a
Cythonized function from Python, there is an automatic conversion from the
Python object of actual arguments to the C value of formal argument, and when
returning a C variable it is converted to corresponding Python object.
Automatic conversions are carried out also in most cases within the Cython
code where both Python objects and C variables are involved.
The table below lists the most common C types and their corresponding Python
Cython documentation.
From Python types To C types
int int, long
int, float float, double
str/bytes char *
From C types To Python types
int, long int
float, double float
char * str/bytes
## Static typing in Mandelbrot kernel
In week 1 we did a performance analysis of Mandelbrot fractal in Step 1.11. The analysis revealed
that the kernel function in module mandelbrot.py was the most time critical one. Thus, let’s make mandelbrot.py into Cython module mandelbrot.pyx and introducing static typing to the
function kernel.
Pure Python version was:
def kernel(zr, zi, cr, ci, lim, cutoff):
''' Computes the number of iterations n such that
|z_n| > lim, where z_n = z_{n-1}**2 + c.
'''
count = 0
while ((zr*zr + zi*zi) < (lim*lim)) and count < cutoff:
zr, zi = zr * zr - zi * zi + cr, 2 * zr * zi + ci
count += 1
return count
We can add type information both to the function signature and to the function
body:
def kernel(double zr, double zi, double cr, double ci, double lim, int cutoff):
''' Computes the number of iterations n such that
|z_n| > lim, where z_n = z_{n-1}**2 + c.
'''
cdef int count = 0
while ((zr*zr + zi*zi) < (lim*lim)) and count < cutoff:
zr, zi = zr * zr - zi * zi + cr, 2 * zr * zi + ci
count += 1
return count
When comparing the performance of pure Python and Cythonized versions, we obtain the following results:
• Pure Python: 0.57 s
• Static type declarations in the kernel: 14 ms
Thus, we obtained a speed up of ~40 !
© CC-BY-NC-SA 4.0 by CSC - IT Center for Science Ltd. | 2022-05-24 21:07:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18671482801437378, "perplexity": 3499.4702767689155}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662577259.70/warc/CC-MAIN-20220524203438-20220524233438-00361.warc.gz"} |
http://ncatlab.org/nlab/show/completely+prime+filter | # nLab completely prime filter
Recall that a filter $F$ on a lattice $L$ is called prime if $\bot \notin F$ and, whenever $x \vee y \in F$, then $x \in F$ or $y \in F$. In other words, for every finite index set $I$, $x_k \in F$ for some $k$ whenever $\bigvee_{k\colon I} x_i \in F$.
We now generalise from finitary joins to arbitrary joins: A filter $F$ on a complete lattice $L$ is completely prime if, for any index set $I$ whatsoever, $x_k \in F$ for some $k$ whenever $\bigvee_{k\colon I} x_i \in F$. Equivalently, a completely prime filter is given by a simlutaneous suplattice and lattice homomorphism from $L$ to the lattice $TV$ of truth values (which is classically the boolean domain $\mathbb{2}$).
In particular, if $L$ is a frame, then a completely prime filter of $L$ is given by a frame homomorphism from $L$ to $TV$. Thinking of $L$ as a locale, this is the same as a point of $L$.
Created on February 5, 2010 22:01:49 by Toby Bartels (173.60.119.197) | 2014-12-22 14:53:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 25, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9592971205711365, "perplexity": 164.96618749193502}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802775392.34/warc/CC-MAIN-20141217075255-00146-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/692088/prove-that-lim-n-rightarrow-infty-frac1n-sum-k-1nf-frackn | # Prove that $\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k = 1}^{n}{f(\frac{k}{n}) }$ $=\int_0^1 f(x)dx.$
Question:
Let $f$ be continuous on $[0,1]$. Prove that
$\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k = 1}^{n}{f(\frac{k}{n}) }$ $=\int_0^1 f(x)dx.$
where $k=0,1,...,n.$
Attempt:
I don't even know where to start. It makes sense reading the sum, as $k\rightarrow n$, and dividing it by the number of partitions, I should reach the definition of the integral. Hoping for a little push to get started.
• Well, what is the definition of integral [you're using]? – Grigory M May 8 '14 at 22:45
• @Brian AFAICS Martin answered the question completely a couple of months ago, no? – Grigory M May 8 '14 at 22:51
A continuous function on a compact set is uniformly continuous.
Given an $\epsilon\gt0$, find an $N$ so that if $|x-y|\le\frac1N$ we have $|f(x)-f(y)|\le\epsilon$. Thus, for any $n\ge N$, $$\int_{\large\frac{k-1}n}^{\large\frac kn}\left|\,f\left(\frac kn\right)-f(x)\,\right|\,\mathrm{d}x\le\frac\epsilon{n}$$ Therefore, for $n\ge N$, \begin{align} \left|\,\sum_{k=1}^nf\left(\frac kn\right)\frac1n-\int_0^1f(x)\,\mathrm{d}x\,\right| &=\left|\,\sum_{k=1}^n\left(\int_{\large\frac{k-1}n}^{\large\frac kn}f\left(\frac kn\right)\,\mathrm{d}x-\int_{\large\frac{k-1}n}^{\large\frac kn}f(x)\,\mathrm{d}x\right)\,\right|\\ &\le\sum_{k=1}^n\int_{\large\frac{k-1}n}^{\large\frac kn}\left|\,f\left(\frac kn\right)-f(x)\,\right|\,\mathrm{d}x\\[6pt] &\le\epsilon \end{align}
• Dear robjohn, (while the answer is technically correct) the result has nothing to do with uniform continuity — the statement is true even if $f$ is not cont. (as long as the RHS exists). – Grigory M May 8 '14 at 22:48
• @GrigoryM: $\int_0^1\log(1-x)\,\mathrm{d}x$ exists, but the sum does not since $\log(1-x)$ does not exist at $x=1$. – robjohn May 8 '14 at 22:52
• Well, yes, if Riemann integral exists (in your example not Riemann integral, but only improper integral exists). – Grigory M May 8 '14 at 22:55
• @GrigoryM: Certainly, if a function is bounded on a bounded domain, and its set of discontinuities is a null set (a set of measure zero), then the function is Riemann Integrable. However, the standard proof is usually done with continuous functions on compact sets, then generalized by using compact intervals where the function is continuous, whose complement is arbitrarily small. Since the domain is a compact interval and the function is given as continuous, why not use these facts to prove convergence? – robjohn May 8 '14 at 23:35
It is exactly as you say. The limit on the left is a limit of Riemann sums of $f$ in the interval $[0,1]$.
Since $f$ is continuous, it is Riemann integrable, so by definition for each $\varepsilon>0$ there exists a partition $P$ such that for all partitions $P'=\{0=x_1<x_2<\dots<x_N=1\}$ which are finer than $P$ we have $$\sum_{k=1}^N \max_{x\in [x_{k-1},x_k]}f(x) \cdot (x_{k}-x_{k-1}) -\int _0^1 f(x) dx<\varepsilon$$ and $$\int _0^1 f(x) dx -\sum_{k=1}^N \min_{x\in [x_{k-1},x_k]}f(x) \cdot (x_{k}-x_{k-1})<\varepsilon$$ The partition $\{0,1/n,2/n,\dots,(n-1)/n,1\}$, $x_k=k/n$ will be finer than $P$ for large $n$, and since in this case we have $x_{k}-x_{k-1} =k/n- (k-1)/n= 1/n$, we have $$\sum_{k=1}^n \max_{x\in [x_{k-1},x_k]}f(x) \frac{1}{n} -\int _0^1 f(x) dx<\varepsilon$$ and $$\int _0^1 f(x) dx -\sum_{k=1}^n \min_{x\in [x_{k-1},x_k]}f(x) \frac{1}{n}<\varepsilon$$ Now, also observe that $$\min_{x\in [x_{k-1},x_k]}f(x)\leq f(k/n)\leq \max_{x\in [x_{k-1},x_k]}f(x)$$ and combining this with the above inequalities you obtain $$\left|\sum_{k=1}^n f(k/n) \frac{1}{n} -\int _0^1 f(x) dx \right|<\varepsilon$$ which is true for large $n$. This gives you the result.
• To me, the question seems to be to show that if $f$ is continuous on $[0,1]$, then it is Riemann Integrable. If we assume that it is Riemann Integrable in the first place, then the answer follows simply from the definition of Riemann Integrability. – robjohn May 8 '14 at 23:33
• That could be true, but then the question should be phrased as "show that the limit exists". – Dimitris May 9 '14 at 2:01
• It appears as if you are using the Darboux integral rather than the Riemann integral. It turns out that a function is Riemann integrable if and only if it is Darboux integrable. However, Riemann integrability is ostensibly weaker and proofs using it as an assumption are a bit more involved. – robjohn May 9 '14 at 2:42
Continuity of $f$ plays no rôle in this game, we only have to assume that the Riemann integral $\int_a^b f(x)\>dx$ exists.
For a function $f:\ [a,b]\to{\mathbb R}$ and a subinterval $Q\subset[a,b]$ write $$|\Delta f|_Q:=\sup_{x\in Q} f(x)-\inf_{x\in Q} f(x)\ .$$ Such a function is Riemann integrable over $[a,b]$ if for any $\epsilon>0$ there is a partition $P$ of $[a,b]$ into subintervals $Q_k=[x_{k-1}, x_k]$ $\>(1\leq k\leq N)$ such that $$D_P(f):=\sum_{k=1}^N |\Delta f|_{Q_k}(x_k-x_{k-1})<\epsilon\ .$$ When $f$ passes this simple test then there is a unique number $S\in{\mathbb R}$ such that $$|R_P-S|\leq D_P(f)\tag{1}$$ for all partitions $P$ and all Riemann sums $R_P=\sum_{k=1}^N f(\xi_k)(x_k-x_{k-1})$ computed using $P$. This $S$ is called the integral of $f$ over $[a,b]$, and is denoted by $\int_a^b f(x)\ dx$.
The following Lemma has been proved several times on MSE: When $f$ is integrable over $[a,b]$ then for each $\epsilon>0$ there is $\delta >0$ such that $D_P(f)<\epsilon$ as soon as $\max_{1\leq k\leq N}(x_k-x_{k-1})<\delta$.
We now argue as follows: Given an $\epsilon>0$ choose a $\delta>0$ according to the Lemma. There is an $n_0$ such that ${b-a\over n_0}<\delta$. Denote the partition considered in the question by $P_n$ and the displayed Riemann sum by $R_n$. When $n>n_0$ then ${b-a\over n}<\delta$. Therefore it follows from the principle $(1)$ that $$\left|R_n-\int_a^b f(x)\ dx\right|\leq D_{P_n}(f)< \epsilon\ .$$
$f(x) =f(0)+\frac{f'(0)x}{1!}+\frac{f''(0)x^2}{2!}+.....=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$
$$\int _0^x {f(t) dt}=\int _0^x(\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} t^n)dt=\sum_{n=0}^{\infty} (\frac{f^{(n)}(0)}{n!}\int _0^x t^n dt)=\sum_{n=0}^{\infty} (\frac{f^{(n)}(0)}{n!}\frac{x^{n+1}}{n+1})$$ $$\int _0^x {f(t) dt}=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)x^{n+1}}{(n+1)!}$$ $$(1)$$
$$f(\frac{kx}{n})=\sum_{m=0}^{\infty} \frac{f^{(m)}(0)}{m!} (\frac{kx}{n})^m$$ $$\sum \limits_{k=1}^{n} k^m=\frac{n^{m+1}}{m+1}+a_mn^m+....+a_1n=\frac{n^{m+1}}{m+1}+\sum \limits_{j=1}^m a_jn^j$$ where $a_j$ are constants. More information about summation http://en.wikipedia.org/wiki/Summation
$$\lim_{n\to\infty} \frac{x}{n}\sum \limits_{k=1}^n f(\frac{kx}{n})=\lim_{n\to\infty} \frac{x}{n}\sum \limits_{k=1}^n \sum_{m=0}^{\infty} \frac{f^{(m)}(0)}{m!} (\frac{kx}{n})^m=\lim_{n\to\infty} \frac{x}{n}\sum_{m=0}^{\infty} \frac{x^m}{n^m} \frac{f^{(m)}(0)}{m!} \sum \limits_{k=1}^n k^m=\lim_{n\to\infty} \frac{x}{n}[f(0)n+\frac{f'(0)x}{n 1!}(\frac{n^2}{2}+\frac{n}{2})+ \frac{f''(0)x^2}{n^2 2!}(\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6})+\frac{f'''(0)x^3}{n^3 3!}(\frac{n^4}{4}+\frac{n^3}{2}+\frac{n^2}{4})+\frac{f^{(4)}(0)x^4}{n^4 4!}(\frac{n^5}{5}+\frac{n^4}{2}+\frac{n^3}{3}-\frac{n}{30})+...... ]= \lim_{n\to\infty} [f(0)x+\frac{f'(0)x^2}{n^2 1!}(\frac{n^2}{2}+\frac{n}{2})+ \frac{f''(0)x^3}{n^3 2!}(\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6})+\frac{f'''(0)x^4}{n^4 3!}(\frac{n^4}{4}+\frac{n^3}{2}+\frac{n^2}{4})+\frac{f^{(4)}(0)x^5}{n^5 4!}(\frac{n^5}{5}+\frac{n^4}{2}+\frac{n^3}{3}-\frac{n}{30})+...... ]= [f(0)x+\frac{f'(0)x^2}{ 2!}+ \frac{f''(0)x^3}{ 3!}+\frac{f'''(0)x^4}{ 4!}+\frac{f^{(4)}(0)x^5}{ 5!}+...... ]$$
$$\lim_{n\to\infty} \frac{x}{n}\sum \limits_{k=1}^n f(\frac{kx}{n})=\sum_{m=0}^{\infty} \frac{f^{(m)}(0)x^{m+1}}{(m+1)!}$$ $$(2)$$
Equation $(1)$ and equation $(2)$ are equal to each other. Thus
$$\lim_{n\to\infty} \frac{x}{n}\sum \limits_{k=1}^n f(\frac{kx}{n})=\int _0^x {f(t) dt}$$
• You are assumning that $f$ is $C^ \infty$. The result is true even for just continuous functions. – Georgy May 14 '14 at 9:36
The first thing I notice is that we will be multiplying by zero as the current formula stands (constant divided by infinity is zero). But, intuitively, we can not say the integral of the two parts will be zero. This leads me to thinking of rewriting the sum using some formula. In fact, if we prove this formula, we have our answer! | 2019-05-20 11:30:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9712708592414856, "perplexity": 118.97565622341268}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232255943.0/warc/CC-MAIN-20190520101929-20190520123929-00016.warc.gz"} |
https://www.openaircraft.com/ccx-doc/cgx/node134.html | ## movi
'movi' [loops <nr>]|[delay <sec>]|[start]|[stop]|
[frames ['auto']|[<nr> [<epilogFile>]]]|
[make [<pic-nr> <pic-nr> [<prolog.gif>]]]|
[clean]
This keyword is used to start or stop the recording of a movie. After ”start” all frames will be stored in single gif files until the ”stop” command is issued. Use the option ”make” to assemble the movie from the individual files. The range consists of the nr of the first and last picture to be used. An existing movie will be copied in front of a range of frames if its name is given. With the option ”delay” a time-delay (in seconds) between frames can be specified. With the option ”loops” a certain nr of loops can be chosen before the animation stops. Without giving a certain nr the default is chosen which is infinite loops. With the option ”clean,” all single gif-files will be erased.
Below is an example command sequence. Do not use this sequence in a file since the start and stop commands will be executed without delay (see option 'frames' for use in a command file). Instead of using the default value of loops here one loop is defined:
movi delay 0.01
movi loops 1
movi start
(let the program run until all frames are recorded)
movi stop
movi make
(or if a certain movie should be extended by the first 500 frames:)
movi make 1 500 prolog.gif
movi clean
When using the “frames” option the recording starts and a given nr of frames will be recorded before the recording stops atomatically. In cases were an animation of a mode shape or a sequence of datasets should be recorded it might be useful to use the argument 'auto' instead of a specific nr of frames. With the 'auto' functionality the program determines how much frames are needed to cover one period of frames and this period is then recorded. The 'make' and 'clean' functionality is included in the 'auto' mode. The 'auto' mode requires that the animation or the sequence is defined and started with the next command line (see ”ds”):
anim real
movi frames auto
ds 3 eh 7
There is a second method available when successive commands after the recording of a given number of frames are needed:
movi frames 90 epilogCommandFile.fbl
This command must be the last command in an eventual command file. After 90 frames the given file 'epilogCommandFile.fbl' will be executed (the records are interpreted as cgx commands).
Further remarks in ”How to change the format of the movie file”. See also the menu options ”Start Recording Gif-Movie”. | 2022-05-23 10:57:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.524001955986023, "perplexity": 2504.7298715851166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662558015.52/warc/CC-MAIN-20220523101705-20220523131705-00581.warc.gz"} |
https://mpboardguru.com/mp-board-class-9th-maths-solutions-chapter-4-ex-4-1-english-medium/ | ## MP Board Class 9th Maths Solutions Chapter 4 Linear Equations in Two Variables Ex 4.1
Question 1.
The cost of notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.
Solution:
Let cost of pen be ₹ x and cost of a notebook be ₹ y
y = 2x
y – 2x = 0.
Question 2.
Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case:
1. 2x + 3y = 9.3$$\overline { 5 }$$
2. x – $$\frac{y}{5}$$ – 10 = 0
3. -2x + 3y = 6
4. x = 3y
5. 2x = – 5y
6. 3x + 2 = 0
7. y – 2 = 0
8. 5 = 2x
Solution:
1. 2x + 3y = 9.3$$\overline { 5 }$$
2x + 3y – 9.3$$\overline { 5 }$$ = 0
a = 2, b = 3, c = – 9.3$$\overline { 5 }$$
2. x – $$\frac{y}{5}$$ – 10 = 0
a = 1, b = – $$\frac{1}{5}$$, c = – 10
3. -2x + 3y = 6
– 2x + 3y – 6 = 0
a = – 2, b = 3, c = – 6
4. x = 3y
1. x – 3y + 0 = 0
a – 1, b = – 3, c = 0
5. 2x = – 5y
2x + 5y + 0 = 0
a = 2, b = 5, c = 0
6. 3x + 2 = 0
3x + 0y + 2 = 0
a = 3, b = 0, c = 2
7. y – 2 = 0
0x + y – 2 = 0
a = 0, b = 1, c = – 2
8. 5 = 2x
– 2x + 0y + 5 = 0
a = – 2, b = 0, c = 5
Solution of a Linear Equation:
Consider a Linear equation x + 2y = 6
Let x = 2 and y = 2.
Then L.H.S. of the equation = x + 2y = 2 + 2 x 2 = 6
and R.H.S. of the equation = 6 (given)
i.e., LHS. = R.H.S. for x = 2 and y = 2.
Therefore, x = 2 and y = 2 i.e., (2, 2) is the solution of the given equation x + 2y = 6.
Any pair of values of x and y which satisfies the given equation is called a solution of the equation. A linear equation in two variables has infinitely many solutions.
Note:
To find the solution of an equation, assure a value of one of the variable and calculate the value of second variable from the given equation. | 2022-05-23 05:42:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45851072669029236, "perplexity": 203.42330129246682}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662555558.23/warc/CC-MAIN-20220523041156-20220523071156-00266.warc.gz"} |
https://eprint.iacr.org/2002/020 | ### Cryptanalysis of stream ciphers with linear masking
Don Coppersmith, Shai Halevi, and Charanjit Jutla
##### Abstract
We describe a cryptanalytical technique for distinguishing some stream ciphers from a truly random process. Roughly, the ciphers to which this method applies consist of a non-linear process'' (say, akin to a round function in block ciphers), and a linear process'' such as an LFSR (or even fixed tables). The output of the cipher can be the linear sum of both processes. To attack such ciphers, we look for any property of the non-linear process'' that can be distinguished from random. In addition, we look for a linear combination of the linear process that vanishes. We then consider the same linear combination applied to the cipher's output, and try to find traces of the distinguishing property. In this report we analyze two specific distinguishing properties''. One is a linear approximation of the non-linear process, which we demonstrate on the stream cipher SNOW. This attack needs roughly $2^{95}$ words of output, with work-load of about $2^{100}$. The other is a low-diffusion'' attack, that we apply to the cipher Scream-0. The latter attack needs only about $2^{43}$ bytes of output, using roughly $2^{50}$ space and $2^{80}$ time.
Available format(s)
Category
Secret-key cryptography
Publication info
Published elsewhere. extended abstract appears in Crypto'02
Keywords
Hypothesis testingLinear cryptanalysisLinear maskingLow-Diffusion attacksStream ciphers
Contact author(s)
shaih @ watson ibm com
History
2002-06-05: last of 2 revisions
See all versions
Short URL
https://ia.cr/2002/020
CC BY
BibTeX
@misc{cryptoeprint:2002/020,
author = {Don Coppersmith and Shai Halevi and Charanjit Jutla},
title = {Cryptanalysis of stream ciphers with linear masking},
howpublished = {Cryptology ePrint Archive, Paper 2002/020},
year = {2002},
note = {\url{https://eprint.iacr.org/2002/020}},
url = {https://eprint.iacr.org/2002/020}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content. | 2022-06-26 01:22:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21666453778743744, "perplexity": 4029.47146318618}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103036363.5/warc/CC-MAIN-20220626010644-20220626040644-00667.warc.gz"} |