url stringlengths 14 2.42k | text stringlengths 100 1.02M | date stringlengths 19 19 | metadata stringlengths 1.06k 1.1k |
|---|---|---|---|
https://e-edukasyonph.com/math/explain-in-a-direct-proportion-as-o-2383047 | , 28.10.2019 15:29 kuanjunjunkuan
# Explainin a direct proportion, as one quantity increases, the other quantityincreases at the same rate and vice versa.can you cite an example of real-life situation that involves directproportion? in an inverse proportion, one quantity increases as the other quan-tity decreases at the same rate, and vice-versa.speed varies inversely with time of travel because the faster wego, the shorter the time of travel.
Answers: 3
### Another question on Math
Math, 28.10.2019 15:28
$$\frac{9}{5} = \frac{a - 3}{2}$$solving linear equationfind the "x"please put the thank : )
Answers: 1
Math, 28.10.2019 16:28
What is the solution and answer in 16÷59?
Answers: 2
Math, 28.10.2019 17:29
Translate each following sentence using mathematical symbol5 is an integery is a multiple of 10a belongs to both sets x and ythe values of y range feom -4 to 5the square of the difference of x and y is not more than 10the square of a number is positive 8 is an even number7 is an odd number1/4 is a rational number fo the answer. its a big
Answers: 3
Math, 28.10.2019 18:29
1. 36x7)x10= pls i am really struggling with this
Answers: 1
You know the right answer?
Explain
in a direct proportion, as one quantity increases, the other quantity
increases at...
Questions
Technology and Home Economics, 25.03.2021 11:15
Araling Panlipunan, 25.03.2021 11:15
Math, 25.03.2021 11:15
Questions on the website: 14018453 | 2021-06-15 06:04:43 | {"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": 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": 0.6249500513076782, "perplexity": 5716.148263029753}, "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-2021-25/segments/1623487617599.15/warc/CC-MAIN-20210615053457-20210615083457-00239.warc.gz"} |
http://mathoverflow.net/questions/126890/on-the-oscillation-of-the-summatory-totient-about-its-average/126902 | # On the oscillation of the summatory totient about its average
Let $$R(x)=\sum_{n\leq x}\phi(n)-\frac{3x^2}{\pi^2}.$$ Montgomery has shown that $R(x)=\Omega_{\pm}(x\sqrt{\log\log x})$, which is the best known lower bound. It seems interesting therefore that $$\int_0^{\infty}\frac{R(x)dx}{x^2}=0,$$ because it tells us that the oscillations (which continue indefinitely) are particularly regular.
I cannot find any references for this integral, so I am wondering if it is known. I would particularly like to find other work of this nature as I cannot prove anything about the rate of convergence of the improper integral (other than $o(1)$ as $X\rightarrow\infty$ where $X$ is the upper limit of integration).
-
If I may ask, how do you prove this? – quid Apr 8 '13 at 20:10
It is quite lengthy, but the essence is that the integral over a finite interval may be written in terms of a uniformly convergent (for $X>1$) sum over the zeros of $\zeta(s)$. The necessary estimates to justify the limit of the contour are available. The uniform convergence and zero free region enables you to arrive at a contradiction supposing the limit as $X\rightarrow\infty$ is not $0$. More can be probably be said- it appears that the Mellin transform converges on the line $\sigma=1$. – Kevin Smith Apr 8 '13 at 20:34
The Mellin transform of $R(x)$, that is. – Kevin Smith Apr 8 '13 at 20:39
Thank you for the explanation! – quid Apr 8 '13 at 23:28
I am not sure if this result is explicitly mentioned in the literature, but it certainly is classical.
Let $$R(x) = \sum_{n \leq x}{\varphi(n)} - \frac{3x^2}{\pi^2}, \qquad H(x) = \sum_{n \leq x}{\frac{\varphi(n)}{n}} - \frac{6x}{\pi^2}.$$
Then by partial summation, $$\int^{x}_{0}{\frac{R(t)}{t^2} \: dt} = H(x) - \frac{R(x)}{x}.$$ A classical result of Chowla states that $$H(x) - \frac{R(x)}{x} = O\left((\log x)^{-4}\right).$$ See Lemma 13 of S. Chowla, "Contributions to the analytic theory of numbers", Mathematische Zeitschrift 35:1 (1932), 279-299. (If you have access to Springer Link then it is available here.)
From a cursory glance of Chowla's proof, the negative powers of a logarithm stem from the prime number theorem applied to the summatory function of the Möbius function, so it is likely that this bound could be improved with more modern estimates for this.
For what it's worth, I answered a question closely related to this here.
-
Do you mean $R(t)$ rather than $E(t)$ in the integral? – Barry Cipra Apr 8 '13 at 21:03
Yep, thanks. All fixed now. – Peter Humphries Apr 8 '13 at 21:09
Marvellous. I knew the partial summation but not the estimate. I think it is sufficient for my purposes. Thank you. – Kevin Smith Apr 8 '13 at 21:12 | 2015-08-30 16:25:42 | {"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": 0.9502593278884888, "perplexity": 259.75863823530136}, "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-2015-35/segments/1440644065324.41/warc/CC-MAIN-20150827025425-00083-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/3rd-derivative-name.80737/ | # 3rd derivative name
#### bomba923
Although my notation is likely incorrect,
Momentum =
$$m\frac{{dx}}{{dt}} = m \cdot v$$
Force =
$$m\frac{{d^2 x}}{{dt^2 }} = m\frac{{dv}}{{dt}} = m \cdot a$$
Then,
$$m\frac{{d^3 x}}{{dt^3 }} = m\frac{{d^2 v}}{{dt^2 }} = m\frac{{da}}{{dt}}$$
But what would/do you call the $$m\frac{{da}}{{dt}}$$ ?
Last edited:
Related Other Physics Topics News on Phys.org
#### rick1138
Don't know the answer in this context, but the derivative of acceleration is called jerk.
Homework Helper
Gold Member
#### bomba923
Well, my question is what is the name of mass*(da/dt); exactly what physical quantity does it represent (is it a useful physical quantity)?
From that, i think, i can give it a name---but if it already has one,
*What would/do we call the product represented by mass*(da/dt) ?
#### Claude Bile
It is just the first time derivative of Force $\frac{dF}{dt}$ (which is also the second time derivative of momentum), which is what the link robphy supplied in his post refers to.
It should be pointed out that using the word 'Yank' to represent this quantity is by no means official, it is more of a tongue in cheek proposition.
Claude.
#### bomba923
Ahh--that's right !
Yank =
$$m\frac{{da}}{{dt}}$$
#### AikenDrum
I can only say that its the quantity of "Yank" that makes you sick in a rollercoaster, because uniform acceleration doesn't disturb our senses very much...
### Physics Forums Values
We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving | 2019-10-17 00:09:49 | {"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": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4594613015651703, "perplexity": 4226.3376111631715}, "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/1570986672431.45/warc/CC-MAIN-20191016235542-20191017023042-00257.warc.gz"} |
https://math.stackexchange.com/questions/1061594/signing-y-from-log-fracxyx-xy | # Signing $y''$ from $\log(\frac{x+y}{x})=x+y$
Suppose that $x,y>0$ are positive reals such that $y$ is defined implicitly in terms of $x$ via: $$\log\left(\frac{x+y}{x}\right)=x+y.\tag{\star}$$ I would like study the sign of $y''$.
Attempt: Write ($\star$) as $$\log(x+y)-\log(x)=x+y.$$ Differentiate both sides w.r.t. $x$ yields $$\frac{1+y'}{x+y}-\frac{1}{x}=1+y'\tag{\star\star}$$ which can be solved to get $$1+y'=\frac{x+y}{x(1-x-y)}\cdot$$ Differentiate both sides of ($\star\star$) w.r.t. $x$ to get $$\frac{(x+y)y''-(1+y')^2}{(x+y)^2}+\frac{1}{x^2}=y''$$ which, after feeding to Mathematica while using $1+y'$ found above, gives $$y''=\frac{(x+y-2) (x+y)^2}{x^2 (x+y-1)^3}$$ which can clearly take on positive and negative values depending on $x+y$. Indeed, looking back at ($\star$), we can freely vary $x+y$: to have $x+y=r>0$, simply set $$x=e^{-r}r,\quad y=(1-e^{-r})r.$$ Is my attempt here reasonable to you? The reason I'm not confident is that if I feed ($\star$) directly to Mathematica, I get $$y=-x-\text{ProductLog}[-x]$$ where (according to Help File) $\text{ProductLog}[z]$ gives the principal solution for $w$ in $z=we^w$. Then I plotted $$\partial_x(\partial_x(-x-\text{ProductLog}[-x]))$$ and saw something that is only positive:
What is going on? Can someone please explain this seeming discrepancy?
• One thing you can do is define $z(x)=y(x)+x$; then $\frac{d^2z}{dx^2}=\frac{d^2y}{dx^2}$ and the calculations are a bit simpler. – Steven Stadnicki Dec 12 '14 at 16:09
The problem is that $y$ isn't globally uniquely determined, i.e. there are multiple functions $y_i(x)$ that verify $(\star)$. In $\mathbb C$ these are infinite, while in $\mathbb R$ there are two, and you have, via Mathematica, found just one of them. The other solution can be seen here to vary in sign as you predicted (to know that this is a solution, see this. If you want a single function, you'll have to work with a local definition of $y(x)$ in neighbourhoods of concrete solutions $(x_0,y_0)$.
If the global vs. local aspect seems confusing, take the unit circumference as an example: there is no function $y(x)$ that traverses the entire curve of solutions to $x^2+y^2 = 1$, but you can define, for every solution $(x_0,y_0)$, a function $y(x)$ that passes through $(x_0,y_0)$ and such that every $(x,y(x))$ is a solution. | 2019-10-19 07:17:54 | {"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": 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": 0.9279865622520447, "perplexity": 177.12584601843386}, "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/1570986692126.27/warc/CC-MAIN-20191019063516-20191019091016-00307.warc.gz"} |
https://iim-cat-questions-answers.2iim.com/quant/arithmetic/ratio-mixtures-averages/ratio-mixtures-averages_37.shtml | CAT Practice : Averages, Ratios, Mixtures
When we mix two mixtures in a particular ratio, we get a third mixture. Given the third mixture how does one find the ratio in which they were mixed.
Alligation
Q.37: What would be the ratio of milk and water in a final mixture formed by mixing milk and water that are present in three vessels of capacity 1l, 2l, and 3l respectively and in the ratios 5:1, 3:2 and 4:3 respectively?
1. 747:443
2. 787:1260
3. 787:473
4. 747:473
Choice C. 787:473
Detailed Solution
Solve this type of questions by taking 2 at a time. Take the first 2 vessels,
In 1st, fraction of milk = $\frac{5}{6}$
In 2nd, fraction of milk = $\frac{3}{5}$
Therefore,
= ) $\frac{\frac{3}{5} - x}{x - \frac{5}{6}} = \frac{1}{2}$
= ) $\frac{6}{5} - 2x = x - \frac{5}{6}$
= ) $\frac{6}{5} + \frac{5}{6} = 3x$
= )$x = \frac{61}{90}$ (And the volume of mixture after mixing = 1 + 2 = 3 l)
Therefore,
= ) $\frac{\frac{4}{7} - x}{x - \frac{61}{90}} = \frac{3}{3}$
= ) $\frac{4}{7} - x = x + \frac{61}{90}$
= ) $\frac{4}{7} + \frac{61}{90} = 2x$
= ) $2x = \frac{787}{630} => x = \frac{787}{1260}$
Therefore, Milk : Water = $\frac{787}{(1260 - 787)} = \frac{787}{473}$
Our Online Course, Now on Google Playstore!
Fully Functional Course on Mobile
All features of the online course, including the classes, discussion board, quizes and more, on a mobile platform.
Cache Content for Offline Viewing
Download videos onto your mobile so you can learn on the fly, even when the network gets choppy!
More questions from Averages, Ratios, Mixtures
Averaages, Ratios and Mixtures XXXXXXXXXXXXXXXXXXXXXXXXXe. | 2018-09-24 05:47:32 | {"extraction_info": {"found_math": true, "script_math_tex": 11, "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": 0.5749896764755249, "perplexity": 4269.000767763469}, "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-2018-39/segments/1537267160145.76/warc/CC-MAIN-20180924050917-20180924071317-00414.warc.gz"} |
https://socratic.org/questions/what-is-the-antiderivative-of-5x-x-2-1-1 | # What is the antiderivative of (5x)/(x^2+1) ?
Jan 10, 2017
$F \left(x\right) = \frac{5}{2} \ln \left({x}^{2} + 1\right) + C$
#### Explanation:
The primitive of a function can be calculated as its indefinite integral:
$F \left(x\right) = \int \frac{5 x}{{x}^{2} + 1} \mathrm{dx}$
This integral can be calculated easily noting that:
$d \left({x}^{2} + 1\right) = 2 x \mathrm{dx}$ so:
$\int \frac{5 x}{{x}^{2} + 1} \mathrm{dx} = \frac{5}{2} \int \frac{2 x \mathrm{dx}}{{x}^{2} + 1} = \frac{5}{2} \int \frac{d \left({x}^{2} + 1\right)}{{x}^{2} + 1} = \frac{5}{2} \ln \left({x}^{2} + 1\right) + C$ | 2022-05-19 21:15:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "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": 0.9992985725402832, "perplexity": 1385.2578156326938}, "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/1652662530066.45/warc/CC-MAIN-20220519204127-20220519234127-00598.warc.gz"} |
https://apboardsolutions.in/ap-board-6th-class-maths-notes-chapter-6/ | # AP Board 6th Class Maths Notes Chapter 6 Basic Arithmetic
Students can go through AP Board 6th Class Maths Notes Chapter 6 Basic Arithmetic to understand and remember the concepts easily.
## AP State Board Syllabus 6th Class Maths Notes Chapter 6 Basic Arithmetic
→ If a comparison is made by finding the difference between two quantities, it is called comparison by difference.
Eg: Age of Harshita is 11 years and age of Srija is 8 years. Harshita is (11 – 8 = 3) 3 years older than Srija or Srija is 3 years younger than Harshita.
→ If a comparison is made by division it makes more sense than the comparison made by taking the difference.
Eg: If cost a key pad cell phone is Rs. 3000 and another smart phone is Rs. 15000, then the cost of the second phone is five times the cost of the first phone.
→ Ratio: Comparison of two quantities of the same type by virtue of division is called ratio. Eg: The weight of Ramu is 24 kg and the weight of the Gopi is 36 kg., then the ratio of weights is 24/36. It can also be written as 24:36 and read as 24 is to 36.
The ratio of two numbers ‘a’ and ‘b’ (b ≠ 0) is a ÷ b or a/b or $$\frac{a}{b}$$ and is denoted as a : b and is read as a is to b.
In the ratio a : b the quantities a and b are called the terms of the ratio.
In the ratio a : b the quantity a is called the first term or antecedent and b is called the second term or the consequent of the ratio.
The value of a fraction remains the same if the numerator and the denominators are multiplied or divided by the same non-zero number so is the ratio.
That is if the first term and the second term of a ratio are multiplied or divided by . the same non-zero number.
3 : 4 = 3 × 5 : 4 × 5 = 15 : 20
Also 36 : 24 = 36 – 4 : 24 – 4 = 9 : 6.
→ Ratio in the simplest form or in the lowest terms:
A ratio a : b is said to be in its simplest form if its terms have no factors in common other than 1. A ratio in the simplest form is also called the ratio in its lowest terms. Generally ratios are expressed in their lowest terms.
To express a given ratio in its simplest term, we cancel H.C.F. from both its terms. To find the ratio of two terms, we express the both terms in the same units.
Eg: Ratio of 3 hours and 120 minutes is 3 : 2 (as 120 minutes = 2 hours) or 180 : 120 (as 3 hours = 180 minutes)
A ratio has no units or it is independent of units used in the quantities compared. The order of terms in a ratio a : b is important a : b ≠ b : a.
→ Equivalent ratio:
A ratio obtained by multiplying or dividing the antecedent and consequent of a given ratio by the same number is called its equivalent ratio.
Eg: 3 : 4 = 3 × 5 : 4 × 5 = 15 : 20. Here 3 : 4 & 15 : 20 are called equivalent ratios.
Also 36 : 24 = 36 ÷ 4 : 24 ÷ 4 = 9 : 6. Here 36 : 24 & 9 : 6 are called equivalent ratios.
→ Comparison of ratios: To compare two ratios
a) First express them as fractions
b) Now convert them to like fractions
c) Compare the like fractions
→ Proportion:
If two ratios are equal, then the four terms of these ratios are said to be in proportion. If a : b = c : d, then a, b, c and d are said to be in proportion.
This is represented as a : b :: c : d and read as a is b is as c is d.
The equality of ratios is called proportion.
Conversely in the proportion a : b :: c : d , the terms a and d are called extremes and b and c are called means.
If four quantities are in proportion, then
Product of extremes = Product of means .
If a : b :: c : d, then a × d = b × c
From this we have
→ Unitary method:
The method in which first we find the value of one unit and then the value of required number of units is known as unitary method.
Eg: If the cost of 8 books Rs.96, then find the cost of 15 books.
Cost of one book = 96/8 = 12 Cost of 15 books = 12 × 15 = 180
Distance travelled in a given time = speed × time From this we have
→ Percentage:
The word per cent means for every hundred or out of hundred. The word percentage is derived from the Latin language. The % symbol is uses to represent percent.
Eg: 5% is read as five percent
5% = $$\frac{5}{100}$$ = 0.05
38% = $$\frac{38}{100}$$ = 0.38
→ To convert a percentage into a fraction:
a) Drop the % symbol
b) Divide the number by 100
Eg: 25% = $$\frac{25}{100}$$ = 0.25 = $$\frac{1}{4}$$
→ To convert a fraction into percentage:
a) Assign the percentage symbol %
b) Multiply the given fraction with 100
Eg: $$\frac{3}{4}$$ = $$\frac{3}{4}$$ × 100% = 75% = 0.75 | 2022-11-26 09:42:12 | {"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": 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": 0.7051289081573486, "perplexity": 635.4815120768205}, "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/1669446706285.92/warc/CC-MAIN-20221126080725-20221126110725-00325.warc.gz"} |
https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2/share/doc/Macaulay2/Macaulay2Doc/html/_creating_spand_spwriting_spfiles.html | # creating and writing files
We can print to a file in essentially the same way we print to the screen. In the simplest case, we create the entire file with one command; we give the file name as the initial left hand operand of <<, and we close the file with close. Files must be closed before they can be used for something else.
i1 : "testfile" << 2^100 << endl << close o1 = testfile o1 : File i2 : value get "testfile" o2 = 1267650600228229401496703205376
More complicated files may require printing to the file multiple times. One way to handle this is to assign the open file created the first time we use << to a variable, so we can use it for subsequent print operations and for closing the file.
i3 : f = "testfile" << "" o3 = testfile o3 : File i4 : f << "hi" << endl o4 = testfile o4 : File i5 : f << "ho" << endl o5 = testfile o5 : File i6 : f << close o6 = testfile o6 : File i7 : get "testfile" o7 = hi ho i8 : removeFile "testfile" | 2021-06-23 21:51:14 | {"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": 0.43498924374580383, "perplexity": 2934.157101151373}, "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/1623488540235.72/warc/CC-MAIN-20210623195636-20210623225636-00399.warc.gz"} |
http://mathoverflow.net/questions/101611/good-covers-on-complex-algebraic-varieties-with-normal-crossings-singularities | # Good covers on complex algebraic varieties with normal crossings singularities
Let $X$ be a topological space. A good cover on $X$ is an open cover such that all finite non-empty intersections are contractible. It is a theorem of Hironaka that (complex) algebraic sets admit triangulations. As a consequence, all algebraic sets admit a good cover: simply take as an open cover the collection of open stars of the vertices in the triangulation. In case $X$ is an algebraic curve with nodes, there is a way of constructing an open cover geometrically.
Consider the resolution of singularities $\pi:\widetilde{X}\rightarrow X$. This is a compact Riemann surface obtained by blowing up the singular points of $X$. The singular locus $S$ of $X$ is finite and $\pi^{-1}(s)$ consists of two points when $s\in S$. If $E = \pi^{-1}(S)$, then $\pi|_{\widetilde{X}\backslash E}: \widetilde{X}\backslash E\rightarrow X\backslash S$ is a biholomorphism. Note that a node is an example of a normal crossings singularity — one which is locally isomorphic to a union of coordinate hyperplanes. In fact, nodes are the only examples of normal crossings singularities on curves.
Since $\widetilde{X}$ is a smooth (paracompact) manifold, it admits a good cover by differential geometry (choose a metric on $\widetilde{X}$, then cover it by geodesically convex balls). There exists a finite cover $(U_j)_{j\in J}$, consisting of convex balls, for which each point of $E$ is contained in exactly one $U_j$. We can assume that the $U_j$ which intersect $E$ are mutually disjoint. Each $\pi(U_j)$ is homeomorphic to an open disc $\mathbb{D}\subset \mathbb{C}$. However, $\pi(U_j)$ is open in $X$ if and only if it does not contain a singular point. Define a subset $K\subset J$ where $j\in K$ if and only if $\pi(U_j)\cap S = \emptyset$ . To get an open neighborhood of a singular point $s\in S$, define $W_s = \pi(U_{j_1})\cup \pi(U_{j_2})$ where the $U_{j_k}$ each contain exactly one point of $\pi^{-1}(s)$. $W_s$ is contractible because it is homeomorphic to a wedge sum $\mathbb{D}\vee \mathbb{D}$, where the two discs are joined at the origin. Then $(W_s)_{s\in S}\cup (\pi(U_k))_{k\in K}$ is a good cover for $X$.
My question is:
Does this construction generalize to varieties with normal crossings singularities? More precisely, does there exist a good (?) open cover $(U_{\alpha})$ of $\widetilde{X}$ such that finite unions of the $\pi(U_{\alpha})$ can be used to build a good cover for $X$? If so, is there an analogously simple description of the open sets which happen to contain singular points?
- | 2015-09-02 11:02:38 | {"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": 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": 0.8874903321266174, "perplexity": 83.7534956842436}, "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-2015-35/segments/1440645261055.52/warc/CC-MAIN-20150827031421-00330-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://experiment-ufa.ru/8z-4-2z=14 | # 8z-4-2z=14
## Simple and best practice solution for 8z-4-2z=14 equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so dont hesitate to use it as a solution of your homework.
If it's not what You are looking for type in the equation solver your own equation and let us solve it.
## Solution for 8z-4-2z=14 equation:
Simplifying
8z + -4 + -2z = 14
Reorder the terms:
-4 + 8z + -2z = 14
Combine like terms: 8z + -2z = 6z
-4 + 6z = 14
Solving
-4 + 6z = 14
Solving for variable 'z'.
Move all terms containing z to the left, all other terms to the right.
Add '4' to each side of the equation.
-4 + 4 + 6z = 14 + 4
Combine like terms: -4 + 4 = 0
0 + 6z = 14 + 4
6z = 14 + 4
Combine like terms: 14 + 4 = 18
6z = 18
Divide each side by '6'.
z = 3
Simplifying
z = 3`
## Related pages
solution to the system of equations calculatorsin 3x derivative2.2kg lbmaths equations calculator3x2 system of equations solverwhat is 0.125 in a fractionx ln x derivative3x3 equation solver81 prime factorization3x 4ygraph 2xdifferentiate e cos x2002 roman numeralsx squared 7xprime factorization of 294what is prime factorization of 52sen 4xfactoring x 32x 5y 6graph 2x y 4taxi tip calculator6x 2x 1sqrt 1000adding three fractions calculatorax3 bx2 cx dsimultaneous equations calculator with stepsfraction decimal percent calculator7.5b1942 roman numerals3y x 6tanx dxpvrnt100 000 dollars in poundsmath derivative solverwhat is the greatest common factor of 12017x29x 2-161.2.3.4what is the prime factorization for 92greatest prime factor of 1000factorization of 147sin2pi2 8thssolution of equation calculatorroman numerals 1000000keiseruniversity blackboard com6x 2x 1to 2j-1-18solve 2x y 3what is the lcm of 4how to solve pv nrt635-100what is 4x95x 7 2xthe coefficient of 6x is6x 2x 1500000 naira in pounds6u8derivative of sin2t1988 roman numerals767-600what is 50000 pounds in dollars18kg in lbs3x 5y 30prime factors of 539calculator for multiplying fractionsroman numerals for 1968737-1000what is 84 in roman numeralswebwork ttu6x 5y3x 2y 2 graphlcm of 150prt equation | 2018-06-22 10:49:24 | {"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": 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": 0.42330479621887207, "perplexity": 11019.57501074305}, "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-2018-26/segments/1529267864391.61/warc/CC-MAIN-20180622104200-20180622124200-00163.warc.gz"} |
https://triangle.mth.kcl.ac.uk/?search=location:City%20U | Notice: Undefined index: location:City U in /var/www/html/triangle.mth.kcl.ac.uk/html/golden/index.php on line 377
Notice: Undefined index: location:City U in /var/www/html/triangle.mth.kcl.ac.uk/html/golden/index.php on line 377
## Seminars at
Found at least 20 result(s)
### 02.04.2019 (Tuesday)
#### TBA
Regular Seminar Seung-Joo Lee (CERN)
at: 15:00 City U.room C309 abstract:
### 25.03.2019 (Monday)
#### TBA
Regular Seminar Sat Gupta (UNC)
at: 15:00 City U.room C318 abstract:
### 19.03.2019 (Tuesday)
#### TBA
Regular Seminar Misha Portnoi (Exeter)
at: 15:00 City U.room AG22 abstract:
### 12.03.2019 (Tuesday)
#### TBA
at: 15:00 City U.room C320 abstract:
### 05.03.2019 (Tuesday)
#### TBA
Regular Seminar Weini Huang (QMUL)
at: 15:00 City U.room C309 abstract:
### 19.02.2019 (Tuesday)
#### From Yang-Mills and Maxwell in de Sitter space electromagnetic knots
Regular Seminar Olaf Lechtenfeld (Hannover)
at: 15:00 City U.room AG21 abstract: I will review analytic SU(2) Yang-Mills solutions with finite action on de Sitter space from a new perspective. As a byproduct, all abelian solutions are classified and related with rational electromagnetic knots. In the Yang-Mills case, the gravitational backreaction is easily taken in to account as well.
### 12.02.2019 (Tuesday)
#### TBA
at: 15:00 City U.room C305 abstract:
### 06.02.2019 (Wednesday)
#### TBA
Triangular Seminar Eliezer Rabinovici (HUJ)
at: 15:00 City U.room BG03 abstract:
### 06.02.2019 (Wednesday)
#### A Worldsheet Dual for the Symmetric Orbifold
Triangular Seminar Rajesh Gopakumar (ICTS-TIFR)
at: 16:30 City U.room BG03 abstract: We will argue that superstring theory on ${\rm AdS}_3\times {\rm S}^3\times \mathbb{T}^4$ with the smallest amount of NS-NS flux ($k=1$'') is dual to the spacetime CFT given by the large $N$ limit of the free symmetric product orbifold $\mathrm{Sym}^N(\mathbb{T}^4)$. The worldsheet theory, at $k=1$, is defined using the hybrid formalism in which the ${\rm AdS}_3\times {\rm S}^3$ part is described by a $\mathfrak{psu}(1,1|2)_1$ WZW model (which is well defined). Unlike the case for $k\geq 2$, it turns out that the string spectrum at $k=1$ does not exhibit a long string continuum, and perfectly matches with the large $N$ limit of the symmetric product. The fusion rules of the symmetric orbifold are also reproduced from the worldsheet perspective. This proposal therefore affords a tractable worldsheet description of a tensionless limit in string theory.
### 05.02.2019 (Tuesday)
#### Mode interactions in complex and disordered patterns
Regular Seminar Alastair Rucklidge (Leeds)
at: 15:00 City U.room BLG07 abstract: Why do some systems organise themselves into well ordered patterns with astonishing symmetry and regularity, while other superficially similar systems produce defects and disorder? In systems where two different length scales are unstable, the nonlinear interaction between the different modes is key: steady complex patterns can be stabilised when the modes act together to reinforce each other. But, if the two types of pattern compete with each other, the outcome can be considerably more complicated: a time-dependent disordered mixture of patterns constantly shifting and changing. In a small domain, the nature of the interaction between a small number of modes on each length scale can readily be computed. In a large domain, each mode can interact with hundreds of other modes, but the overall behaviour still appears to be guided by small-domain considerations.
### 29.01.2019 (Tuesday)
#### Q-operators for rational spin chains
Regular Seminar Rouven Frassek (MPIM Bonn)
at: 15:00 City U.room C320 abstract: I plan to discuss how Q-operators for rational spin chains can be constructed in the framework of the quantum inverse scattering method. The presentation will include open and closed XXX type spin chains with compact and non-compact representations of sl(2) in the quantum space. Further I plan to elaborate on the generalisation to spin chains of higher rank and in particular u(2,2|4) which underlies N=4 super Yang-Mills theory at weak coupling. Finally I will discuss the classification of the oscillator type solutions to the Yang-Baxter equation that are relevant to build Q-operators and give an outlook.
### 22.01.2019 (Tuesday)
#### Graphene and Boundary Conformal Field Theory
Regular Seminar Chris Herzog (King's)
at: 15:00 City U.room C320 abstract: The infrared fixed point of graphene under the renormalization group flow is a relatively under studied yet important example of a boundary conformal field theory with a number of remarkable properties. It has a close relationship with three dimensional QED. It maps to itself under electric-magnetic duality. Moreover, it along with its supersymmetric cousins all possess an exactly marginal coupling -- the charge of the electron -- which allows for straightforward perturbative calculations in the weak coupling limit. I will review past work on this model and also discuss my own contributions, which focus on understanding the boundary contributions to the anomalous trace of the stress tensor and their role in helping to understand the structure of boundary conformal field theory.
### 03.04.2018 (Tuesday)
#### tba
Regular Seminar Luca Tagliacozzo (Strathclyde)
at: 15:00 City U.room C320 abstract:
### 27.03.2018 (Tuesday)
#### CANCELLED
at: 15:00 City U.room C320 abstract:
### 20.03.2018 (Tuesday)
#### tba
Regular Seminar Nuno Freitas (Warwick and KCL)
at: 15:00 City U.room C320 abstract:
### 13.03.2018 (Tuesday)
#### tba
Regular Seminar Attila Csikasz-Nagy (KCL)
at: 15:00 City U.room B103 abstract:
### 27.02.2018 (Tuesday)
#### Topological strings and 6d SCFTs
Regular Seminar Amir-Kian Kashani-Poor (ENS)
at: 15:00 City U.room B103 abstract: The topological string is a simplified version of physical string theory. It is of interest because it computes the BPS spectrum of relevant string theory compactifications, but also because it shares structural properties of physical string theory, Dualities and symmetries which often must be argued for arduously in the physical string can often be verified by computation in the topological setting. The central observable of the theory is the topological string partition function Z_top. This quantity has an eerie habit of making surprise appearances in many areas of mathematical physics. Numerous techniques exist for its computation in various expansions in parameters of the theory, yet to date, no satisfactory closed form for this quantity is known. In this talk, after reviewing notions of topological string theory with an emphasis on the interplay between worldsheet and target space physics (one of the structural similarities between the physical and the topological string alluded to above), I will report on progress in computing Z_top in settings where it is related to enigmatic 6d theories.
### 20.02.2018 (Tuesday)
#### A review of Double and Exceptional Field Theory
Regular Seminar David Berman (QMUL)
at: 15:00 City U.room C312 abstract: Recently a new formulation for supergravity has emerged inspired by the presence of duality symmetries in reduced theories. These new theories generalise ideas of Riemannian geometry and lead to new ways of looking at string and M-theory.
### 13.02.2018 (Tuesday)
#### Emergent hydrodynamics in integrable systems out of equilibrium
Regular Seminar Benjamin Doyon (King's)
at: 15:00 City U.room B103 abstract: The hydrodynamic approximation is an extremely powerful tool to describe the behavior of many-body systems such as gases. At the Euler scale (that is, when variations of densities and currents occur only on large space-time scales), the approximation is based on the idea of local thermodynamic equilibrium: locally, within fluid cells, the system is in a Galilean or relativistic boost of a Gibbs equilibrium state. This is expected to arise in conventional gases thanks to ergodicity and Gibbs thermalization, which in the quantum case is embodied by the eigenstate thermalization hypothesis. However, integrable systems are well known not to thermalize in the standard fashion. The presence of infinitely-many conservation laws preclude Gibbs thermalization, and instead generalized Gibbs ensembles emerge. In this talk I will introduce the associated theory of generalized hydrodynamics (GHD), which applies the hydrodynamic ideas to systems with infinitely-many conservation laws. It describes the dynamics from inhomogeneous states and in inhomogeneous force fields, and is valid both for quantum systems such as experimentally realized one-dimensional interacting Bose gases and quantum Heisenberg chains, and classical ones such as soliton gases and classical field theory. I will give an overview of what GHD is, how its main equations are derived and its relation to quantum and classical integrable systems. If time permits I will touch on the geometry that lies at its core, how it reproduces the effects seen in the famous quantum Newton cradle experiment, and how it leads to exact results in transport problems such as Drude weights and non-equilibrium currents. This is based on various collaborations with Alvise Bastianello, Olalla Castro Alvaredo, Jean-Sébastien Caux, Jérôme Dubail, Robert Konik, Herbert Spohn, Gerard Watts and my student Takato Yoshimura, and strongly inspired by previous collaborations with Denis Bernard, M. Joe Bhaseen, Andrew Lucas and Koenraad Schalm.
### 07.02.2018 (Wednesday)
#### Triangle Seminar: Exploring EPR=ER with LLM
Triangular Seminar Joan Simon (Edinburgh)
at: 16:00 City U.room A130 abstract: The extremal limit of single R-charged AdS5 black holes in type IIB is known to be described by a system of N free fermions in a one dimensional harmonic oscillator potential. Since the quantum mechanical problem is solvable and its phase space formulation appears in the gravity dual (LLM geometries), it allows us to explore the relation between entanglement, quantum correlation design and connectivity in space in this set-up, both in a single and a two boundary situation. | 2019-02-18 19:53:29 | {"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": 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": 0.445010781288147, "perplexity": 3022.8477644611735}, "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-2019-09/segments/1550247487624.32/warc/CC-MAIN-20190218175932-20190218201932-00610.warc.gz"} |
https://eduzip.com/ask/question/the-current-in-a-conductor-and-the-potential-difference-across-it-274189 | Physics
# The current in a conductor and the potential difference across its ends are measured by an ammeter and a voltmeter. The meters draw negligible currents. The ammeter is accurate but the voltmeter has a zero error (that is, it does not read zero when no potential difference is applied). Calculate the zero error if the readings for two different conditions are $1.75\ A, 14.4\ V$ and $2.75\ A, 22.4\ V.$
##### SOLUTION
Let the voltmeter reading when, the voltage is $0$ be $V$.
$\dfrac{l_1R}{l_2R} = \dfrac{V_1}{V_2}$
$\Rightarrow \dfrac{1.75}{2.75} = \dfrac{14.4-V}{22.4-V} \Rightarrow \dfrac{0.35}{0.55} = \dfrac{14.4-V}{22.4 - V}$
$\Rightarrow \dfrac{0.07}{0.11} = \dfrac{14.4-V}{22.4-V}\Rightarrow \dfrac{7}{11} = \dfrac{14.4-V}{22.4 - V}$
$\Rightarrow 7(22.4 - V) = 11(14.4 - V) \Rightarrow 156.8 - 7V = 158.4 - 11V$
$\Rightarrow (7 - 11)V = 156.8 - 158.4 \Rightarrow -4V = -1.6$
$\Rightarrow V = 0.4V$
You're just one step away
Subjective Medium Published on 18th 08, 2020
Questions 244531
Subjects 8
Chapters 125
Enrolled Students 199
#### Realted Questions
Q1 Single Correct Medium
Which of the following have the same dimensions?
• A. impulse and momentum
• B. specific heat and latent heat
• C. moment of inertia and force
• D. thrust and surface tension
Asked in: Physics - Units and Measurement
1 Verified Answer | Published on 18th 08, 2020
Q2 Single Correct Medium
If the units of mass, length and time are doubled, then what happen to the unit of 'relative density'?
• A. doubled
• B. remains same
• C. tripled
• D. equatrepled
Asked in: Physics - Units and Measurement
1 Verified Answer | Published on 18th 08, 2020
Q3 Subjective Medium
Find the percentage error in kinetic energy of a body of mass $m=50.0 \pm 0.5g$ and moving with a velocity of $v= 10.0\pm 0.1 cm/s$
Asked in: Physics - Units and Measurement
1 Verified Answer | Published on 18th 08, 2020
Q4 Single Correct Medium
Pressure depends on distance as, $P=\dfrac {\alpha}{\beta}exp\left (-\dfrac {\alpha z}{k\theta}\right )$, where $\alpha, \beta$ are constants, $z$ is distance, $k$ is Boltzman's constant and $\theta$ is temperature. The dimensions of $\beta$ are
• A. $[M^0L^0T^0]$
• B. $[M^{-1}L^{-1}T{-1}]$
• C. $[M^0L^2T^0]$
• D. $[M^{-1}L^1T^2]$
Asked in: Physics - Units and Measurement
1 Verified Answer | Published on 18th 08, 2020
Q5 Subjective Medium
The distance of a galaxy is $56 \times10^{25}m$. Assume the speed of light to be $3\times10^8 m s^{-1}$. Express order of magnitude of time taken by light travelled to the galaxy.
Asked in: Physics - Units and Measurement
1 Verified Answer | Published on 18th 08, 2020 | 2021-11-30 13:00:45 | {"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": 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": 0.6348621845245361, "perplexity": 3552.23162023758}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "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-2021-49/segments/1637964358973.70/warc/CC-MAIN-20211130110936-20211130140936-00462.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition/chapter-2-graphs-and-functions-summary-exercises-on-graphs-circles-functions-and-equations-exercises-page-248/11 | Precalculus (6th Edition)
$(x-2)^2+(y+1)^2=9$ Refer to the graph below.
RECALL: A circle whose equation is of the form $(x-h)^2+(y-k)^2=r^2$ has its center at $(h, k)$ and has a radius of $r$. The given circle has its center $(2, -1)$ and a radius of $3$ units. Substituting these values into the equation above gives: $(x-2)^2+(y-(-1))^2=3^2 \\(x-2)^2+(y+1)^2=9$ Refer to the graph in the answer part above. | 2018-07-22 10:29:35 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8611499667167664, "perplexity": 114.32457802350423}, "config": {"markdown_headings": false, "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-2018-30/segments/1531676593208.44/warc/CC-MAIN-20180722100513-20180722120513-00281.warc.gz"} |
https://socratic.org/questions/566932b47c01494ac22e269a | # How is the limiting reagent assessed in chemical reactions?
Dec 10, 2015
For the reaction, $A + B \rightarrow C$, clearly there must be an equal number of atoms (or molecules) of $A$ and $B$; if these undergo complete reaction, an equal number of $C$ particles result.
If there are not equal numbers of $A$ and $B$, then ONE of these reagents will be in excess, and ONE will be in deficiency. Since the stoichiometry of the reaction demands 1:1 equivalence, the reagent in excess is along for the ride and will not undergo reaction.
This leads to the calculation of the number of moles of $A$ and $B$. The mass of an atom or molecule corresponds to the NUMBER of those molecules. That is why chemists speak of equivalent masses. The masses ARE different, but these different masses will contains the SAME number of particles. (SAYS he for some reason capitalizing the words HE thinks ARE important!). | 2019-12-05 19:31:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 8, "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": 0.6189911365509033, "perplexity": 489.00269044636843}, "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-2019-51/segments/1575540482038.36/warc/CC-MAIN-20191205190939-20191205214939-00357.warc.gz"} |
https://www.clutchprep.com/chemistry/practice-problems/133787/which-of-the-molecules-in-the-choices-given-is-not-isoelectronic-with-the-others-1 | Lewis Dot Structures: Ions Video Lessons
Concept
# Problem: Which of the molecules in the choices given is not isoelectronic with the others? The choices are: SO3, NF3, NO3–, CO32–?a. SO3b. NF3c. NO3–d. CO32-e. all of the above are isoelectronic molecules
###### FREE Expert Solution
Determine the group of each atom present in every formula and calculate for the total # of valence electrons to find which is not isoelectronic
Isoelectronic compounds mean that they have the same number of valence electrons.
Recall that the # of valence electron is determined by the Group # of each atom.
Analyzing each compound:
79% (336 ratings)
###### Problem Details
Which of the molecules in the choices given is not isoelectronic with the others? The choices are: SO3, NF3, NO3, CO32–?
a. SO3
b. NF3
c. NO3
d. CO32-
e. all of the above are isoelectronic molecules | 2021-12-08 09:45:50 | {"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": 0.8853760957717896, "perplexity": 9429.398557864892}, "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-2021-49/segments/1637964363465.47/warc/CC-MAIN-20211208083545-20211208113545-00617.warc.gz"} |
https://www.gradesaver.com/textbooks/math/prealgebra/prealgebra-7th-edition/chapter-6-section-6-2-proportions-exercise-set-page-426/9 | ## Prealgebra (7th Edition)
$\frac{22}{1}=\frac{55}{2.5}$
x is to y is the same as $\frac{x}{y}$ So 22 vanilla wafers is to 1 cup of cookie crumbs as 55 vanilla wafers is to 2.5 cups of cookie crumbs is the same as $\frac{22}{1}=\frac{55}{2.5}$ | 2018-04-27 03:04:40 | {"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": 0.6226804852485657, "perplexity": 4349.928497714383}, "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-2018-17/segments/1524125948950.83/warc/CC-MAIN-20180427021556-20180427041556-00080.warc.gz"} |
https://martin.kleppmann.com/2020/12/02/bloom-filter-hash-graph-sync.html | # Using Bloom filters to efficiently synchronise hash graphs
This blog post uses MathJax to render mathematics. You need JavaScript enabled for MathJax to work.
In some recent research, Heidi and I needed to solve the following problem. Say you want to sync a hash graph, such as a Git repository, between two nodes. In Git, each commit is identified by its hash, and a commit may include the hashes of predecessor commits (a commit may include more than one hash if it’s a merge commit). We want to figure out the minimal set of commits that the two nodes need to send to each other in order to make their graphs the same.
You might wonder: isn’t this a solved problem? Git has to do this every time you do git pull or git push! You’re right, and some cases are easy, but other cases are a bit trickier. What’s more, the algorithm used by Git is not particularly well-documented, and in any case we think that we can do better.
For example, say we have two nodes, and each has one of the following two hash graphs (circles are commits, arrows indicate one commit referencing the hash of another). The blue part (commit A and those to the left of it) is shared between the two graphs, while the dark grey and light grey parts exist in only one of the two graphs.
We want to reconcile the two nodes’ states so that one node sends all of the dark-grey-coloured commits, the other sends all of the light-grey-coloured commits, and both end up with the following graph:
How do we efficiently figure out which commits the two nodes need to send to each other?
## Traversing the graph
First, some terminology. Let’s say commit A is a predecessor of commit B if B references the hash of A, or if there is some chain of hash references from B leading to A. If A is a predecessor of B, then B is a successor of A. Finally, define the heads of the graph to be those commits that have no successors. In the example above, the heads are B, C, and D. (This is slightly different from how Git defines HEAD.)
The reconciliation algorithm is easy if it’s a “fast-forward” situation: that is, if one node’s heads are commits that the other node already has. In that case, one node sends the other the hashes of its heads, and the other node replies with all commits that are successors of the first node’s heads. However, the situation is tricker in the example above, where one node’s heads B and C are unknown to the other node, and likewise head D is unknown to the first node.
In order to reconcile the two graphs, we want to figure out which commits are the latest common predecessors of both graphs’ heads (also known as common ancestors, marked A in the example), and then the nodes can send each other all commits that are successors of the common predecessors.
As a first attempt, we can try this: the two nodes send each other their heads; if those contain any unknown predecessor hashes, they request those, and repeat until all hashes resolve to known commits. Thus, the nodes gradually work their way from the heads towards the common predecessors. This works, but it is slow if your graph contains long chains of commits, since the number of round trips required equals the length of the longest path from a head to a common predecessor.
The “smart” transfer protocol used by Git essentially works like this, except that it sends 32 hashes at a time in order to reduce the number of round trips. Why 32? Who knows. It’s a trade-off: send more hashes to reduce the number of round trips, but each request/response is bigger. Presumably they decided that 32 was a reasonable compromise between latency and bandwidth.
Recent versions of Git also support an experimental “skipping” algorithm, which can be enabled using the fetch.negotiationAlgorithm config option. Rather than moving forward by a fixed number of predecessors in each round trip, this algorithm allows some commits to be skipped, so that it reaches the common predecessors faster. The skip size grows similarly to the Fibonacci sequence (i.e. exponentially) with each round trip. This reduces the number of round trips to $$O(\log n)$$, but you can end up overshooting the common predecessors, and thus the protocol may end up unnecessarily transmitting commits that the other node already has.
## Bloom filters to the rescue
In our new paper draft, which we are making available on arXiv today, Heidi and I propose a different algorithm for performing this kind of reconciliation. It is quite simple if you know how Bloom filters work.
In addition to sending the hashes of their heads, each node constructs a Bloom filter containing the hashes of the commits that it knows about. In our prototype, we allocate 10 bits (1.25 bytes) per commit. This number can be adjusted, but note that it is a lot more compact than sending the full 16-byte (for SHA-1, used by Git) or 32-byte (for SHA-256, which is more secure) hash for each commit. Moreover, we keep track of the heads from the last time we reconciled our state with a particular node, and then the Bloom filter only needs to include commits that were added since the last reconciliation.
When a node receives such a Bloom filter, it checks its own commit hashes to see whether they appear in the filter. Any commits whose hash does not appear in the Bloom filter, and its successors, can immediately be sent to the other node, since we can be sure that the other node does not know about those commits. For any commits whose hash does appear in the Bloom filter, it is likely that the other node knows about that commit, but due to false positives it is possible that the other node actually does not know about those commits.
After receiving all the commits that did not appear in the Bloom filter, we check whether we know all of their predecessor hashes. If any are missing, we request them in a separate round trip using the same graph traversal algrorithm as before. Due to the way the false positive probabilities work, the probability of requiring n round trips decreases exponentially as n grows. For example, you might have a 1% chance of requiring two round trips, a 0.01% chance of requiring three round trips, a 0.0001% chance of requiring four round trips, and so on. Almost all reconciliations complete in one round trip.
Unlike the skipping algorithm used by Git, our algorithm never unnecessarily sends any commits that the other side already has, and the Bloom filters are very compact, even for large commit histories.
## Practical relevance
In the paper we also prove that this algorithm allows nodes to sync their state even in the presence of arbitrarily many malicious nodes, making it immune to Sybil attacks. We then go on to prove a theorem that shows which types of applications can and cannot be implemented in this Sybil-immune way, without requiring any Sybil countermeasures such as proof-of-work or the centralised control of permissioned blockchains.
All of this is directly relevant for local-first peer-to-peer applications in which apps running on different devices need to sync up their state without necessarily trusting each other or relying on any trusted servers. I assume it’s also relevant for blockchains that use hash graphs, but I don’t know much about them. So, syncing a Git commit history is just one of many possible use cases – I just used it because most developers will be at least roughly familiar with it!
The details of the algorithm and the theorems are in the paper, so I won’t repeat them here. Instead, I will briefly mention a few interesting things that didn’t make it into the paper.
## Why Bloom filters?
One thing you might be wondering: rather than creating a Bloom filters with 10 bits per commit, can we not just truncate the commit hashes to 10 bits and send those instead? That would use the same amount of network bandwidth, and intuitively it may seem like it should be equivalent.
However, that is not the case: Bloom filters perform vastly better than truncated hashes. I will use a small amount of probability theory to explain why.
Say we have a hash graph containing $$n$$ distinct items, and we want to use $$b$$ bits per item (so the total size of the data structure is $$m=bn$$ bits). If we are using truncated hashes, there are $$2^b$$ possible values for each $$b$$-bit hash. Thus, given two independently chosen, uniformly distributed hashes, the probability that they are the same is $$2^{-b}$$.
If we have $$n$$ uniformly distributed hashes, the probability that they are all different from a given $$b$$-bit hash is $$(1-2^{-b})^n$$. The false positive probability is therefore the probability that a given $$b$$-bit hash equals one or more of the $$n$$ hashes:
$P(\text{false positive in truncated hashes}) = 1 - (1 - 2^{-b})^n$
On the other hand, with a Bloom filter, we start out with all $$m$$ bits set to zero, and then for each item, we set $$k$$ bits to one. After one uniformly distributed bit-setting operation, the probability that a given bit is zero is $$1 - 1/m$$. Thus, after $$kn$$ bit-setting operations, the probability that a given bit is still zero is $$(1 - 1/m)^{kn}$$.
A Bloom filter has a false positive when we check $$k$$ bits for some item and they are all one, even though that item was not in the set. The probability of this happening is
$P(\text{false positive in Bloom filter}) = (1 - (1 - 1/m)^{kn})^k$
It’s not obvious from those expressions which of the two is better, so I plotted the false positive probabilities of truncated hashes and Bloom filters for varying numbers of items $$n$$, and with parameters $$b=10$$, $$k=7$$, $$m=bn$$:
For a Bloom filter, as long as we grow the size of the filter proportionally to the number of items (here we have 10 bits per item), the false positive probability remains pretty much constant at about 0.8%. But truncated hashes of the same size behave much worse, and with more than about 1,000 items the false positive probability exceeds 50%.
The reason for this: with 10-bit truncated hashes there are only 1,024 possible hash values, and if we have 1,000 different items, then most of those 1,024 possible values are already taken. With truncated hashes, if we wanted to keep the false positive probability constant, we would have to use more bits per item as the number of items grows, so the total size of the data structure would grow faster than linearly in the number of items.
Viewing it like this, it is quite remarkable that Bloom filters work as well as they do, using only a constant number of bits per item!
## Further details
The Bloom filter false positive formula given above is the one that is commonly quoted, but it’s actually not quite correct. To be precise, it is a lower bound on the exact false positive probability (open access paper).
Out of curiosity I wrote a little Python script that calculates the false positive probability for truncated hashes, Bloom filters using the approximate formula, and Bloom filters using the exact formula. Fortunately, for the parameter values we are interested in, the difference between approximate and exact probability is very small. The gist also contains a Gnuplot script to produce the graph above.
Peter suggested that a Cockoo filter may perform even better than a Bloom filter, but we haven’t looked into that yet. To be honest, the Bloom filter approach already works so well, and it’s so simple, that I’m not sure the added complexity of a more sophisticated data structure would really be worth it.
That’s all for today. Our paper is at arxiv.org/abs/2012.00472. Hope you found this interesting, and please let us know if you end up using the algorithm! | 2021-12-09 12:59:49 | {"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": 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": 0.5754363536834717, "perplexity": 626.5884668741126}, "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-2021-49/segments/1637964364169.99/warc/CC-MAIN-20211209122503-20211209152503-00480.warc.gz"} |
https://themusegarden.wordpress.com/tag/programming/ | # First Steps With Luigi Data Pipelines
Luigi makes managing your data pipelines easy and fun.
Are you a data scientist or engineer that deals with large amounts of data in your daily work? Do you have an unmanageable mess of cron jobs and scripts to run queries and ETL tasks? Does dependency management make your head spin?
Enter Luigi. Named after the famous character, Luigi is a Python utility that allows you to manage and modularize data pipelines in a consistent fashion. Luigi was originally created by Spotify to handle their data processing needs (billions of log messages and terabytes of files per day). They later released into the world as an open source project. Any service that has python bindings (from SQL to MongoDB to Hadoop to Spark to graphing and analysis in Pandas) can be tied together as a Task in Luigi.
To install, simply run pip install luigi and it should install the necessary dependencies. You can also find the latest source at Luigi and build it yourself.
## How does it work?
Building a data pipeline in Luigi is similar to creating a makefile. Actions or steps of your process are contained in Tasks, and each Task has a few simple methods to tell Luigi what to do with it.
• run()
• This tells Luigi what you want the task to do when it is run. This could be something like submitting a Map-Reduce job, querying a database, or running an external script in Python. The main logic of the task goes here. Since Luigi is highly modular, we break up our work into chunks that we can move around, edit, and maintain later. This way, if one part of the process goes down (say, if the database is unreachable), then it won’t lose the progress on other parts of the pipeline and you can pinpoint what exactly failed.
• output()
• This describes the method of output of your Task. It could be writing to a file, HDFS, or simply updating a database.
• requires()
• This function defines the dependencies of your task. Maybe you need to wait for something else to run first. Maybe you need to make sure that a certain dataset is updated before your task runs. The requires() function is what ties your tasks together in Luigi and adds a sense of chronology to the execution.
Let’s start with a simple Task and then we’ll grow it out to be a more complex pipeline.
import luigi
def run(self):
with self.output().open('w') as f:
def output(self):
# tell luigi where to ouptut to
if __name__ == '__main__':
# it will check for the requirements first, then run
https://gist.github.com/nelsonam/3a17d1f540a749dbff89
All right, so this does the very simple task of printing a line to a file. But that’s easy. Let’s add in some dependencies. Run this at the command line by typing python mytask.py. This is a very simple example just to give you the structure. Let’s add some dependencies.
# MyFirstTask() needs to run first
def requires(self):
def run(self):
# here we're going to count the words in the first file
# and output it to a second file
numwords = 0
# once MyFirstTask() has run, this file has the info we need
# it could just as easily be database output in here
for line in f:
words = line.split(None) # this splits on whitespace of any length
numwords += len(words)
# print our results
with open('words.txt', 'w') as out:
out.write(str(numwords)+"\n")
def output(self):
return luigi.LocalTarget('words.txt')
https://gist.github.com/nelsonam/5dc77ffcdfa5556d47a8
As you can see here, we utilize the method to define a dependency. This means that when you run the second method, MySecondTask(), it will check to see if an output exists for MyFirstTask(). If it’s not there, it will run the task under requires() before proceeding. Additionally, if the dependent task fails for any reason, an error email will be sent to the address of your choosing and the workflow will start up where you left off the next time you run it.
## The Central Scheduler
All right, let’s take this online! The Central Scheduler is a Luigi daemon that runs on your server. To set it up, run luigid.
The official docs go into more detailed documentation about configuring the daemon here.
When you go to http://localhost:8082(or whatever the ip of your server is), then you should see Luigi running like this:
Let’s run our tasks and see what happens. Run python mytask.py again while you have luigid running. Make sure you change the luigi.run statement so that it no longer includes ['--local-scheduler']. This will tell it to go to the central scheduler on our server.
After it runs, you should see this:
One of the best features of the central scheduler is that it provides this visualization of running tasks. It shows tasks in queue, tasks running, tasks failed, and where all the dependencies point. Our example is small, but as your workflows grow more complex, this is a very useful tool indeed.
## Extensions
Luigi isn’t just for straight Python, either! Spotify uses Luigi to run thousands of Hadoop jobs per day, and has built in other extensions since then as well.
According to the official Luigi docs, you can now implement tasks that talk to all kinds of data technologies big and small. Just a few things you can do once you get the hang of the basics are submit Hadoop jobs, interface with mySQL, and store data in Redis.
Best of all, Luigi is open source and contributions are welcomed. Go check it out today!
# Pretty Print JSON From The Command Line
JSON can come in all shapes and sizes, and sometimes you want to see it in a structured format that’s easier on the eyes. This is called pretty printing. But how do you accomplish that, especially if you have a really large JSON file? While there are some converter tools online to show json files in a pretty format, they can get very slow and even freeze if your file is too large. Let’s do it locally.
Requirements: Have Python installed, and a terminal environment.
cat file.json | python -m json.tool > prettyfile.json
And that’s all there is to it! Python comes built in with a JSON encoding/decoding library, and you can use it to your advantage to get nice formatted output. Alternatively, if you are receiving JSON from an API or HTTP request, you can pipe your results from a curl call directly into this tool as well.
Hope this helps!
# Chopsticks Game – A Combinatorial Challenge
So I don’t know if anyone else is familiar with this game, but I just remember playing it with friends in middle school and it occurred to me the other day that it would be an interesting game to analyze combinatorially, and perhaps write a game playing algorithm for. This game can be found in more detail here: http://en.wikipedia.org/wiki/Chopsticks_(hand_game)
Players: 2+.
Rules of Play: Players each begin with two “piles” of points, and each pile has 1 point to begin with. We used fingers to represent this, one finger on each hand.
On each turn, a player can choose to do one of two things:
1. Send points from one of the player’s pile to one of the opponents piles. So if Player 1 wanted to send 1 point to Player 2’s left pile, then Player 2 would have 2 points in the left pile and Player 1 would still have 1 point in his left pile. Player 1 does not lose points, they are simply “cloned” over to the opponents pile.
2. If the player has an even number of points in one pile and zero points in the other pile, the player may elect to split his points evenly between the two piles. This consumes the player’s turn. Example: if Player 1 has (0 4) then he can use his turn to split his points, giving him (2 2).
If a player gets exactly 5 points in either pile, that pile loses all of its points and reverts to 0. However, if points applied goes over 5 (such as adding 2 points to a 4 point pile), then the remainder of points are added. (meaning that 4 + 2 = 1). The opponent simply gets points mod 5.
If a player gets to 0 points in both their piles, then they lose. The last person that has points remaining wins.
$=========================$
Okay, so let’s break this down. Here’s an example game for those of you that are more visually oriented (follow the turns by reading left to right, moves are marked with red arrows):
• On turn 4, player 2 adds 3 points to player 1’s 2 points, making 5. The rules state that any pile with exactly 5 points reverts to zero.
• On turn 7, player 2 adds 3 points to 3 points. $3+3=6$ as we all know, but $6 \equiv 1 \mod 5$, so player 1 now has one point in his pile.
• On turn 13, player 2 decides to split his points, turning his one pile of 4 into two piles of 2. This consumes his turn.
• On turn 15, player 2 adds 2 points to player 1’s 3, thus reverting his pile to zero. Player 1 now has no more points to play with, so player 2 wins.
We can think about this game as a combinatorial problem. What are the optimal positions to play? How would one program a computer to play this game? I plan to create an interactive web game where players can try this for themselves.
# The Collatz Conjecture and Hailstone Sequences: Deceptively Simple, Devilishly Difficult
Here is a very simple math problem:
If a number is even, divide the number by two. If a number is odd, multiply the number by three, and then add one. Do this over and over with your results. Will you always get down to the value 1 no matter what number you choose?
Go ahead and test this out for yourself. Plug a few numbers in. Try it out. I’ll wait.
Back? Good. Here’s my example: I start with the number $12$.
$12$ is even, so I divide it by $2$.
$12/2=6$.
$6/2=3$.
$3$ is odd, so we multiply it by $3$ then add $1$.
$(3*3)+1=10$.
$10/2=5$.
$(5*3)+1=16$.
$16/2=8$.
$8/2=4$.
$4/2=2$.
$2/2=1$.
And we have arrived at one, just like we thought we would. But is this ALWAYS the case, for ANY number I could possibly dream of?
This may sound easy, but in fact it is an unsolved mathematics problem, with some of the best minds in the field doing research on it. It’s easy enough to check low values, but numbers are infinite, and can get very, very, very large. How do we KNOW that it works for every single number that exists?
That is what is so difficult about this problem. While every single number we have checked (which is a LOT) ends up at 1 sooner or later (some numbers can bounce around for a very long time, hence why they are called “hailstone sequences”), we still have no method to prove that it works for every number. Mathematicians have been looking for some method to predict this, but despite getting into some pretty heavy mathematics in an attempt to attack this problem, we still do not know for sure.
This problem is known as the Collatz Conjecture and is very interesting to mathematicians young and old because it is so easy to explain and play with, yet so tough to exhaustively prove. What do you think? Will this problem ever be solved? And what would be the implications if it was?
Here is some simple Python code that can display the cycles for any number you type in:
#!/usr/bin/python
num = int(input("Enter a number: "))
while num!=1:
if num%2 == 0:
num = num/2
else:
num = (num*3)+1
print num
# A Genetic Algorithm for Computing Ramsey Numbers: Update
All the 78 possible friends-strangers graphs with 6 nodes. For each graph the red/blue nodes shows a sample triplet of mutual friends/strangers.
In my last post on this topic, I discussed how I was working on a genetic algorithm to search mathematical graphs for elusive properties called Ramsey Numbers. (For a refresher on genetic algorithms, visit here, and for a refresher on Ramsey Numbers, visit here). I’ve been doing some work on it since then (check out the code here), and I thought I would describe some improvements and further progress I’ve made in this area.
New features:
• colorings dumped to a file at the end of each run
• ability to load in data sets from file, further refining of the data than starting from scratch each time
The next problem I ran up against while working through this was that even if I am able to load in previously analyzed data, I still only have one fitness function that checks a static set of edges. As I see it, there are two ways to solve this:
• Make the current fitness function dynamic; that is, it tests a different set of edges every time. However, this is counterproductive to the purpose of the program “eliminating” certain sets of edges in each “round”. However, this would be easier to maintain than the other option, which is
• Make a “FitnessHandler” method that takes in a value for which method to run, and uses that to determine what set of edges to test. However, this would lead to a lot of extra code and overhead. I’m thinking having a static variable at the beginning of each run with what “fitness method” to start on, so that it doesn’t have to start on round one each time.
I haven’t fully decided which of these I will go with. I feel like the second one fills my purpose of methodically “weeding out” the improbable graphs, but its going to be a lot of extra work. Oh well, nothing worthwhile ever came easy…
Leave a note here or on my github if you have suggestions!
Today I’m going to talk about a personal project I’ve been working on recently. I was trying to come up with some way to make a cool project with natural language processing and I had also noticed that with the rise of social networks, there is a treasure trove of data out there waiting to be analyzed. I’m a fairly active user of Twitter, and its a fun way to get short snippets of info from people or topics you’re interested in. I know personally, I have found a lot of math, science, and tech people that have twitter accounts and post about their work or the latest news in the field. I find just on a short inspection that the people I follow tend to fall into certain “groups”:
• math
• computer science
• general science | 2017-08-22 16:50:19 | {"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": 0, "mathjax_asciimath": 0, "img_math": 19, "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": 0.3057064712047577, "perplexity": 899.5987475571036}, "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-2017-34/segments/1502886112533.84/warc/CC-MAIN-20170822162608-20170822182608-00684.warc.gz"} |
https://ai-scholar.tech/en/articles/self-supervised-learning/Material_Texture | Self-Supervised Material Texture Representation Learning For Remote Sensing
3 main points
✔️ We proposed a novel material texture-based self-supervised learning method to obtain features with the high inductive bias required for downstream tasks on remote sensing data.
✔️ The models pre-trained with our method recorded SOTA in supervised and unsupervised change detection, segmentation, and land cover classification experiments.
✔️ Provided a multi-time spatially adjusted, atmospherically processed remote sensing dataset in an unchanging domain used for self-supervised learning.
written by Peri AkivaMatthew PurriMatthew Leotta
(Submitted on 3 Dec 2021)
Subjects: Computer Vision and Pattern Recognition (cs.CV); Artificial Intelligence (cs.AI); Image and Video Processing (eess.IV)
code：
The images used in this article are from the paper, the introductory slides, or were created based on them.
first of all
Self-supervised learning aims to learn the representation features of an image without annotation data. In addition, by initializing the network weights to the weights learned by self-supervised learning in a downstream task, faster convergence and higher performance may be obtained. However, self-supervised learning requires a high inductive bias. In this paper, we proposed material-texture-based self-supervised learning called MATTER (MATerial and TExture Representation Learning).In MATTER, to obtain the luminance and viewing angle invariants, multiple spatially adjusted remote sensing data in time to obtain luminance and viewing angle invariants.
technique
A schematic diagram of MATTER is shown below. Given an anchor image $x_a\in {\cal R}^{B\times H \times W}$ in an unchanged region, we obtain a positive image $x_p\in {\cal R}^{B\times H \times W}$ in the same region and a negative image $x_n\in {\cal R}^{B\times H \times W}$. Where $B, H, W$ are the number of bands, height, and width of the input image. Tile all the images into P patches of size $H\times W$ and denote them as $c_a, c_p, c_n$ respectively. To learn material-texture-centric features, we propose a Texture Refinement Network (TeRN) and a patch-wise Surface Residual Encoder.TeRN aims to increase the activation of low-level features required for the texture representativeness Surface Residual Encoder is a patch-wise adaptation of prior work Deep-TEN to learn surface-based residual representation quantities. The network is trained by minimizing the feature distance of positive patch pairs $c_a, c_p$ and maximizing the feature distance of negative patch pairs $c_a, c_n$. Here the features are the learned residual representation quantities. We used the noise-contrast loss as the objective function.
$${\cal L}_{NCE}=-{\mathbb E}_C\left[{\log}\frac{\exp (f(c_a)\cdot (f(c_p))}{\sum_{c_j \in C}\exp (f(c_a)\cdot f(c_j))}\right]$$
where $f(c_j)$ is the feature of patch $c_j$ and $C$ is the set of positive and negative patches.
Texture Refinement Network
TeRN refines texture features, which are often low-level in satellite images. It employs the recently proposed pixel adaptive convolution layer, and given a kernel $k^{i,j}$ centered at position $(i,j)$, calculates the cosine similarity between pixel $x_{i.j}$ and the neighborhood point ${\cal N}(i,j)$, and divides by the square of the standard deviation $\sigma_{{\ cal N}(i,j)}$ and divide by the square of $\sigma_{{\cal N}(i,j)}$.
$$k^{i,j}=-\frac{1}{\sigma^2_{{\cal N}(i,j)}}\frac{x_{i,j}\cdot x_{p,q}}{||x_{i,j}||_2\cdot ||x_{p,q}||_2}, \forall p,q \in {\cal N}(i,j)$$
The above equation represents the similarity to the center pixel and the gradient strength in the kernel. Since the texture represents the spatial distribution of the structure, it is directly related to the gradient strength. A single kernel layer is $K$ and a refinement network of $L$ layers is constructed.
Surface Residual Agreement Learning
This method performs patchwise clustering. It learns residuals of small patches and enforces consistency with the corresponding multi-time patch residuals. Given a feature vector $z_i^{1\times D}$ and $\Upsilon$ learned cluster centers $Q=\{q_0,q_1,\cdots,q_{\Upsilon-1}\}$ for some crop $c_i$, the residuals between $z_i$ and cluster center $q_v$ are $r_{i,v}^{1\times D}=z_i-q_v$. By repeating this for all clusters and taking the weighted average, the final residual vector is obtained as follows.
$$r_i = \frac{1}{\Upsilon}\sum_{v=0}^{\Upsilon-1}\theta_{i,v}r_{i,v}$$
where $\theta_v$ is the learned cluster weights. By using this, we can consider the affinity between clusters.
experiment
prior learning
For self-supervised learning, we used ortho-rectified, atmospherically processed Sentinel-2 data from a non-urbanized region. A total of $1217 km^2$ of $27$ area was finally collected, yielding $14857$ $1096^times 1096$ tiles.
change detection
We used the Onera Satellite Change Detection (OSCD) dataset. We evaluated two types of models: a self-supervised learning model with only a pre-training model and a supervised model that was fine-tuned with the supervised data afterward. The change point was the case where the Euclidean distance between the residual features of the before and after images exceeded a threshold value. The results are shown in the table below. The F1 score recorded SOTA for both self-supervised and supervised models.
land subsidence classification
As a dataset, we used the BigEarthNet dataset with 19 classes of land cover labels. The results are shown in the table below. The method recorded an average accuracy SOTA.
segmentation
As a dataset, we used SpaceNet for building segmentation. The results are shown in the table below. The method recorded a SOTA of average IoU.
result
By implementing our method, we achieved faster convergence and higher accuracy for downstream tasks and found that texture and material are important features. We also compared the visual word map (pixel-wise clustering) to qualitatively evaluate whether the material and texture are well represented. The results are shown in the figure below: Textons evaluate pixel values and are therefore more sensitive to small changes in texture, leading to multiple clusters of the same class; Patch-wise Backbone loses information from low-level features, leading to multiple classes being clustered into a single cluster. On the other hand, our method classifies very close to the input image.
summary
In this paper, we proposed a self-supervised learning method called MATTER, which learns texture and material features that are strongly correlated with surface changes and could be applied to pre-training for remote sensing tasks.
If you have any suggestions for improvement of the content of the article, | 2022-09-24 19:36:01 | {"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": 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": 0.3573801815509796, "perplexity": 1253.3455951076535}, "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-2022-40/segments/1664030333455.97/warc/CC-MAIN-20220924182740-20220924212740-00484.warc.gz"} |
http://www.numdam.org/articles/10.1051/ita:2007048/ | Arithmetization of the field of reals with exponentiation extended abstract
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 105-119.
(1) Shepherdson proved that a discrete unitary commutative semi-ring ${A}^{+}$ satisfies $I{E}_{0}$ (induction scheme restricted to quantifier free formulas) iff $A$ is integral part of a real closed field; and Berarducci asked about extensions of this criterion when exponentiation is added to the language of rings. Let $T$ range over axiom systems for ordered fields with exponentiation; for three values of $T$ we provide a theory ${}_{⌞}{T}_{⌟}$ in the language of rings plus exponentiation such that the models ($A$, exp${}_{A}$) of ${}_{⌞}{T}_{⌟}$ are all integral parts $A$ of models $M$ of $T$ with ${A}^{+}$ closed under exp${}_{M}$ and ${exp}_{A}={exp}_{M}↾{A}^{+}$. Namely $T$ = EXP, the basic theory of real exponential fields; $T$ = EXP+ the Rolle and the intermediate value properties for all ${2}^{x}$-polynomials; and $T$ = ${T}_{exp}$, the complete theory of the field of reals with exponentiation. (2) ${}_{⌞}$${T}_{exp}$${}_{⌟}$ is recursively axiomatizable iff ${T}_{exp}$ is decidable. ${}_{⌞}$${T}_{exp}$${}_{⌟}$ implies $L{E}_{0}\left({x}^{y}\right)$ (least element principle for open formulas in the language $<,+,×,-1,{x}^{y}$) but the reciprocal is an open question. ${}_{⌞}$${T}_{exp}$${}_{⌟}$ satisfies “provable polytime witnessing”: if ${}_{⌞}$${T}_{exp}$${}_{⌟}$ proves ${\forall x\exists y:|y|<|x|}^{k}\right)R\left(x,y\right)$ (where $|y|:{=}_{⌞}$$log\left(y\right)$${}_{⌟}$, $k<\omega$ and $R$ is an NP relation), then it proves $\forall x\phantom{\rule{4pt}{0ex}}R\left(x,f\left(x\right)\right)$ for some polynomial time function $f$. (3) We introduce “blunt” axioms for Arithmetics: axioms which do as if every real number was a fraction (or even a dyadic number). The falsity of such a contention in the standard model of the integers does not mean inconsistency; and bluntness has both a heuristic interest and a simplifying effect on many questions - in particular we prove that the blunt version of ${}_{⌞}$${T}_{exp}$${}_{⌟}$ is a conservative extension of ${}_{⌞}$${T}_{exp}$${}_{⌟}$ for sentences in $\forall {\Delta }_{0}\left({x}^{y}\right)$ (universal quantifications of bounded formulas in the language of rings plus ${x}^{y}$). Blunt Arithmetics - which can be extended to a much richer language - could become a useful tool in the non standard approach to discrete geometry, to modelization and to approximate computation with reals.
DOI : https://doi.org/10.1051/ita:2007048
Classification : 03H15
Mots clés : computation with reals, exponentiation, model theory, o-minimality
@article{ITA_2008__42_1_105_0,
author = {Boughattas, Sedki and Ressayre, Jean-Pierre},
title = {Arithmetization of the field of reals with exponentiation extended abstract},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {105--119},
publisher = {EDP-Sciences},
volume = {42},
number = {1},
year = {2008},
doi = {10.1051/ita:2007048},
zbl = {1144.03027},
mrnumber = {2382546},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ita:2007048/}
}
TY - JOUR
AU - Boughattas, Sedki
AU - Ressayre, Jean-Pierre
TI - Arithmetization of the field of reals with exponentiation extended abstract
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2008
DA - 2008///
SP - 105
EP - 119
VL - 42
IS - 1
PB - EDP-Sciences
UR - http://www.numdam.org/articles/10.1051/ita:2007048/
UR - https://zbmath.org/?q=an%3A1144.03027
UR - https://www.ams.org/mathscinet-getitem?mr=2382546
UR - https://doi.org/10.1051/ita:2007048
DO - 10.1051/ita:2007048
LA - en
ID - ITA_2008__42_1_105_0
ER -
Boughattas, Sedki; Ressayre, Jean-Pierre. Arithmetization of the field of reals with exponentiation extended abstract. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 105-119. doi : 10.1051/ita:2007048. http://www.numdam.org/articles/10.1051/ita:2007048/
[1] S. Boughattas, Trois Théorèmes sur l'induction pour les formules ouvertes munies de l'exponentielle. J. Symbolic Logic 65 (2000) 111-154. | MR 1782110 | Zbl 0959.03023
[2] L. Fuchs, Partially Ordered Algebraic Systems. Pergamon Press (1963). | MR 171864 | Zbl 0137.02001
[3] M.-H. Mourgues and J.P. Ressayré, Every real closed field has an integer part. J. Symbolic Logic 58 (1993) 641-647. | MR 1233929 | Zbl 0786.12005
[4] S. Priess-Crampe, Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen. Springer-Verlag, Berlin (1983). | MR 704186 | Zbl 0558.51012
[5] A. Rambaud, Quasi-analycité, o-minimalité et élimination des quantificateurs. PhD. Thesis. Université Paris 7 (2005). | MR 2241948
[6] J.P. Ressayre, Integer Parts of Real Closed Exponential Fields, Arithmetic, Proof Theory and Computational Complexity, edited by P. Clote and J. Krajicek, Oxford Logic Guides 23. | Zbl 0791.03018
[7] J.P. Ressayre, Gabrielov's theorem refined. Manuscript (1994).
[8] J.C. Shepherdson, A non-standard model for a free variable fragment of number theory. Bulletin de l'Academie Polonaise des Sciences 12 (1964) 79-86. | MR 161798 | Zbl 0132.24701
[9] L. Van Den Dries, Exponential rings, exponential polynomials and exponential functions. Pacific J. Math. 113 (1984) 51-66. | MR 745594 | Zbl 0603.13019
[10] A. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996) 1051-1094. | MR 1398816 | Zbl 0892.03013
Cité par Sources : | 2022-01-18 07:00:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 51, "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": 0.4953123927116394, "perplexity": 3286.2645955586167}, "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-2022-05/segments/1642320300805.79/warc/CC-MAIN-20220118062411-20220118092411-00692.warc.gz"} |
http://blog.carlesmateo.com/category/performance/page/2/ | # Improving performance in PHP
This year I was invited to speak at the PHP Conference at Berlin 2014.
It was really nice, but I had to decline as I was working hard in a Start up, and I hadn’t the required time in order to prepare the nice conference I wanted and that people deserves.
However, having time, I decided to write an article about what I would had speak at the conference.
I will cover improving performance in a single server, and Scaling out multi-Server architecture, focusing on the needs of growing and Start up projects. Many of those techniques can be used to improve performance with other languages, not just with PHP.
Many of my friends are very good Developing, but know nothing about Architecture and Scaling. Hope this approach the two worlds, Development ad Operatings, into a DevOps bridge.
# Improving performance on a single server
## Hosting
Choose a good hosting. And if you can afford it choose a dedicated server.
Shared hostings are really bad. Some of them kill your http and mysql instances if you reach certain CPU use (really few), while others share the same hardware between 100+ users serving your pages sloooooow. Others cap the amount of queries that your MySql will handle per hour at so ridiculous few amount that even Drupal or WordPress are unable to complete a request in development.
Other ISP (Internet Service Providers) have poor Internet bandwidth, and so you web will load slow to users.
Some companies invest hundreds of thousands in developing a web, and then spend 20 € a year in the hosting. Less than the cost of a dinner.
You can use a decent dedicated server from 50 to 99 €/month and you will celebrate this decision every day.
Take in count that virtualization wastes between 20% and 30% of the CPU power. And if there are several virtual machines the loss will be more because you loss the benefits of the CPU caching for optimizing parallel instructions execution and prediction. Also if the hypervisor host allows to allocate more RAM than physically available and at some point it swaps, the performance of all the VM’s will be much worst.
If you have a VM and it swaps, in most providers the swap goes over the network so there is an additional bottleneck and performance penalty.
To compare the performance of dedicated servers and instances from different Cloud Providers you can take a look at my project cmips.net
If your Sever has few RAM, add more. And if your project is running slow and you can afford a better Server, do it.
Using SSD disk will incredibly improve the performance on I/O operations and on swap operations. (but please, do backups and keep them in another place)
If you use a CMS like ezpublish with http_cache enabled probably you will prefer to have a Server with faster cores, rather tan a Server with one or more CPU’s plenty of cores, but slower cores, and that last for a longer time to render the page to the http cache.
That may seem obvious but often companies invest 320 hours in optimizing the code 2%, at a cost of let’s say 50 €/h * 320 hours = 16.000 €, while hiring a better Server would had bring between a 20% to 1000% improvement at a cost of additional 50€/month only or at the cost of 100 € of increasing the RAM memory.
The point here is that the hardware is cheap, while the time of the Engineers is expensive. And good Engineers are really hard to find.
And you probably, as a CEO or PO, prefer to use the talent to warranty a nice time to market for your project, or adding more features, rather than wasting this time in refactorizing.
Even with the most optimal code in the universe, if your project is successful at certain point you’ll have to scale. So adding more Servers. To save a Server now at the cost of slowing the business has not any sense.
Many projects still use PHP 5.3, and 5.4.
Latest versions of PHP bring more and more performance. If you use old versions of PHP you can have a Quick Win by just upgrading to the last PHP version.
## Use OpCache (or other cache accelerator)
OpCache is shipped with PHP 5.5 by default now, so it is the recommended option. It is though to substitute APC.
To activate OpCache edit php.ini and add:
Linux/Unix:
zend_extension=/path/to/opcache.so
Windows:
zend_extension=C:\path\to\php_opcache.dll
It will greatly improve your PHP performance.
Ensure that OpCache in Production has the optimal config for Production, that will be different from Development Environment.
Note: If you plan to use it with XDebug in Development environments, load OpCache before XDebug.
## Disable Profiling and xdebug in Production
In Production disable the profiling, xdebug, and if you use a Framework ensure the Development/Debug features are disabled in Production.
## Ensure your logs are not full of warnings
Check that Production logs are not full of warnings.
I’ve seen systems were every seconds 200 warnings were written to logs, the same all the time, and that obviously was slowing down the system.
Typical warnings like this can be easily fixed:
Message: date() [function.date]: It is not safe to rely on the system’s timezone settings. You are required to use the date.timezone setting or the date_default_timezone_set() function. In case you used any of those methods and you are still getting this warning, you most likely misspelled the timezone identifier. We selected ‘UTC’ for ‘8.0/no DST’ instead
## Profile in Development
To detect where your slow code is, profile it in Development to see where it is spent the most CPU/time.
Check the slow-queries if you use MySql.
## Cache html to disk
Imagine you have a sort of craigslist and you are displaying all the categories, and the number of new messages in this landing page. To do that you are performing many queries to the database, SELECT COUNTs, etc… every time a user visits your page. That certainly will overload your database with actually few concurrent visitors.
Instead of querying the Database all the time, do cache the generated page for a while.
This can be achieved by checking if the cache html file exists, and checking the TTL, and generating a new page if needed.
A simple sample would be:
<?php
// Cache pages for 5 minutes
$i_cache_TTL = 300;$b_generate_cache = false;
$s_cache_file = '/tmp/index.cache.html'; if (file_exists($s_cache_file)) {
// Get creation date
$i_file_timestamp = filemtime($s_cache_file);
$i_time_now = microtime(true); if ($i_time_now > ($i_file_timestamp +$i_cache_TTL)) {
$b_generate_cache = true; } else { // Up to date, get from the disk$o_fh = fopen($s_cache_file, "rb");$s_html = stream_get_contents($o_fh); fclose($o_fh);
// If the file was empty something went wrong (disk full?), so don't use it
if (strlen($s_html) == 0) {$b_generate_cache = true;
} else {
// Print the page and exit
echo $s_html; exit(); } } } else {$b_generate_cache = true;
}
ob_start();
// Render your page normally here
// ....
$s_html = ob_get_clean(); if ($b_generate_cache == true) {
// Create the file with fresh contents
$o_fp = fopen($s_cache_file, 'w');
if (fwrite($o_fp,$s_html) === false) {
// Error. Impossible to write to disk
// throw new Exception('CacheCantWrite');
}
fclose($o_fp); } // Send the page to the browser echo$s_html;
This sample is simple, and works for many cases, but presents problems.
Imagine for example that the page takes 5 seconds to be generated with a single request, and you have high traffic in that page, let’s say 500 requests per second.
What will happen when the cache expires is that the first user will trigger the cache generation, and the second, and the third…. so all of the 500 requests * 5 seconds will be hitting the database to generate the cache, but… if creating the page per one requests takes 5 seconds, doing this 2,500 times will not last 5 seconds… so your process will enter in a vicious state where the first queries have not ended after minutes, and more and more queries are being added to the queue until:
a) Apache runs out of childs/processes, per configuration
b) Mysql runs out of connections, per configuration
c) Linux runs out of memory, and processes crashes/are killed
Not to mention the users or the API client, waiting infinitely for the http request to complete, and other processes reading a partial file (size bigger than 0 but incomplete).
Different strategies can be used to prevent that, like:
a) using semaphores to lock access to the cache generation (only one process at time)
b) using a .lock file to indicate that the file is being generated, and so next requests serving from the cache until the cache generation process ends the task, also writing to a buffer like acachefile.buffer (to prevent incomplete content being read) and finally when is complete renaming to the final name and removing the .lock
c) using memcached, or similar, to keep an index in memory of what pages are being generated now, and why not, keeping the cached files there instead of a filesystem
d) using crons to generate the cache files, so they run hourly and you ensure only one process generates the cache files
If you use crons, a cheap way to generate the .html content is that the crons curls/wget your webpage. I don’t recommend this as has some problems, like if that web request fails for any reason, you’ll have cached an error instead of content.
I prefer preparing my projects to being able of rendering the content being invoked from HTTP/S or from command line. But if you use curl because is cheap and easy and time to market is important for your project, then be sure that you check that your backend code writes an Status OK in the HTML that the cron can check to ensure that the content has been properly generated. (some crons only check for http status, like 200, but if your database or a xml gateway you use fails you will likely get a 200 and won’t detect that you’re caching pages with “error I can’t connect to the database” instead)
Many Frameworks have their own cache implementation that prevent corruption that could come by several processes writing to the same file at the same time, or from PHP dying in the middle of the render.
You can see a more complex MVC implementation, with Views, from my Framework Catalonia here:
By serving .html files instead of executing PHP with logic and performing queries to the database you will be able to serve hundreds of thousands requests per day with a single machine and really fast -that’s important for SEO also-.
I’ve done this in several Start ups with wonderful results, and my Framework Catalonia also incorporates this functionality very easily to use.
Note: This is only one of the techniques to save the load of the Database Servers. Many more come later.
## Cache languages to disk
If you have an application that is multi-language, or if your point for the Strings (sections, pages, campaigns..) to be edited by Marketing is the Database, there is no need to query it all the time.
Simply provide a tool to “generate language files”.
Your languages files can be Javascript files loaded by the page, or can be PHP files generated.
For example, the file common_footer_en.php could be generated reading from Database and be like that:
<php
/* Autogenerated English translations file common_footer_en.php
on 2014-08-10 02:22 from the database */
$st_translations['seconds'] = 'seconds';$st_translations['Time'] = 'Time';
$st_translations['Vars used'] = 'Vars used in these templates';$st_translations['Total Var replacements'] = 'Total replaced';
$st_translations['Exec time'] = 'Execution time';$st_translations['Cached controller'] = 'Cached controller';
So the PHP file is going to be generated when someone at your organization updates the languages, and your code is including it normally like with any other PHP file.
## Use the Crons
You can set cron jobs to do many operations, like map reduce, counting in the database or effectively deleting the data that the user selected to delete.
Imagine that you have classified portal, and you want to display the number of announces for that category. You can have a table NUM_ANNOUNCES to store the number of announces, and update it hourly. Then your database will only do the counting once per hour, and your application will be reading the number from the table NUM_ANNOUNCES.
The Cron can also be used to make expire old announces. That way you can avoid a user having to wait for that clean up taking process when you have a http request to PHP.
A cron file can be invoked by:
php -f cron.php
By:
./cron.php
If you give permissions of execution with chmod +x and set the first line in cron.php as:
#!/usr/bin/env php
Or you can do a trick, that is emulate a http request from bash, by invoking a url with curl or with wget. Set the .htaccess so the folder for the cron tasks can only be executed from localhost for adding security.
This last trick has the inconvenient that the calling has the same problems as any http requests: restarting Apache will kill the process, the connection can be closed by timed out (e.g. if process is taking more seconds than the max. execution time, etc…)
## Use Ramdisk for PHP files
With Linux is very easy to setup a RamDisk.
You can setup a RamDisk and rsync all your web .PHP files at system boot time, and when deploying changes, and config Apache to use the Ramdisk folder for the website.
That way for every request to the web, PHP files will be served from RAM directly, saving the slow disk access. Even with OpCache active, is a great improvement.
At these times were one Gigabyte of memory is really cheap there is a huge difference from reading files from disk, and getting them from memory. (Reading and writing to RAM memory is many many many times faster than magnetic disks, and many times faster than SSD disks)
Also .js, .css, images… can be served from a Ram disk folder, depending on how big your web is.
## Ramdisk for /tmp
If your project does operations on disk, like resizing images, compressing files, reading/writing large CSV files, etcetera you can greatly improve the performance by setting the /tmp folder to a Ramdisk.
If your PHP project receives file uploads they will also benefit (a bit) from storing the temporal files to RAM instead to the disk.
## Use Cache Lite
Cache Lite is a Pear extension that allows you to keep data in a local cache of the Web Server.
You can cache .html pages, or you can cache Queries and their result.
<?php
require_once "Cache/Lite.php";
$options = array( 'cacheDir' => '/tmp/', 'lifeTime' => 7200, 'pearErrorMode' => CACHE_LITE_ERROR_DIE );$cache = new Cache_Lite($options); if ($data = $cache->get('id_of_the_page')) { // Cache hit ! // Content is in$data
echo $data; } else { // No valid cache found (you have to make and save the page)$data = '<html><head><title>test</title></head><body><p>this is a test</p></body></html>';
echo $data;$cache->save($data); } It is nice that Cache Lite handles the TTL and keeps the info stored in different sub-directories in order to keep a decent performance. (As you may know many files in the same directory slows the access much). ## Use HHVM (HipHop Virtual Machine) from Facebook Facebook Engineers are always trying to optimize what is run on the Servers. Faster code means, less machines. Even 1% of CPU use improvement means a lot of Servers less. Less Servers to maintain, less money wasted, less space on the Data Centers… So they created the HHVM HipHop Virtual Machine that is able to run PHP code, much much faster than PHP. And is compatible with most of the Frameworks and Open Source projects. They also created the Hack language that is an improved PHP, with type hinting. So you can use HHVM to make your code run faster with the same Server and without investing a single penny. ## Use C extensions You can create and use your own C extensions. C extensions will bring really fast execution. Just to get the idea: I built a PHP extension to compare the performance from calculating the Bernoulli number with PHP and with the .so extension created in C. In my Core i7 times were: PHP: Computed in 13.872583150864 s PHP calling the C compiled extension: Computed in 0.038495063781738 s That’s 360.37 times faster using the C extension. Not bad. ## Use Zephir Zephir is a an Open Source language, very similar to PHP, that allows to create and maintain easily extensions for PHP. ## Use Phalcon Phalcon is a Web MVC Framework implemented as C extension, so it offers a high performance. The views syntax are very very similar to Twig. ## Check if you’re using the correct Engine for MySql Many Developers create the tables and never worry about that. And many are using MyIsam by default. It was the by default Engine prior to MySql 5.5. While MyIsam can bring good performance in some certain cases, my recommendation is to use InnoDb. Normally you’ll have a gain in performance with MyIsam if you’ve a table were you only write or only read, but in all the other cases InnoDb is expected to be much more performant and safe. MyIsam tables also get corruption from time to time and need manually fixing and writing to disks are not so reliable than InnoDb. As MyIsam uses table-locking for updates and deletes to any existing row, it is easy to see that if you’re in a web environment with multiple users, blocking the table -so the other operations have to wait- will make things be slow. If you have to use Joins clearly you will benefit from using InnoDb also. ## Use InMemory Engine from MySql MySql has a very powerful Engine called InMemory. The InMemory Engine will store things in RAM and loss the data when MySql is restarted. However is very fast and very easy to use. Imagine that you have a travel application that constantly looks at which country belongs the city specified by user. A Quickwin would be to INSERT all this data in the InMemory Engine of MySql when it is started, and do just one change in your code: to use that Table. Really easy. Quick improvement. ## Use curl asynchronously If your PHP has to communicate with other systems using curl, you can do the http/s call, and instead of waiting for a response let your PHP do more things in the meantime, and then check the results. You can also call to multiple curl calls in parallel, and so avoid doing one by one in serial. ## Serialize Guess that you have a query that returns 1000 results. Then you add one by one to an array. Probably you’re going to have substantial gain if you keep in the database a single row, with the array serialized. So an array like:$st_places = Array(‘Barcelona’, ‘Dublin’, ‘Edinburgh’, ‘San Francisco’, ‘London’, ‘Berlin’, ‘Andorra la Vella’, ‘Prats de Lluçanès’);
Would be serialized to an string like:
a:8:{i:0;s:9:”Barcelona”;i:1;s:6:”Dublin”;i:2;s:9:”Edinburgh”;i:3;s:13:”San Francisco”;i:4;s:6:”London”;i:5;s:6:”Berlin”;i:6;s:16:”Andorra la Vella”;i:7;s:19:”Prats de Lluçanès”;}
This can be easily stored as String and unserialized later back to an array.
Note: In Internet we have a lot of encodings, Hebrew, Japanese… languages. Be careful with encodings when serializing, using JSon, XML, storing in databases without UTF support, etc…
## Use Memcached to store common things
Memcached is a NoSql database in memory that can run in cluster.
The idea is to keep things there, in order to offload the load of the database. And as everything is in RAM it really runs fast.
You can use Memcached to cache Queries and their results also.
For example:
You have query SELECT * FROM translations WHERE section=’MAIN’.
Then you look if that String exists as key in the Memcached, and if it exists you fetch the results (that are serialized) and you avoid the query. If it doesn’t exist, you do normally the query to the database, serialize the array and store it in the Memcached with a TTL (Time to Live) using the Query (String) as primary key. For security you may prefer to hash the query with MD5 or SHA-1 and use the hash as key instead of using it plain.
When the TTL is reached the validity of the data would have expired and so it’s time to reinsert the contents in the next query.
Be careful, I’ve seen projects that were caching private data from users without isolating the key properly, so other users were getting the info from other users.
For example, if the key used was ‘Name’ and the value ‘Carles Mateo’ obviously the next user that fetch the key ‘Name’ would get my name and not theirs.
If you store private data of users in Memcache, it is a nice idea to append the owner of that info to the hash. E.g. using key: 10701577-FFADCEDBCCDFFFA10C
Where ‘10701577’ would be the user_id of the owner of the info, and ‘FFADCEDBCCDFFFA10C’ a hash of the query.
Before I suggested that you can keep a table of counting for the announces in a classified portal. This number can be stored in the Memcached instead.
You can store also common things, like translations, or cities like in the example before, rate of change for a currency exchanging website…
The most common way to store things there is serialized or Json encoded.
Be aware of the memory limits of Memcached and contrl the cache hitting ratio to avoid inserting data, and losing it constantly because is used few and Memcached has few memory.
You can also use Redis.
## Use jQuery for Production (small file) and minimized files for js
Use the Production jQuery library in Production, I mean do not use the bigger file Development jQuery library for Production.
There are product that eliminate all the necessary spaces in .js and .css files, and so are served much faster. These process is called minify.
It is important to know that in many emerging markets in the world, like Brazil, they have slow DSL lines. Many 512 Kbit/secons, and even modem connections!.
## Activate compression in the Server
If you send large text files, or Jsons, you’ll benefit from activating the compression at the Server.
It consumes some CPU, but many times it brings an important improvement in speed serving the pages to the users.
## Use a CDN
You can use a Content Delivery Network to offload your Servers from sending plain texts, html, images, videos, js, css…
You can delegate this to the CDN, they have very speedy Internet lines and Servers, so your Servers can concentrate into doing only BackEnd operations.
The most well known are Akamai and Amazon Cloud Front.
Please take attention to the documentation, a common mistake is to send Cache Headers to the CDN servers, while they’ll use this headers to set the cache TTL and ignore their web configuration parameters. (For example s-maxage, like: Cache-Control: public, s-maxage=600)
HTTP/1.1 200 OK
Server: nginx
Date: Wed, 20 Aug 2014 10:50:21 GMT
Content-Type: text/html; charset=UTF-8
Connection: close
Vary: Accept-Encoding
Cache-Control: max-age=0, public, s-maxage=10800
Vary: X-User-Hash,Accept-Encoding
X-Location-Id: 2
X-Content-Digest: ezlocation/2/end5139244ced4b25606ef0a39235982b1662d01cc
Content-Length: 68250
Age: 3
You can take a look at any website by telneting to the port 80 and doing the request manually or easily by using lynx:
## Do you need a Framework?
If you’re processing only BackEnd petitions, like in the video games industry, serving API’s, RESTful, etc… you probably don’t need a Framework.
The Frameworks are generic and use much more resources than you’re really need for a fast reply.
Many times using a heavy Framework has a cost of factor times, compared to use simply PHP.
## Save database connections until really needed
Many Frameworks create a connection to the Database Server by default. But certain parts of your code application do not require to connect to the database.
For example, validating the data from a form. If there are missing fields, the PHP will not operate with the Database, just return an error via JSon or refreshing the page, informing that the required field is missing.
If a not logged user is requesting the dashboard page, there is no need to open a connection to the database (unless you want to write the access try to an error log in the database).
In fact opening connections by default makes easier for attackers to do DoS attacks.
With a Singleton pattern you can easily implement a Db class that handles this transparently for you.
## Memcached session
When you have several Web Servers you’ll need something more flexible than the default PHP handler (that stores to a file in the Web Server).
The most common is to store the Session, serialized, in a Memcached Cluster.
## Use Cassandra
Apache Cassandra is a NoSql database that allows to Scale out very easily.
The main advantage is that scales linearly. If you have 4 nodes and add 4 more, your performance will be doubled. It has no single point of failure, is also resilient to node failures, it replicates the data among the nodes, splits the load over the nodes automatically and support distributed datacenter architectures.
To know more abiut NoSql and Cassandra, read my article: Upgrade your scalability with NoSql. And to start developing with Cassandra in PHP, python or Java read my contributed article: Begin developing with Cassandra.
## Use MySql primary and secondaries
A easy way to split the load is to have a MySql primary Server, that handles the writes, and MySql secondary (or Slave) Servers handling the reads.
Every write sent to the Master is replicated into the Slaves. Then your application reads from the slaves.
You have to tell your code to do the writes to database to the primary Server, and the reads to the secondaries. You can have a Load Balancer so your code always ask the Load Balancer for the reads and it makes the connection to the less used server.
## Do Database sharding
To shard the data consist into splitting the data according to a criteria.
For example, imagine we have 8 MySql Servers, named mysql0 to mysql7. If we want to insert or read data for user 1714, then the Server will be chosen from dividing the user_id, so 1714, between the number of Servers, and getting the MOD.
So 1714 % 8 gives 2. This means that the MySql Server to use is the mysql2.
For the user_id 16: 16 & 8 gives 0, so we would use mysql0. And so.
You can shard according to the email, or other fields as well. And you can have the same master and secondaries for the shards also.
When doing sharding in MySql you cannot do joins to data in other Servers. (but you can replicate all the data from the several shards in one big server in house, in your offices, and so query it and join if you need that for marketing purposes).
I always use my own sharding, but there is a very nice product from CodeFutures called dbshards. It handles the traffics transparently. I used it when in a video games Start up with very satisfying result.
## Use Cassandra assync queries
Cassandra support asynchronous queries. That means you can send the query to the Server, and instead of waiting, do other jobs. And check for the result later, when is finished.
## Consider using Hadoop + HBASE
A Cluster alternative to Cassandra.
You can put a Load Balancer or a Reverse Proxy in front of your Web Servers. The Load Balancer knows the state of the Web Servers, so it will remove a Web Server from the Array if it stops responding and everything will continue being served to the users transparently.
There are many ways to do Load Balancing: Round Robin, based on the load on the Web Servers, on the number of connections to each Web Server, by cookie…
To use a Cookie based Load Balancer is a very easy way to split the load for WordPress and Drupal Servers.
Imagine you have 10 Web Servers. In the .htaccess they set a rule to set a Cookie like:
SERVER_ID=WEB01
That was in the case of the first Web Server.
SERVER_ID=WEB02
Etcetera
When for first time an user connects to the Load Balancer it sends the user to one of the 10 Web Servers. Then the Web Server sends its cookie to the browser of the Client. E.g. WEB07
After that, in the next requests from the client it will be redirected to the server by the Load Balancer to the Server that set the Cookie, so in this example WEB07.
The nice thing of this way of splitting the traffic is that you don’t have to change your code, nor handling the Sessions different.
If you use two Load Balancers you can have a heartbeat process in them and a Virtual Ip, and so in case your main Load Balancer become irresponsible the Virtual Ip will be mapping to the second Load Balancer in milliseconds. That provides HA.
## Use http accelerators
Nginx, varnish, squid… to serve static content and offload the PHP Web Servers.
## Auto-Scale in the Cloud
If you use the Cloud you can easily set Auto-Scaling for different parts of your core.
A quick win is to Scale the Web Servers.
As in the Cloud you pay per hour using a computer, you will benefit from cost reduction in you stop using the servers when you don’t need them, and you add more Servers when more users are coming to your sites.
Video game companies are a good example of hours of plenty use and valleys with few users, although as users come from all the planet it is most and most diluted.
Some cool tools to Auto-Scaling are: ECManaged, RightScale, Amazon CloudWatch.
Actually the Performance of the Google Cloud to Scale without any precedent is great.
Opposite to other Clouds that are based on instances, Google Cloud offers the platform, that will spawn your code across so many servers as needed, transparently to you. It’s a black box.
## Schedule operations with RabbitMQ
Or other Queue Manager.
The idea is to send the jobs to the Queue Manager, the PHP will continue working, and the jobs will be performed asynchronously and notify the end.
RabbitMQ is cool also because it can work in cluster and HA.
## Use GlusterFs for NAS
GlusterFs (and other products) allow you to have a Distributed File System, that splits the load and the data across the Servers, and resist node failures.
If you have to have a shared folder for the user’s uploads, for example for the profile pictures, to have the PHP and general files locally in the Servers and the Shared folder in a GlusterFs is a nice option.
## Avoid NFS for PHP files and config files
As told before try to have the PHP files in a RAM disk, or in the local disk (Linux caches well and also OpCache), and try to not write code that reads files from disk for determining config setup.
I remember a Start up incubator that had a very nice Server, but the PHP files were read from a mounted NFS folder.
That meant that on every request, the Server had to go over the network to fetch the files.
Sadly for the project’s performance the PHP was reading a file called ENVIRONMENT that contained “PROD” or “DEVEL”. And this was done in every single request.
Even worst, I discovered that the switch connecting the Web Server and the NFS Server was a cheap 10 Mbit one. So all the traffic was going at 10 Mbit/s. Nice bottleneck.
You can use 10 GbE (10 Gigabit Ethernet) to connect the Servers. The Web Servers to the Databases, Memcached Cluster, Load Balancers, Storage, etc…
You will need 10 GbE cards and 10 GbE switchs supporting bonding.
Use bonding to aggregate 10 + 10 so having 20 Gigabit.
You can also use Fibre Channel, for example 10 Gb and aggregate them, like 10 + 10 so 20 Gbit for the connection between the Servers and the Storage.
The performance improvements that your infrastructure will experiment are amazing.
# Begin developing with Cassandra
We architects, developers and start ups are facing new challenges.
We have now to create applications that have to scale and scale at world-wide level.
That puts over the table big and exciting challenges.
To allow that increasing level of scaling, we designed and architect tools and techniques and tricks, but fortunately now there are great products born to scale out and to deal with this problems: like NoSql databases like Cassandra, MongoDb, Riak, Hadoop’s Hbase, Couchbase or CouchDb, NoSql in-Memory like Memcached or Redis, big data solutions like Hadoop, distributed files systems like Hadoop’s HDFS, GlusterFs, Lustre, etc…
In this article I will cover the first steps to develop with Cassandra, under the Developer point of view.
As a first view you may be interested in Cassandra because:
• Is a Database with no single point of failure
• Where all the Database Servers work in Peer to Peer over Tcp/Ip
• Fault-tolerance. You can set replication factor, and the data will be sharded and replicated over different servers and so being resilient to node failures
• Because the Cassandra Cluster splits and balances the work across the Cluster automatically
• Because you can scale by just adding more nodes to the Cluster, that’s scaling horizontally, and it’s linear. If you double the number of servers, you double the performance
• Because you can have cool configurations like multi-datacenter and multi-rack and have the replication done automatically
• You can have several small, cheap, commodity servers, with big SATA disks with better result than one very big, very expensive, and unable-to-scale-more server with SSD or SAS expensive disks.
• It has the CQL language -Cassandra Query Language-, that is close to SQL
• Ability to send querys in async mode (the CPU can do other things while waiting for the query to return the results)
Cassandra is based in key/value philosophy but with columns. It supports multiple columns. That’s cool, as theoretically it supports 2 GB per column (at practical level is not recommended to go with data so big, specially in multi-user environments).
I will not lie to you: It is another paradigm, and comes with a lot of knowledge to acquire, but it is necessary and a price worth to pay for being able of scaling at nowadays required levels.
Cassandra only offers native drivers for: Java, .NET, C++ and Python 2.7. The rest of solutions are contributed, sadly most of them are outdated and unmantained.
You can find all the drivers here:
http://planetcassandra.org/client-drivers-tools/
# To develop with PHP
Cassandra has no PHP driver officially, but has some contributed solutions.
By myself I created several solutions: CQLSÍ uses cqlsh to perform queries and interfaces without needing Thrift, and Cassandra Universal Driver is a Web Gateway that I wrote in Python that allows you to query Cassandra from any language, and recently I contributed to a PHP driver that speaks the Cassandra binary protocol (v1) directly using Tcp/Ip sockets.
That’s the best solution for me by now, as it is the fastest and it doesn’t need any third party library nor Thrift neither.
You can git clone it from:
https://github.com/uri2x/php-cassandra
Here we go with some samples:
## Create a keyspace
KeySpace is the equivalent to a database in MySQL.
<?php
require_once 'Cassandra/Cassandra.php';
$o_cassandra = new Cassandra();$s_server_host = '127.0.0.1'; // Localhost
$i_server_port = 9042;$s_server_username = ''; // We don't use username
$s_server_password = ''; // We don't use password$s_server_keyspace = ''; // We don't have created it yet
$o_cassandra->connect($s_server_host, $s_server_username,$s_server_password, $s_server_keyspace,$i_server_port);
// Create a Keyspace with Replication factor 1, that's for a single server
$s_cql = "CREATE KEYSPACE cassandra_tests WITH REPLICATION = { 'class': 'SimpleStrategy', 'replication_factor': 1 };";$st_results = $o_cassandra->query($s_cql);
We can run it from web or from command line by using:
php -f cassandra_create.php
## Create a table
<?php
require_once 'Cassandra/Cassandra.php';
$o_cassandra = new Cassandra();$s_server_host = '127.0.0.1'; // Localhost
$i_server_port = 9042;$s_server_username = ''; // We don't use username
$s_server_password = ''; // We don't use password$s_server_keyspace = 'cassandra_tests';
$o_cassandra->connect($s_server_host, $s_server_username,$s_server_password, $s_server_keyspace,$i_server_port);
$s_cql = "CREATE TABLE carles_test_table (s_thekey text, s_column1 text, s_column2 text,PRIMARY KEY (s_thekey));";$st_results = $o_cassandra->query($s_cql);
If we don’t plan to insert UTF-8 strings, we can use VARCHAR instead of TEXT type.
## Do an insert
In this sample we create an Array of 100 elements, we serialize it, and then we store it.
<?php
require_once 'Cassandra/Cassandra.php';
// Note this code uses the MT notation http://blog.carlesmateo.com/maria-teresa-notation-for-php/
$i_start_time = microtime(true);$o_cassandra = new Cassandra();
$s_server_host = '127.0.0.1'; // Localhost$i_server_port = 9042;
$s_server_username = ''; // We don't have username$s_server_password = ''; // We don't have password
$s_server_keyspace = 'cassandra_tests';$o_cassandra->connect($s_server_host,$s_server_username, $s_server_password,$s_server_keyspace, $i_server_port);$s_time = strval(time()).strval(rand(0,9999));
$s_date_time = date('Y-m-d H:i:s'); // An array to hold a emails$st_data_emails = Array();
for ($i_bucle=0;$i_bucle<100; $i_bucle++) { // Add a new email$st_data_emails[] = Array('datetime' => $s_date_time, 'id_email' =>$s_time);
}
// Serialize the Array
$s_data_emails = serialize($st_data_emails);
$s_cql = "INSERT INTO carles_test_table (s_thekey, s_column1, s_column2) VALUES ('first_sample', '$s_data_emails', 'Some other data');";
$st_results =$o_cassandra->query($s_cql);$o_cassandra->close();
print_r($st_results);$i_finish_time = microtime(true);
$i_execution_time =$i_finish_time-$i_start_time; echo 'Execution time: '.$i_execution_time."\n";
echo "\n";
This insert took Execution time: 0.0091850757598877 seconds executed from CLI (Command line).
If the INSERT works well you’ll have a [result] => ‘success’ in the resulting array.
## Do some inserts
Here we do 9000 inserts.
<?php
require_once 'Cassandra/Cassandra.php';
// Note this code uses the MT notation http://blog.carlesmateo.com/maria-teresa-notation-for-php/
$i_start_time = microtime(true);$o_cassandra = new Cassandra();
$s_server_host = '127.0.0.1'; // Localhost$i_server_port = 9042;
$s_server_username = ''; // We don't have username$s_server_password = ''; // We don't have password
$s_server_keyspace = 'cassandra_tests';$o_cassandra->connect($s_server_host,$s_server_username, $s_server_password,$s_server_keyspace, $i_server_port);$s_date_time = date('Y-m-d H:i:s');
for ($i_bucle=0;$i_bucle<9000; $i_bucle++) { // Add a sample text, let's use time for example$s_time = strval(time());
$s_cql = "INSERT INTO carles_test_table (s_thekey, s_column1, s_column2) VALUES ('$i_bucle', '$s_time', 'http://blog.carlesmateo.com');"; // Launch the query$st_results = $o_cassandra->query($s_cql);
}
$o_cassandra->close();$i_finish_time = microtime(true);
$i_execution_time =$i_finish_time-$i_start_time; echo 'Execution time: '.$i_execution_time."\n";
echo "\n";
Those 9,000 INSERTs takes 6.49 seconds in my test virtual machine, executed from CLI (Command line).
## Do a Select
<?php
require_once 'Cassandra/Cassandra.php';
// Note this code uses the MT notation http://blog.carlesmateo.com/maria-teresa-notation-for-php/
$i_start_time = microtime(true);$o_cassandra = new Cassandra();
$s_server_host = '127.0.0.1'; // Localhost$i_server_port = 9042;
$s_server_username = ''; // We don't have username$s_server_password = ''; // We don't have password
$s_server_keyspace = 'cassandra_tests';$o_cassandra->connect($s_server_host,$s_server_username, $s_server_password,$s_server_keyspace, $i_server_port);$s_cql = "SELECT * FROM carles_test_table LIMIT 10;";
// Launch the query
$st_results =$o_cassandra->query($s_cql); echo 'Printing 10 rows:'."\n"; print_r($st_results);
$o_cassandra->close();$i_finish_time = microtime(true);
$i_execution_time =$i_finish_time-$i_start_time; echo 'Execution time: '.$i_execution_time."\n";
echo "\n";
Printing 10 rows passing the query with LIMIT:
$s_cql = "SELECT * FROM carles_test_table LIMIT 10;"; echoing as array with print_r takes Execution time: 0.01090407371521 seconds (the cost of printing is high). If you don’t print the rows, it takes only Execution time: 0.00714111328125 seconds. Selecting 9,000 rows, if you don’t print them, takes Execution time: 0.18086194992065. # Java The official driver for Java works very well. The only initial difficulties may be to create the libraries required with Maven and to deal with the different Cassandra native data types. To make that travel easy, I describe what you have to do to generate the libraries and provide you with a Db Class made by me that will abstract you from dealing with Data types and provide a simple ArrayList with the field names and all the data as String. Datastax provides the pom.xml for maven so you’ll create you jar files. Then you can copy those jar file to Libraries folder of any project you want to use Cassandra with. My Db class: /* * By Carles Mateo blog.carlesmateo.com * You can use this code freely, or modify it. */ package server; import java.util.ArrayList; import java.util.List; import com.datastax.driver.core.*; /** * @author carles_mateo */ public class Db { public String[] s_cassandra_hosts = null; public String s_database = "cchat"; public Cluster o_cluster = null; public Session o_session = null; Db() { // The Constructor this.s_cassandra_hosts = new String[10]; String s_cassandra_server = "127.0.0.1"; this.s_cassandra_hosts[0] = s_cassandra_server; this.o_cluster = Cluster.builder() .addContactPoints(s_cassandra_hosts[0]) // More than 1 separated by comas .build(); this.o_session = this.o_cluster.connect(s_database); // This is the KeySpace } public static String escapeApostrophes(String s_cql) { String s_cql_replaced = s_cql.replaceAll("'", "''"); return s_cql_replaced; } public void close() { // Destructor calles by the garbagge collector this.o_session.close(); this.o_cluster.close(); } public ArrayList query(String s_cql) { ResultSet rows = null; rows = this.o_session.execute(s_cql); ArrayList st_results = new ArrayList(); List<String> st_column_names = new ArrayList<String>(); List<String> st_column_types = new ArrayList<String>(); ColumnDefinitions o_cdef = rows.getColumnDefinitions(); int i_num_columns = o_cdef.size(); for (int i_columns = 0; i_columns < i_num_columns; i_columns++) { st_column_names.add(o_cdef.getName(i_columns)); st_column_types.add(o_cdef.getType(i_columns).toString()); } st_results.add(st_column_names); for (Row o_row : rows) { List<String> st_data = new ArrayList<String>(); for (int i_column=0; i_column<i_num_columns; i_column++) { if (st_column_types.get(i_column).equals("varchar") || st_column_types.get(i_column).equals("text")) { st_data.add(o_row.getString(i_column)); } else if (st_column_types.get(i_column).equals("timeuuid")) { st_data.add(o_row.getUUID(i_column).toString()); } else if (st_column_types.get(i_column).equals("integer")) { st_data.add(String.valueOf(o_row.getInt(i_column))); } // TODO: Implement other data types } st_results.add(st_data); } return st_results; } public static String getFieldFromRow(ArrayList st_results, int i_row, String s_fieldname) { List<String> st_column_names = (List)st_results.get(0); boolean b_column_found = false; int i_column_pos = 0; for (String s_column_name : st_column_names) { if (s_column_name.equals(s_fieldname)) { b_column_found = true; break; } i_column_pos++; } if (b_column_found == false) { return null; } int i_num_columns = st_results.size(); List<String> st_data = (List)st_results.get(i_row); String s_data = st_data.get(i_column_pos); return s_data; } } # Python 2.7 There is no currently driver for Python 3. I requested Datastax and they told me that they are working in a new driver for Python 3. To work with Datastax’s Python 2.7 driver: 1) Download the driver from http://planetcassandra.org/client-drivers-tools/ or git clone from https://github.com/datastax/python-driver 2) Install the dependencies for the Datastax’s driver ### Install python-pip (Installer) sudo apt-get install python-pip ### Install python development tools sudo apt-get install python-dev This is required for some of the libraries used by original Cassandra driver. ### Install Cassandra driver required libraries sudo pip install futures sudo pip install blist sudo pip install metrics sudo pip install scales ## Query Cassandra from Python The problem is the same as with Java, the different data types are hard to deal with. So I created a function convert_to_string that converts known data types to String, and so later we will only deal with Strings. In this sample, the results of the query are rendered in xml or in html. #!/usr/bin/env python # -*- coding: UTF-8 -*- # Use with Python 2.7+ __author__ = 'Carles Mateo' __blog__ = 'http://blog.carlesmateo.com' import sys from cassandra import ConsistencyLevel from cassandra.cluster import Cluster from cassandra.query import SimpleStatement s_row_separator = u"||*||" s_end_of_row = u"//*//" s_data = u"" b_error = 0 i_error_code = 0 s_html_output = u"" b_use_keyspace = 1 # By default use keyspace b_use_user_and_password = 1 # Not implemented yet def return_success(i_counter, s_data, s_format = 'html'): i_error_code = 0 s_error_description = 'Data returned Ok' return_response(i_error_code, s_error_description, i_counter, s_data, s_format) return def return_error(i_error_code, s_error_description, s_format = 'html'): i_counter = 0 s_data = '' return_response(i_error_code, s_error_description, i_counter, s_data, s_format) return def return_response(i_error_code, s_error_description, i_counter, s_data, s_format = 'html'): if s_format == 'xml': print ("Content-Type: text/xml") print ("") s_html_output = u"<?xml version='1.0' encoding='utf-8' standalone='yes'?>" s_html_output = s_html_output + '<response>' \ '<status>' \ '<error_code>' + str(i_error_code) + '</error_code>' \ '<error_description>' + '<![CDATA[' + s_error_description + ']]>' + '</error_description>' \ '<rows_returned>' + str(i_counter) + '</rows_returned>' \ '</status>' \ '<data>' + s_data + '</data>' \ '</response>' else: print("Content-Type: text/html; charset=utf-8") print("") s_html_output = str(i_error_code) s_html_output = s_html_output + '\n' + s_error_description + '\n' s_html_output = s_html_output + str(i_counter) + '\n' s_html_output = s_html_output + s_data + '\n' print(s_html_output.encode('utf-8')) sys.exit() return def convert_to_string(s_input): # Convert other data types to string s_output = s_input try: if value is not None: if isinstance(s_input, unicode): # string unicode, do nothing return s_output if isinstance(s_input, (int, float, bool, set, list, tuple, dict)): # Convert to string s_output = str(s_input) return s_output # This is another type, try to convert s_output = str(input) return s_output else: # is none s_output = "" return s_output except Exception as e: # Were unable to convert to str, will return as empty string s_output = "" return s_output def convert_to_utf8(s_input): return s_input.encode('utf-8') # ******************** # Start of the program # ******************** s_format = 'xml' # how you want this sample program to output s_cql = 'SELECT * FROM test_table;' s_cluster = '127.0.0.1' s_port = "9042" # default port i_port = int(s_port) b_use_keyspace = 1 s_keyspace = 'cassandra_tests' if s_keyspace == '': b_use_keyspace = 0 s_user = '' s_password = '' if s_user == '' or s_password == '': b_use_user_and_password = 0 try: cluster = Cluster([s_cluster], i_port) session = cluster.connect() except Exception as e: return_error(200, 'Cannot connect to cluster ' + s_cluster + ' on port ' + s_port + '.' + e.message, s_format) if (b_use_keyspace == 1): try: session.set_keyspace(s_keyspace) except: return_error(210, 'Keyspace ' + s_keyspace + ' does not exist', s_format) try: o_results = session.execute_async(s_cql) except Exception as e: return_error(300, 'Error executing query. ' + e.message, s_format) try: rows = o_results.result() except Exception as e: return_error(310, 'Query returned result error. ' + e.message, s_format) # Query returned values i_counter = 0 try: if rows is not None: for row in rows: i_counter = i_counter + 1 if i_counter == 1 and s_format == 'html': # first row is row titles for key, value in vars(row).iteritems(): s_data = s_data + key + s_row_separator s_data = s_data + s_end_of_row if s_format == 'xml': s_data = s_data + '' for key, value in vars(row).iteritems(): # Convert to string numbers or other types s_value = convert_to_string(value) if s_format == 'xml': s_data = s_data + '<' + key + '>' + '<![CDATA[' + s_value + ']]>' + '' else: s_data = s_data + s_value s_data = s_data + s_row_separator if s_format == 'xml': s_data = s_data + '' else: s_data = s_data + s_end_of_row except Exception as e: # No iterable data return_success(i_counter, s_data, s_format) # Just print the data return_success(i_counter, s_data, s_format) If you did not create the namespace like in the samples before, change those lines to: s_cql = 'CREATE KEYSPACE cassandra_tests WITH REPLICATION = { \'class\': \'SimpleStrategy\', \'replication_factor\': 1 };' s_cluster = '127.0.0.1' s_port = "9042" # default port i_port = int(s_port) b_use_keyspace = 1 s_keyspace = '' Run the program to create the Keyspace and you’ll get: carles@ninja8:~/Desktop/codi/python/test$ ./lunacloud-create.py
Content-Type: text/xml
<error_code>0<error_description>
Then you can create the table simply by setting:
s_cql = 'CREATE TABLE test_table (s_thekey text, s_column1 text, s_column2 text,PRIMARY KEY (s_thekey));'
s_cluster = '127.0.0.1'
s_port = "9042" # default port
i_port = int(s_port)
b_use_keyspace = 1
s_keyspace = 'cassandra_tests'
Cassandra Universal Driver
As mentioned above if you use a language Tcp/Ip enabled very new, or very old like ASP or ColdFusion, or from Unix command line and you want to use it with Cassandra, you can use my solution http://www.cassandradriver.com/.
It is basically a Web Gateway able to speak XML, JSon or CSV alike. It relies on the official Datastax’s python driver.
It is not so fast as a native driver, but it works pretty well and allows you to split your architecture in interesting ways, like intermediate layers to restrict even more security (For example WebServers may query the gateway, that will enstrict tome permissions instead of having direct access to the Cassandra Cluster. That can also be used to perform real-time map-reduce operations on the amount of data returned by the Cassandras, so freeing the webservers from that task and saving CPU).
Tip: If you use Cassandra for Development only, you can limit the amount of memory used by editing the file /etc/cassandra/cassandra-env.sh and hardcoding:
# limit the memory for development environment
# --------------------------------------------
system_memory_in_mb="512"
system_cpu_cores="1"
# --------------------------------------------
Just before the line:
# set max heap size based on the following
That way Cassandra will believe your system memory is 512 MB and reserve only 256 MB for its use.
# Troubleshooting apps in Linux
Let’s say you are on a system and a program stops working.
You check the space on disk, check that no one has modified the config files, check things like dns, etc… everything seems normal and you don’t know what else to check.
It could be that the filesystem got corrupted after a powerdown, for example, and one file or more are corrupted and this would be hard to figure out.
To find whats going wrong then you can use strace.
In the simplest case strace runs the specified command until it exits. It intercepts and records the system calls which are called by a process and the signals which are received by a process. The name of each system call, its arguments and its return value are printed on standard error or to the file specified with the -o option.
http://linux.die.net/man/1/strace
As you may know the programs request system calls, and get signals from the Operating System/Kernel.
strace will show all those requests done by the program, and the signals received. That means that you will see the requests from the program to the kernel to open a file, for example a config file.
Executing:
strace /usr/bin/ssh
That is the sample output:
strace /usr/bin/ssh
execve("/usr/bin/ssh", ["/usr/bin/ssh"], [/* 61 vars */]) = 0
brk(0) = 0x7fc71509c000
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
mmap(NULL, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc713cb2000
access("/etc/ld.so.preload", R_OK) = -1 ENOENT (No such file or directory)
open("/etc/ld.so.cache", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=126104, ...}) = 0
mmap(NULL, 126104, PROT_READ, MAP_PRIVATE, 3, 0) = 0x7fc713c93000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libselinux.so.1", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=134224, ...}) = 0
mmap(NULL, 2234088, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc713870000
mprotect(0x7fc71388f000, 2097152, PROT_NONE) = 0
mmap(0x7fc713a8f000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x1f000) = 0x7fc713a8f000
mmap(0x7fc713a91000, 1768, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_ANONYMOUS, -1, 0) = 0x7fc713a91000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libcrypto.so.1.0.0", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=1934816, ...}) = 0
mmap(NULL, 4045240, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc713494000
mprotect(0x7fc713646000, 2097152, PROT_NONE) = 0
mmap(0x7fc713846000, 155648, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x1b2000) = 0x7fc713846000
mmap(0x7fc71386c000, 14776, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_ANONYMOUS, -1, 0) = 0x7fc71386c000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libdl.so.2", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=14664, ...}) = 0
mmap(NULL, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc713c92000
mmap(NULL, 2109736, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc713290000
mprotect(0x7fc713293000, 2093056, PROT_NONE) = 0
mmap(0x7fc713492000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x2000) = 0x7fc713492000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libz.so.1", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=100728, ...}) = 0
mmap(NULL, 2195784, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc713077000
mprotect(0x7fc71308f000, 2093056, PROT_NONE) = 0
mmap(0x7fc71328e000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x17000) = 0x7fc71328e000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libresolv.so.2", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=97144, ...}) = 0
mmap(NULL, 2202280, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc712e5d000
mprotect(0x7fc712e73000, 2097152, PROT_NONE) = 0
mmap(0x7fc713073000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x16000) = 0x7fc713073000
mmap(0x7fc713075000, 6824, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_ANONYMOUS, -1, 0) = 0x7fc713075000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/usr/lib/x86_64-linux-gnu/libgssapi_krb5.so.2", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=252704, ...}) = 0
mmap(NULL, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc713c91000
mmap(NULL, 2348608, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc712c1f000
mprotect(0x7fc712c5a000, 2097152, PROT_NONE) = 0
mmap(0x7fc712e5a000, 12288, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x3b000) = 0x7fc712e5a000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libc.so.6", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0755, st_size=1853400, ...}) = 0
mmap(NULL, 3961912, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc712857000
mprotect(0x7fc712a14000, 2097152, PROT_NONE) = 0
mmap(0x7fc712c14000, 24576, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x1bd000) = 0x7fc712c14000
mmap(0x7fc712c1a000, 17464, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_ANONYMOUS, -1, 0) = 0x7fc712c1a000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libpcre.so.3", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=256224, ...}) = 0
mmap(NULL, 2351392, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc712618000
mprotect(0x7fc712655000, 2097152, PROT_NONE) = 0
mmap(0x7fc712855000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x3d000) = 0x7fc712855000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
fstat(3, {st_mode=S_IFREG|0755, st_size=135757, ...}) = 0
mmap(NULL, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc713c90000
mmap(NULL, 2212936, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc7123fb000
mprotect(0x7fc712412000, 2097152, PROT_NONE) = 0
mmap(0x7fc712612000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x17000) = 0x7fc712612000
mmap(0x7fc712614000, 13384, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_ANONYMOUS, -1, 0) = 0x7fc712614000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/usr/lib/x86_64-linux-gnu/libkrb5.so.3", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=848672, ...}) = 0
mmap(NULL, 2944608, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc71212c000
mprotect(0x7fc7121f1000, 2093056, PROT_NONE) = 0
mmap(0x7fc7123f0000, 45056, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0xc4000) = 0x7fc7123f0000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/usr/lib/x86_64-linux-gnu/libk5crypto.so.3", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=158136, ...}) = 0
mmap(NULL, 2257008, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc711f04000
mprotect(0x7fc711f2a000, 2093056, PROT_NONE) = 0
mmap(0x7fc712129000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x25000) = 0x7fc712129000
mmap(0x7fc71212b000, 112, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_ANONYMOUS, -1, 0) = 0x7fc71212b000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libcom_err.so.2", O_RDONLY|O_CLOEXEC) = 3
mmap(NULL, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc713c8f000
fstat(3, {st_mode=S_IFREG|0644, st_size=14592, ...}) = 0
mmap(NULL, 2109896, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc711d00000
mprotect(0x7fc711d03000, 2093056, PROT_NONE) = 0
mmap(0x7fc711f02000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x2000) = 0x7fc711f02000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/usr/lib/x86_64-linux-gnu/libkrb5support.so.0", O_RDONLY|O_CLOEXEC) = 3
read(3, "\177ELF\2\1\1\0\0\0\0\0\0\0\0\0\3\0>\0\1\0\0\0@ \0\0\0\0\0\0"..., 832) = 832
fstat(3, {st_mode=S_IFREG|0644, st_size=31160, ...}) = 0
mmap(NULL, 2126632, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc711af8000
mprotect(0x7fc711aff000, 2093056, PROT_NONE) = 0
mmap(0x7fc711cfe000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x6000) = 0x7fc711cfe000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libkeyutils.so.1", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=14256, ...}) = 0
mmap(NULL, 2109456, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc7118f4000
mprotect(0x7fc7118f6000, 2097152, PROT_NONE) = 0
mmap(0x7fc711af6000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x2000) = 0x7fc711af6000
close(3) = 0
mmap(NULL, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc713c8e000
mmap(NULL, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc713c8d000
mmap(NULL, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc713c8b000
arch_prctl(ARCH_SET_FS, 0x7fc713c8b840) = 0
munmap(0x7fc713c93000, 126104) = 0
set_robust_list(0x7fc713c8bb20, 24) = 0
futex(0x7fff5c43f09c, FUTEX_WAIT_BITSET_PRIVATE|FUTEX_CLOCK_REALTIME, 1, NULL, 7fc713c8b840) = -1 EAGAIN (Resource temporarily unavailable)
rt_sigaction(SIGRTMIN, {0x7fc7124017e0, [], SA_RESTORER|SA_SIGINFO, 0x7fc71240abb0}, NULL, 8) = 0
rt_sigaction(SIGRT_1, {0x7fc712401860, [], SA_RESTORER|SA_RESTART|SA_SIGINFO, 0x7fc71240abb0}, NULL, 8) = 0
rt_sigprocmask(SIG_UNBLOCK, [RTMIN RT_1], NULL, 8) = 0
getrlimit(RLIMIT_STACK, {rlim_cur=8192*1024, rlim_max=RLIM64_INFINITY}) = 0
statfs("/sys/fs/selinux", 0x7fff5c43f090) = -1 ENOENT (No such file or directory)
statfs("/selinux", 0x7fff5c43f090) = -1 ENOENT (No such file or directory)
brk(0) = 0x7fc71509c000
brk(0x7fc7150bd000) = 0x7fc7150bd000
open("/proc/filesystems", O_RDONLY) = 3
fstat(3, {st_mode=S_IFREG|0444, st_size=0, ...}) = 0
mmap(NULL, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc713cb1000
close(3) = 0
munmap(0x7fc713cb1000, 4096) = 0
open("/dev/null", O_RDWR) = 3
close(3) = 0
openat(AT_FDCWD, "/proc/13672/fd", O_RDONLY|O_NONBLOCK|O_DIRECTORY|O_CLOEXEC) = 3
getdents(3, /* 6 entries */, 32768) = 144
getdents(3, /* 0 entries */, 32768) = 0
close(3) = 0
getuid() = 1000
geteuid() = 1000
setresuid(-1, 1000, -1) = 0
socket(PF_LOCAL, SOCK_STREAM|SOCK_CLOEXEC|SOCK_NONBLOCK, 0) = 3
connect(3, {sa_family=AF_LOCAL, sun_path="/var/run/nscd/socket"}, 110) = -1 ENOENT (No such file or directory)
close(3) = 0
socket(PF_LOCAL, SOCK_STREAM|SOCK_CLOEXEC|SOCK_NONBLOCK, 0) = 3
connect(3, {sa_family=AF_LOCAL, sun_path="/var/run/nscd/socket"}, 110) = -1 ENOENT (No such file or directory)
close(3) = 0
open("/etc/nsswitch.conf", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=513, ...}) = 0
mmap(NULL, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc713cb1000
read(3, "# /etc/nsswitch.conf\n#\n# Example"..., 4096) = 513
close(3) = 0
munmap(0x7fc713cb1000, 4096) = 0
open("/etc/ld.so.cache", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=126104, ...}) = 0
mmap(NULL, 126104, PROT_READ, MAP_PRIVATE, 3, 0) = 0x7fc713c93000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libnss_compat.so.2", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=35728, ...}) = 0
mmap(NULL, 2131288, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc7116eb000
mprotect(0x7fc7116f3000, 2093056, PROT_NONE) = 0
mmap(0x7fc7118f2000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x7000) = 0x7fc7118f2000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libnsl.so.1", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=97296, ...}) = 0
mmap(NULL, 2202360, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc7114d1000
mprotect(0x7fc7114e8000, 2093056, PROT_NONE) = 0
mmap(0x7fc7116e7000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x16000) = 0x7fc7116e7000
mmap(0x7fc7116e9000, 6904, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_ANONYMOUS, -1, 0) = 0x7fc7116e9000
close(3) = 0
munmap(0x7fc713c93000, 126104) = 0
open("/etc/ld.so.cache", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=126104, ...}) = 0
mmap(NULL, 126104, PROT_READ, MAP_PRIVATE, 3, 0) = 0x7fc713c93000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libnss_nis.so.2", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=47760, ...}) = 0
mmap(NULL, 2143616, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc7112c5000
mprotect(0x7fc7112d0000, 2093056, PROT_NONE) = 0
mmap(0x7fc7114cf000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0xa000) = 0x7fc7114cf000
close(3) = 0
access("/etc/ld.so.nohwcap", F_OK) = -1 ENOENT (No such file or directory)
open("/lib/x86_64-linux-gnu/libnss_files.so.2", O_RDONLY|O_CLOEXEC) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=52160, ...}) = 0
mmap(NULL, 2148504, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc7110b8000
mprotect(0x7fc7110c4000, 2093056, PROT_NONE) = 0
mmap(0x7fc7112c3000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0xb000) = 0x7fc7112c3000
close(3) = 0
munmap(0x7fc713c93000, 126104) = 0
open("/etc/passwd", O_RDONLY|O_CLOEXEC) = 3
lseek(3, 0, SEEK_CUR) = 0
fstat(3, {st_mode=S_IFREG|0644, st_size=1823, ...}) = 0
mmap(NULL, 1823, PROT_READ, MAP_SHARED, 3, 0) = 0x7fc713cb1000
lseek(3, 1823, SEEK_SET) = 1823
munmap(0x7fc713cb1000, 1823) = 0
close(3) = 0
write(2, "usage: ssh [-1246AaCfgKkMNnqsTtV"..., 466usage: ssh [-1246AaCfgKkMNnqsTtVvXxYy] [-b bind_address] [-c cipher_spec]
[-D [bind_address:]port] [-e escape_char] [-F configfile]
[-I pkcs11] [-i identity_file]
[-l login_name] [-m mac_spec] [-O ctl_cmd] [-o option] [-p port]
[-W host:port] [-w local_tun[:remote_tun]]
[user@]hostname [command]
) = 466
exit_group(255) = ?
+++ exited with 255 +++
You can also generate a log with that info:
strace -o test_log.txt /usr/bin/ssh
Let’s pay attention to the open files:
Here we can see what files were open, the mode and the result.
So if your program failen opening a certain file you will see it on the traces.
Also we can review the access:
cat test_log.txt | grep access --after-context=2
You can specify to trace only certain set of system calls by passing parameter -e trace=open,close,read,write,stat,chmod,unlink or -e trace=network or -e trace=process or -e trace=memory or -e trace=ipc or -e trace=signal etcetera.
Can also dump data read -e read=set or -e write=set for a full hexadecimal and ASCII dump of all the data written to file descriptors listed in the specified set… or -e signal=set (default signal=all) or even by negation -e signal =! SIGIO (or signal=!io)…
You can also trace libraries with ltrace or processes with ptrace.
And see the open files with lsof.
You can use lsof to see the TCP connections:
lsof -iTCP:80
You can also know information of what process is owner of a tcp/udp connection:
netstat -tnp
Take a look at ss for advanced sockets inspecting.
Of course you will find very interesting info on /proc pseudo-filesystem.
You can troubleshoot the environment for the process by doing:
strings /proc/1714/environ`
Where 1714 is the process id, whatever.
/proc/[pid]/fd/ is a subdirectory containing one entry for each file open by the process, named by its file descriptor, being a symbolic link to the actual file.
/proc/[pid]/fdinfo/ will show information on the flags for the access mode of the open files and /proc/[pid]/io contains input/outputs statistics for the process.
# The Cloud is for Scaling
The Cloud is for Startups, and for Scaling. Nothing more.
In the future will be used by phone operators, to re-dimension their infrastructure and bandwidth in real time according to demand, but nowadays the Cloud is for Startups.
Examine the prices in my post in cmips, take a look, examine the performance also of the different CPU. You see that according to CMIPS v.1.03 a Desktop Processor Intel i7-4770S, worth USD $300, performs better than an Amazon M2 High Memory Quadruple Extra Large and than a Rackspace First gen. 30 GB RAM 8 Cores?. Today the public cost of an Amazon M2 High Memory Quadruple Extra Large running for a month is USD$1,180.80 so USD $1.64 per hour and the Rackspace First Generation 30 GB RAM 8 Cores 1200 GB of disk costs is USD$1,425.60 so USD $1.98 per hour running. And that’s the key, the cost per hour. Because the greatness, the majesty of the Cloud is that you pay per hour, you pay as you need, or as you go. No attaching contracts. All on demand. I had my company at a time where the hosting companies and the Data Centers were forcing customers to sign yearly contracts. What if a company only needs to host their Servers for three months? What if they have to close?. No options. You take it or you leave it. Even renting a dedicated hosting was for at least a month or more, and what if the latency was not good? What if the bandwidth of the provider was not enough?. Amazon irrupted in the market with strength. I really like that company because they grew the best eCommerce company for buying books, they did a system that really worked, and was able to recommend very useful computer books, and the delivery, logistics was so good, also post-sales service. They simply started to rent the same infrastructure they were using to attend their millions of customers and was a total success. And for a while few people knew about Amazon deep technologies and functionalities, but later became a fashion. Now people is using Amazon or whatever provider/Service that contains the word “Cloud” because the Cloud is in the mouth of everyone. Magazines and newspapers speak about the Cloud, so many many companies use it simply because everyone is talking about the Cloud. And those ISP that didn’t had a Cloud have invested heavily to create a Cloud, just because they didn’t want to be the ones without a Cloud, since everyone was asking for it and all the ISP companies were offering their “Clouds”. Every company claims to have “Cloud” where the only many of them have is Vmware servers, Xen servers, Open Stack… running the tenants or instances of the customers always on the same host servers. No real Cloud, professional Cloud, abstract layered in a Professional way like Amazon, only the traditional “shared hosting” with another name, sharing CPU and RAM and Disk storage using virtual machines called instances. So, Cloud fashion has become a confusing craziness where no one knows why they are in the Cloud but they believe they have to be in. But do companies need the Cloud?. Cloud instances? It depends. The best would be to ask that companies Why you choose the Cloud?. If you compare the cost of having an instance in the Cloud, is much much more expensive than having a dedicated server. And for that high cost you don’t get more performance. Virtualization is always slower and disk speed is always an issue in Cloud providers, where all the data travels via network from the disk cabins NAS to the Host servers running the guest instances. Data cannot be at local disks, since every time you start an instance, the resources like CPU and RAM are provisioned, and your instance run in totally different hardware. Only your data remain in the NAS (Network Attached Storage). So unless you run your in-the-Cloud instance in a special provider that offers local disks, like DigitalOcean that offers SSD but monthly paying, (and so you pay the price by losing the hardware abstraction capability because you’re attached to the CPU that has the disk connected, and also you loss the flexibility of paying per hour of use, as you go), then you’ll face a bottleneck that is the hard disk performance (that for real takes all the data from NAS, where is stored, through the local network). So what are the motivations to use the Cloud?. I try to put some examples, out of these it has no much sense, I think. You can send me your happy-in-Cloud scenarios if you found other good uses. Example A) Saving initial costs, avoid contract attachment and grow easily own-made Imagine a Developer that start its own project. May be it works, may be not, but instead of having a monthly contract for a dedicated server, he starts with an Amazon Free Tier (better not, use Small instance at least) and runs a web. If it does not work, simply stop the instance and pay no more. If the project works and has more and more users he can re-dimension the server with a click. Just stop the instance, change the type of instance, start it again with more RAM and more CPU power. Fast. Hiring a dedicated server implies at least monthly contracts, average of USD$100 per month, and is not easy to move to a bigger server, not fast and is expensive as it requires the ISP tech guys to move the data, to migrate from a Server to another.
Also the available bandwidth is to be taken in consideration. Bandwidth is expensive and Amazon can offer 150 Mbit to smaller machines. Not all the Internet Service Providers can offer that bandwidth even with most advanced packets.
If the project still grows, with a click, in seconds, 20 instances with a lot of bandwidth can be deployed and serving traffic to your customers very quick.
You save the init costs of buying Servers, and the time to deal with hardware, bandwidth limitations and avoid contracts, but you pay an hourly rate a lot more expensive. So in the long run is much much expensive using Amazon and less powerful than having dedicated servers. That happened to Zynga, that was paying $63M annually to Amazon and decided to step back from Amazon to their own Data Centers again. (another fortune tech link) The limited CPU power was also a deal breaker for many companies that needed really powerful CPU and gigs of RAM for their Database Servers. Now this situation is much better with the introduction of the new Servers. This developer can benefit from doing bacups with a click, cloning, starting instances from an image, having more static ip’s with a click, deploying built-in (from the Cloud provider) load balancers, using monitoring services like CloudWatch, creating Volumes and attaching to the servers for additional space… Example B) An Startup with fluctuating number of users and hopes of growing Imagine an Startup with a wonderful Facebook Application. During 80% of the day has few visits, may be only need 3 Servers, but during 20% of the hours of the day from 10:00 to 15:00 users connect like hell, so they need 20 servers to attend this traffic and workload, and may be tomorrow needs 30 servers. With the Cloud they pay for 3 servers 24 hours per day and for the other 17 servers only pay the hours they are on, that’s 5 hours per day. Doing that they save money and they have an unlimited * amount of power. (* There are limits for real, you have to specially request authorisation to run more than default max. servers for the zone, that is normally 20 instances for Amazon. Also it can happen theoretically that when you request new instances the Zone has no instances available). So well, for an Startup growing, avoiding hiring 20 dedicated servers and instead running into the Cloud as many as they need, for just the time they need, Auto-Scaling up and down, and can use the servers NOW and pay the next month with Visa card, all of that can make a difference for a growing Startup. If the servers chosen are not powerful enough that is solved with a click, changing instance type. So fast. A minute. It’s only a matter of money. Example C) e-Learning companies and online universities e-Learning platforms also get benefits from the Auto-Scaling for the full occupation hours. The built-in functionalities of the Cloud to clone instances is very useful to deploy new web servers, or new environments for students doing practices, in the case of teaching Information Technology subjects, where the users need to practice against a real server (Linux or windows). Those servers can be created and destroyed, cloned from the main -ready to go- template. And also servers can be scheduled to stop at a certain hour and to start also, so saving the money from the hours not needed. Example D) Digital agencies, sports and other events When there is an Special event, like motorcycle running, when a Football Team scores, when there is an spot in tv announcing a product… At those moments the traffic to the site can multiply, so more servers and more bandwidth have to be deployed instantly. That cannot be done with physical servers, hardware, but is very easy to provision instances from the Cloud. Mass mailing email campaigns can also benefit from creating new Servers when needed. Example E) Proximity and SEO Cloud providers have Data Centres everywhere. If you want to have servers in Asia, or static content to be deployed faster, or in South-America, or in Europe… the Cloud providers have plenty of Data Centers all over the world. Example F) Game aficionado and friends sharing contents People that loves cooperative games can find the needed hungry bandwidth and at a moderate price. If they run their private server few hours, at night, from 22:00 to 01:00 as example, they will benefit from a great bandwidth from the big Cloud provider and pay only 3 hours per day (the exceed of traffic uses to be paid in most providers, but price of additional GB uses to be really really competitive). Friends sharing contents in an Ftp also, can benefit from this Cloud servers, but probably they will find more easy to use services like Dropbox. Example G) Startup serving contents An Startup serving videos, images, or books, can benefit not only from the great bandwidth of big Cloud providers (this has been covered before), but for a very cheap price for exceeding Gigabyte transferred. Local ISP can’t offer 150 Mbit for an instance of USD$20 and USD \$0.12 per additional GB transferred.
Many Cloud providers also allow unlimited incoming traffic from the Internet, and from Server to Server through private ip’s.
Other cases
For other cases Dedicated Servers are much more Powerful, faster and cheaper, at the price of being “static” in the sense of attached, not layer abstracted, but all the aspects of your Project have to be taken in count before deciding stepping into or out of the Cloud.
In general terms I would say that the Cloud is for Scaling.
# NAS and Gigabit
I’ve found this problem in several companies, and I’ve had to show their error and convince experienced SysAdmins, CTOs and CEO about the erroneous approach. Many of them made heavy investments in NAS, that they are really wasting, and offering very poor performance.
Normally the rack servers have their local disks, but for professional solutions, like virtual machines, blade servers, and hundreds of servers the local disk are not used.
NAS – Network Attached Storage- Servers are used instead.
This NAS Servers, when are powerful (and expensive) offer very interesting features like hot backups, hot backups that do not slow the system (the most advanced), hot disk replacement, hot increase of total available space, the Enterprise solutions can replicate and copy data from different NAS in different countries, etc…
Smaller NAS are also used in configurations like Webservers’ Webfarms, were all the nodes has to have the same information replicated, and when a used uploads a new profile image, has to be available to all the webservers for example.
In this configurations servers save and retrieve the needed data from the NAS Servers, through LAN (Local Area Network).
The main error I have seen is that no one ever considers the pipe where all the data is travelling, so most configurations are simply Gigabit, and so are bottleneck.
Imagine a Dell blade server, like this in the image on the left.
This enclosure hosts 16 servers, hot plugable, with up to two CPU’s each blade, we also call those blade servers “pizza” (like we call before to rack servers).
A common use is to use those servers to have Vmware, OpenStack, Xen or other virtualization software, so the servers run instances of customers. In this scenario the virtual disks (the hard disk of the virtual machines) are stored in the NAS Server.
So if a customer shutdown his virtual server, and start it later, the physical server where its virtual machine is running will be another, but the data (the disk of the virtual server) is stored in the NAS and all the data is saved and retrieved from the NAS.
The enclosure is connected to the NAS through a Gigabit connection, as 10 Gigabit connections are still too expensive and not yet supported in many servers.
Once we have explained that, imagine, those 16 servers, each with 4 or 5 virtual machines, accessing to their disks through a Gigabit connection.
If only one of these 80 virtual machines is accessing to disk, the will be no problem, but if more than one is accessing the Gigabit connection, that’s a maximum of 125 MB (Megabytes) per second, will be shared among all the virtual machines.
So imagine, 70 virtual machines are accessing NAS to serve web pages, with not much traffic, OK, but the other 10 virtual machines are doing heavy data transmission: for example one is serving data through FTP server, the other is broadcasting video, the other is copying heavy log files, and so… Imagine that scenario.
The 125 MB per second is divided between the 80 servers, so those 10 servers using extensively the disk will monopolize the bandwidth, but even those 10 servers will have around 12,5 MB each, that is 100 Mbit each and is very slow.
Imagine one of the virtual machines broadcast video. To broadcast video, first it has to get it from the NAS (the chunks of data), so this node serving video will be able to serve different videos to few customers, as the network will not provide more than 12,5 MB under the circumstances provided.
This is a simplified scenario, as many other things has to be taken in count, like the SATA, SCSI and SAS disks do not provide sustained speeds, speed depends on locating the info, fragmentation, etc… also has to be considered that NAS use protocol iSCSI, a sort of SCSI commands sent through the Ethernet. And Tcp/Ip uses verifications in their protocol, and protocol headers. That is also an overhead. I’ve considered only traffic in one direction, so the servers downloading from the NAS, as assuming Gigabit full duplex, so Gigabit for sending and Gigabit for receiving.
So instead of 125 MB per second we have available around 100 MB per second with a Gigabit or even less.
Also the virtualization servers try to handle a bit better the disk access, by keeping a cache in memory, and not writing immediately to disk.
So you can’t do dd tests in virtual machines like you would do in any Linux with local disks, and if you do go for big files, like 10 GB with random data (not just 0, they have optimizations for that).
Let’s recalculate it now:
70 virtual machines using as low as 0.10 MB/second each, that’s 7 MB/second. That’s really optimist as most webservers running PHP read many big files for attending a simple request and webservers server a lot of big images.
10 virtual machines using extensively the NAS, so sharing 100 MB – 7 MB = 93 MB. That is 9.3 MB each.
So under these circumstances for a virtual machine trying to read from disk a file of 1 GB (1000 MB), this operation will take 107 seconds, so 1:47 minutes.
So with this considerations in mind, you can imagine that the performance of the virtual machines under those configurations are leaved to the luck. The luck that nobody else of the other guests in the servers are abusing the disk I/O.
I’ve explained you in a theoretical plan. Sadly reality is worst. A lot worst. Those 70 web virtual machines with webservers will be so slow that they will leave your company very disappointed, and the other 10 will not even be happier.
One of the principal problems of Amazon EC2 has been always disk performance. Few months ago they released IOPS, high performance disks, that are more expensive, but faster.
It has to be recognized that in Amazon they are always improving.
They have also connection between your servers at 10 Gbit/second.
Returning to the Blades and NAS, an easy improvement is to aggregate two Gigabits, so creating a connection of 2 Gbit. This helps a bit. Is not the solution, but helps.
Probably different physical servers with few virtual machines and a dedicated 1 Gbit connection (or 2 Gbit by 1+1 aggregated if possible) to the NAS, and using local disks as much as possible would be much better (harder to maintain at big scale, but much much better performance).
But if you provide infrastructure as a Service (IaS) go with 2 x 10 Gbit Fibre aggregated, so 20 Gigabit, or better aggregate 2 x 20 Gbit Fibre. It’s expensive, but crucial.
Now compare the 9.3 MB per second, or even the 125 MB theoretical of Gigabit of the average real sequential read of 50 MB/second that a SATA disk can offer when connected on local, or nearly the double for modern SAS 15.000 rpm disks… (writing is always slower)
… and the 550 MB/s for reading and 550 MB/s for writing that some SSD disks offer when connected locally. (I own two OSZ SSD disks that performs 550 MB).
I’ve seen also better configuration for local disks, like a good disk controller with Raid 5 and disks SSD. With my dd tests I got more than 900 MB per second for writing!.
So if you are going to spend 30.000 € in your NAS with SATA disks (really bad solution as SATA is domestic technology not aimed to work 24×7 and not even fast) or SAS disks, and 30.000 € more in your blade servers, think very well what you need and what configuration you will use. Contact experts, but real experts, not supposedly real experts.
Otherwise you’ll waste your money and your customers will have very very poor performance on these times where applications on the Internet demand more and more performance. | 2018-11-18 16:19:52 | {"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": 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": 0.23338930308818817, "perplexity": 6227.333405307159}, "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/1542039744513.64/warc/CC-MAIN-20181118155835-20181118181835-00512.warc.gz"} |
https://ask.libreoffice.org/en/question/285380/if-replacing-text-how-do-you-add-an-extra-row-underneath/ | # if replacing text, how do you add an extra row underneath?
Eg I wanted to replace all cells that contain “mike007” with “mike007” and then adding an extra row under that cell that has “moppy” writtenin it, how would you go about doing this? https://i.imgur.com/Gva61j7.png so i want to change it to like this https://i.imgur.com/wlWrcqB.png
edit retag close merge delete
1
how would you go about doing this
Write a macro.
( 2021-01-02 12:38:32 +0100 )edit
Sort by » oldest newest most voted
Hi @michaely,
as Opaque said, you'll most likely need a macro for this.
Here is something that already mostly does what you want. But you might have to patch something to match your exact requirements.
Sub SearchAndDoStuff
Const needle = "mike007"
Const searchRange = "A1:Z100"
Set oSheet = ThisComponent.CurrentController.ActiveSheet
Set oRange = oSheet.getCellRangeByName(searchRange)
For i = oRange.Rows.getCount() - 1 To 0 Step -1 rem bottom up
For j = 0 To oRange.Columns.getCount() - 1
Set oCell = oRange.getCellByPosition( j, i )
idx = InStr(oCell.String,needle)
If ( idx > 0) Then
oCell.String = needle
oSheet.Rows.insertByIndex(i+1,j) REM Add Row below
oCell = oRange.getCellByPosition( j, i+1 )
oCell.String = "moppy"
Endif
Next
Next
End Sub
Result:
If one need further help, just ask in the comments. Hope it helps.
Update: 2021-01-21
To add a row above the found term, one can use this:
Sub SearchAndDoStuff2
Const needle = "mike007"
Const searchRange = "A1:C6"
Set oSheet = ThisComponent.CurrentController.ActiveSheet
Set oRange = oSheet.getCellRangeByName(searchRange)
jump = False
For i = 0 To oRange.Rows.getCount() - 1
For j = 0 To oRange.Columns.getCount() - 1
Set oCell = oRange.getCellByPosition( j, i )
idx = InStr(oCell.String,needle)
If ( idx > 0) Then
oCell.String = needle
oSheet.Rows.insertByIndex(i,1)
oRange = oSheet.getCellRangeByName(searchRange)
oCell = oRange.getCellByPosition( j, i )
oCell.String = "moppy"
jump=True
Endif
Next
if jump then
i=i+1
endif
jump=False
Next
End Sub
Result:
If one need further help, just ask in the comments. Hope it helps.
more
1
Many thanks!
( 2021-01-02 22:56:09 +0100 )edit
@michaely If an answer solved your problem or answerd your question, please mark it as correct. Helps to keep the site usable for everybody. Thanks.
( 2021-01-03 23:06:00 +0100 )edit
The macro runs almost perfectly, one small problem though - for some reason the macro only runs on cells up to row 99, anything past row 100 it's not inserting any new content
( 2021-01-14 15:16:03 +0100 )edit
1
You can change the searchRange from A1:Z100 for example to A1:Z1000 to include rows up to 999, and so on.
( 2021-01-14 17:15:00 +0100 )edit
one quick last question buddy, if I wanted to add an extra row ABOVE (Rather than underneath) can you tell me what change I'd make to the script?
( 2021-01-21 14:32:35 +0100 )edit
1
buddy ... is an unusual way to address someone, but ... i dont mind. I updated my answer, but my solution is a little more complex than i personally think it should be, but ... it works. Improvements welcome.
( 2021-01-21 15:57:46 +0100 )edit
## Stats
Asked: 2021-01-02 10:31:36 +0100
Seen: 75 times
Last updated: Jan 21 | 2021-03-03 22:26:54 | {"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": 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": 0.18326328694820404, "perplexity": 7395.401185250946}, "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-2021-10/segments/1614178367790.67/warc/CC-MAIN-20210303200206-20210303230206-00158.warc.gz"} |
https://quantumcomputing.stackexchange.com/questions/17385/hhl-algorithm-for-linear-systems-with-a-real-matrix-and-a-real-right-side | # HHL algorithm for linear systems with a real matrix and a real right side
HHL algorithm can be used for solving linear system $$A|x\rangle=|b\rangle$$. If we put $$|b\rangle$$ (to be precise its normalized version) into the algorithm and measuring ancilla to be $$|1\rangle$$ we are left with the state $$\sum_{i=1}^n \beta_i\frac{C}{\lambda_i}|x_i\rangle,$$ where $$|x_i\rangle$$ is ith eigenvector of matrix $$A$$, $$\lambda_i$$ is respecitve eigenvalue and $$\beta_i = \langle b|x_i\rangle$$ is ith coordinate of $$|b\rangle$$ in basis composed of eigenvectors of $$A$$.
It is known that HHL brings exponential speed-up, however, to get whole state $$|x\rangle$$ we need to do a tomography which cancels the speed-up completely.
However, let us assume that $$A$$ is real matrix and $$|b\rangle$$ is real vector. Since HHL assumes that $$A$$ is Hermitian, $$\lambda_i$$ are real. Eigenvalues satisfy relation $$A|x_i\rangle = \lambda_i|x_i\rangle$$ and since they are real and $$A$$ is real, it follows that we can find $$|x_i\rangle$$ to be real (despite that fact that they are orthogonal basis of $$\mathbb{C}^n$$). As a result, coeficients $$\beta_i$$ are also real as they are inner product of two real vectors. In the end we are left with real probability amplitudes $$\beta_i\frac{C}{\lambda_i}$$ in the state above.
This all means that in case of real matrix and real right side, we can simply measure probabilities of possible outcomes in an output register and do not have to employ tomography. Hence, for real systems $$Ax=b$$, we are able to get whole solution and at the same time the exponential speed-up is preserved.
Is my reasoning right or am I missing something?
• Aren’t there still an exponential number of $\lambda_i$? May 7 '21 at 12:17
• @Mark S: Exponentials should be eliminated by inverse Fourier in phase estimation. May 7 '21 at 13:22
• What do you mean by "we are able to get whole solution and at the same time the exponential speed-up is preserved?" Are you trying to prepare the state $\vert x\rangle$, or get some expectation? You could always execute the Hamiltonian simulation/perform QPE/calculate the inverse in superposition, and then measure your last register to get $\vert x_i\rangle$ having a large probability. HHL's paper describes a "simple example" on page 2 - therein all of $A$ and $\vert b\rangle$ are real I think. May 7 '21 at 14:36
• There are still $2^n$ different eigenvalues. If you need to know all of them then you'd still need tomography, right? May 7 '21 at 14:51
• @MarkS: I meant that in case we are not sure that the solution of our linear system is real, we have to do complete tomography to take phases into account. In case we know that that the solution is real, we need to measure the output in computational basis only. As a result, we save a resources for doing the tomography and speed-up given by HHL is not hindered. May 7 '21 at 21:58
If you mean by this sentence:
we are able to get whole solution
to know all the components of vector $$x$$, then I'm afraid you are wrong.
Given a sparse $$N \times N$$ matrix with a condition number $$\kappa$$, HHL algorithm has a runtime $$O(\log(N)\kappa ^{2})$$. This offers an exponential speedup over the best known classical algorithm, which has a runtime $$O(N\kappa )$$.
Now $$x$$ is $$N$$-dimensional vector, which means we need $$O(N)$$ measurements to get all its components. That would kill the exponential speedup.
In his paper, "Equation solving by simulation", Andrew Childs said:
Producing a quantum state proportional to $$A^{−1}|b〉$$ does not, by itself, solve the task at hand. To extract information from a quantum state, we must perform a measurement. Learning all $$N$$ amplitudes of an $$N$$-dimensional quantum state requires a number of measurements at least proportional to $$N$$. Thus, if our goal is to completely reconstruct a solution $$x$$, there is no hope for a quantum algorithm to offer a significant advantage over classical methods.
• Thank you, now I see where is flaw in my reasoning. May 8 '21 at 17:27 | 2022-01-18 19:47:14 | {"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": 35, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7700439691543579, "perplexity": 365.58589616152267}, "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/1642320300997.67/warc/CC-MAIN-20220118182855-20220118212855-00406.warc.gz"} |
http://cerco.cs.unibo.it/changeset/1455/Deliverables | # Changeset 1455 for Deliverables
Ignore:
Timestamp:
Oct 23, 2011, 5:51:04 PM (9 years ago)
Message:
ratios changed
File:
1 edited
### Legend:
Unmodified
r1454 call_dest_for_main: call_dest p; succ_pc: succ p $\rightarrow$ address $\rightarrow$ res address; greg_store_: generic_reg p $\rightarrow$ beval $\rightarrow$ regsT $\rightarrow$ res regsT; greg_retrieve_: regsT $\rightarrow$ generic_reg p $\rightarrow$ res beval; ... pair_reg_move_: regsT $\rightarrow$ pair_reg p $\rightarrow$ res regsT; pointer_of_label: label $\rightarrow$ $\Sigma$p:pointer. ptype p = Code }. \end{lstlisting} Here, the fields \texttt{empty\_framesT}, \texttt{empty\_regsT}, \texttt{call\_args\_for\_main} and \texttt{call\_dest\_for\_main} are used for state initialisation. The field \texttt{succ\_pc} takes an address, and a successor' label, and returns the address of the instruction immediately succeeding the one at hand. The fields \texttt{greg\_store\_} and \texttt{greg\_retrieve\_} store and retrieve values from a generic register, respectively. We extend \texttt{more\_sem\_params} with yet more parameters via \texttt{more\_sem\_params2}: \begin{lstlisting} record more_sem_params2 (globals: list ident) (p: params globals) : Type[1] := record more_sem_params1 (globals: list ident) (p: params globals) : Type[1] := { more_sparams1 :> more_sem_params p; succ_pc: succ p $\rightarrow$ address $\rightarrow$ res address; pointer_of_label: label $\rightarrow$ $\Sigma$p:pointer. ptype p = Code; ... fetch_statement: genv ... p $\rightarrow$ state (mk_sem_params ... more_sparams1) $\rightarrow$ }. \end{lstlisting} The field \texttt{succ\_pc} takes an address, and a successor' label, and returns the address of the instruction immediately succeeding the one at hand. Here, \texttt{fetch\_statement} fetches the next statement to be executed. The fields \texttt{save\_frame} and \texttt{pop\_frame} manipulate stack frames. Description & Matita & Lines & O'Caml & Lines & Ratio \\ \hline Semantics of the abstracted languages & \texttt{joint/semantics.ma} & 64 & N/A & N/A & N/A \\ Generic utilities used in semantics joint' languages & \texttt{joint/SemanticUtils.ma} & 77 & N/A & N/A & N/A \\ Semantics of the abstracted languages & \texttt{joint/semantics.ma} & 434 & N/A & N/A & N/A \\ Generic utilities used in semantics joint' languages & \texttt{joint/SemanticUtils.ma} & 70 & N/A & N/A & N/A \\ Semantics of RTLabs & \texttt{RTLabs/semantics.ma} & 223 & \texttt{RTLabs/RTLabsInterpret.ml} & 355 & 0.63 \\ Semantics of RTL & \texttt{RTL/semantics.ma} & 121 & \texttt{RTL/RTLInterpret.ml} & 324 & 1.88\tnote{a} \\ Semantics of ERTL & \texttt{ERTL/semantics.ma} & 125 & \texttt{ERTL/ERTLInterpret.ml} & 504 & 1.22\tnote{a} \\ Semantics of the joint LTL-LIN language & \texttt{LIN/joint\_LTL\_LIN\_semantics.ma} & 64 & N/A & N/A & N/A \\ Semantics of LTL & \texttt{LTL/semantics.ma} & 6 & \texttt{LTL/LTLInterpret.ml} & 416 & 1.25\tnote{b} \\ Semantics of LIN & \texttt{LIN/semantics.ma} & 22 & \texttt{LIN/LINInterpret.ml} & 379 & 1.52\tnote{b} Semantics of RTL & \texttt{RTL/semantics.ma} & 173 & \texttt{RTL/RTLInterpret.ml} & 324 & 2.01\tnote{a} \\ Semantics of ERTL & \texttt{ERTL/semantics.ma} & 130 & \texttt{ERTL/ERTLInterpret.ml} & 504 & 1.26\tnote{a} \\ Semantics of the joint LTL-LIN language & \texttt{LIN/joint\_LTL\_LIN\_semantics.ma} & 67 & N/A & N/A & N/A \\ Semantics of LTL & \texttt{LTL/semantics.ma} & 5 & \texttt{LTL/LTLInterpret.ml} & 416 & 1.38\tnote{b} \\ Semantics of LIN & \texttt{LIN/semantics.ma} & 43 & \texttt{LIN/LINInterpret.ml} & 379 & 1.62\tnote{b} \end{tabular} \begin{tablenotes} \item{b} Includes \texttt{joint/semantics.ma}, \texttt{joint/SemanticUtils.ma} and \texttt{joint/joint\_LTL\_LIN\_semantics.ma}. \\ \begin{tabular}{ll} Total lines of Matita code for the above files:& 1125 \\ Total lines of Matita code for the above files:& 1145 \\ Total lines of O'Caml code for the above files:& 1978 \\ Ration of total lines:& 0.57 Ration of total lines:& 0.58 \end{tabular} \end{tablenotes} | 2020-10-31 11:45:19 | {"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": 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": 0.9999432563781738, "perplexity": 12502.59894408941}, "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/1603107917390.91/warc/CC-MAIN-20201031092246-20201031122246-00235.warc.gz"} |
https://minimalsurfaces.blog/category/minimal-history/ | Björling Surfaces I
Plateau’s problem asks to find a minimal surface that spans a given closed curve. This is a global question, and the answers are delicate. In contrast, there is a much simpler local problem, posed by Emanuel Gabriel Björling in 1844:
For a given space curve and normal field along the curve, find a minimal surfaces that contains the curve and has the given normal as surface normal. For instance, for a circle with outer normal we expect a catenoid, and for a straight line with a normal that rotates with constant speed a helicoid.
In his 1844 paper, Björling proved that for any given real analytic space curve and real analytic normal field along the curve, there is a unique local solution. Later, Hermann Amandus Schwarz gave an explicit formula for the Weierstrass data of the solution.
The fact that the solution of Björling’s problem is rather simple has been quite fertile. For instance, one can let a normal rotate about a circle and get Pablo Mira’s circular helicoids. However, one quickly finds oneself in the default situation of the wizard’s apprentice who has learned how to use spell but not acquired knowledge about the consequences.
For instance, who would have thought that if one starts with a planar cycloid as a curve and takes as normal the normal vector to the curve, the resulting surface (already known to Eugène Charles Catalan) contains parabolas as geodesics? Also, the formulas that Schwarz gave us are not always easy to integrate. The helicoids winding along logarithmic helices below (due to Christine Breiner and Stephen Kleene) can be explicitly described, but the formulas span an entire page.
So it would be desirable to have some control about the global nature of the solutions, and also some insight when to expect explicit formulas. We will earn about this next week.
Resources
E.-G. Björling: In integrationem aequationis Derivatarum partialium superficiei, cujus in puncto unoquoque principales ambo radii curvedinis aequales sunt signoque contrario, Archiv der Mathematik IV (1844), 290-315
Mathematica Notebook with Examples
Björling Surfaces Repository Page
Wrapped Packages
Another fascinating minimal surface from Alan Schoen’s NASA 1970 report is his I-WP surface.
The name I-WP indicates the two skeletal graphs of the complement: The I-graph and the WP-graph. WP stands for wrapped package. You can see 8 copies of the surface below.
Because it is cut by symmetry planes into simply connected pieces, the conjugate surface is tiles with minimal polygons. This is Steßmann’s surface, discovered 1931, about 40 years earlier.
Berthold Steßman’s thesis determined the Enneper-Weierstrass data of all minimal quadrilaterals such that rotations about the edges generate a discrete group, completing work begun by Riemann and Enneper another 70 years earlier. One might wonder why Schoen’s I-WP surface was not discovered much earlier. Likewise, one might wonder why Steßmann (and Carl Ludwig Siegel, his advisor in Frankfurt), was interested in the Enneper-Weierstrass representation when Jesse Douglas and Tibor Radó had established the existence of much more general Plateau solutions by 1931.
The reasons for progress or the lack of it often lie in human fate. I could find only little about Berthold Steßmann. A short biographical note by the German Mathematical Society mentions that he was born on August 4, 1906 in Hüllenberg, Germany, studied in Göttingen and Frankfurt to become a high school teacher, which he completed in 1933. Then, a year later, he received his PhD about periodic minimal surfaces, with Carl Ludwig Siegel as advisor. The note also mentions that Steßmann was Jewish. I have some hope that his doctoral degree helped him to emigrate in time. Another historical note mentions that he received the Golden Doctoral Certificate at the 50th anniversary of his doctorate in Frankfurt.
Such were the times: such are the times.
The story of I-WP continues a little further. Hermann Karcher found a tetragonal cousin which he called T-WP:
A final riddle: Sven Lidin, Stephen Hyde and Barry Ninham discovered that the associate family of the I-WP surface contains several embedded triply periodic minimal surfaces at angles that are multiples of 60º. These are, however, all isometric to the I-WP surface. It is conceivable however, that they possess different kinds of deformations.
Alan Schoen’s Nasa Report 1970
Alan Schoen celebrated his 94th birthday earlier this month, so it is only fitting to write a little about his NASA report in my series of blog posts of highly influential papers about the construction of minimal surfaces.
This task is not easy, because there is too much worth discussing, so I decided to split this into multiple posts, beginning today with his simplest surfaces. These are H’-T, H”-R, S’-S”, and T’-R’. I add to these P and H, found much earlier by Hermann Amandus Schwarz, because these six surfaces share enough properties so that a single piece of code can be used to compute all of them.
They all have a reflectional fundamental cell consisting of a right prism over a triangle, which must be of one of the types (3,3,3), (2,4,4), or (2,3,6), the numbers representing the triangle angles as fractions of 180º. Below is a piece of the T’-R’ surface in a (6,2,3) prism, and next to it how this prism fits into a translational fundamental piece.
If we remove from the skeleton of the prism the edges that intersect the surface, we obtain the two skeletal graphs of the surface, shown below in extreme wide angle perspective from above. That one of the graph has triangle layers and the other rhombic layers is the reason for Schoen’s naming convention: The letters T and R stand for triangle and rhombus.
The piece in a prism can also be used to effectively parametrize these surfaces. To do this, note that the vertical faces of the prism meet the surfaces in two arcs: One is a segment without corners, the other has two 30º corners where the arc switches from one face of the prism to another. These corners are also the only points with vertical normal. We can therefore conformally parametrize a surface piece in a prism by the shaded rectangle below.
The vertical edges correspond to horizontal symmetry lines, and the horizontal edges to symmetry lines in the vertical prism faces. In this way, the shaded rectangle corresponds to the flat structure of the height differential. The points marked a and -a correspond to the two corners, where the Gauss map as a pole and zero, respectively. The value of a for an (r,s,t)-prism is determined (a consequence of Abel’s theorem) as
$a = \frac12\frac{s}{s+t}$.
This has the curious consequence that the height of the corners of the surface piece are determined relative to the height of the prism. For instance, for T’-R’, the value of a is 1/5, meaning that if the prism has top and bottom face at height +1/2 and -1/2, then the two corners are at height +1/5 and -1/5.
The integrals of the Enneper-Weierstrass forms G dh and 1/G dh become Schwarz-Christoffel maps that map the horizontal gray strip to a periodic polygon or freeze pattern. For T’-R’ this looks like
For other surfaces in this group, the angles will change. The Schwarz P-surface corresponds to the (2,4,4) prism, a will be 1/4, and the frieze pattern has an additional symmetry:
The red lines correspond the the horizontal straight lines on the P-surface. We will see in a later post that the same method can be used to generate many more surfaces.
Finally, all the surfaces in today’s group come in a 1-parameter family and have similar limits. The catenoidal neck pattern and the singly periodic Scherk surface arrangement is encoded by the skeletal graphs:
Resources
Universal Mathematica Notebook
Derived from Scherk’s Examples
During my last semester as an undergraduate student at the Technical University in Berlin in 1984, Dirk Ferus mentioned in his Algebraic Topology class that there would be a geometry conference over the weekend, which he recommended attending. Stupid me, I didn’t go. I could have met my future advisor (Hermann Karcher), and I could have seen a future collaborator (David Hoffman) present the first images of the Costa surface.
This conference is also mentioned in the introduction of another paper from my list of highly influential papers with new examples of minimal surfaces, namely Hermann Karcher’s 1988 Embedded Minimal Surfaces Derived from Scherk’s Examples.
During the academic year 1984/85, I had attended two semesters of Karcher’s Differential Geometry. At the end of the second term he announced that while the third semester would only be for those specializing in geometry, we all should come for the first two weeks, because he intended to spend them with explaining the basics about minimal surfaces, which he had completely neglected. I was a little disappointed, because I was eager to learn about the darker arts – symmetric spaces, Einstein manifolds, Finiteness Theorems…
Karcher didn’t just spend the first two weeks on minimal surfaces, but about half of the semester, developing and presenting what would become the paper mentioned above.
The images here represent only a selection of the surfaces described in that paper: There are the saddle towers, the toroidal half plane layers, and the helicoidal saddle towers. Besides all these example Karcher develops a method to derive the complex analytic Enneper-Weierstraß data from geometric features of the surface, which, ultimately, has led to the enormous zoo of examples we are dealing with today.
Not Just a Special Surface
If I had to sum up the content of Hermann Amandus Schwarz’ price winning monograph Bestimmung einer speciellen Minimalfläche from 1867, I would do so using figures from plate VI from the Nachtrag, conveniently compiled in his Collected Works in a single figure:
What is shown here are polyhedra whose vertices are the branched values of the Gauß map of five families of triply periodic minimal surfaces that Schwarz is investigating.
Schwarz spends most of the over 100 pages discussing a single surface, now called the Diamond or D-surface. It solves the Plateau problem for four consecutive edges of a regular tetrahedron. The details Schwarz provides are overwhelming, and it is easy to overlook that the methods Schwarz develops reach far beyond this special surface, and that he was fully aware of it.
What was keeping mathematicians busy these days? Bernhard Riemann had died in 1866 and left a legacy of new concepts and open problems. Complex analysts and geometers were working towards proofs of the Riemann mapping theorem, the uniformization theorem, and the Plateau problem. Schwarz had its own approach: Solve simple cases first, understand them as well as possible, and then apply the developed methods to solve the general case. Both for the Riemann mapping theorem and the Plateau problem, Schwarz looks at polygonal boundaries. He develops the Schwarz-Christoffel formula, and tries something similar for minimal surfaces.
Schwarz uses cutting edge technology: The Weierstraß representation for minimal surfaces, the language of Riemann surfaces, and elliptic integrals. He realizes that he can do more than just solve Plateau problems: In addition to straight lines, he can also prescribe symmetry planes. This leads to a differential equation which he can solve if the branched values of the Gauß map are sufficiently symmetric.
Competition was fierce, in particular between Göttingen (Riemann and Enneper) and Berlin (Weierstraß and Schwarz). Riemann had left a few pages of notes that hint at what Schwarz discovers. Schwarz must have been shocked when he saw the posthumous paper, with details added by Hattendorf. He also learns that Enneper had used a version of the Weierstraß representation in 1864, maybe without quite grasping its scope, two years before Weierstraß’ note from 1866. It appears that Riemann knew about this, too, as usual. How much did Enneper and Riemann talk in Göttingen?
With the exception of Schwarz’ figure 47, representing the H-surface, all vertices are antipodally symmetric. I suspect that Schwarz would have instantly nodded if somebody had told him that his differential equation can be solved just under this symmetry assumption, an observation made by Bill Meeks in his 1975 thesis. How the differently symmetric H-surface fits into the picture, together with other, more recently found surfaces like Alan Schoen’s Gyroid, is one of the big open problems of the area.
Scherk’s Fourth Surface
In his second paper about minimal surfaces from 1835, Heinrich Ferdinand Scherk summarizes his earlier findings from 1830 and gives equations for five new minimal surfaces, the first new ones since the catenoid and helicoid.
Equation 7 describes the doubly periodic Scherk surface in general form (the orthogonal case is equation 6). This is the first non-trivial deformation family of minimal surfaces.
Equation 9 is easily recognized as the associate family deformation of catenoid to helicoid, parametrized as screw motion invariant surfaces. These parametrizations are not conformal, and no complex analysis is involved. If only someone had realized that these surfaces share the same Gauß map, the discovery of the Enneper-Weierstraß representation could have happened decades earlier.
Equation 16 is a mystery to me, I couldn’t verify that it satisfies the minimal surface equation.
Equation 20, Scherk’s fourth surface, is also quite complicated, but one of the components of the implicitly given surface does satisfy the minimal surface equation.
Using
$t = 4\sin(x/2)^2+y^2\cos(x)\quad\text{and}\quad \rho^2 = t^2 + y^4 \sin(x)^2$
the equation reads (slightly modernized)
$\cosh\left( z+\sqrt{(t+\rho)/2} \csc(x/2)\right) = \frac{4 \sin(x/2)^2 + \rho}{y^2}$
To find its Enneper-Weierstraß representation and make a decent image, I looked at the level curve for x=π, which simplifies to
$1+\cosh\left(\sqrt{4-y^2}\right) = \frac{8}{y^2} \ .$
This turns out to be a symmetry curve of the surface, so its normal lies in the plane x=0, and the Schwarz-Björling formula can be used to find the Enneper-Weierstraß representation:
$G(z) =\frac{z-1}{z+1} \quad\text{and}\quad dh = i\frac{z}{z^4-1} \ .$
From here we can see that the surface is singly periodic with two annular and two helicoidal ends, and is also singular (at the points corresponding to 0 and infinity).
Above you can see one half of the surface, with (parts of) both helicoidal ends and one of the annular ends. The singular point is where the horizontal symmetry curve in the middle meets the intersection of the two helicoidal ends, which is a straight line on the surface. Rotating about it gives a fundamental piece; below are three copies of it.
For details, see the notebook under the resource below.
Amusingly, there is a simpler surface with the same type of ends that I accidentally discovered a while ago.
Finally, there is equation 30, giving the orthogonal case of Scherk’s singly periodic surface. Scherk does note some similarities to his doubly periodic surface.
Resources
Mathematica Notebook for Scherk IV | 2019-05-23 00:00:26 | {"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": 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": 0.6963192224502563, "perplexity": 1124.192660457268}, "config": {"markdown_headings": false, "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-2019-22/segments/1558232256980.46/warc/CC-MAIN-20190522223411-20190523005411-00146.warc.gz"} |
https://cs.stackexchange.com/questions/78118/equivalence-of-dfferent-tm-definitions | # Equivalence of dfferent TM definitions
I stumbled upon these two defintions of a turing machine:
http://www.cs.um.edu.mt/gordon.pace/Teaching/Complexity/CoursePage/Notes/chapter5.pdf
http://scholar.harvard.edu/files/harrylewis/files/s6_sols.pdf
The difference which bothers me is that the second one allows only left and right movements whereas the first allows left, right and not moving the head at all.
First definition: $\delta_1: Q_1 \times \Gamma_1 \mapsto Q_1 \times \Gamma_1 \times \{L,R,S\}$
Second definition: $\delta_2: Q_2 \times \Gamma_2 \mapsto Q_2 \times \Gamma_2 \times \{L,R\}$
Are they equivalent? If they are equivalent it should be possible to transform a TM that can do $\{L,R,S\}$ into a TM which always has to move left or right (does only $\{L,R\}$)?
How is this possible?
Yes they are equivalent and it is always possible. Informally: you can exchange any $S$ move state with pair of states that goes $L$ then $R$ (or in the other direction).
Formally:
Prove that standard Turing Machine is equivalent to class of Turing Machines with $S$ (no move).
Let $M$ denote standard TM, there exists a TM $M'$ with no move option such that $L(M) = L(M')$. This way is trivial, simply do not use $S$ move in the extended version.
In the other direction:
Given TM $M'$ with no move option, construct equivalent TM $M$ such that $L(M') = L(M)$.
Construct new transition function $\delta'$ for $M$ such that for each state with move $\in \{L, R\}$ in $\delta$ there is equivalent state in $\delta'$.
For each $S$ move in $\delta$ put two states in $\delta'$.
So for $\delta(q_x, a, b, S, q_y)$ put two states to $\delta'$, $\delta(q_x, a, b, L, q_n)$ and $\delta(q_n, *, *, R, q_y)$. The $q_n$ matches the acceptance with $q_x$. If the $S$ move was accepting state, make the second state accepting. This gives the equivalence in the accepting states and the head (heads) position.
$L(M') = L(M)$, so these classes are equivalent.
• Sounds reasonable. Do you have a suggestion about how to write this idea down in terms of the $\delta$ functions of both definitions? – Anna Vopureta Jul 20 '17 at 7:00
• This doesn't work if $q_1$ is an accepting state, your construction skips it. – Anna Vopureta Jul 20 '17 at 12:08
• @AnnaVopureta I have changed whole post, as I understood you want formal method of equivalence not manual method. I hope it is ok now. – Evil Jul 21 '17 at 20:49 | 2020-02-20 12:24:57 | {"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": 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": 0.7236372232437134, "perplexity": 538.1090630484579}, "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/1581875144722.77/warc/CC-MAIN-20200220100914-20200220130914-00035.warc.gz"} |
http://theinfolist.com/php/SummaryGet.php?FindGo=Electronvolt | TheInfoList
Mass
By mass–energy equivalence, the electronvolt is also a unit of mass. It is common in particle physics, where units of mass and energy are often interchanged, to express mass in units of eV/c2, where c is the speed of light in vacuum (from E = mc2). It is common to simply express mass in terms of "eV" as a unit of mass, effectively using a system of natural units with c set to 1.[6] The mass equivalent of 1 eV/c2 is
${\displaystyle 1\;{\text{eV}}/c^{2}={\frac {(1.60217646\times 10^{-19}\;{\text{C}})\cdot 1\;{\text{V}}}{(2.99792458\times 10^{8}\;{\text{m}}/{\text{s}})^{2}}}=1.783\times 10^{-36}\;{\text{kg}}.}$
For example, an electron and a positron, each with a mass of 0.511 MeV/c2, can annihilate to yield 1.022 MeV of energy. The proton has a mass of 0.938 GeV/c2. In general, the masses of all hadrons are of the order of 1 GeV/c2, which makes the GeV (gigaelectronvolt) a convenient unit of mass for particle physics:
1 GeV/c2 = 1.783×10−27 kg.
The unified atomic mass unit (u), 1 gram divided by Avogadro's number, is almost the mass of a hydrogen atom, which is mostly the mass of the proton. To convert to megaelectronvolts, use the formula:
1 u = 931.4941 MeV/c2 = 0.9314941 GeV/c2.
Momentum
In high-energy physics, the electronvolt is often used as a unit of momentum. A potential difference of 1 volt causes an electron to gain an amount of energy (i.e., 1 eV). This gives rise to usage of eV (and keV, MeV, GeV or TeV) as units of momentum, for the energy supplied results in acceleration of the particle.
The dimensions of momentum units are LMT−1. The dimensions of energy units are L2MT−2. Then, dividing the units of energy (such as eV) by a fundamental constant that has units of velocity (LT−1), facilitates the required conversion of using energy units to describe momentum. In the field of high-energy particle physics, the fundamental velocity unit is the speed of light in vacuum c.
By dividing energy in eV by the speed of light, one can describe the momentum of an electron in units of eV/c.[7] [8]
The fundamental velocity constant c is often dropped from the units of momentum by way of defining units of length such that the value of c is unity. For example, if the momentum p of an electron is said to be 1 GeV, then the conversion to MKS can be achieved by:
${\displaystyle p=1\;{\text{GeV}}/c={\frac {(1\times 10^{9})\cdot (1.60217646\times 10^{-19}\;{\text{C}})\cdot (1\;{\text{V}})}{(2.99792458\times 10^{8}\;{\text{m}}/{\text{s}})}}=5.344286\times 10^{-19}\;{\text{kg}}\cdot {\text{m}}/{\text{s}}.}$
Distance
In particle physics, a system of "natural units" in which the speed of light in vacuum c and the reduced Planck constant ħ are dimensionless and equal to unity is widely used: c = ħ = 1. In these units, both distances and times are expressed in inverse energy units (while energy and mass are expressed in the same units, see mass–energy equivalence). In particular, particle scattering lengths are often presented in units of inverse particle masses.
Outside this system of units, the conversion factors between electronvolt, second, and nanometer are the following:
${\displaystyle \hbar ={{h} \over {2\pi }}=1.054\ 571\ 726(47)\times 10^{-34}\ {\mbox{J s}}=6.582\ 119\ 28(15)\times 10^{-16}\ {\mbox{eV s}}.}$
The above relations also allow expressing the mean lifetime τ of an unstable particle (in seconds) in terms of its decay width Γ (in eV) via Γ = ħ/τ. For example, the B0 meson has a lifetime of 1.530(9) picoseconds, mean decay length is = 459.7 μm, or a decay width of (4.302±25)×10−4 eV.
Conversely, the tiny meson mass differences responsible for meson oscillations are often expressed in the more convenient inverse picoseconds.
Energy in electronvolts is sometimes expressed through the wavelength of light with photons of the same energy: 1 eV = 8065.544005(49) cm−1.
Temperature
In certain fields, such as plasma physics, it is convenient to use the electronvolt as a unit of temperature. The conversion to the Kelvin scale is defined by using kB, the Boltzmann constant:
${\displaystyle {1 \over k_{\text{B}}}={1.602\,176\,53(14)\times 10^{-19}{\text{ J/eV}} \over 1.380\,6505(24)\times 10^{-23}{\text{ J/K}}}=11\,604.505(20){\text{ K/eV}}.}$
For example, a typical magnetic confinement fusion plasma is 15 keV, or 170 MK.
As an approximation: kBT is about 0.025 eV (≈ 290 K/11604 K/eV) at a temperature of 20 °C.
Properties
Energy of photons in the visible spectrum in eV
Graph of wavelength (nm) to energy (eV)
The energy E, frequency v, and wavelength λ of a photon are related by
${\displaystyle E=h\nu ={\frac {hc}{\lambda }}}$ ${\displaystyle ={\frac {(4.13566\,7516\times 10^{-15}\,{\mbox{eV}}\,{\mbox{s}})(299\,792\,458\,{\mbox{m/s}})}{\lambda }}}$
where h is the Planck constant, c is the speed of light. This reduces to
${\displaystyle E{\mbox{(eV)}}=4.13566\,7516\,{\mbox{feVs}}\cdot \nu \ {\mbox{(PHz)}}}$ ${\displaystyle ={\frac {1\,239.84193\,{\mbox{eV}}\,{\mbox{nm}}}{\lambda \ {\mbox{(nm)}}}}.}$[9]
A photon with a wavelength of 532 nm (green light) would have an energy of approximately 2.33 eV. Similarly, 1 eV would correspond to an infrared photon of wavelength 1240 nm or frequency 241.8 THz.
Scattering experiments
In a low-energy nuclear scattering experiment, it is conventional to refer to the nuclear recoil energy in units of eVr, keVr, etc. This distinguishes the nuclear recoil energy from the "electron equivalent" recoil energy (eVee, keVee, etc.) measured by scintillation light. For example, the yield of a phototube is measured in phe/keVee (photoelectrons per keV electron-equivalent energy). The relationship between eV, eVr, and eVee depends on the medium the scattering takes place in, and must be established empirically for each material.
Energy comparisons
Photon frequency vs. energy particle in electronvolts. The energy of a photon varies only with the frequency of the photon, related by speed of light constant. This contrasts with a massive particle of which the energy depends on its velocity and rest mass.[10][11][12] Legend
γ: Gamma rays MIR: Mid infrared HF: High freq. HX: Hard X-rays FIR: Far infrared MF: Medium freq. SX: Soft X-rays Radio waves LF: Low freq. EUV: Extreme ultraviolet EHF: Extremely high freq. VLF: Very low freq. NUV: Near ultraviolet SHF: Super high freq. VF/ULF: Voice freq. Visible light UHF: Ultra high freq. SLF: Super low freq. NIR: Near Infrared VHF: Very high freq. ELF: Extremely low freq. Freq: Frequency
Per mole
One mole of particles given 1 eV of energy has approximately 96.5 kJ of energy – this corresponds to the Faraday constant (F96485 C mol−1) where the energy in joules of N moles of particles each with energy X eV is X·F·N.
Notes and references
1. ^ IUPAC Gold Book Archived 2009-01-03 at the Wayback Machine., p. 75
2. ^ SI brochure, Sec. 4.1 Table 7 Archived July 16, 2012, at the Wayback Machine.
3. ^ "CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-22. 2014 CODATA recommended values
4. ^ "CODATA Value: electron volt". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-22. 2014 CODATA recommended values
5. ^ "Definitions of the SI units: Non-SI units". Archived from the original on 2009-10-31.
6. ^ Barrow, J. D. "Natural Units Before Planck." Quarterly Journal of the Royal Astronomical Society 24 (1983): 24.
7. ^ "Units in particle physics". Associate Teacher Institute Toolkit. Fermilab. 22 March 2002. Archived from the original on 14 May 2011. Retrieved 13 February 2011.
8. ^ "Special Relativity". Virtual Visitor Center. SLAC. 15 June 2009. Retrieved 13 February 2011.
9. ^ "CODATA Value: Planck constant in eV s". Archived from the original on 22 January 2015. Retrieved 30 March 2015.
10. ^ What is Light? Archived December 5, 2013, at the Wayback Machine. – UC Davis lecture slides
11. ^ Elert, Glenn. "Electromagnetic Spectrum, The Physics Hypertextbook". hypertextbook.com. Archived from the original on 2016-07-29. Retrieved 2016-07-30.
12. ^ "Definition of frequency bands on". Vlf.it. Archived from the original on 2010-04-30. Retrieved 2010-10-16.
13. ^ Open Questions in Physics. Archived 2014-08-08 at the Wayback Machine. German Electron-Synchrotron. A Research Centre of the Helmholtz Association. Updated March 2006 by JCB. Original by John Baez.
14. ^ "A growing astrophysical neutrino signal in IceCube now features a 2-PeV neutrino". Archived from the original on 2015-03-19.
15. ^ Glossary Archived 2014-09-15 at the Wayback Machine. - CMS Collaboration, CERN
16. ^ ATLAS; CMS (26 March 2015). "Combined Measurement of the Higgs Boson Mass in pp Collisions at √s=7 and 8 TeV with the ATLAS and CMS Experiments". Physical Review Letters. 114 (19): 191803. arXiv:. Bibcode:2015PhRvL.114s1803A. doi:. PMID 26024162. | 2020-05-25 05:21:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "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": 0.8735604882240295, "perplexity": 2098.6435395405974}, "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/1590347387219.0/warc/CC-MAIN-20200525032636-20200525062636-00033.warc.gz"} |
http://www.exampleproblems.com/wiki/index.php/MvCalc49 | # MvCalc49
Let the center of the sphere be at the origin and let the axis of the hole be along the z-axis. The volume V of the sphere is ${\frac {4}{3}}\pi a^{3}\,$ and that of the circular hole is obtained as follows.
Volume of the upper half of the hole=$\iint _{R}f(x,y)dxdy=\iint _{R}zdxdy\,$ where z is obtained from the equation $x^{2}+y^{2}+z^{2}=a^{2}\,$ and R is the circle in the XY-plane,that is $x^{2}+y^{2}=b^{2}\,$.Hence,the volume V1 of the circular hole is
$V_{1}=2\iint _{R}{\sqrt {a^{2}-x^{2}-y^{2}}}dxdy\,$ where R is given by $x^{2}+y^{2}=b^{2}\,$. Changing into polar coordinates,we obtain,
$v_{1}=2\int _{{0}}^{{2\pi }}\int _{0}^{b}{\sqrt {a^{2}-r^{2}}}rdrd\theta ={\frac {4\pi }{3}}[a^{3}-(a^{2}-b^{2})^{{{\frac {3}{2}}}}]\,$
Hence the remaining portion =$V-V_{1}={\frac {4\pi }{3}}a^{3}-{\frac {4\pi }{3}}[a^{3}-(a^{2}-b^{2})^{{{\frac {3}{2}}}}]={\frac {4\pi }{3}}(a^{2}-b^{2})^{{{\frac {3}{2}}}}\,$ | 2018-12-11 01:08:04 | {"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": 0, "img_math": 8, "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": 0.7737404108047485, "perplexity": 315.0538915591604}, "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/1544376823516.50/warc/CC-MAIN-20181210233803-20181211015303-00399.warc.gz"} |
https://holooly.com/solutions-v20/figure-10-33-shows-a-compound-beam-loaded-at-its-free-end-if-the-flexural-rigidity-is-constant-throughout-the-beam-calculate-the-total-strain-energy-stored-using-this-strain-energy-find-the-deflec/ | ## Textbooks & Solution Manuals
Find the Source, Textbook, Solution Manual that you are looking for in 1 click.
## Tip our Team
Our Website is free to use.
To help us grow, you can support our team with a Small Tip.
## Holooly Tables
All the data tables that you may search for.
## Holooly Help Desk
Need Help? We got you covered.
## Holooly Arabia
For Arabic Users, find a teacher/tutor in your City or country in the Middle East.
Products
## Textbooks & Solution Manuals
Find the Source, Textbook, Solution Manual that you are looking for in 1 click.
## Holooly Arabia
For Arabic Users, find a teacher/tutor in your City or country in the Middle East.
## Holooly Help Desk
Need Help? We got you covered.
## Q. 10.18
Figure 10.33 shows a compound beam loaded at its free end. If the flexural rigidity is constant throughout the beam, calculate the total strain energy stored. Using this strain energy, find the deflection at point E.
## Verified Solution
Let us draw the free-body diagram of the beams as shown in Figure 10.34.
From Figure 10.34(b) above
$\sum M_{ C }=0 \Rightarrow R_{ D }=2 P(\uparrow)$
Therefore,
$\sum F_y=0 \Rightarrow R_{ C }=P(\downarrow)$
Again from Figure 10.34(a)
$\sum M_{ A }=0 \Rightarrow R_{ B }=2 R_{ C }=2 P(\downarrow)$
and $\sum F_y=0 \Rightarrow R_{ A }=P(\uparrow)$
So, both beams are symmetrically and identically loaded. Thus, the total strain energy of the system is
$U_{\text {bending }}=2\left[\left\lgroup \frac{1}{2 E I} \right\rgroup \int_0^a M_x^2 d x\right]_{ AC }+2\left[\left\lgroup \frac{1}{2 E I} \right\rgroup \int_0^a M_x^2 d x\right]_{ CE }$
$=\left\lgroup \frac{2}{E I} \right\rgroup \int_0^a M_x^2 d x$
$=\frac{2}{E I} \int_0^a P^2 x^2 d x=\frac{2}{3} \frac{P^2 a^2}{E I}$
The total strain energy of the system is $2 P^2 a^3 / 3 E I$
To calculate the deflection at point E, we can apply Castigliano’s second theorem [refer Eq. (10.66)].
$\frac{\partial U}{\partial Q_i}=\Delta_i ; \quad 1 \leq i \leq n$ (10.66)
Therefore,
$\Delta_{ E }=\frac{\partial U}{\partial P}=\frac{4 P a^3}{3 E I}$
Thus, vertical deflection at E is
$\Delta_{ E }=\frac{4 P a^3}{3 E I}(\downarrow)$ | 2023-02-02 17:16:22 | {"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": 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": 11, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27314862608909607, "perplexity": 6856.368741449623}, "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-2023-06/segments/1674764500035.14/warc/CC-MAIN-20230202165041-20230202195041-00557.warc.gz"} |
https://plainmath.net/5420/write-augmented-matrix-system-linear-equations-equal-plus-equal-equal | # Write the augmented matrix for the system of linear equations: {(2y-z=7),(x+2y+z=17),(2x-3y+2z=-1):}
Write the augmented matrix for the system of linear equations:
$\left\{\begin{array}{c}2y-z=7\\ x+2y+z=17\\ 2x-3y+2z=-1\end{array}$
You can still ask an expert for help
• Questions are typically answered in as fast as 30 minutes
Solve your problem for the price of one coffee
• Math expert for every subject
• Pay only if we can solve it
Maciej Morrow
Step 1:We have to write the augmented matrix for the given system of linear equations.
An augmented matrix for the system of equations is a matrix of numbers in which each row represents the constants from one equation(both the coefficients and the constant on the other side of the equal sign.
and each column represents all the coefficients for a single variable
Given system of linear equation is,
2y−z = 7
x+2y+z=17
2x−3y+2z=−1
Step 2:
Steps to write augmented matrix,
Step 1) Write the coefficients of the x-terms as the numbers down the first column.
Step 2) Write the coefficients of the y-terms as the numbers down the second column.
Step 3) Write the coefficients of the z-terms as the numbers down the third column.
Step 4) Draw a vertical line and write the constants to the right of the line.
Therefore we get,
$\left[\begin{array}{cccc}0& 2& -1& 7\\ 1& 2& 1& 17\\ 2& -3& 2& -1\end{array}\right]$
This is the augmented matrix for given system of linear equations. | 2022-08-16 04:49:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 41, "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": 0.894121527671814, "perplexity": 550.163022674653}, "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/1659882572220.19/warc/CC-MAIN-20220816030218-20220816060218-00094.warc.gz"} |
https://math.stackexchange.com/questions/1371756/quadratic-form-as-generalized-distance/2453593 | # Quadratic form as generalized distance?
In the book A Linear Systems Primer (by Antsaklis and others), they first mention squared distance of a point x from the origin:
$$x^{T}x = ||{x}||^2$$ which represents the square of the Euclidean distance of the state from the equilibrium $x=0$.
So far, so good this is basic linear algebra. Then they go on to say:
In the following discussion, we will employ as a "generalized distance function" the quadratic form given by $${x}^TPx , P={P}^T$$ where $P$ is a real $n\times n$ matrix.
I am familiar with this definition of a quadratic form from linear algebra : we interpose a symmetric matrix to weight the variables in different ways.
But I am not familiar with this as a distance function. Is that mainly to say that it satisfies the requirements of a metric space? Is there a geometric (intuitive) discussion of the sense in which the quadratic form generalizes the more basic notion of a distance?
• In general, it might not satisfy the requirements of a metric: unless $P$ is positive definite, distance between $x$ and $y \neq x$ may be zero or negative. – Budenn Jul 23 '15 at 20:03
• @neuronet: Thanks for that post, it helped me with a Problem I have. Would you mind taking a look at my question regarding this Topic? I cant find a way to clearify this to me... math.stackexchange.com/questions/1634485/… – Benvorth Jan 31 '16 at 15:39
• @Benvorth I still don't feel I grasped this problem yet, but see my comment to the answer below. The wikipedia page it links to I find largely impenetrable, unfortunately. I think a close study of mahalanobis distance is the way to go, as that has been addressed a lot more, at an intuitive level, and is pretty much a special case of this. I just haven't had time to undertake it yet. – neuronet Jan 31 '16 at 16:32
This determines a norm $\|\_\|_P$ iff $P$ is positive definite, then it naturally defines a metric by $d(x,y):=\|y-x\|_P$.
Else the above $d$ function would not be defined or would fail to be a metric, as e.g. there could be $x\ne 0$ with $x^TPx\le0$.
All in all, what it basically says is that the quadratic form $x\mapsto x^TPx$ can be viewed as a generalisation of $x\mapsto \|x\|^2$.
For more details and geometric insights, see for example Pseudo Euclidean spaces.
• That is helpful, I wonder if there is a good discussion of this in a linear algebra context? That wikipedia page is tantalizing if a bit general, its connection to this specific measure gets lost to me fairly quickly. I am starting to think that the mahalanobis distance may be a good route to massage intuitions, as it seems to have been fairly well explored and is the same form as the quadratic form. – neuronet Jul 24 '15 at 15:08
I think some examples from physics might help provide the geometric (intuitive) sense you seek, in which quadratic forms generalize distance, though I doubt whether it’s useful to think of quadratic forms as providing a “more basic notion of distance” in quite the way that I think you're expecting. I see them as providing measures of distance like the usual (Pythagorean/Euclidean) notion of distance, but applied in slightly more general spaces.
A simple example of a generalisation of distance encoded by a quadratic form is given by the Minkowski metric for space-time; for the squared length of a vector from the origin:
$$d^2 = x^2 + y^2 + z^2 –(ct)^2$$
… which I’m sure you can see can be expressed as $X^TPX$ where $P$ is a $4\times 4$ matrix with all off-diagonal elements zero and the diagonal $\{1,1,1,-1\}$. See also http://mathworld.wolfram.com/MinkowskiMetric.html where the matrix is illustrated.
And so for the slightly less-clear case of a distance between the endpoints of vectors v1 and v2, I think you will now see this also as a quadratic form (using an overly-simplistic notation):
$$d^2(v1-v2) = (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2 –(ct1-ct2)^2$$
In General Relativity, however, the matrix $P$ is called the Metric Tensor, and can have all non-zero components. As these components can change from point to point, this metric provides for a quadratic form that can encode distances in curved space-times, which is a generalisation of the "flat" Minkowski space-time.
So, as Berci pointed out via https://en.wikipedia.org/wiki/Pseudo-Euclidean_space , there’s a clear geometric interpretation, but it’s “simply” a generalisation of the familiar types of distances that apply in Pythagorean/Euclidean/pseudo-Euclidean spaces, rather than a generalisation of the more “disjointed” type that would be needed to include, say, Manhattan, Chessboard or Mahalanobis distances.
(Clearly, I’m not a mathematician, but I think the physics examples help provide the geometric insight you seek, and if I’m wrong in this, would appreciate correction from those more expert. Put into MathJax with assistance acknowledged below.)
• Appreciate the formatting edit, Siong Thye Goh. . – iSeeker Oct 1 '17 at 23:22 | 2019-05-21 14:52:22 | {"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": 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": 0.8400835394859314, "perplexity": 207.0093006180872}, "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/1558232256426.13/warc/CC-MAIN-20190521142548-20190521164548-00141.warc.gz"} |
https://www.physicsforums.com/threads/relationship-between-division-subtraction-and-limits.712437/ | # Relationship between division, subtraction, and limits
1. Sep 24, 2013
### stoopkid
Ok, so I'm not really too good at group theory and that kind of math, so I hope I can explain my question:
I tried to evaluate $\frac{d}{dx}e^{x}$:
$\frac{d}{dx}e^{x} = \frac{e^{x+h}-e^{x}}{h}$, h -> 0
$= \frac{e^{x}e^{h}-e^{x}}{h}$, h-> 0
$= e^{x}(\frac{e^{h}-1}{h})$, h-> 0
So I figured if I could show that: $\frac{e^{h}-1}{h}$ goes to 1 as h goes to 0, then the proof would be done. So I reasoned that $e^{h}$ goes to 1 as h goes to 0, so $e^{h}-1$ must go to 0 as h goes to 0. But does $e^{h}-1$ "go to h" as h goes to 0?
It took me a little bit to phrase this mathematically, but I reasoned that if the difference between $e^{h}-1$ and h went to 0, as h went to 0, then their ratio would get closer and closer to 1, as h went to 0, and the proof would be done. So I tried it:
$e^{h} - 1 - h = 1 - 1 - 0 = 0$, as h goes to 0.
So I figure that this means that $\frac{e^h-1}{h} = 1$, as h goes to 0. I don't know whether or not this is true, but I assume that it is because I don't see how else the derivative would end up as $e^{x}$. So this brings me to my question:
So the difficult part of my problem began with two functions, which I will assume to be elements of some ring or defined on some ring (I'm not sure what ring theorists would say here), they are: $e^{h}-1$, and h.
I encountered a problem where the multiplicative inverse of h was multiplied to $e^{h}-1$. I did not know how to take the limit of this as h went to 0.
I translated this into a problem where the additive inverse of h was added to $e^{h}-1$. I DID know how to take the limit of this.
Taking the limit as h goes to 0 of this additive version of the problem, I discovered it was equal to 0. This (as far as I know) just happens to be the additive identity.
Using this information, I concluded that the limit as h goes to 0 of the multiplicative version must be 1. This (as far as I know) just happens to be the multiplicative identity.
... This is where I'm not sure how to phrase my question ... This was a nifty little trick to solve this one problem, but there seems to be a lot of structure and patterns here, and, not knowing very much group theory, I don't know what to make of it.
1) Is there some deeper thing going on here that group theory explains?
2) I.e. is this some kind of relationship between the addition part and multiplication part of a single RING?
3) Or is this like a "homomorphism" between a GROUP where the operation is addition, and another GROUP where the operation is multiplication?
4) Does this apply more generally to other groups/rings/fields, etc?
5) Is there some more general procedure to translate difficult problems involving multiplication into easier ones involving addition? How would I know when something like this can be applied?
6) Does the fact that I'm taking limits have anything to do with it?
7) Does the fact that the limit points are the additive and multiplicative identities have anything to do with it?
8) Does the continuity of the real numbers and the existence of limits have anything to do with it?
Thanks in advance for any help in understanding this
2. Sep 24, 2013
### fzero
The type of limit (of the form $0/0$) that you are taking is called an indeterminate form and these are often tricky to evaluate. I'm afraid that you've gotten the correct answer using somewhat specious reasoning.
The problem with an indeterminate form is that its value strongly depends on the way that the numerator and denominator go to their individual limits. This is explained on the wiki, but deserves a separate demonstration here. The limit involved here is indeed
$$\lim_{h\rightarrow 0} \frac{e^h -1}{h} = \lim_{h\rightarrow 0} \frac{h}{h} = 1.$$
However, you concluded that
$$\lim_{h\rightarrow 0} (e^h -1) =\lim_{h\rightarrow 0} h$$
because you found that
$$\lim_{h\rightarrow 0} (e^h -1 -h ) =0.$$
Unfortunately, this is not enough, because one can see that
$$\lim_{h\rightarrow 0} (e^h -1 ) = 0 ~~~~\mathrm{and} ~~~~ \lim_{h\rightarrow 0} h =0.$$
independently. So we are really just adding 0 to 0 and haven't proven anything.
The further problem with the indeterminate form is that
$$\lim_{h\rightarrow 0} (e^{h^{1/2}} -1 ) = 0$$
as well, so we also have
$$\lim_{h\rightarrow 0} (e^{h^{1/2}} -1 -h ) = 0.$$
But
$$\lim_{h\rightarrow 0} \frac{e^{h^{1/2}} -1}{h} = \lim_{h\rightarrow 0} \frac{h^{1/2}}{h} = \lim_{h\rightarrow 0} \frac{1}{h^{1/2}} = \infty$$
is not defined. Similarly,
$$\lim_{h\rightarrow 0} \frac{e^{h^2} -1}{h} = \lim_{h\rightarrow 0} \frac{h^2}{h} = \lim_{h\rightarrow 0} h=0.$$
We are led to conclude that the rate at which the numerator and denominator go to zero are very important in taking a limit of this type, so we must actually use a better method to evaluate the limit in the numerator. One valid method to define the limit (without using calculus, since the calculus result is what we wanted to prove) is to use the limit definition of the exponential function
$$e^h = \lim_{n\rightarrow \infty} \left( 1 + \frac{h}{n} \right)^n.$$
We can use the binomial formula to compute the terms in this expression and take the limit $h\rightarrow 0$ to obtain the correct result.
3. Sep 25, 2013
### D H
Staff Emeritus
A perhaps more intuitive method is to use the power series definition of the exponential function, $e^h = \sum_{n=0}^{\infty} \frac {h^n}{n!}$, or $1 + h + h^2/2 + h^3/6 + \cdots$. Subtracting one and dividing by h yields $(e^h-1)/h = \sum_{n=0}^{\infty} \frac {h^n}{(n+1)!}$, or $1+h/2+h^2/6+\cdots$. All of those terms involving hn (n>0) vanish as h→0, leaving just 1 in the limit as h→0.
Last edited: Sep 25, 2013
4. Sep 25, 2013
### fzero
The problem is asking to derive an expression for the derivative of the exponential. The power series for the exponential function arises in one of two ways. Either we use the Taylor series, which depends on already knowing the derivative, or we use the definition of the exponential in terms of the limit in the same manner that I suggested. So your suggestion is valid (barring the typos in the expression $e^h = 1 + h + h/2 + h^2/6 + \cdots$), but the pedagogy is a bit mixed up. I gave the method most appropriate to a typical course presentation, where the limit definition of the exponential comes before the discussion of the derivative, with infinite series for arbitrary functions being discussed after differential calculus.
5. Sep 25, 2013
### Office_Shredder
Staff Emeritus
$$e^h = \lim_{n\to \infty} (1+h/n)^n$$
and there's just way too much math involved. I prefer the alternative (below is not 100% rigorous thing, but I think it's clear that it works): Suppose there exists some number e such that
$$\lim_{h\to 0} \frac{ e^h - 1}{h} = 1$$.
In this case, when h is very small
$$e^h - 1 \approx h$$
$$e \approx (1+h)^{1/h}$$
So e is going to be
$$\lim_{h\to 0} (1+h)^{1/h}$$
which is easily seen to be equivalent to the usual limit definition of e. So instead of taking this mysterious number and being amazed that it works (and being unsure of why), we start by trying to find the right base to work with and finding out that it's the Euler number.
Also, as the thread has very little to do with algebra, I'm going to move it to the calculus forum.
Last edited: Sep 25, 2013 | 2017-08-22 23:32:24 | {"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": 2, "mathjax_display_tex": 2, "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": 0.9490706324577332, "perplexity": 193.56255176295798}, "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/1502886116921.7/warc/CC-MAIN-20170822221214-20170823001214-00318.warc.gz"} |
https://leanprover-community.github.io/archive/stream/116395-maths/topic/Construction.20of.20Algebraic.20Closure.html | ## Stream: maths
### Topic: Construction of Algebraic Closure
#### Kenny Lau (May 29 2018 at 18:03):
I saw many constructions of the algebraic closure of a field k using direct limit, but I have a different construction in mind:
The set k-bar is { (f,n) in k[X] x N | f is irreducible and n < deg f }. The n represents the n-th root of the polynomial.
Addition and multiplicatoin can be defined using resultant.
Is this construction valid? Would this be a better construction?
#### Johan Commelin (May 29 2018 at 18:06):
Hmm, maybe I'm being silly. But how do you order the roots?
I don't
#### Johan Commelin (May 29 2018 at 18:06):
Ok, so how do you do addition and multiplication?
#### Kenny Lau (May 29 2018 at 18:06):
For f and g, I use resultant to construct h that contains all the roots
#### Kenny Lau (May 29 2018 at 18:07):
then just, you know, do the thing
#### Kenny Lau (May 29 2018 at 18:07):
if f has deg m and g has deg n, then h has deg mn
#### Kenny Lau (May 29 2018 at 18:07):
no this doesn't work
#### Johan Commelin (May 29 2018 at 18:08):
I mean, your approach looks very constructive. But we know that you need choice for k-bar
#### Johan Commelin (May 29 2018 at 18:08):
So that makes me suspicious
#### Kenny Lau (May 29 2018 at 18:09):
do you need choice for the direct limit construction?
#### Johan Commelin (May 29 2018 at 18:12):
Yes, you want to use Zorn to pick a maximal element
#### Johan Commelin (May 29 2018 at 18:15):
Does this mean you are going to refuse the project that Kevin gave you?
#### Kenny Lau (May 29 2018 at 18:15):
I think the problem is when I add 1+sqrt(2) and -sqrt(2)
no, that doesn't
#### Kenny Lau (May 29 2018 at 18:15):
and how do you know about the project
#### Johan Commelin (May 29 2018 at 18:15):
Kevin mentioned somewhere that you were working on some algebraic stuff
#### Johan Commelin (May 29 2018 at 18:15):
for a project that he gave you
#### Johan Commelin (May 29 2018 at 18:16):
Anyway, I think it is very cool. I have been thinking about Galois theory. But I was daunted by defining the algebraic closure.
#### Johan Commelin (May 29 2018 at 18:16):
I haven't worked with Choice yet in Lean
nice
#### Johan Commelin (May 29 2018 at 18:17):
But we really need Galois theory
#### Kenny Lau (May 29 2018 at 18:17):
stop before you are corrupted by choice
#### Kenny Lau (May 29 2018 at 18:17):
I mean, your approach looks very constructive. But we know that you need choice for k-bar
we all know that you don't need choice for F_p-bar or Q-bar or R-bar
#### Johan Commelin (May 29 2018 at 18:17):
Yes, I also reject infinity (-;
#### Kevin Buzzard (May 29 2018 at 18:23):
Yes you can't do add this way Kenny
#### Kevin Buzzard (May 29 2018 at 18:24):
The problem is that what you are doing in your head, is this:
#### Kevin Buzzard (May 29 2018 at 18:24):
if you have have two polynomials f(X) and g(X), irreducible in k[X] say
#### Kevin Buzzard (May 29 2018 at 18:24):
then you are doing mathematics in the ring k[X]/(f) tensor_k k[X]/(g)
#### Kevin Buzzard (May 29 2018 at 18:24):
and unfortunately this is not in general a field
#### Kevin Buzzard (May 29 2018 at 18:25):
Consider the polynomials f(X)=X^3-2 and g(X)=(X+1)^3-2. Both are irredudible over Q
#### Kevin Buzzard (May 29 2018 at 18:25):
You order the roots of both of them
#### Mario Carneiro (May 29 2018 at 18:25):
If you do this construction, I would like to have a computable algebraic numbers construction from it
#### Kevin Buzzard (May 29 2018 at 18:25):
but who is to say that if a,b,c was the first order then a-1,b-1,c-1 was the second one
#### Kenny Lau (May 29 2018 at 18:26):
but we all know that Q-bar is computable
#### Kevin Buzzard (May 29 2018 at 18:26):
so who can possibly tell when (root 1 of f) - (root 1 of g) is 1 or not?
#### Kevin Buzzard (May 29 2018 at 18:26):
The problem is that whilst g is irreducible over Q
#### Kevin Buzzard (May 29 2018 at 18:27):
it is not irreducible over the larger field Q[X]/(f)
#### Kevin Buzzard (May 29 2018 at 18:27):
indeed, it factors into a linear and an irreducible quadric over this larger field
#### Kevin Buzzard (May 29 2018 at 18:27):
so now all of a sudden the roots are not as indistinguishable as they used to be
#### Assia Mahboubi (May 29 2018 at 21:02):
Hi @Kenny Lau here is a formalized construction of the algebraic closure of countable fields. It heavily relies on this, the existence of an algebraically closed field with an automorphism of order 2. Here is an abstract construction of algebraic numbers. I can help deciphering the statements and proofs if you're interested. But several of these files have long headers describing what's done in them.
thanks
#### Assia Mahboubi (May 29 2018 at 21:04):
And all this is constructive. It only relies on the fact that there is choice operator on countable types with a decidable equality. This is provable in Coq without extra axioms, but using a subtle singleton elimination argument. I do not know if the same holds in Lean.
#### Kenny Lau (May 29 2018 at 21:05):
we don't have the axiom of unique choice in Lean, if that's what you mean
#### Kenny Lau (May 29 2018 at 21:05):
I suppose we can look at the preimage under the bijection from N and find the minimum element
#### Assia Mahboubi (May 29 2018 at 21:05):
No this is not what I mean, unique choice does not hold in Coq either.
#### Kenny Lau (May 29 2018 at 21:05):
then it should still be constructive in Lean
#### Mario Carneiro (May 29 2018 at 21:07):
There is a choice operator on countable types in lean
#### Mario Carneiro (May 29 2018 at 21:08):
encodable.choose in data.encodable
#### Patrick Massot (May 29 2018 at 21:08):
Noooo! Assia, please don't encourage Kenny in his constructive deviance
#### Assia Mahboubi (May 29 2018 at 21:09):
Ah thanks @Mario Carneiro, I was trying to dig into Lean to see if I could find it.
#### Mario Carneiro (May 29 2018 at 21:09):
The axiomatically basic one is nat.find
#### Assia Mahboubi (May 29 2018 at 21:15):
Hi again @Patrick Massot! Don't worry, I am just saying that for countable fields, classical proofs are constructive, in fact. I don't think that constructivism is the difficult issue here but I may well have forgotten how easy classical life is.
#### Junyan Xu (Aug 15 2020 at 17:24):
Mario Carneiro said:
The axiomatically basic one is nat.find
What do you mean by axiomatically basic? #print axioms nat.find shows classical.choice quot.sound and propext, so essentially all the axioms.
In ZF set theory it doesn't require any choice. I wonder if adding the axiom of unique choice makes Lean suitable for formalizing implications between different forms of choice in set theory, like dependent choice or ultrafilter lemma.
The use of choice seems to make nat.find noncomputable, which is confusing as it's just the mu-operator (as in mu-recursive functions, even with a termination guarantee). Is Lean using a stronger notion of computability somewhere between primitive recursive and recursive? To be clear, nat.find itself isn't labeled noncomputable, but the following code
def bld_extr (p : ℕ → ℕ → Prop)
(h : ∀ m N, ∃ n ≥ N, p m n) : ℕ → ℕ
-- build a extraction (indices for increasing subsequence) satisfying condition p
| 0 := nat.find (h 0 0)
| (m+1) := nat.find (h (m+1) (bld_extr m + 1))
yields the message
equation compiler failed to generate bytecode for 'bld_extr._main'
nested exception message:
code generation failed, VM does not have code for 'classical.choice'
which disappears when I add the noncomputable prefix.
However, the code def f := nat.find (by use 0; trivial : ∃n:ℕ, n=n) yields a different message definition 'f' is noncomputable, it depends on 'classical.prop_decidable'.
#### Kenny Lau (Aug 15 2020 at 17:25):
h is not decidable
#### Junyan Xu (Aug 15 2020 at 17:34):
The first message points to choice as a problem, while the second message is strange:
def nat.find {p : ℕ → Prop} [decidable_pred p] : (∃ (n : ℕ), p n) → ℕ only requires p to be decidable. In this case, p is n=n, which Lean should know/be able to infer is decidable.
#### Kenny Lau (Aug 15 2020 at 17:35):
no, p is ∃ n ≥ 0, p 0 n
#### Junyan Xu (Aug 15 2020 at 17:36):
The second example is def f := nat.find (by use 0; trivial : ∃n:ℕ, n=n)
#### Junyan Xu (Aug 15 2020 at 17:36):
you can replace it with def f := nat.find (by use 0 : ∃n:ℕ, true) and get the same message
#### Kenny Lau (Aug 15 2020 at 17:37):
if you have open_locale classical then the decidable instance is by default the classical one
#### Junyan Xu (Aug 15 2020 at 17:44):
Good point! I was working in a file in the tutorial exercises and tuto_lib was imported. I switched to an empty file and def f := nat.find (⟨0,true.intro⟩ : ∃n:ℕ, true) and no error message appears.
#### Junyan Xu (Aug 15 2020 at 17:48):
tuto_lib contains attribute [instance] classical.prop_decidable
#### Mario Carneiro (Aug 15 2020 at 17:53):
What do you mean by axiomatically basic? #print axioms nat.find shows classical.choice quot.sound and propext, so essentially all the axioms.
What? That should not be, nat.find should require no axioms
lean#446
#### Mario Carneiro (Aug 15 2020 at 18:08):
What the hell? I just noticed that the section https://github.com/leanprover-community/mathlib/blob/55d430ca22bad157799afdf5e3d9a26597ee37f9/src/algebra/order.lean#L125-L180 of choiceless proofs about decidable linear orders was completely destroyed by the decidable classical linter
#### Bryan Gin-ge Chen (Aug 15 2020 at 18:18):
Looks like the discussion was in this thread: https://leanprover.zulipchat.com/#narrow/stream/144837-PR-reviews/topic/.232332/near/193211152
#### Mario Carneiro (Aug 15 2020 at 18:22):
Yes, I didn't realize that they actually went forward with the theorem defacement. In any case I've restored the theorems in #3799
#### Mario Carneiro (Aug 15 2020 at 18:22):
They have been marked @[nolint classical_decidable] this time, so hopefully this won't happen again
#### Mario Carneiro (Aug 15 2020 at 18:23):
If we decide these don't deserve to exist in mathlib, then they should be deleted, not trivialized
#### Bryan Gin-ge Chen (Aug 15 2020 at 18:24):
It'd be good to add a library note or some other documentation about the decidable namespace too.
#### Mario Carneiro (Aug 15 2020 at 18:25):
I was actually thinking of moving these theorems to core so that they can support things like lean#446 (which was the original reason I wrote them)
#### Mario Carneiro (Aug 15 2020 at 18:25):
but this means they can't keep the nolint
#### Bryan Gin-ge Chen (Aug 15 2020 at 18:26):
We can always add nolint in mathlib.
I thought we were planning to move order-stuff out of core rather than into it though?
that too
#### Mario Carneiro (Aug 15 2020 at 18:28):
I would like the proof that int is a ring to not use the axiom of choice. If all the lemmas come out of core then these can come with
#### Mario Carneiro (Aug 15 2020 at 18:28):
Like does init.data.nat.lemmas need to exist?
#### Junyan Xu (Aug 15 2020 at 18:40):
I think the three axioms are not so bad as a single invocation of classical.em in the proof will make the theorem depend on all three, because of how classical.em is proved. Would be nice to have a command that tells whether a proof uses only excluded middle and not choice (as is the case for most of the incidence geometry problems). Maybe designate certain important theorems from where you don't trace further back.
#### Junyan Xu (Aug 15 2020 at 18:40):
I think core.init.data.nat.lemmas contains many useful lemmas. It's the go-to place when I can't find what I need in data.nat.basic.
#### Mario Carneiro (Aug 15 2020 at 18:42):
Of course the contents of init.data.nat.lemmas are important, I'm not saying to delete them! I mean to move everything to mathlib
#### Mario Carneiro (Aug 15 2020 at 18:43):
the question is really what is the minimal substrate that lean needs to pass its test suite
#### Mario Carneiro (Aug 15 2020 at 18:43):
and what is needed to make VM builtins work
#### Junyan Xu (Aug 17 2020 at 02:28):
I studied the proof of nat.find to convince myself it works and to learn the terse proof style, and I just came up with a different, shorter proof. Unfortuantely it uses nat.decreasing_induction which is not in the core, but it doesn't use addition. Let me know if it is worth PRing or can be further simplified.
parameter {p : ℕ → Prop}
variables [decidable_pred p] (H : ∃n, p n)
private def lbp (m n : ℕ) : Prop := m = n + 1 ∧ ¬p n
protected def find_x : {n // p n ∧ ∀m < n, ¬p m} :=
@well_founded.fix_F _ lbp (λk, (∀n<k, ¬p n) → {n // p n ∧ ∀m<n, ¬p m})
(λk h al, if pk: p k then ⟨k,pk,al⟩ else h (k+1) ⟨rfl,pk⟩
(λn pn, (lt_or_eq_of_le (nat.le_of_lt_succ pn)).elim
(λh, al n h) (λh, by rwa h))) 0
(let ⟨n,pn⟩ := H in suffices ∀m≤n, acc lbp m, from this 0 n.zero_le,
λ_ hm, nat.decreasing_induction
(λ_ hn, ⟨_, λ_ hy, by rwa hy.1⟩) hm ⟨_, λ_ hy, absurd pn hy.2⟩)
(λ_ h, absurd h (nat.not_lt_zero _))
Last updated: May 11 2021 at 15:12 UTC | 2021-05-11 15:56:01 | {"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": 0.5166760683059692, "perplexity": 4273.155568471591}, "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/1620243991648.10/warc/CC-MAIN-20210511153555-20210511183555-00092.warc.gz"} |
http://mathoverflow.net/questions/108491/about-an-argument-in-koszul-duality-patterns-in-representation-theory-by-beili | # About an argument in Koszul duality patterns in representation theory' by Beilinson-Ginzburg-Soergel.
I am trying to understand Proposition 3.4.2 in Koszul duality patterns in representation theory' by Beilinson-Ginzburg-Soergel [BGS]. A copy of the paper can be found at http://home.mathematik.uni-freiburg.de/soergel/
I outline the setup below. Text in bold are my own commentary. I have taken the liberty to change some confusing notation in the paper. However, it is entirely possible that the reason I am confused is that one of my "changes of notation" isn't correct. I have tried to be careful in my changes, but apologies in advance for any additional confusion this may contribute to.
My question is that indicated by the last bold face text below.
Let $X$ be a complex variety with an algebraic stratification by affine linear spaces $X = \sqcup_{w\in W} X_w$. Let $IC_w$ denote the intersection cohomology complex on $X$ corresponding to the constant sheaf on $X_w$.
Let $X = Y_0 \supset Y_1 \supset \cdots \supset Y_r = \emptyset$ be the corresponding filtration by closed subvarieties so that $Y_{p}- Y_{p+1} = X_p$ for some strata $X_p$.
Let
$j_w\colon X_w \to X$
be the inclusion of the strata. Assume parity vanishing, i.e., assume
$H^ij_v^*IC_w = 0$ unless $i = dim(X_v) + dim(X_w) \mod 2$, for all $v,w\in W$.
Here $H^*$ denotes perverse cohomology.
Proposition: Under the assumption of parity vanishing, hypercohomology induces an injection
$Hom^{\bullet}_{D^b(X)}(IC_x, IC_y) \to Hom_{\mathbb{C}}(\mathbb{H}^{\bullet}IC_x, \mathbb{H}^{\bullet}IC_y)$.
Proof: By parity vanishing the spectral sequence $\mathbb{H}^{p+q}j_p^!IC_x \implies \mathbb{H}^n IC_x$ is degenerate (the spectral sequence is defined via the filtration by local hypercohomology along the strata, for details see Section 3.4 of [BGS]). So if $f\in Hom^{\bullet}_{D^b(X)}(IC_x, IC_y)$ is given such that $\mathbb{H}^{\bullet}f = 0$, then necessarily $0 = j_p^!f \in Hom^{\bullet}_{D^b(X)}(j_p^!IC_x, j_p^!IC_y)$ for all $p$. Let
$a_p\colon Y_p \to X$
be the closed inclusion. We have a decomposition
$u\colon X_p = Y_p - Y_{p+1} \to Y_p$,
$i\colon Y_{p+1}\to Y_p$
in an open and a closed subset and a distinguished triangle
Edit: the original distinguished triangle (as stated in [BGS]) wasn't correct, I have now made the fix
$i_*i^!a_p^! \to a_p^! \to u_*u^!a_p^!$
(so this distinguished triangle is the same as $i_*a_{p+1}^! \to a_p^! \to u_*j_p^!$ )
which shows that $a_{p+1}^!f = 0 = j_{p}^!f$ implies $a_p^!f = 0$ (it is this implication that I don't understand, related to my confusion is an earlier question of mine Showing morphism of sheaves is zero )
Hence by induction $j_p^!f = 0$ for all $p$ implies $f = a_0^!f = 0$.
Any comments that would clarify the above would be most appreciated!
- | 2016-02-11 11:05:38 | {"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": 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": 0.9783921241760254, "perplexity": 305.97506460535914}, "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/1454701161942.67/warc/CC-MAIN-20160205193921-00133-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://dataspace.princeton.edu/jspui/handle/88435/dsp01f7623g226 | Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01f7623g226
Title: Exploration of Novel Intermetallic Superconductors Authors: Srivichitranond, Laura Advisors: Cava, Robert J. Department: Chemistry Class Year: 2017 Abstract: I report the discovery of the new superconductor TaIr$$_{2}$$Ge$$_{2}$$, which has a critical temperature around 3.5 K. This material crystallizes in an entirely new orthorhombic structure type consisting of endohedral Ta@Ir$$_{7}$$Ge$$_{4}$$ clusters, one that is more complex than those of the more commonly observed 1:2:2 intermetallic phases. The superconducting transition of this compound is characterized by temperature-dependent resistivity, magnetic susceptibility, and specific heat measurements, and is of the weak coupling BCS type with $$\Delta \text{C/}\gamma \text{T}_c$$ =1.55. In addition, I explore known structure types with the goal of generating new superconducting compounds, either by elemental substitution or by the addition of a small element to an already existing structure. This search led to the synthesis of three previously unreported phases: LuPd$$_3$$Si$$_x$$, TmB$$_3$$B$$_{1-x}$$Si$$_x$$, and Zr$$_4$$Mn$$_4$$Si$$_7$$. Further work on these systems may lead to the identification of other new superconducting materials. URI: http://arks.princeton.edu/ark:/88435/dsp01f7623g226 Type of Material: Princeton University Senior Theses Language: en_US Appears in Collections: Chemistry, 1926-2017 | 2018-05-21 02:52:55 | {"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": 0, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.39655768871307373, "perplexity": 3932.023476260757}, "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-2018-22/segments/1526794863923.6/warc/CC-MAIN-20180521023747-20180521043747-00032.warc.gz"} |
https://www.physicsforums.com/threads/a-little-push-on-this-trig-identity.65117/ | # A little push on this trig identity
A little push on this trig identity plz
$$\tan^2(x)= \frac {1-\cos(2x)} {1+cos(2x)}$$
I need a little push I know from my other post that $$\cos(2x)=\cos^2 (x) - \sin^2 (x)$$ (can someone explain why?)
what is your questioin? proving the tan identity or the cos one?
I have to make the left hand side equal the right hand side I dont think it matters if you use cos or tan which ever one is easier.
It would be easier to work with the right side.
So it would be like:
$$\frac {1-\cos(2x)} {1+cos(2x)}$$
$$\frac {1 - (1 - 2\sin^2x)}{1 + 2\cos^2x - 1}$$ Double angle identities
$$\frac{2\sin^2x}{2\cos^2x}$$ 2's cancel out
$$\tan^2x$$
Thanks sooo much BLUE SODA Im not good with the double angle identity thanks again | 2021-11-26 23:42:30 | {"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": 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": 0.8933249711990356, "perplexity": 874.9569325742882}, "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-2021-49/segments/1637964358074.14/warc/CC-MAIN-20211126224056-20211127014056-00399.warc.gz"} |
https://blender.stackexchange.com/questions/92471/can-i-calculate-the-surface-of-a-part-of-a-mesh-confined-by-a-square-boundary | # Can I calculate the surface of a part of a mesh, confined by a square boundary?
I have one single mesh object that consists of many little, randomly placed but non-overlapping circles, like so (top view):
The object was created by looping the bmesh.ops.create_cone(... ) -operator. I need to calculate the surface of the (parts of) circles that are within the square boundaries drawn in the image, which is in this case a square of radius 2 centered around the origin. I found this command in related questions
area = sum(f.calc_area() for f in bm.faces if ...),
which works well if I leave out the if statement, but I don't know how I can include a statement to cut away all pieces of circles that are outside of the bounding box.
Edit: @batFINGER I made a minimal working example of two non-overlapping circles of radius 2. For the sake of clarity I did not include the part of the code that puts them in random places, but I just defined the centers manually in ccValids. Each circle has 64 segments, so since both circles belong to the same mesh data, Blender considers this single mesh object, consisting of two circles, to have 128 faces that are all shaped like pieces of pie.
ccValids = [(-1,2,0), (2,-2,0)]
bm = bmesh.new()
for c in ccValids:
m = Matrix()
bmesh.ops.create_cone(
bm,
diameter2 = 0,
depth = 0,
segments = 64,
matrix = m.Translation( c )
)
l_d = bpy.data.meshes.new('LeafsM')
bm.to_mesh( l_d )
l_o = bpy.data.objects.new('LeafsO', l_d )
Now let's say I want to know what part of the total surface of these two circles lies inside the square of radius 2, that is shown in the camera view below:
Clearly the parts of interest are not segment-shaped, so it won't be useful to base the calculation on the counting on segments. Is there a way to perhaps "split" segments (thus the 128 faces) in an inside and an outside part first, and then count the surfaces of the parts that are inside?
• @batFINGER I try to understand but I don't get it. There seem to be 128 faces in these two circles but I don't know what's their shape and how they are used to build up the circles. I am trying to remove those faces of which I suspect that they're outside of the box, in order to discover what the faces look like, but it seems I cannot use a f.remove() command. – Gnub Oct 17 '17 at 16:07
• I meant remove(f), does not work – Gnub Oct 17 '17 at 16:21
• I made them by looping bmesh.create_cone like the example lTousky showed in blender.stackexchange.com/questions/91679/… , with segments = 64. Since I made 2 circles to keep an overview, there are now 128 of them and they all belong to the same mesh data. I only want the surface of those parts within some square boundaries, but to be sure I get the correct result I now try to remove the parts of circles that are supposed to be outside, to first see if I would really calculate the correct surface. – Gnub Oct 17 '17 at 19:21
• I can add a sample file tomorrow morning when I get back to work, I don't have the file here right now. – Gnub Oct 17 '17 at 19:24
Boolean Intersection modifier.
Given you are chopping aligned cylinders with an aligned box, the boolean modifier would be a much better way to go IMO.
Add a cube to the scene as the bounding box, scale in z to make sure it is taller than cylinders in z. Add a boolean intersection modifier to cylinders using cube as bounds. A copy of the cylinders mesh with the modifiers applied is calculated using Object.to_mesh(...) This mesh can be loaded into a bmesh, to do face area calculations.
To only calculate the areas of the circular tops, would need to look at face normals being parallel to z axis or similar.
Result of running script. The result mesh on left in place, result in edit mode on right. The top face area is 2.5704, which is 64.26% of the square area (4).
Test script
import bpy
import bmesh
from mathutils import Vector
TOL = 0.0001 # anle test
up = Vector((0, 0, 1))
context = bpy.context
scene = context.scene
obj = context.object
# add a new default cube
cube = context.object
cube.scale.z = 4
# add a boolean modifer to the obj
bmod = obj.modifiers.new(name="bbox", type='BOOLEAN')
bmod.operation = 'INTERSECT'
bmod.object = cube
scene.update()
# create a copy
copy_mesh = obj.to_mesh(scene, True, settings='PREVIEW')
bm = bmesh.new()
bm.from_mesh(copy_mesh)
area = sum(f.calc_area() for f in bm.faces
if f.normal.angle(up) < TOL)
print("Top face area:", area)
bm.free()
# clean up...
obj.modifiers.remove(bmod)
Test run on cones. Bottom faces normals matched with -Z axis. The sum of the areas of bottom edges of the mesh is 3.04279, 70.70% of a 2 x 2 square.
• The script did work first time I ran it, but now I keep getting an error message on bmod = obj.modifiers.new(name="bbox", type='BOOLEAN') : val: BooleanModifier.object ID type does not support assignment to its self. – Gnub Oct 19 '17 at 9:54
• Remove all the boolean modifiers on spotty cone object. (unremoved from crashing edit mode runs) If the circle face of the cone is on the bottom, you can match it against down = (0, 0, -1) instead of up. – batFINGER Oct 19 '17 at 14:38 | 2019-11-13 22:50:43 | {"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": 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": 0.45926034450531006, "perplexity": 1084.6043064237197}, "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/1573496667442.36/warc/CC-MAIN-20191113215021-20191114003021-00038.warc.gz"} |
https://ambp.centre-mersenne.org/articles/10.5802/ambp.232/ | Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions
Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 2, pp. 187-197.
A famous theorem of E. Gagliardo gives the characterization of traces for Sobolev spaces ${W}^{1,\phantom{\rule{0.166667em}{0ex}}p}\left(\Omega \right)$ for $1\le p<\infty$ when $\Omega \subset {ℝ}^{N}$ is a Lipschitz domain. The extension of this result to ${W}^{m,\phantom{\rule{0.166667em}{0ex}}p}\left(\Omega \right)$ for $m\ge 2$ and $1 is now well-known when $\Omega$ is a smooth domain. The situation is more complicated for polygonal and polyhedral domains since the characterization is given only in terms of local compatibility conditions at the vertices, edges, .... Some recent papers give the characterization for general Lipschitz domains for m=2 in terms of global compatibility conditions. Here we give the necessary compatibility conditions for $m\ge 3$ and we prove how the local compatibility conditions can be derived.
DOI: 10.5802/ambp.232
Giuseppe Geymonat 1
1 Laboratoire de Mécanique et de Génie Civil, UMR 5508 CNRS, Université Montpellier II Place Eugène Bataillon 34695 Montpellier Cedex 5 France
@article{AMBP_2007__14_2_187_0,
author = {Giuseppe Geymonat},
title = {Trace {Theorems} for {Sobolev} {Spaces} on {Lipschitz} {Domains.} {Necessary} {Conditions}},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {187--197},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {14},
number = {2},
year = {2007},
doi = {10.5802/ambp.232},
mrnumber = {2369871},
zbl = {1161.46019},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.232/}
}
TY - JOUR
AU - Giuseppe Geymonat
TI - Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions
JO - Annales mathématiques Blaise Pascal
PY - 2007
DA - 2007///
SP - 187
EP - 197
VL - 14
IS - 2
PB - Annales mathématiques Blaise Pascal
UR - https://ambp.centre-mersenne.org/articles/10.5802/ambp.232/
UR - https://www.ams.org/mathscinet-getitem?mr=2369871
UR - https://zbmath.org/?q=an%3A1161.46019
UR - https://doi.org/10.5802/ambp.232
DO - 10.5802/ambp.232
LA - en
ID - AMBP_2007__14_2_187_0
ER -
%0 Journal Article
%A Giuseppe Geymonat
%T Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions
%J Annales mathématiques Blaise Pascal
%D 2007
%P 187-197
%V 14
%N 2
%I Annales mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.232
%R 10.5802/ambp.232
%G en
%F AMBP_2007__14_2_187_0
Giuseppe Geymonat. Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions. Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 2, pp. 187-197. doi : 10.5802/ambp.232. https://ambp.centre-mersenne.org/articles/10.5802/ambp.232/
[1] R. A. Adams; J. J. F. Fournier Sobolev spaces. Second edition, Academic Press, New York, 2003 | Zbl
[2] A. Buffa; G. Geymonat On the traces of functions in ${W}^{1,\phantom{\rule{0.166667em}{0ex}}p}\left(\Omega \right)$ for Lipschitz domains in ${ℝ}^{3}$, C. R. Acad. Sci. Paris, Série I, Volume 332 (2001), pp. 699-704 | MR | Zbl
[3] A. Buffa; Jr P. Ciarlet On traces for functional spaces related to Maxwell’s equations. Part I: an integration by parts formula in Lipschitz Polyedra, Math. Meth. Appl. Sci., Volume 24 (2001), pp. 9-30 | DOI | Zbl
[4] Zhonghai Ding A proof of the trace theorem of Sobolev spaces on Lipschitz domains, Proc. A. M. S., Volume 124 (1996), pp. 591-600 | DOI | MR | Zbl
[5] R. G. Durán; M. A. Muschietti On the traces of ${W}^{2,\phantom{\rule{0.166667em}{0ex}}p}\left(\Omega \right)$ for a Lipschitz domain, Rev. Mat. Complutense, Volume XIV (2001), pp. 371-377 | MR | Zbl
[6] E. Gagliardo Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n-variabili, Rend. Sem. Mat. Univ. Padova, Volume 27 (1957), pp. 284-305 | Numdam | MR | Zbl
[7] G. Geymonat; F. Krasucki On the existence of the airy function in Lipschitz domains. Application to the traces of ${H}^{2}$, C. R. Acad. Sci. Paris, Série I, Volume 330 (2000), pp. 355-360 | MR | Zbl
[8] P. Grisvard Elliptic boundary value problems in nonsmooth domains, Pitman, London, 1985 | Zbl
[9] J. Nečas Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967 | MR
Cited by Sources: | 2023-01-29 06:39:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 12, "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": 0.5739142298698425, "perplexity": 1277.4336521429204}, "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-2023-06/segments/1674764499700.67/warc/CC-MAIN-20230129044527-20230129074527-00055.warc.gz"} |
http://metafor-project.org/doku.php/tips?do=diff | # The metafor Package
A Meta-Analysis Package for R
tips
# Differences
This shows you the differences between two versions of the page.
— tips [2019/06/30 11:46] (current) Line 1: Line 1: + ===== Tips and Notes ===== + + The links below point to pages illustrating various tips and notes that may be useful when working with the metafor package. In addition, some features of the package that may not be readily apparent from the documentation are explained in more detail. + + * [[tips:handling_missing_data|Handling Missing Data in Output/Figures]]: An illustration/discussion of how to show studies in figures and output that were actually excluded from model fitting due to missing data. + + * [[tips:assembling_data_smd|Assembling Data for a Meta-Analysis of Standardized Mean Differences]]: An illustration of how a dataset for a meta-analysis of standardized mean differences (Cohen's d values) can be assembled/constructed from various pieces of information. + + * [[tips:assembling_data_or|Assembling Data for a Meta-Analysis of (Log) Odds Ratios]]: An illustration of how a dataset for a meta-analysis of (log) odds ratios can be assembled/constructed from various pieces of information. + + * [[tips:regression_with_rma|Linear Regression and the Mixed-Effects Meta-Regression Model]]: An illustration of the relationship between the linear regression model (fitted by the ''lm()'' function) and the mixed-effects meta-regression model (fitted by the ''rma()'' function). + + * [[tips:two_stage_analysis|Two-Stage Analysis versus Linear Mixed-Effects Models for Longitudinal Data]]: An illustration of two different approaches to analyzing longitudinal data: A two-stage analysis (which the ''rma.mv()'' function can be used for) and linear mixed-effects models (e.g., using the ''lme()'' function). + + * [[tips:rma.uni_vs_rma.mv|A Comparison of the rma.uni() and rma.mv() Functions]]: A comparison of the ''rma.uni()'' and ''rma.mv()'' functions for fitting fixed- and random-effects models. + + * [[tips:rma_vs_lm_lme_lmer|A Comparison of the rma() and the lm(), lme(), and lmer() Functions]]: An illustration of the difference between the models fitted by the ''rma()'' function and the models fitted by the ''lm()'', ''lme()'', and ''lmer()'' functions (or: why the ''lm()'', ''lme()'', and ''lmer()'' functions cannot be used to fit meta-analytic models). + + * [[tips:models_with_or_without_intercept|Meta-Regression Models With or Without an Intercept]]: A discussion of what happens when we fit meta-regression models with or without an intercept. + + * [[tips:testing_factors_lincoms|Testing Factors and Linear Combinations of Parameters]]: An illustration of how to test factors and linear combinations of parameters in (mixed-effects) meta-regression models. + + * [[tips:multiple_factors_interactions|Models with Multiple Factors and Their Interaction]]: An illustration of how to examine and conduct tests of models involving multiple factors and their interaction. + + * [[tips:bootstrapping_with_ma|Bootstrapping with Meta-Analytic Models]]: An example showing how to conduct parametric and non-parametric bootstrapping with meta-analytic models. + + * [[tips:comp_mh_different_software|Comparison of the Mantel-Haenszel Method in Different Software]]: A comparison of the results obtained with the Mantel-Haenszel method as implemented in metafor and other software. + + * [[tips:comp_two_independent_estimates|Comparing Estimates of Independent Meta-Analyses or Subgroups]]: An illustration of how to compare two estimates from two independent meta-analyses or subgroups of studies. + + * [[tips:model_selection_with_glmulti_and_mumin|Model Selection using the glmulti and MuMIn Packages]]: An illustration of how to use the metafor package in combination with the glmulti and MuMIn packages for model selection and multimodel inference based on an information-theoretic approach. + + * [[tips:convergence_problems_rma|Convergence Problems with the rma() Function]]: A discussion and illustration of convergence problems that can rise when fitting random/mixed-effects (meta-regression) models with the ''rma()'' function. + + * [[tips:clogit_paired_binary_data|Conditional Logistic Regression for Paired Binary Data]]: An illustration of how to fit the conditional logistic regression model for paired binary data. + + * [[tips:i2_multilevel_multivariate|I^2 for Multilevel and Multivariate Models]]: A discussion of how one can compute $I^2$-type statistics in multilevel and multivariate models. + + * [[tips:hunter_schmidt_method|Hunter and Schmidt Method]]: A discussion of how one can conduct meta-analyses according to the Hunter & Schmidt method. + + * [[tips:speeding_up_model_fitting|Speeding Up Model Fitting]]: A discussion of some methods and strategies for speeding up model fitting with complex models. + + * [[tips:multiple_imputation_with_mice_and_metafor|Multiple Imputation with the mice and metafor Packages]]: An illustration of how to do multiple imputation together with the mice and metafor packages.
tips.txt · Last modified: 2019/06/30 11:46 (external edit) | 2020-01-29 01:55:46 | {"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": 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": 0.21820995211601257, "perplexity": 3517.4742785120825}, "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-2020-05/segments/1579251783621.89/warc/CC-MAIN-20200129010251-20200129040251-00310.warc.gz"} |
https://zbmath.org/?q=an:0878.17029 | ## Alternative loop rings.(English)Zbl 0878.17029
North-Holland Mathematics Studies. 184. Amsterdam: North-Holland. xv, 387 p. (1996).
Since the beginning of this century group algebras have been of permanent interest. In a similar way as for groups, quasigroups and loop algebras may be defined. However, this construction is too general to produce something remarkable. But the restriction to loop algebras satisfying the left and right alternative laws created a new fertile subject. This book is now the first survey of the theory of alternative loop rings written by the protagonists of this new area.
In the first three chapters the authors present important results on the fundamental algebraic structures the book is concerned with, namely alternative rings, Moufang loops, and loop rings.
Chapter IV investigates the properties of those (finite) loops which, in characteristic different from two, have alternative, but not associative loop rings. Such loops are called RA (ring alternative) loops. Chapter V establishes the classification of finite RA loops.
The following chapters contain an analysis of the loop rings themselves and of the group rings of certain groups which are closely related to RA loops. In Chapter VI, the authors describe the Jacobson and prime radicals of an alternative loop ring. Moreover, it turns out that over many fields, alternative loop algebras are direct sums of simple algebras (i.e., a Maschke type theorem holds). Chapter VII describes the simple components of a (semisimple) alternative loop algebra over the rationals by establishing concrete isomorphisms with Zorn’s vector matrix algebras. In this chapter the authors also treat the primitive idempotents of the group algebra of a finite abelian group. This enables them to characterize the primitive central idempotents of the rational loop algebra $$QL$$ of a finite RA loop $$L$$ and thus the loop algebra $$QL$$ itself concretely.
With Chapter VIII a study of the units of an integral alternative loop ring starts. The authors investigate under what conditions all the units and all the torsion units are trivial. This yields new proofs of well known theorems of G. Higman and S. D. Berman for group rings.
Chapters IX to XI are concerned with isomorphism problems. In Chapter IX it is shown that any finite RA loop is determined by its integral loop ring. Moreover, for a finite RA loop $$L$$, every normalized automorphism of $$ZL$$ is the composition of an automorphism of $$L$$ and an inner automorphism of the rational loop algebra $$QL$$. Also variations for loop rings of three conjectures of H. Zassenhaus for group rings are presented.
The entire Chapter X is devoted to the isomorphism problem for group algebras of finite abelian groups since abelian groups play an important role in the structure of RA loops. These results are applied in Chapter XI to loop algebras of RA loops over arbitrary fields.
In Chapter XII the authors collect some more results on the units of an integral alternative loop ring. The problem of finding all the units of an integral alternative loop ring is very hard. Thus the authors confine themselves to more accessible questions, e.g., they exhibit a certain subloop of finite index in the unit loop of $$ZL$$.
The theme of the last Chapter XIII is the question under what conditions each element of an alternative loop algebra over a field has only finitely many conjugates. In solving this problem, the authors give conditions under which all idempotents of a loop algebra are central and determine when all the nilpotent elements are trivial.
This book is a competent source for all themes concerning RA loops and their loop rings. It will certainly become a standard reference for specialists working in this area. But the topics of this book may also be interesting for other disciplines. So group and ring theorists will find some new insights and some new proofs of well known theorems for group rings. Since the text is nearly self-contained, the book serves also as an excellent introduction to the theory of alternative loop rings.
### MSC:
17D05 Alternative rings 20N05 Loops, quasigroups 17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras 16S36 Ordinary and skew polynomial rings and semigroup rings 16U60 Units, groups of units (associative rings and algebras) | 2022-09-27 03:53:23 | {"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": 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": 0.6426241397857666, "perplexity": 577.5095733877032}, "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/1664030334987.39/warc/CC-MAIN-20220927033539-20220927063539-00743.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=305760 | Linear Algebra - Diagonalizable and Eigenvalue Proof
by B_Phoenix
Tags: algebra, diagonalizable, eigenvalue, linear, proof
P: 1 1. The problem statement, all variables and given/known data "Let A be a diagonalizable n by n matrix. Show that if the multiplicity of an eigenvalue lambda is n, then A = lambda i" 2. Relevant equations 3. The attempt at a solution I had no idea where to start.
P: 111 Since $$A$$ is diagonalizable, we can choose some invertible matrix $$S$$ such that $$A = S D S^{-1}$$, where $$D$$ is diagonal and the diagonal entries of $$D$$ are the eigenvalues of $$A$$. We can translate the assumption regarding the multiplicity of $$\lambda$$ into a statement about $$D$$, after which the result follows by using $$A = S D S^{-1}$$.
Related Discussions Calculus & Beyond Homework 1 Calculus & Beyond Homework 5 Linear & Abstract Algebra 2 Calculus & Beyond Homework 1 Calculus & Beyond Homework 1 | 2013-12-08 20:08:25 | {"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": 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": 0.8790624141693115, "perplexity": 274.44626973776474}, "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-2013-48/segments/1386163806278/warc/CC-MAIN-20131204133006-00093-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/359550/if-cup-mathcalf-a-then-a-in-mathcalf-prove-that-a-has-exactly-one | # If $\cup \mathcal{F}=A$ then $A \in \mathcal{F}$. Prove that $A$ has exactly one element.
I'm reading through How to Prove It by Velleman and I'm having trouble with this exercise in the section about Existence and Uniqueness proofs. Here is the exercise:
Suppose $A$ is a set and for every family of sets $\mathcal{F}$, if $\cup \mathcal{F}=A$ then $A \in \mathcal{F}$. Prove that $A$ has exactly one element.
He hints that for both the existence and uniqueness parts of the proof it would be a good idea to use contradiction. I've been playing around with this proof for a while but I can't seem to make any substantial progress.
I current idea is considering some cases where for some family of sets $\mathcal{G}$, $A \in \mathcal{G}$ and $A \notin \mathcal{G}$. I thought if I could show that if $A= \varnothing$ lead to contradictions, I could at least say that there is something in $A$, and try to prove that it is unique from there. I haven't been able to make any progress with this though. Any help with this problem would be greatly appreciated!
-
Hints: For $A=\varnothing$, try $\mathcal{F}=\varnothing$. Then conclude $\exists x\in A$ and consider $\mathcal{F} = \{\{x\},A\backslash\{x\}\}$.
If $\mathcal F=\emptyset$, then $\bigcup\mathcal F = \emptyset$, but $\emptyset\notin\mathcal F$, hence clearly $A\ne\emptyset$.
Assume $a\in A$. Let $\mathcal F=\{A\setminus\{a\},\{a\}\}$. Then $\bigcup F=A$ implies $A\in \mathcal F$. Since $a\in A\ne A\setminus\{a\}\not\ni a$, we conclude $A=\{a\}$. | 2016-05-01 06:29:54 | {"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": 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": 0.9446254372596741, "perplexity": 46.99942993548817}, "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-2016-18/segments/1461860114285.77/warc/CC-MAIN-20160428161514-00126-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://myschool.ng/classroom/mathematics?page=8 | ### Mathematics Past Questions
36
Evaluate $$\frac{(2.813 \times 10^{-3} \times 1.063)}{(5.637 \times 10^{-2})}$$ reducing each number to two significant figures and leaving your answer in two significant figures.
• A. 0.056
• B. 0.055
• C. 0.054
• D. 0.54
37
A man wishes to keep his money in a savings deposit at 25% compound interest so that after three years he can buy a car for N150,000. How much does he need to deposit?
• A. N112,000.50
• B. N96,000.00
• C. N85,714.28
• D. N76,800.00
38
If 31410 - 2567 = 340x, find x.
• A. 7
• B. 8
• C. 9
• D. 10
39
Simplify $$\frac{3(2^{n+1}) - 4(2^{n-1})}{2^{n+1} - 2^n}$$
• A. 2n+1
• B. 2n-1
• C. 4
• D. 1/4
40
If $$P344_{6} - 23P2_{6} = 2PP2_{6}$$, find the value of the digit P.
• A. 2
• B. 3
• C. 4
• D. 5 | 2020-02-20 02:54:45 | {"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": 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": 0.2964131236076355, "perplexity": 12444.707746227929}, "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-2020-10/segments/1581875144498.68/warc/CC-MAIN-20200220005045-20200220035045-00375.warc.gz"} |
https://math.stackexchange.com/questions/1586213/knowing-which-factorization-algorithm-to-use | # Knowing which factorization algorithm to use
There are many ways of factorization available, e.g. trial division, Pollard rho, elliptic curve factorisation, the general number field sieve. But for what ranges of numbers are such algorithms appropriate? Obviously, using the general number field sieve to factor 15 is using a sledgehammer to crack a walnut, and using trial division to factorise a large Mersenne prime would take longer than the age of the universe, but at what points should one stop using one algorithm and use another?
My current understanding is as follows:
$\underline{2 \leq n \leq 100000}$
Use trial division
$\underline{10^5 \leq n \leq 10^{10}}$
Use Pollard rho, or Pollard $p-1$
$\underline{10^{10} \leq n \leq 10^{20}}$
Elliptic curve factorisation
$\underline{10^{20} \leq n}$
The general number field sieve
Of course, this is all very dependent of your algorithm implementation, but roughly speaking, is this analysis correct, or should I be considering other special subcases/using other algorithms? If I list a bunch of algorithms, like the ones above, we can give their known computational complexities, but in practise, when do they start to outperform each other?
• Regarding your last sentence, that's roughly what I was trying to get at with this question. I'll try and rephrase it as such. – MadMonty Dec 23 '15 at 1:33
• You'll always want to do a little trial division. Under 64 bits it's a bit more complicated for optimal performance, but consider SQUFOF. p-1 and ecm are good at all sizes to find small factors, and all you need for ~40 digits. QS is good for roughly 30 to 100 digits. GNFS is quite a bit more complicated than the rest, and typically the crossover with QS will be 90-110 digits. While it's quite old now, I wrote up some practical experiments with crossover graphs at diamond.boisestate.edu/~liljanab/BOISECRYPTFall09/Jacobsen.pdf. – DanaJ Jan 3 '16 at 8:25
Well it is very dependent on what hardware you have, but two categories of algorithm need to be distinguished.
### Deterministic
These are algorithms that will, eventually, always find the factors. They will systematically search for them. At one end you have trial division and at the other GNFS.
### Probabilistic
These are algorithms that are designed to be quick for most numbers, but are never guaranteed to find a solution. They might run forever, for longer than trial division, and never find a solution. In practice they'll probably work faster than that, but they're typically not fast for the most difficult factoring problems ( products of large primes ).
### Special Numbers
There are some algorithms for special types of numbers. For example, there are algorithms for checking if a Fermat number is prime. These allow the discovery of factors that are absolutely enormous, well beyond what would be workable with even GNFS.
Typically you would employ basic trial division "for a while", then move to a good probabilistic method "for a while", then move to something heavyweight like GNFS when your patience is exhausted.
### Implementation difficulty
Another factor in deciding what to use is how difficult it is to code, bug free, maintain and refine an algorithm.
Another practical consider is that complex, highly tuned algorithms like GNFS are very difficult to write. If a library is available, it may, for particular purposes, be better to use something pre-packaged that doesn't require you to make your entire career factoring theory just to debug it, than try and take on the task of maintaining your own.
There is no one way to build an approach. | 2019-07-21 16:55:16 | {"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": 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": 0.6097173094749451, "perplexity": 452.2562867590249}, "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/1563195527089.77/warc/CC-MAIN-20190721164644-20190721190644-00343.warc.gz"} |
https://socratic.org/questions/how-do-you-differentiate-y-ln-3-9-x-2-x | # How do you differentiate y =-ln [ 3+ (9+x^2) / x]?
Dec 24, 2017
$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{3 + \frac{9}{x} + x} \cdot \left(1 - \frac{9}{x} ^ 2\right)$
#### Explanation:
$y = - \ln \left(3 + \frac{9 + {x}^{2}}{x}\right) = - \ln \left(3 + \frac{9}{x} + x\right)$
Use the chain rule $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$
Let $y = - \ln \left(u\right)$ then $\frac{\mathrm{dy}}{\mathrm{du}} = - \frac{1}{u}$
And $u = 3 + \frac{9}{x} + x$ then $\frac{\mathrm{du}}{\mathrm{dx}} = - \frac{9}{x} ^ 2 + 1$
$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{u} \cdot \left(1 - \frac{9}{x} ^ 2\right)$
Substitute $u = 3 + \frac{9}{x} + x$
$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{3 + \frac{9}{x} + x} \cdot \left(1 - \frac{9}{x} ^ 2\right)$ | 2019-06-19 11:04:57 | {"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": 0.9960921406745911, "perplexity": 6249.559875896051}, "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-2019-26/segments/1560627998959.46/warc/CC-MAIN-20190619103826-20190619125826-00302.warc.gz"} |
https://www.math.bgu.ac.il/en/teaching/spring2021/courses/algebraic-geometry-schemes-2 | ## Prof. Amnon Yekutieli
#### Time and Place:
Wed 12:00 - 14:00 Israel time
See pdf files
## Course topics
1. Sheaves on topological spaces
2. Affine schemes
3. Schemes and morphisms between them.
4. Quasi-coherent sheaves
5. Separated and proper morphisms.
6. Vector bundles and the Picard group of a scheme.
7. The functor of points and moduli spaces.
8. Morphisms to projective space and blow-ups.
9. Smooth morphisms and differential forms.
10. Sheaf cohomology.
11. Group schemes. | 2021-04-21 14:42:03 | {"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": 0.9721634387969971, "perplexity": 14195.123877341346}, "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-2021-17/segments/1618039544239.84/warc/CC-MAIN-20210421130234-20210421160234-00518.warc.gz"} |
https://community.stenci.la/t/missing-inserted-in-alignment-preamble-xlsx-to-pdf/206/5 | # "Missing # inserted in alignment preamble" - xlsx to pdf
#1
## This is my problem:
Can’t convert directly from .xlsx to .pdf (first to .md then to .pdf works, bad results). Error: “Missing # inserted in alignment preamble”. Version 0.30.1, Win7 64-bit. Pandoc version 2.2.3.2. “pdflatex --version” reports “pdfTeX 3.14159265-2.6-1.40.19 (TeX Live 2018/W32TeX)”. Conversion from .xlsx to .md works, from .md to .pdf works, but results are not acceptable.
EDIT: Identical results with .ODS file to .PDF.
## This is what I am trying to do:
I would like to be able to directly convert .xlsx to .pdf.
## My experience level is with Stencila is:
low
NOTE: When someone posts a solution which answers your question. Click the ‘solved’ button.
#2
Hi @burque50, welcome to the Stencila community!
Thanks for trying out the converter tool. The cross-document-type conversion that you’re trying e.g. spreadsheet to Markdown or PDF) is a bit of an experiment in seeing if we can use a single, internal JSON-based format for executable documents (i.e spreadsheets and notebooks like RMarkdown and Jupyter). Being experimental, this feature is still buggy.
Can you share your .xlsx file here (or some simplified version of it) so I can see if I can create a bug fix that will work for your specific case?
Nokome
#3
Thanks, @nokome, for responding. There doesn’t appear to be an option to upload document files to the forum, only images. I’ll be happy to upload the files. How might I do it?
Even a spreadsheet with only 6 cells, each containing only text, fails with the error mentioned. The json created, however, contains all the data. I posted copied console output in another message, but it came through quite badly. There appears to be no way to format code properly in the forum, at least that I’ve discovered so far. The option from the editor (</>) hammers the code, at least for me anyway, I’ve tried in both Firefox and Chrome.
EDIT:
Code problem solved with bbcode tag (trying spoiler out also). By the way, I get a useable conversion by first converting xlxs -> md and then md -> pdf, but no charts so far. Once again, I’d love to share the files if I can figure out how to upload them.
? Error converting "TM.xlsx" to "TM.pdf": Error calling Pandoc:
message: Error producing PDF.
! Missing # inserted in alignment preamble.
<to be read again>
\cr
l.62 \begin{longtable}[]{@{}@{}}
args: --from json --output TM.pdf --data-dir=C:\Users\Winter\AppDa
tencila\data\pandoc
content: {
"pandoc-api-version": [
1,
17,
5
],
"meta": {
"name": {
"t": "MetaString",
"c": "Monkeys"
}
},
"blocks": [
{
"t": "Table",
"c": [
[],
[],
[],
[],
[
[
[
{
"t": "Plain",
"c": [
{
"t": "Str",
"c": "# Monkeys"
}
]
}
],
[
{
"t": "Plain",
"c": [
{
"t": "Str",
"c": "Tame?"
}
]
}
]
],
[
[
{
"t": "Plain",
"c": [
{
"t": "Str",
"c": "10"
}
]
}
],
[
{
"t": "Plain",
"c": [
{
"t": "Str",
"c": "Partly"
}
]
}
]
],
[
[
{
"t": "Plain",
"c": [
{
"t": "Str",
"c": "30"
}
]
}
],
[
{
"t": "Plain",
"c": [
{
"t": "Str",
"c": "Completely"
}
]
}
]
],
[
[
{
"t": "Plain",
"c": [
{
"t": "Str",
"c": "28"
}
]
}
],
[
{
"t": "Plain",
"c": [
{
"t": "Str",
"c": "Utterly intransigent"
}
]
}
]
]
]
]
}
]
}
That works better
Regards,
burque505
#4
Hi @burque505,
Thanks for your messages and sorry for slow reply. Can you try to upload the xlsx file now? It should be possible.
I will have a look. Though as @nokome said, the converters are still in their development-experimental phase.
Cheers,
Aleksandra
#5
I am now going through your posts in Stencila Converter - probably best to move/continue the discussion there
#6
Moving there now, thanks! | 2019-01-21 08:25:03 | {"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": 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": 0.38932931423187256, "perplexity": 12225.26944413761}, "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-2019-04/segments/1547583763839.28/warc/CC-MAIN-20190121070334-20190121092334-00626.warc.gz"} |
http://crypto.stackexchange.com/questions/12021/can-the-xor-of-two-non-collision-resistant-hashes-be-collision-resistant/12297 | # Can the XOR of two non-collision-resistant hashes be collision resistant?
Suppose I have two hash functions, of which neither (or only one) is collision resistant, and I want to create a new hash function by taking the bitwise exclusive or (XOR) of the results of those two functions.
Is it possible for this new function to be collision resistant? I suppose not, but I'm not sure why.
-
The answer to your edited question is "yes, it is possible".
As a trivial example, let $H$ be an ideal $k$-bit hash function. Due to the existence of the generic birthday attack, $H$ provides only about $k/2$ bits of collision resistance — that is, an attack can, on average, find a collision after about $2^{k/2}$ hash function evaluations. Denote the output of $H$, given the input $x$, as $$H(x) = (b_1(x), b_2(x), b_3(x), \dotsc, b_k(x)),$$ where $b_i$ denotes the $i$-th bit of the output.
Now define the $k$-bit hash functions $$H_L(x) = (b_1(x), b_2(x), b_3(x), \dotsc, b_{k/2}(x), 0, 0, \dotsc, 0)$$ and $$H_R(x) = (0, 0, \dotsc, 0, b_{k/2+1}(x), b_{k/2+2}(x), \dotsc, b_k(x)).$$
That is, $H_L$ is the same as $H$, except that the last $k/2$ bits of the output are replaced by zeros, and $H_R$ is the same as $H$ except that the first $k/2$ bits are replaced by zeros.
Now, clearly, either of $H_L$ or $H_R$ alone only provides $k/4$ bits of collision resistance. If, say $k = 128$, then $H$ may still be considered practically collision-resistant (effort to break = $2^{64}$), but finding collisions for $H_L$ or $H_R$ would be trivial (effort to break = $2^{32}$).
However, by construction, $H_L(x) \oplus H_R(x) = H(x)$. Thus, the hash function obtained by XORing the outputs of $H_L$ and $H_R$ is identical to, and thus equally strong as, the original hash $H$.
That said, this doesn't necessarily mean that the XOR of two hash function always has better collision resistance than the original hashes — in fact, it's easy to construct examples where the collision resistance of the XORed hashes is worse than that of either original hash. (For a simple example, let $H$ be a collision-resistant hash, and let $H_A = H_B = H$. The either of $H_A$ or $H_B$ alone is collision-resistant, but $H_A(x) \oplus H_B(x) = 0$ for all $x$!)
More to the point, for real-world hash functions, it depends on why and how the collision resistance of the original hashes is compromised. In some cases, XORing two hashes might improve their collision resistance; in other cases, it might not, or it might even make it worse.
-
To add some more perspective to this: this question has been studied quite extensively in the context of hash function combiners. A combiner is simply a function that gets (black-box) access to two hash functions and implements a new hash function. The question there is: does a combiner with "short output" exist that is robust for collision resistant hash functions? Here robust means that the combiner should be collision resistant if at least one of the two input functions is. Short in this context means that the combiner's output length is more than super-logarithmically shorter than the combined output of the two input hash functions. Your XOR example falls into this category, as the output is only $n$ bits, in contrast to $2n$ bits for the concatenation combiner (assuming the two input hash functions have $n$-bit outputs).
In a 2008 Crypto paper by Pietrzak it was shown that no such combiner exists that is good for "arbitrary" input functions, where good is defined as "collisions on the combiner can be reduced to collisions on both input functions".
To sum up, the XOR can work for specific functions. However, there can be situations where you start with two perfectly collision resistant functions and end up with a function that is not secure.
- | 2014-10-25 04:37:15 | {"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": 0.49554720520973206, "perplexity": 463.44581004497627}, "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-2014-42/segments/1414119647629.9/warc/CC-MAIN-20141024030047-00255-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://www.ams.org/mathscinet-getitem?mr=0164375 | MathSciNet bibliographic data MR164375 60.65 Blackwell, David; Freedman, David The tail $\sigma$$\sigma$-field of a Markov chain and a theorem of Orey. Ann. Math. Statist. 35 1964 1291–1295. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews. | 2017-05-25 03:31:17 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "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": 0.9985577464103699, "perplexity": 6350.284152619872}, "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-2017-22/segments/1495463607963.70/warc/CC-MAIN-20170525025250-20170525045250-00111.warc.gz"} |
https://stackoverflow.com/questions/44965/what-is-a-monad?page=2&tab=votes | # What is a monad?
Having briefly looked at Haskell recently, what would be a brief, succinct, practical explanation as to what a monad essentially is?
I have found most explanations I've come across to be fairly inaccessible and lacking in practical detail.
I'm trying to understand monads as well. It's my version:
Monads are about making abstractions about repetitive things. Firstly, monad itself is a typed interface (like an abstract generic class), that has two functions: bind and return that have defined signatures. And then, we can create concrete monads based on that abstract monad, of course with specific implementations of bind and return. Additionally, bind and return must fulfill a few invariants in order to make it possible to compose/chain concrete monads.
Why create the monad concept while we have interfaces, types, classes and other tools to create abstractions? Because monads give more: they enforce rethinking problems in a way that enables to compose data without any boilerplate.
In the Coursera "Principles of Reactive Programming" training - Erik Meier describes them as:
"Monads are return types that guide you through the happy path." -Erik Meijer
• The course doesn't exist on Coursera any more – icc97 Dec 15 '16 at 13:26
• That's a surprising - and v poor - reflection on Coursera. – javadba Dec 5 '17 at 1:30
A monad is a thing used to encapsulate objects that have changing state. It is most often encountered in languages that otherwise do not allow you to have modifiable state (e.g., Haskell).
An example would be for file I/O.
You would be able to use a monad for file I/O to isolate the changing state nature to just the code that used the Monad. The code inside the Monad can effectively ignore the changing state of the world outside the Monad - this makes it a lot easier to reason about the overall effect of your program.
• As I understand, monads are more than that. Encapsulating mutable state in a "pure" functional languages is only one application of monads. – thSoft Dec 30 '10 at 2:17
A very simple answer is:
Monads are an abstraction that provide an interface for encapsulating values, for computing new encapsulated values, and for unwrapping the encapsulated value.
What's convenient about them in practice is that they provide a uniform interface for creating data types that model state while not being stateful.
It's important to understand that a Monad is an abstraction, that is, an abstract interface for dealing with a certain kind of data structure. That interface is then used to build data types that have monadic behavior.
You can find a very good and practical introduction in Monads in Ruby, Part 1: Introduction.
Essentially, and Practically, monads allow callback nesting
(with a mutually-recursively-threaded state (pardon the hyphens))
(in a composable (or decomposable) fashion)
(with type safety (sometimes (depending on the language)))
)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
E.G. this is NOT a monad:
//JavaScript is 'Practical'
var getAllThree =
bind(getFirst, function(first){
return bind(getSecond,function(second){
return bind(getThird, function(third){
var fancyResult = // And now make do fancy
// with first, second,
// and third
return RETURN(fancyResult);
});});});
But monads enable such code.
The monad is actually the set of types for:
{bind,RETURN,maybe others I don't know...}.
Which is essentially inessential, and practically impractical.
So now I can use it:
var fancyResultReferenceOutsideOfMonad =
getAllThree(someKindOfInputAcceptableToOurGetFunctionsButProbablyAString);
//Ignore this please, throwing away types, yay JavaScript:
// RETURN = K
// bind = \getterFn,cb ->
// \in -> let(result,newState) = getterFn(in) in cb(result)(newState)
Or Break it up:
var getFirstTwo =
bind(getFirst, function(first){
return bind(getSecond,function(second){
var fancyResult2 = // And now make do fancy
// with first and second
return RETURN(fancyResult2);
});})
, getAllThree =
bind(getFirstTwo, function(fancyResult2){
return bind(getThird, function(third){
var fancyResult3 = // And now make do fancy
// with fancyResult2,
// and third
return RETURN(fancyResult3);
});});
Or ignore certain results:
var getFirstTwo =
bind(getFirst, function(first){
return bind(getSecond,function(second){
var fancyResult2 = // And now make do fancy
// with first and second
return RETURN(fancyResult2);
});})
, getAllThree =
bind(getFirstTwo, function(____dontCare____NotGonnaUse____){
return bind(getThird, function(three){
var fancyResult3 = // And now make do fancy
// with three only!
return RETURN(fancyResult3);
});});
Or simplify a trivial case from:
var getFirstTwo =
bind(getFirst, function(first){
return bind(getSecond,function(second){
var fancyResult2 = // And now make do fancy
// with first and second
return RETURN(fancyResult2);
});})
, getAllThree =
bind(getFirstTwo, function(_){
return bind(getThird, function(three){
return RETURN(three);
});});
To (using "Right Identity"):
var getFirstTwo =
bind(getFirst, function(first){
return bind(getSecond,function(second){
var fancyResult2 = // And now make do fancy
// with first and second
return RETURN(fancyResult2);
});})
, getAllThree =
bind(getFirstTwo, function(_){
return getThird;
});
Or jam them back together:
var getAllThree =
bind(getFirst, function(first_dontCareNow){
return bind(getSecond,function(second_dontCareNow){
return getThird;
});});
The practicality of these abilities doesn't really emerge,
or become clear until you try to solve really messy problems
Can you imagine thousands of lines of indexOf/subString logic?
What if frequent parsing steps were contained in little functions?
Functions like chars, spaces,upperChars, or digits?
And what if those functions gave you the result in a callback,
without having to mess with Regex groups, and arguments.slice?
And what if their composition/decomposition was well understood?
Such that you could build big parsers from the bottom up?
So the ability to manage nested callback scopes is incredibly practical,
especially when working with monadic parser combinator libraries.
(that is to say, in my experience)
DON'T GET HUNG UP ON:
- CATEGORY-THEORY
- !!!!
Let the below "{| a |m}" represent some piece of monadic data. A data type which advertises an a:
(I got an a!)
/
{| a |m}
Function, f, knows how to create a monad, if only it had an a:
(Hi f! What should I be?)
/
(You?. Oh, you'll be /
that data there.) /
/ / (I got a b.)
| -------------- |
| / |
f a |
|--later-> {| b |m}
Here we see function, f, tries to evaluate a monad but gets rebuked.
(Hmm, how do I get that a?)
o (Get lost buddy.
o Wrong type.)
o /
f {| a |m}
Funtion, f, finds a way to extract the a by using >>=.
(Muaahaha. How you
like me now!?)
(Better.) \
| (Give me that a.)
(Fine, well ok.) |
\ |
{| a |m} >>= f
Little does f know, the monad and >>= are in collusion.
(Yah got an a for me?)
(Yeah, but hey |
listen. I got |
something to |
tell you first |
...) \ /
| /
{| a |m} >>= f
But what do they actually talk about? Well, that depends on the monad. Talking solely in the abstract has limited use; you have to have some experience with particular monads to flesh out the understanding.
For instance, the data type Maybe
data Maybe a = Nothing | Just a
has a monad instance which will acts like the following...
Wherein, if the case is Just a
(Yah what is it?)
(... hm? Oh, |
forget about it. |
Hey a, yr up.) |
\ |
(Evaluation \ |
time already? \ |
Hows my hair?) | |
| / |
| (It's |
| fine.) /
| / /
{| a |m} >>= f
But for the case of Nothing
(Yah what is it?)
(... There |
is no a. ) |
| (No a?)
(No a.) |
| (Ok, I'll deal
| with this.)
\ |
\ (Hey f, get lost.)
\ | ( Where's my a?
\ | I evaluate a)
\ (Not any more |
\ you don't. |
| We're returning
| Nothing.) /
| | /
| | /
| | /
{| a |m} >>= f (I got a b.)
| (This is \
| such a \
| sham.) o o \
| o|
|--later-> {| b |m}
So the Maybe monad lets a computation continue if it actually contains the a it advertises, but aborts the computation if it doesn't. The result, however is still a piece of monadic data, though not the output of f. For this reason, the Maybe monad is used to represent the context of failure.
Different monads behave differently. Lists are other types of data with monadic instances. They behave like the following:
(Ok, here's your a. Well, its
a bunch of them, actually.)
|
| (Thanks, no problem. Ok
| f, here you go, an a.)
| |
| | (Thank's. See
| | you later.)
| (Whoa. Hold up f, |
| I got another |
| a for you.) |
| | (What? No, sorry.
| | Can't do it. I
| | have my hands full
| | with all these "b"
| | I just made.)
| (I'll hold those, |
| you take this, and /
| come back for more /
| when you're done /
| and we'll do it /
| again.) /
\ | ( Uhhh. All right.)
\ | /
\ \ /
{| a |m} >>= f
In this case, the function knew how to make a list from it's input, but didn't know what to do with extra input and extra lists. The bind >>=, helped f out by combining the multiple outputs. I include this example to show that while >>= is responsible for extracting a, it also has access to the eventual bound output of f. Indeed, it will never extract any a unless it knows the eventual output has the same type of context.
There are other monads which are used to represent different contexts. Here's some characterizations of a few more. The IO monad doesn't actually have an a, but it knows a guy and will get that a for you. The State st monad has a secret stash of st that it will pass to f under the table, even though f just came asking for an a. The Reader r monad is similar to State st, although it only lets f look at r.
The point in all this is that any type of data which is declared itself to be a Monad is declaring some sort of context around extracting a value from the monad. The big gain from all this? Well, its easy enough to couch a calculation with some sort of context. It can get messy, however, when stringing together multiple context laden calculations. The monad operations take care of resolving the interactions of context so that the programmer doesn't have to.
Note, that use of the >>= eases a mess by by taking some of the autonomy away from f. That is, in the above case of Nothing for instance, f no longer gets to decide what to do in the case of Nothing; it's encoded in >>=. This is the trade off. If it was necessary for f to decide what to do in the case of Nothing, then f should have been a function from Maybe a to Maybe b. In this case, Maybe being a monad is irrelevant.
Note, however, that sometimes a data type does not export it's constructors (looking at you IO), and if we want to work with the advertised value we have little choice but to work with it's monadic interface.
Another attempt at explaining monads, using just Python lists and the map function. I fully accept this isn't a full explanation, but I hope it gets at the core concepts.
I got the basis of this from the funfunfunction video on Monads and the Learn You A Haskell chapter 'For a Few Monads More'. I highly recommend watching the funfunfunction video.
At it's very simplest, Monads are objects that have a map and flatMap functions (bind in Haskell). There are some extra required properties, but these are the core ones.
flatMap 'flattens' the output of map, for lists this just concatenates the values of the list e.g.
concat([[1], [4], [9]]) = [1, 4, 9]
So in Python we can very basically implement a Monad with just these two functions:
def flatMap(func, lst):
return concat(map(func, lst))
def concat(lst):
return sum(lst, [])
func is any function that takes a value and returns a list e.g.
lambda x: [x*x]
## Explanation
For clarity I created the concat function in Python via a simple function, which sums the lists i.e. [] + [1] + [4] + [9] = [1, 4, 9] (Haskell has a native concat method).
I'm assuming you know what the map function is e.g.:
>>> list(map(lambda x: [x*x], [1,2,3]))
[[1], [4], [9]]
Flattening is the key concept of Monads and for each object which is a Monad this flattening allows you to get at the value that is wrapped in the Monad.
Now we can call:
>>> flatMap(lambda x: [x*x], [1,2,3])
[1, 4, 9]
This lambda is taking a value x and putting it into a list. A monad works with any function that goes from a value to a type of the monad, so a list in this case.
I think the question of why they're useful has been answered in other questions.
## More explanation
Other examples that aren't lists are JavaScript Promises, which have the then method and JavaScript Streams which have a flatMap method.
So Promises and Streams use a slightly different function which flattens out a Stream or a Promise and returns the value from within.
The Haskell list monad has the following definition:
instance Monad [] where
return x = [x]
xs >>= f = concat (map f xs)
fail _ = []
i.e. there are three functions return (not to be confused with return in most other languages), >>= (the flatMap) and fail.
Hopefully you can see the similarity between:
xs >>= f = concat (map f xs)
and:
def flatMap(f, xs):
return concat(map(f, xs))
This is the video you are looking for.
Demonstrating in C# what the problem is with composition and aligning the types, and then implementing them properly in C#. Towards the end he displays how the same C# code looks in F# and finally in Haskell.
Mathematial thinking
For short: An Algebraic Structure for Combining Computations.
return data: create a computation who just simply generate a data in monad world.
(return data) >>= (return func): The second parameter accept first parameter as a data generator and create a new computations which concatenate them.
You can think that (>>=) and return won't do any computation itself. They just simply combine and create computations.
Any monad computation will be compute if and only if main trigs it.
• Big Mistake : monad computation can be triggerred wo. main. – Titou Jul 13 '15 at 13:28
If you are asking for a succinct, practical explanation for something so abstract, then you can only hope for an abstract answer:
a -> b
is one way of representing a computation from as to bs. You can chain computations, aka compose them together:
(b -> c) -> (a -> b) -> (a -> c)
More complex computations demand more complex types, e.g.:
a -> f b
is the type of computations from as to bs that are into fs. You can also compose them:
(b -> f c) -> (a -> f b) -> (a -> f c)
It turns out this pattern appears literally everywhere and has the same properties as the first composition above (associativity, right- and left-identity).
One had to give this pattern a name, but then would it help to know that the first composition is formally characterised as a Semigroupoid?
"Monads are just as interesting and important as parentheses" (Oleg Kiselyov)
A Monad is a box with a special machine attached that allows you to make one normal box out of two nested boxes - but still retaining some of the shape of both boxes.
Concretely, it allows you to perform join, of type Monad m => m (m a) -> m a.
It also needs a return action, which just wraps a value. return :: Monad m => a -> m a
You could also say join unboxes and return wraps - but join is not of type Monad m => m a -> a (It doesn't unwrap all Monads, it unwraps Monads with Monads inside of them.)
So it takes a Monad box (Monad m =>, m) with a box inside it ((m a)) and makes a normal box (m a).
However, usually a Monad is used in terms of the (>>=) (spoken "bind") operator, which is essentially just fmap and join after each other. Concretely,
x >>= f = join (fmap f x)
(>>=) :: Monad m => (a -> m b) -> m a -> m b
Note here that the function comes in the second argument, as opposed to fmap.
Also, join = (>>= id).
Now why is this useful? Essentially, it allows you to make programs that string together actions, while working in some sort of framework (the Monad).
The most prominent use of Monads in Haskell is the IO Monad.
Now, IO is the type that classifies an Action in Haskell. Here, the Monad system was the only way of preserving (big fancy words):
• Referential Transparency
• Lazyness
• Purity
In essence, an IO action such as getLine :: IO String can't be replaced by a String, as it always has a different type. Think of IO as a sort of magical box that teleports the stuff to you.
However, still just saying that getLine :: IO String and all functions accept IO a causes mayhem as maybe the functions won't be needed. What would const "üp§" getLine do? (const discards the second argument. const a b = a.) The getLine doesn't need to be evaluated, but it's supposed to do IO! This makes the behaviour rather unpredictable - and also makes the type system less "pure", as all functions would take a and IO a values.
Enter the IO Monad.
To string to actions together, you just flatten the nested actions.
And to apply a function to the output of the IO action, the a in the IO a type, you just use (>>=).
As an example, to output an entered line (to output a line is a function which produces an IO action, matching the right argument of >>=):
getLine >>= putStrLn :: IO ()
-- putStrLn :: String -> IO ()
This can be written more intuitively with the do environment:
do line <- getLine
putStrLn line
In essence, a do block like this:
do x <- a
y <- b
z <- f x y
w <- g z
h x
k <- h z
l k w
... gets transformed into this:
a >>= \x ->
b >>= \y ->
f x y >>= \z ->
g z >>= \w ->
h x >>= \_ ->
h z >>= \k ->
l k w
There's also the >> operator for m >>= \_ -> f (when the value in the box isn't needed to make the new box in the box) It can also be written a >> b = a >>= const b (const a b = a)
Also, the return operator is modeled after the IO-intuituion - it returns a value with minimal context, in this case no IO. Since the a in IO a represents the returned type, this is similar to something like return(a) in imperative programming languages - but it does not stop the chain of actions! f >>= return >>= g is the same as f >>= g. It's only useful when the term you return has been created earlier in the chain - see above.
Of course, there are other Monads, otherwise it wouldn't be called Monad, it'd be called somthing like "IO Control".
For example, the List Monad (Monad []) flattens by concatenating - making the (>>=) operator perform a function on all elements of a list. This can be seen as "indeterminism", where the List is the many possible values and the Monad Framework is making all the possible combinations.
For example (in GHCi):
Prelude> [1, 2, 3] >>= replicate 3 -- Simple binding
[1, 1, 1, 2, 2, 2, 3, 3, 3]
Prelude> concat (map (replicate 3) [1, 2, 3]) -- Same operation, more explicit
[1, 1, 1, 2, 2, 2, 3, 3, 3]
Prelude> [1, 2, 3] >> "uq"
"uququq"
Prelude> return 2 :: [Int]
[2]
Prelude> join [[1, 2], [3, 4]]
[1, 2, 3, 4]
because:
join a = concat a
a >>= f = join (fmap f a)
return a = [a] -- or "= (:[])"
The Maybe Monad just nullifies all results to Nothing if that ever occurs. That is, binding auto-checks if the function (a >>= f) returns or the value (a >>= f) is Nothing - and then returns Nothing as well.
join Nothing = Nothing
join (Just Nothing) = Nothing
join (Just x) = x
a >>= f = join (fmap f a)
or, more explicitly:
Nothing >>= _ = Nothing
(Just x) >>= f = f x
The State Monad is for functions that also modify some shared state - s -> (a, s), so the argument of >>= is :: a -> s -> (a, s).
The name is a sort of misnomer, since State really is for state-modifying functions, not for the state - the state itself really has no interesting properties, it just gets changed.
For example:
pop :: [a] -> (a , [a])
pop (h:t) = (h, t)
sPop = state pop -- The module for State exports no State constructor,
-- only a state function
push :: a -> [a] -> ((), [a])
push x l = ((), x : l)
sPush = state push
swap = do a <- sPop
b <- sPop
sPush a
sPush b
get2 = do a <- sPop
b <- sPop
return (a, b)
getswapped = do swap
get2
then:
Main*> runState swap [1, 2, 3]
((), [2, 1, 3])
Main*> runState get2 [1, 2, 3]
((1, 2), [1, 2, 3]
Main*> runState (swap >> get2) [1, 2, 3]
((2, 1), [2, 1, 3])
Main*> runState getswapped [1, 2, 3]
((2, 1), [2, 1, 3])
also:
Prelude> runState (return 0) 1
(0, 1)
According to What we talk about when we talk about monads the question is wrong:
The short answer to the question "What is a monad?" is that it is a monoid in the category of endofunctors or that it is a generic data type equipped with two operations that satisfy certain laws. This is correct, but it does not reveal an important bigger picture. This is because the question is wrong. In this paper, we aim to answer the right question, which is "What do authors really say when they talk about monads?"
While that paper does not directly answer what a monad is it helps understanding what people with different backgrounds mean when they talk about monads and why.
A Monad is an Applicative (ie. something that you can lift binary -- hence, "n-ary" -- functions to,(1) and inject pure values into(2)) Functor (i.e. something that you can map over,(3) i.e. lift unary functions to(3)) with the added ability to flatten the nested datatype. In Haskell, this flattening operation is called join.
The general (generic, parametric) type of this operation is:
join :: Monad m => m (m a) -> m a
for any monad m (NB all ms are the same!).
A specific m monad defines its specific version of join working for any value type a "carried" by the monadic values of type m a. Some specific types are:
join :: [[a]] -> [a] -- for lists, or nondeterministic values
join :: Maybe (Maybe a) -> Maybe a -- for Maybe, or optional values
join :: IO (IO a) -> IO a -- for I/O-produced values
The join operation converts an m-computation producing an m-computation of a-type values into one combined m-computation of a-type values. This allows for combination of computation steps into one larger computation.
This computation steps-combining "bind" (>>=) operator simply uses fmap and join together, i.e.
(ma >>= k) == join (fmap k ma)
{-
ma :: m a -- m-computation which produces a-type values
k :: a -> m b -- create new m-computation from an a-type value
fmap k ma :: m ( m b ) -- m-computation of m-computation of b-type values
(m >>= k) :: m b -- m-computation which produces b-type values
-}
Conversely, join can be defined via bind, join mma == mma >>= id where id ma = ma -- whichever is more convenient for a given type m.
For monads, both the do-notation and its equivalent code written with the bind operator,
do { x <- mx ; y <- my ; return (f x y) } -- x :: a , mx :: m a
-- y :: b , my :: m b
mx >>= (\x -> -- nested
my >>= (\y -> -- lambda
return (f x y) )) -- functions
can be read as
first "do" mx, and when it's done, get its "result" as x and let me use it to "do" something else.
In a given do block, each of the values to the right of binding arrow <- is of type m a for some type a and the same monad m throughout the do block. return is a neutral m-computation which just produces the pure value it is given, such that combining any m-computation with it does not change that computation at all.
(1) with liftA2 :: Applicative m => (a -> b -> c) -> m a -> m b -> m c
(2) with pure :: Applicative m => a -> m a
(3) with fmap :: Functor m => (a -> b) -> m a -> m b
There's also the equivalent Monad methods,
liftM2 :: Monad m => (a -> b -> c) -> m a -> m b -> m c
return :: Monad m => a -> m a
liftM :: Monad m => (a -> b) -> m a -> m b
Given a monad, the other definitions could be made as
pure a = return a
fmap f ma = do { a <- ma ; return (f a) }
liftA2 f ma mb = do { a <- ma ; b <- mb ; return (f a b) }
(ma >>= k) = do { a <- ma ; b <- k a ; return b }
## Explanation
It's quite simple, when explained in C#/Java terms:
1. A monad is a function that takes arguments and returns a special type.
2. The special type that this monad returns is also called monad. (A monad is a combination of #1 and #2)
3. There's some syntactic sugar to make calling this function and conversion of types easier.
## Example
A monad is useful to make the life of the functional programmer easier. The typical example: The Maybe monad takes two parameters, a value and a function. It returns null if the passed value is null. Otherwise it evaluates the function. If we needed a special return type, we would call this return type Maybe as well. A very crude implementation would look like this:
object Maybe(object value, Func<object,object> function)
{
if(value==null)
return null;
return function(value);
}
This is spectacularly useless in C# because this language lacks the required syntactic sugar to make monads useful. But monads allow you to write more concise code in functional programming languages.
Oftentimes programmers call monads in chains, like so:
var x = Maybe(x, x2 => Maybe(y, y2 => Add(x2, y2)));
In this example the Add method would only be called if x and y are both non-null, otherwise null will be returned.
To answer the original question: A monad is a function AND a type. Like an implementation of a special interface.
following your brief, succinct, practical indications:
The easiest way to understand a monad is as a way to apply/compose functions within a context. Let's say you have two computations which both can be seen as two mathematical functions f and g.
• f takes a String and produces another String (take the first two letters)
• g takes a String and produces another String (upper case transformation)
So in any language the transformation "take the first two letter and convert them to upper case" would be written g(f("some string")). So, in the world of pure perfect functions, composition is just: do one thing and then do the other.
But let's say we live in the world of functions which can fail. For example: the input string might be one char long so f would fail. So in this case
• f takes a String and produces a String or Nothing.
• g produces a String only if f hasn't failed. Otherwise, produces Nothing
So now, g(f("some string")) needs some extra checking: "Compute f, if it fails then g should return Nothing, else compute g"
This idea can be applied to any parametrized type as follows:
Let Context[Sometype] be a computation of Sometype within a Context. Considering functions
• f:: AnyType -> Context[Sometype]
• g:: Sometype -> Context[AnyOtherType]
the composition g(f()) should be readed as "compute f. Within this context do some extra computations and then compute g if it has sense within the context"
## protected by Tats_innitMay 29 '14 at 2:17
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered questions instead? | 2019-06-20 20:52:35 | {"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": 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": 0.389897882938385, "perplexity": 3485.3759094265415}, "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-2019-26/segments/1560627999273.24/warc/CC-MAIN-20190620190041-20190620212041-00528.warc.gz"} |
https://math.stackexchange.com/questions/858792/a-question-on-a-lemma-about-the-product-map | # A question on a lemma about the product map
Here is a Lemma in the book “C*-algebras and Finite-Dimensional Approximations”:
Lemma 3.8.4. Let $A$ be a C*-algebra, $M\subset B(H)$ be a con Neumann algebra and $\phi: A\rightarrow M$ be a completely positive map. Assume that the product map $$\phi \times \iota_{M}: A\odot M’ \rightarrow B(H),~\phi(\sum\limits_{i}a_{i}\otimes m_{i}’)=\sum\limits_{i}\phi(a_{i})m_{i}’.$$ is continuous with respect to the spatial (or minimal) tensor product norm and let $\pi: M\rightarrow B(K)$ be any normal representation. Then the product map $$(\pi \circ \phi)\times \iota_{\pi(M)’}: A\odot \pi(M)’\rightarrow B(K).$$ is also min-continuous (That is, continuous with respect to the spatial (or minimal) tensor product norm).
Proof. Any normal representation of $M$ can be identified with the cut-down by a projection in the commutant of the representation $M\otimes 1_{K}\subset B(H\otimes K)$. Hence it suffices to show that the product map with the commutant in this particular representation is min-continuous.
Since $(M\otimes 1_{K})'\cap B(H\otimes K)=M'\bar{\otimes} B(K)$ (Here, the $\bar{\otimes}$ denote the tensor product of two von Neumann algebra) -- just think of $B(H\otimes K)$ as matrices with entries in $B(H)$ -- we thus have to show that $$(\phi\otimes1_{B(K)})\times\iota_{M'\bar{\otimes}B(K)}: A\odot(M'\bar{\otimes B(K)}~)\rightarrow B(H\otimes K)$$ is min-continuous. But, excepet for the horrific notation required, this is easy since $(\phi\otimes1_{B(K)})\times\iota_{M'\bar{\otimes}B(K)}$ is a point-strong limit of min-continuous maps (with uniformly bounded norms). More precisely, if $P\in B(K)$ is a finite-rank projection, then the map $$(\phi\otimes1_{B(PK)})\times\iota_{M'\bar{\otimes}B(PK)}: A\odot(M'\bar{\otimes B(PK)}~)\rightarrow B(H\otimes PK)$$ is min-continuous and its norm is bounded by $||\phi\times \iota_{M'}||$ because it can be identified with $$(\phi\times\iota_{M'})\otimes id_{B(PK)}: (A\odot M')\odot B(PK)\rightarrow B(H\otimes PK)$$ (Here, it use the Exercise 3.5.1 in this book). Finally, taking a net $\{P_{\lambda}\}$ of finite-rank projections which converge to $1_{K}$ in the strong operator topology and fixing $$x=\sum a_{i}\otimes T_{i}\in A\odot (M'\bar{\otimes}B(K)),$$ it is easy to check that $$(\phi\otimes 1_{B(P_{\lambda}K)})\times \iota_{M'\otimes B(P_{\lambda}K)}((1_{H}\otimes P_{\lambda})x(1_{H}\otimes P_{\lambda}))\rightarrow (\phi \otimes 1_{B(K)})\times \iota_{M'\bar{\otimes}B(K)}(x).$$ in the strong operator topology. This completes the proof.
I have three questions on the proof above:
1. How to comprehend the first sentence "Any normal representation of $M$ can be identified with the cut-down by a projection in the commutant of the representation $M\otimes 1_{K}\subset B(H\otimes K)$."
2. Why does $(M\otimes 1_{K})'\cap B(H\otimes K)=M'\bar{\otimes} B(K)$ hold?
3. How to check the last srong operator topology $$(\phi\otimes 1_{B(P_{\lambda}K)})\times \iota_{M'\otimes B(P_{\lambda}K)}((1_{H}\otimes P_{\lambda})x(1_{H}\otimes P_{\lambda}))\rightarrow (\phi \otimes 1_{B(K)})\times \iota_{M'\bar{\otimes}B(K)}(x).$$
1. It is a general theorem about von Neumann algebras (I know it from Dixmier's vN algebra book, but it should appear in other places too) that any normal representation $\pi:M\to N$ (for some vN algebra $N$) is of the form $$\pi(x)=V^*[(x\otimes 1_{B(K)})\,P\,]\,V$$ for some Hilbert space $K$ (which is not the same $K$ from the statement), $P\in(M\otimes 1_{B(K)})'$ a projection, and $V$ a unitary. We have $$\pi(M)'=[V^*(M\otimes1_{B(K)})PV]'=V^*P(M\otimes1_{B(K)})'PV=V^*P(M'\otimes B(K))PV.$$ So, for $y\in M'$, $z\in B(K)$, $$(\pi\circ\phi)\times\iota_{\pi(M)'}[x\otimes(V^*P(y\otimes z)PV] =V^*(\phi(x)\otimes1_{B(K)})PVV^*P(y\otimes z)PV =V^*P(\phi(x)\otimes1_{B(K)})(y\otimes z)PV=V^*P[(\phi\otimes1_{B(K)})\times\iota_{M'\otimes B(K)}(x\otimes(y\otimes z)) ]PV.$$ As conjugating with a unitary and with a projection is min-continuous, one only needs to check the min-continuity "inside", which is what the authors do.
2. If $M\subset B(H)$, $N\subset B(K)$, then $(M\otimes N)'=M'\otimes N'$ in $B(H\otimes K)$. This is again a general result about tensor products of von Neumann algebras.
3. It should say $1_A\otimes1_H\otimes P_\lambda$. Then $$(\phi\otimes 1_{B(P_{\lambda}K)})\times \iota_{M'\otimes B(P_{\lambda}K)}((1_A\otimes1_{H}\otimes P_{\lambda})(a\otimes T)(1_A\otimes1_{H}\otimes P_{\lambda})) =(\phi(a)\otimes1_{B(K)}(1_H\otimes P_\lambda)T(1_H\otimes P_\lambda)\\ \longrightarrow(\phi(a)\otimes1_{B(K)})T= (\phi \otimes 1_{B(K)})\times \iota_{M'\bar{\otimes}B(K)}(a\otimes T).$$
• Yes. ${\ \ \ }$ Jul 13 '14 at 14:54 | 2022-01-20 12:29:42 | {"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": 0.9675140976905823, "perplexity": 168.09308905324258}, "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/1642320301737.47/warc/CC-MAIN-20220120100127-20220120130127-00621.warc.gz"} |
https://motls.blogspot.com/2010/02/new-york-times-on-ipccs-and-pachauris.html?m=1 | ## Tuesday, February 09, 2010
### New York Times on IPCC's and Pachauri's scandals
As I have mentioned several times, most of the revelations about the U.N. climate panel and its boss, Rajendra Pachauri, were first published in the British newspapers, especially The Telegraph and The Times. A more limited coverage has been available to the readers and viewers of FoxNews.
But the gospel may finally be coming to the U.S. mainstream media, too. Elizabeth Rosenthal named her article
U.N. Climate Panel and Chief Face Credibility Siege
Although it includes some bizarre alarmist comments such as
"The general consensus among mainstream scientists is that the errors are in any case minor and do not undermine the report’s conclusions,"
it actually says enough true stuff about the GlacierGate and especially various conflicts of interest of the IPCC boss. I feel that they find it easier to sacrifice particular individuals, such as Pachauri, than the core elements of the orthodoxy. That's an explanation why the financial interests are being given so much space while the discussion of the errors and sub-par references in the 2007 report remains limited to the GlacierGate and is not too detailed, anyway.
Elizabeth Rozenthal and others should perhaps be told that the GlacierGate is far from being the only scandal of this type that was recently unconvered. A partial list of these scandals looks like this:
ClimateGate
GlacierGate
TeriGate and PepsiHondaTeriGate and TeriProtectedForestGate
AmazonGate
AussieDroughtGate
DisasterGate
HollandGate
AfricaGate
WaveEnergyGate, new DissertationGates and EcoterrorReferenceGates | 2019-07-19 22:02:02 | {"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": 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": 0.20043157041072845, "perplexity": 4210.665434507178}, "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-2019-30/segments/1563195526359.16/warc/CC-MAIN-20190719202605-20190719224605-00052.warc.gz"} |
http://www.logique.jussieu.fr/archive/semgen/sgen14-15.html | Retour à la page du séminaire général. Archives: Années 00-01, 01-02, 02-03, 03-04, 04-05, 05-06, 06-07, 07-08, 08-09, 09-10, 10-11, 11-12, 12-13, 13-14.
### UNIVERSITE DE PARIS VII, UFR DE MATHEMATIQUES Séminaire général de Logique - Année 2014 - 2015
Résumés
(Ordre chronologique inverse)
15 juin 2015 : Guillaume Malod ( Equipe de Logique Mathématique, IMJ-PRG) Bornes inférieures pour les circuits skew non-commutatifs
Nisan (STOC 1991) a donné un exemple de polynôme calculable par un circuit non-commutatif de taille linéaire mais nécessitant un branching program non-commutatif de taille exponentielle. Cet exemple est en fait calculable par un circuit skew (où chaque porte de multiplication a un argument qui est réduit à une variable ou à une constante). Nous montrons que tout circuit skew calculant le carré du polynôme de Nisan doit voir une taille exponentielle et généralisons ce résultat aux circuits où toute porte de multiplication a un argument de degré au plus d/5, si d est le degré du polynôme calculé. Ces modèles non-commutatifs deviennent ainsi les plus forts pour lesquels nous ayons des bornes inférieures super-polynomiales. L'exposé ne suppose aucune connaissance particulière en complexité algébrique et les techniques employées sont élémentaires. (Travail en commun avec Nutan Limaye et Srikanth Srinivasan)
8 juin 2015 : Didier Caucal (CNRS et Université Paris-Est Marne la Vallée) Recognizability for infinite automata
The recognizability by inverse morphism defined by Eilenberg is extended to infinite automata. Given an automaton family F and an automaton H, we recognize the set of languages accepted by all possible automata of F that can be mapped by morphism into H. We introduce restrictions on morphisms to get various boolean algebras of context-free languages, (higher-order) indexed languages and contextual languages. It is a joint work with Chloé Athenosy.
1er juin 2015 : Michał Skrzypczak ( LIAFA, Université Paris 7, et Université de Varsovie, Pologne ) On uniformisability in monadic second-order logic
The property of uniformisation is one of the important concepts studied in the classical descriptive set theory. From the point of view of definability, it can be seen as a possibility to choose unique witnesses. During this talk I will focus on the uniformisation property for first-order (FO) logic and monadic second-order (MSO) logic. I will try to provide a summary of known results for various classes of structures; mainly finite and infinite words and trees. One of the crucial results in this area is the negative answer given by Gurevich and Shelah to the question of uniformisability in MSO over infinite trees (called Rabin's uniformisation question). Along with the overview of the classical results, I plan to present some recent advances and open questions. Probably the most intriguing of them is the question of MSO-definability of a choice function without parameters over scattered trees (i.e. trees having only countably manybranches).
11 mai 2015: Eugène Asarin ( LIAFA, Université Paris 7 ) Entropy of regular timed languages
Recall first that a timed word is constituted by letters and time delays (e.g. 2.43 a 1.17 b 5.112 a), a timed language is a set of timed words; and timed automata and timed regular languages are studied since early 90s. In this work, in order to study the size of regular timed languages, we generalize a classical approach introduced by Chomsky and Miller for discrete automata: count words having n symbols, and compute the exponential growth rate of their number (entropy). For timed automata, we replace cardinality by volume and define (volumetric) entropy similarly. It represents the average quantity of information per event in a timed word of the language. We exhibit a criterion for telling apart “thick” timed automata with non-vanishing entropy, for which typical runs are non-Zeno and discretizable, from “thin” automata for which all runs behave in a Zeno-like way, implying a quick volume collapse. The main technical tool comes from the functional analysis: we associate to every timed automaton a positive integral operator; the entropy equals the logarithm of its spectral radius. This operator has a spectral gap, thus allowing for fast converging numerical procedures to approximate entropy. In a special case, entropy is even characterized symbolically. (joint work with A. Degorre and N. Basset).
13 avril 2015 : Phillip Wesolek ( Université Catholique de Louvain-la-Neuve, Belgique ) Elementary totally disconnected locally compact Polish groups
The class of elementary groups is the smallest class of totally disconnected locally compact Polish groups that contains the profinite Polish groups and the countable discrete groups, that is closed under group extensions, and that is closed under countable increasing unions. In this talk, we discuss the permanence properties of the class of elementary groups and a characterization of elementary groups. As an application, we show that all compactly generated t.d.l.c. Polish groups can be decomposed into finitely many elementary groups and topologically characteristically simple groups via group extension. We conclude by considering a number of open questions related to elementary groups.
30 mars 2015: André Nies ( University of Auckland, New Zealand ) Descriptions de structures au sein d'une classe
We want to describe a structure in a class up to isomorphism, using an appropriate formal language. Containment in the class is given as an additiona (external) condition. For finite structures in a fixed finite signature, there is always a description in first-order logic of length comparable to the size of the structure. An interesting question is how short such a description can be. In joint work with Katrin Tent we answer this question for groups, compressing the group to a description of length polylogarithmic in the size of the group. Within the class of finitely generated groups, an interesting question is whether a group can be described at all by a single first-order sentence. For instance, this is the case for the Heisenberg group over $\mathbb Z$, and for the restricted wreath product of a finite cyclic group with $\mathbb Z$ (the latter example is not finitely presented). Within the class of countable structures over a countable signature $S$, there is always a description in $L_{\omega_1, \omega}(S)$, the extension of first-order language that allows countable disjunctions over a set of formulas with a shared finite reservoir of free variables (Scott). For the class of separable complete metric spaces, a similar result holds. The most natural language here is an extension of continuous logic for $S$ that allows countable disjunctions.
23 mars 2015 : Laura Fontanella ( Hebrew University of Jerusalem, Israel ) Generalized compactness and large cardinals
Applications of large cardinals arise in many fields, including group theory, general topology and others. Some of these results follow from various generalizations of Compactness theorem to infinitary logics. We discuss a particular principle, called Delta-reflection, that entails a version of this generalized compactness. We will investigate some of its applications to the study of abelian groups, Hausdorff spaces and others, and we will show that, despite its strength, the Delta-reflection can consistently hold even at small cardinals together with another compactness principle known as the tree property. This is a joint work with M. Magidor
16 mars 2015 : Rizos Sklinos ( Université de Lyon 1 ) Some model theory of torsion-free hyperbolic groups
The profound work of Z.Sela has radically changed the picture concerning stable groups. In a series of papers of great volume and complexity he proved that every non-cyclic torsion-free hyperbolic group is stable. The only families of groups whose members were already known to be stable were the family of algebraic groups (over algebraically closed fields) and the family of Abelian groups. In this talk I will survey recent results about the first order theories of torsion-free hyperbolic groups as to give our current understanding of their model theory. I will give the ideas of the proofs of the most important results without the technical details in order to reveal the beautiful interplay between logic and geometric group theory. No prior knowledge of stability or geometric group theory will be assumed. "
9 mars 2015 : Nathanael Mariaule ( Université de Naples, Italie ) Le corps des nombres p-adiques avec un prédicat pour les puissances d'un nombre entier
Dans un article datant de 1985, L. van den Dries axiomatise la théorie du corps des réels avec un prédicat pour les puissances d'un entier n. Dans cet exposé, je vais présenter une version p-adique de ce résultat. En particulier, nous verrons que les méthodes à utiliser sont différentes selon l'entier choisi: Si la valuation p-adique de n est non-nulle, le groupe multiplicatif généré par n est discret et ce cas s'approche de celui étudié par van den Dries. Dans le cas contraire, le groupe est dense dans une partie (définissable) du corps des nombres p-adiques et de nouvelles idées sont nécessaires comparé au cas réel.
2 mars 2015 : Daniel Palacin (Université de Münster, Allemagne) Stabilité, groupes et internalité
Dans cet exposé, je vais faire une introduction à la théorie de la stabilité et des groupes définissables dans ce contexte. Je donnerai des exemples et définitions de base, et aussi des notions plus élaborées comme monobasé et internalité. Enfin, je vais prouver un résultat en collaboration avec Rizos Sklinos sur des expansions de la théorie des nombres entiers.
16 février 2015 : Martino Lupini (York University, Toronto, Canada) Generic spaces of operators
We will present an overview of the construction and study of canonicalgeneric objects in the categories of operator spaces, operator systems, and operator algebras from the perspective of Fraisse theory and model theory for metric structures.
9 février 2015 : Olivia Caramello ( Université Paris 7) Les topos de Grothendieck comme ponts unifiants en logique et mathématiques
On présentera une nouvelle vision des topos de Grothendieck comme des espaces capables de servir efficacement de 'ponts' pour transférer des concepts et résultats entre différentes théories mathématiques. Cette approche a déjà engendré plusieurs applications dans différents domaines des mathématiques et le potentiel de cette théorie a juste commencé à être exploré. On expliquera les principes fondamentaux qui caractérisent cette vision des topos comme 'ponts unifiants' et on illustrera l'utilité technique de ces méthodologies en discutant quelques applications en logique, algèbre et topologie.
2 février 2015 : Veronica Becher ( Université de Buenos Aires & CONICET, Argentine) On normal numbers
Flip a coin a large number of times and roughly half of the flips will come up heads and half will come up tails. Normality makes similar assertions about the sequences of digits in the expansions of a real number. For b an integer greater than or equal to 2, a real number x is simply normal to base b if every digit d in {0, 1, . . . , b-1} occurs in the base b expansion of x with asymptotic frequency 1/b; a real number x is normal to base b if it is simply normal to all powers of b; and a real number x is absolutely normal if it is simply normal to all integer bases greater than or equal to 2. More than one hundred years ago E. Borel showed that almost all real numbers are absolutely normal, and he asked for one example. He would have liked some fundamental mathematical constant such as pi or e, but this remains as the most famous open problem on normality. As for other examples, there have been several constructions of normal numbers since Borel's time, with varying levels of effectivity (computability). I will summarize the latest results, including our constructions of numbers normal to selected bases, a fast algorithm to compute an absolutely normal number which runs in nearly quadratic time, and an algorithm to compute an absolutely normal Liouville number. This is joint work with Theodore Slaman and Pablo Heiber. Verónica Becher is an Associate Professor at the University of Buenos Aires and researcher at CONICET. She is part of the Laboratoire International Associé INFINIS Universidad de Buenos Aires-CONICET/Université Paris Diderot-CNRS.
19 janvier 2015 : François Le Maître ( Université Catholique de Louvain-la-Neuve, Belgique ) Propriété de petit indice et groupes localement compacts
Un groupe topologique satisfait la propriété de petit indice si tous ses sous-groupes d'indice dénombrable sont ouverts. Dans le cas d'un groupe polonais non archimédien, les sous-groupes ouverts sont tous d'indice dénombrable et forment une base voisinages de l'identité: la propriété de petit indice garantit alors que la topologie du groupe est alors encodée algébriquement. D'après un résultat de Dixon-Neumann-Thomas, le groupe des permutations des entiers satisfait la propriété de petit indice. Leur résultat a ensuite été étendu à des groupes d'automorphismes de structures dénombrables plus riches, comme les groupes d'automorphismes de structures dénombrables omega-stables et omega-catégoriques. On s'intéresse ici à l'existence de groupes localement compacts satisfaisant la propriété de petit indice, et on donnera des exemples de groupes d'automorphismes de l'arbre n-régulier qui la vérifient. Ce résultat a été obtenu en collaboration avec Wesolek.
12 janvier 2015 : Vincenzo Mantova (University of Camerino, Italie) Surreal numbers, derivations and transseries
Conway's surreal numbers are a class "No" of inductive objects, originally thought as moves in a game, possessing a natural structure of ordered field and an exponential function which make it a monster model of the theory of (R,exp). It has been conjectured several times that No has also a derivation mimicking the differential structure of Hardy fields, and that No could be described, in some appropriate sense, as a field of transseries. I will discuss the interplay between surreal numbers, derivations of Hardy fields and transseries, and the most recent results regarding the above conjectures. This is joint work with A. Berarducci.
1er décembre 2014 : Jérémie Cabessa (Université Panthéon-Assas Paris II) Computational capabilities of recurrent neural networks
We provide a survey of major results concerning the computational capabilities of recurrent neural networks. In particular, we show that some basic neural models happen to be computationally equivalent to finite automata, Turing machines, probabilistic Turing machines with logarithmic advice, or Turing machines with polynomial advice.
24 novembre 2014 : Friedrich Martin Schneider ( Université de Dresde, Allemagne ) Invariants of group actions and their connection to amenability
Having its roots in the beginnings of modern measure theory, the concept of amenability has become of central importance to topological group theory and topological dynamics. The class of amenable topological groups is vast and allows to define dynamical invariants by means of a well-behaved averaging process. We will present characterizations of amenability in terms of several novel invariants for continuous group actions and discuss some related problems.
10 novembre 2014 : Eli Glasner (Tel-Aviv University) On tame dynamical systems
According to a dynamical version of the Bourgain-Fremlin-Talagrand dichotomy, an enveloping semigroup of a dynamical system is either tame: has cardinality $\le 2^{\aleph_0}$, or it is topologically wild and contains a copy of $\beta\mathbb{N}$, the \v{C}ech-Stone compactificaltion of a discrete countable set. As a general principle one can measure the usefulness of a new mathematical notion by the number of seemingly unrelated ways by which it can be characterized. According to this principle the notion of tameness stands rather high. Tameness can also be characterized by the lack of a certain independence property --- where combinatorial Ramsey type arguments take a leading role --- by the fact that the elements of the enveloping semigroup of a tame system are Baire class 1 maps, and, in terms of eventual non-sensitivity".As an application of the latter we obtain the following characterization of tame subshifts $X \subset \{0,1\}^{\mathbb Z}$: for every infinite subset $L \subseteq {\mathbb Z}$ there exists an infinite subset $K \subseteq L$ such that $\pi_{K}(X)$ is a countable subset of $\{0,1\}^K$. Finally, a dynamical system is tame iff it can be represented on a Banach space which does not contain an isomorphic copy of $\ell_1$. I will review some of these results and indicate some applications. Mostly these are joint works with Michael Megrelishvily.
3 novembre 2014 : Silvain Rideau (ENS Paris et Paris-Sud Orsay) Comptage en théorie des groupes et imaginaires p-adiques
Soit G un groupe nilpotent sans torsion finiment engendré. Le nombre de sous-groupes de G d'indice donné $p^n$ est fini et on le note a_n. Pour étudier le comportement des a_n, on s’intéresse à la série $\zeta_{p,G}(s) = \sum_n a_n t^n$ dont la rationalité a été démontrée par Grunewald, Segal et Smith (1988). Leur preuve consiste à réécrire cette somme comme une intégrale p-adique à paramètres et à utiliser un résultat de Denef (1984) sur la rationalité des telles intégrales. On peut aussi démontrer que de tels résultats de rationalité sont uniformes en p. Le but de cet exposé sera d'exposer une méthode générale pour démontrer de tels résultats de comptage qui permet de retrouver les résultats connus en en simplifiant la preuve et de prouver de nouveaux résultats. Cette méthode générale utilise toujours le résultat de rationalité de Denef et un nouvel ingrédient modèle théorique: l'élimination des imaginaires p-adiques. Tous les objets dont il est question dans ce résumé seront redéfinis. (En commun avec Ehud Hrushovski et Ben Martin.)
29 septembre 2014 : Ivan Tomasic (University of London) Applications of difference algebraic geometry
The term Difference algebraic geometry' refers to the range of methods of difference algebra and the corresponding geometry', originally developed to study definable sets in ACFA (the theory of existentially closed difference fields), as well as sets definable over fields with powers of Frobenius. We will apply these tools in the discussion of a class of exceptionally nice difference polynomials.
22 septembre 2014 : Joerg Brendle ( Kobe University ) Ultrafilters on the natural numbers
Various classes of ultrafilters on the natural numbers, like the P-points or the Ramsey ultrafilters, have found important applications in other areas of mathematics, like combinatorics or topology. We will present a unified treatment of such classesof ultrafilters, in terms of ideals on countable sets, and then discuss the problem of existence and that of generic existence of ultrafilters in these classes. Here we say that ultrafilters in a class exist generically if every filter base of size less than continuum can be extended to an ultrafilter in the class. | 2017-11-23 01:31:37 | {"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": 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": 0.6523823738098145, "perplexity": 2481.5106841505963}, "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-2017-47/segments/1510934806715.73/warc/CC-MAIN-20171123012207-20171123032207-00684.warc.gz"} |
https://mathematica.stackexchange.com/questions/189100/good-practice-about-numerical-precision | # Good practice about numerical precision
In one of my calculations, I get -1.11022*10^-16 as one of my eigenvalues for a matrix. It's essentially zero and I suppose I could use SetPrecision to make it zero but I wonder what's a good practice here? Should I maybe include something at the beginning of the code to specify precision for everything? Or do I just set arbitrary precision when I feel my expression is not simplified enough?
I'm pretty new to Mathematica.
As others have commented, after trying in different calculations, I think use \Chop after each calculation is the best and safest way.One can also explicitly specify the digits to keep.
• You could apply Chop. – Chris K Jan 9 '19 at 4:31
• I do not believe that you really want to ess with precision here. You just want to push values close to 0. to 0.. This can be done by Threshold and by specifying a suitable tolerance as second argument. – Henrik Schumacher Jan 9 '19 at 8:17
• Thank you guys. Is there a way to Chop/Threshold the whole notebook? Or I just apply these functions after whatever output I get? – Histoscienology Jan 11 '19 at 17:07
Mathematica has three different types of numbers.
Exact numbers, e.g.
5, Sqrt[2], 4/3, ...
Machine precision numbers
1.23, 5., .4566, ...
And numbers with a set precision
5.16, 1.23455, ...
You can tell what kind of number something is by applying Precision to it.
If all your input number have an set precision, then the output of you eigenvalue computation should also have a finite precision attached to it. This will allow you to distinguish between an eigenvalue that is really small an one that is consistent with zero within the expected accuracy.
• Thanks. So you mean I should set every input with a precision as " `some number?" And how can Sqrt[2] and 4/3 be exact?? – Histoscienology Jan 11 '19 at 4:22 | 2021-04-20 23:49:42 | {"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": 0.743283212184906, "perplexity": 1062.0791669505554}, "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-2021-17/segments/1618039491784.79/warc/CC-MAIN-20210420214346-20210421004346-00196.warc.gz"} |
http://math.stackexchange.com/questions/244496/problems-where-the-solution-hinges-on-the-correct-definition | # Problems where the solution hinges on the correct definition
Recently, I realized that there are some famous problems in mathematics whose solution depended heavily on the right formulation of an intuitive concept.
For example, there was no precise definition of an algebraic integer before Dedekind. As Milne says in his book on algebraic number theory, Euler's proof of Fermat's Last theorem for the exponent $3$ does only become correct when you replace $\Bbb Z[\sqrt{-3}]$ by $\Bbb Z[\frac{1+\sqrt{-3}}{2}]$.
Do you know any other examples of theories where the correct formulation of a concept was an important step in the evolution of that theory?
(As usual, one example per answer.)
-
community wiki? – Jason DeVito Nov 25 '12 at 20:55
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is called Darboux continuous if for any $[x,y]$ , for each $z$ in the interval bounded by $f(x)$ and $f(y)$ there exists $\theta \in [x,y]$such that $f(\theta) = z$. i.e. the function obeys the "intermediate value property". It then follows from the intermediate value theorem that any continuous function is Darboux continuous. The theorem of Darboux states that that if $f$ is differentiable with derivative $f'$ then $f'$ is Darboux continuous. Darboux himself also gave examples of continuous functions (in the usual sense) with discontinuos derivative, hence the derivatives are Darboux continuous but not continuous, so the two definitions are not the same. | 2015-12-01 06:00:05 | {"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": 0.9369493126869202, "perplexity": 99.90882343567306}, "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-2015-48/segments/1448398464536.35/warc/CC-MAIN-20151124205424-00059-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus_-_An_Investigation_of_Functions_(Lippman_and_Rasmussen)/2%3A_Linear_Functions/2.4%3A_Fitting_Linear_Models_to_Data |
# 2.4: Fitting Linear Models to Data
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$
$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$
In the real world, rarely do things follow trends perfectly. When we expect the trend to behave linearly, or when inspection suggests the trend is behaving linearly, it is often desirable to find an equation to approximate the data. Finding an equation to approximate the data helps us understand the behavior of the data and allows us to use the linear model to make predictions about the data, inside and outside of the data range.
Example 1
The table below shows the number of cricket chirps in 15 seconds, and the air temperature, in degrees Fahrenheit (Selected data from http://classic.globe.gov/fsl/scientistsblog/2007/10/. Retrieved Aug 3, 2010). Plot this data, and determine whether the data appears to be linearly related.
chirps 44 35 20.4 33 31 35 18.5 37 26 Temp 80.5 70.5 57 66 68 72 52 73.5 53
Solution
Plotting this data, it appears there may be a trend, and that the trend appears roughly linear, though certainly not perfectly so.
The simplest way to find an equation to approximate this data is to try to “eyeball” a line that seems to fit the data pretty well, then find an equation for that line based on the slope and intercept.
You can see from the trend in the data that the number of chirps increases as the temperature increases. As you consider a function for this data you should know that you are looking at an increasing function or a function with a positive slope.
flashback
a. What descriptive variables would you choose to represent Temperature & Chirps?
b. Which variable is the independent variable and which is the dependent variable?
c. Based on this data and the graph, what is a reasonable domain & range?
d. Based on the data alone, is this function one-to-one, explain?
1. a. T = Temperature, C = Chirps (answers may vary)
b. Independent (Chirps) , Dependent (Temperature)
c. Reasonable Domain (18.5, 44) , Reasonable Range (52, 80.5) (answers may vary)
d. NO, it is not one-to-one, there are two different output values for 35 chirps.
Example 2
Using the table of values from the previous example, find a linear function that fits the data by “eyeballing” a line that seems to fit.
Solution
On a graph, we could try sketching in a line. Note the scale on the axes have been adjusted to start at zero to include the vertical axis and vertical intercept in the graph.
Using the starting and ending points of our “hand drawn” line, points (0, 30) and (50, 90), this graph has a slope of $$m=\dfrac{60}{50} =1.2$$ and a vertical intercept at 30, giving an equation of
$$T(c)=30+1.2c$$
where $$c$$ is the number of chirps in 15 seconds, and $$T(c)$$ is the temperature in degrees Fahrenheit.
This linear equation can then be used to approximate the solution to various questions we might ask about the trend. While the data does not perfectly fall on the linear equation, the equation is our best guess as to how the relationship will behave outside of the values we have data for. There is a difference, though, between making predictions inside the domain and range of values we have data for, and outside that domain and range.
Definition: Interpolation and Extrapolation
Interpolation: When we predict a value inside the domain and range of the data
Extrapolation: When we predict a value outside the domain and range of the data
For the Temperature as a function of chirps in our hand drawn model above,
• Interpolation would occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44.
• Extrapolation would occur if we used our model to predict temperature when the values for chirps are less than 18.5 or greater than 44.
Example 3
a) Would predicting the temperature when crickets are chirping 30 times in 15 seconds be interpolation or extrapolation? Make the prediction, and discuss if it is reasonable.
b) Would predicting the number of chirps crickets will make at 40 degrees be interpolation or extrapolation? Make the prediction, and discuss if it is reasonable.
Solution
With our cricket data, our number of chirps in the data provided varied from 18.5 to 44. A prediction at 30 chirps per 15 seconds is inside the domain of our data, so would be interpolation. Using our model:
$$T(3)=30+1.2(30)=66$$ degrees.
Based on the data we have, this value seems reasonable.
The temperature values varied from 52 to 80.5. Predicting the number of chirps at 40 degrees is extrapolation since 40 is outside the range of our data. Using our model:
$$\begin{array} {rcl} {40} &= & {30 + 1.2c} \\ {10} &= & {1.2c} \\ {c} &\approx & {8.33} \end{array}$$
Our model predicts the crickets would chirp 8.33 times in 15 seconds. While this might be possible, we have no reason to believe our model is valid outside the domain and range. In fact, generally crickets stop chirping altogether below around 50 degrees.
When our model no longer applies after some point, it is sometimes called model breakdown.
try it now
What temperature would you predict if you counted 20 chirps in 15 seconds?
Add answer text here and it will automatically be hidden if you have a "AutoNum" template active on the page.
Fitting Lines with Technology
While eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data values(Technically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values.). This technique is called least-square regression, and can be computed by many graphing calculators, spreadsheet software like Excel or Google Docs, statistical software, and many web-based calculators(For example, http://www.shodor.org/unchem/math/lls/leastsq.html).
Example 4
Find the least-squares regression line using the cricket chirp data from above.
Solution
Using the cricket chirp data from earlier, with technology we obtain the equation:
$$T(c)=30.281+1.143c$$
Notice that this line is quite similar to the equation we “eyeballed”, but should fit the data better. Notice also that using this equation would change our prediction for the temperature when hearing 30 chirps in 15 seconds from 66 degrees to:
$$T(30) =30.281+1.143(30)=64.571 \approx 64.6$$ degrees.
Most calculators and computer software will also provide you with the correlation coefficient, a measure of how closely the line fits the data.
Definition: correlation coefficient
The correlation coefficient is a value, $$r$$, between -1 and 1.
$$r > 0$$ suggests a positive (increasing) relationship
$$r < 0$$ suggests a negative (decreasing) relationship
The closer the value is to 0, the more scattered the data
The closer the value is to 1 or -1, the less scattered the data is
The correlation coefficient provides an easy way to get some idea of how close to a line the data falls.
We should only compute the correlation coefficient for data that follows a linear pattern; if the data exhibits a non-linear pattern, the correlation coefficient is meaningless. To get a sense for the relationship between the value of $$r$$ and the graph of the data, here are some large data sets with their correlation coefficients:
Examples of Correlation Coefficient Values
Example 5
Calculate the correlation coefficient for our cricket data.
Solution
Because the data appears to follow a linear pattern, we can use technology to calculate r = 0.9509. Since this value is very close to 1, it suggests a strong increasing linear relationship.
Example 6
Gasoline consumption in the US has been increasing steadily. Consumption data from 1994 to 2004 is shown below.(http://www.bts.gov/publications/nati...ble_04_10.html) Determine if the trend is linear, and if so, find a model for the data. Use the model to predict the consumption in 2008.
Year '94 '95 '96 '97 '98 '99 '00 '01 '02 '03 '04 Consumption (billions of gallons) 113 116 118 119 123 125 126 128 131 133 136
Solution
To make things simpler, a new input variable is introduced, $$t$$, representing years since 1994.
Using technology, the correlation coefficient was calculated to be 0.9965, suggesting a very strong increasing linear trend.
The least-squares regression equation is:
$$C(t)=113.318+2.209t.$$
Using this to predict consumption in 2008 (t = 14),
$$C(14)=113.318+2.209(14)=144.244$$ billions of gallons
The model predicts 144.244 billion gallons of gasoline will be consumed in 2008.
try it now
Use the model created by technology in example 6 to predict the gas consumption in 2011. Is this an interpolation or an extrapolation? | 2019-07-18 18:09:46 | {"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": 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": 0.5532817840576172, "perplexity": 666.518615942067}, "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-2019-30/segments/1563195525699.51/warc/CC-MAIN-20190718170249-20190718192249-00421.warc.gz"} |
https://mathematica.stackexchange.com/questions/79066/mathematical-morphology-removing-text-features-from-image-while-keeping-connec | # Mathematical morphology: removing text features from image, while keeping connectivity
I have this image of London's road networks.
img = Image[
7Ccolor:0x000000%7Cweight:1%7Cvisibility:on"], ImageSize -> Medium]
After Binarize[] and ColorNegate[]:
I want to take out:
a) The text, without disrupting the edge connectivity.
b) The rectangles, again, without breaking edges.
How can I do this?
After binarizing and negating the image, I tried using MorphologicalGraph
Image[MorphologicalGraph[blackLondon, EdgeStyle -> Black,
VertexStyle -> White]]
and got this unsatisfactory result:
I also tried binarizing from 0 (the roads that I care about are pure black) and got this:
binarizedLondon2= Binarize[img, 0]
At this image, I applied morphological closing with a DiskMatrix, and managed to identify the rectangular elements.
rectangularElementsMask3 =
ColorNegate[Closing[binarized2, DiskMatrix[3]]]
I clean it up with an Opening.
rectangularElementsMask4 =
then Inpaint, on the binarized image
Inpaint[binarized2, rectangularElementsMask4]
and get this result, which is still disconnected.
• +1 on an interesting question - there are some IP wizards here, s/b interesting to see what they come up with... – ciao Apr 4 '15 at 23:26
• Try ContourDetect[Threshold[img, .9]] where img is your second attached image - gets mighty close to keeping roads cleanly connected. I'd venture with some masking for the rectangles, and then a filter run over the rectangle-removed version, with the rectangles again as masks, that looks for "dangling ends" and connects them with a line would be quite nice... – ciao Apr 5 '15 at 4:39
• MorphologicalPerimeter[img, .9] on same second image is also looking like a pretty good start. – ciao Apr 5 '15 at 4:47
• – dr.blochwave Apr 5 '15 at 15:11
MorphologicalBinarize and ColorNegate it. We use Manipulate to choose the finest parameter.
img = Image[
7Ccolor:0x000000%7Cweight:1%7Cvisibility:on"], ImageSize -> Medium];
Manipulate[
img2 = DeleteSmallComponents@
ColorNegate[MorphologicalBinarize[img, {x, y}]], {x, 0, 1}, {y, 0,
1}]
Then we dilate img2 and find the largest connected component. Manipulate is used here again for parameter issues. To better show the result, we combine the detected roads with the original map.
Manipulate[
r = Dilation[img2, x] //
MorphologicalComponents[#, CornerNeighbors -> False] &; | 2020-04-02 16:57:22 | {"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": 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": 0.18745605647563934, "perplexity": 7091.209792851466}, "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/1585370506988.10/warc/CC-MAIN-20200402143006-20200402173006-00473.warc.gz"} |
https://math.stackexchange.com/questions/2969094/proving-a-differential-equation-has-a-unique-solution-where-ft-x-is-decreas | # Proving a differential equation has a unique solution, where $f(t,x)$ is decreasing.
Let $$f:[t_0,t_1] \times \mathbb{R} \longrightarrow{\mathbb{R}}$$ a continuous function. Suppose that $$f$$ is a decreasing function on $$x$$, how can I prove that for every $$x_0\in \mathbb{R}$$ the problem $$\begin{cases} x'=f(t,x) \\ x(t_0)=x_0 \end{cases}$$ has an unique solution?
I suppose that I should use Picard's theorem, but I dont know how to take advantage of $$f$$ being a decreasing function.
• Use that every contraction mapping is uniformly continuous and apply Picard's theorem Oct 24 '18 at 12:59
Suppose $$x_1$$ and $$x_2$$ are two solutions on $$[t_0,t_1]$$. Let $$h(t)=(x_1(t)-x_2(t))^2$$ and prove that $$h$$ is decreasing (that is, $$h'\le0$$.)
• And since you assumed that $x_1$ and $x_2$ are two solutions, that should get you to a contradiction? I don't understand the reasoning in this answer Oct 24 '18 at 19:13
• $h\ge0$, $h(0)=0$, $h$ decreasing. There are not many possibilities for $h$. Oct 24 '18 at 19:50
• Differentiate $h$, substitute $x’_i$ by $f(t,x_i)$ and use that $f(t,x)$ is decreasing in $x$ to deduce that $h’\le0$. Oct 25 '18 at 13:27 | 2021-09-18 22:33:28 | {"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": 12, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9045742750167847, "perplexity": 79.90946900415537}, "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/1631780056578.5/warc/CC-MAIN-20210918214805-20210919004805-00165.warc.gz"} |
https://math.stackexchange.com/questions/1402235/simple-question-on-closed-sets | # Simple question on closed sets
A closed set is one which contains all its limit points. Why is $[a, \infty)$ closed? Specifically I don't understand how $\infty$ which is a limit point, but it is not in the set.
• Do you agree that $(-\infty, a)$ is open? Aug 19, 2015 at 2:22
• Yes, but I don't want an answer to be that the complement of an open set is closed. Aug 19, 2015 at 2:24
• $\infty$ is not a limit point of $[a,\infty)$ in $\mathbb R$! Aug 19, 2015 at 2:29
• $\infty$ is not a limit point. Your set lives in $\mathbb{R}$. A limit point must also belong to the same space. Aug 19, 2015 at 2:29
• Since "closedness" is relative to the topology, you might wonder if there's a related notion that doesn't depend on the topology. There is: compactness. Aug 19, 2015 at 4:28
The question of why $[a, \infty)$ is closed even though the limit point $\infty$ isn't in it is actually a really good question. Here is the answer:
It depends on what the topological space is. If our space is $\Bbb R$ (i.e., $(-\infty, \infty)$) with its usual topology, then since $\infty$ is not in $\Bbb R$, it's not a limit point of $[a, \infty)$.
Now, if our topological space is the extended reals $\Bbb R^{*}$, which is defined as $[-\infty, \infty]$ with topology generated by all usual open intervals in $\Bbb R$ but also all intervals of the form $(a, \infty]$ and $[-\infty, b)$, then $\infty$ is a limit point of $[a,\infty)$, so $[a,\infty)$ is not closed in $\Bbb R^{*}$.
So, a set being closed is a relative property. A set can be closed in one topological space, but not in another, as we just saw. As a subset of $\Bbb R$, $[a, \infty)$ is closed, but as a subset of $\Bbb R^{*}$, it's not. So when you talk about openness and closedness, you really need to be aware of which topological space you are talking about first.
• Beat me by six sectiods, drat. :-)
– user231101
Aug 19, 2015 at 2:34
• @MikeHaskel Yet you still deserve an upvote. :) Aug 19, 2015 at 2:34
The key point is that whether a set is open or closed depends on what space we're talking about. $[a,\infty)$ isn't "closed" by itself, it's closed in $\mathbb{R}$. There is no point "$\infty$" in $\mathbb{R}$, so it doesn't count for deciding whether or not $[a,\infty)$ is closed in $\mathbb{R}$. In contrast, sometimes we do talk about "$\overline{\mathbb{R}}$", which is a notational convention for $\mathbb{R}$ together with the extra points $\pm \infty$. In $\overline{\mathbb{R}}$, the set $[a,\infty)$ is not closed.
(Don't confuse the $\overline{\mathbb{R}}$ notation with the notation for the closure of a set; they're different concepts.)
• Fixed the error @user251257 pointed out.
– user231101
Aug 19, 2015 at 2:49
• Thanks this is helpful too. Unfortunately I could only mark one answer correct, and the other one provided me with more details I needed. Aug 19, 2015 at 3:15
Take $x\notin [a,\infty)$. It must be that $x<a$. Then consider any open ball of radius less than $\vert a-x \vert$. This ball will not contain any points in $[a,\infty)$. Hence, $x$ is not a limit point by the definition of limit point, because every ball around a limit point must contain another element of the set. Therefore, every limit point of $[a,\infty)$ is also contained in this set.
For $[a,\infty)$ the complement is $(-\infty,a)$ with a point at infinity. This set is clearly open since adding an arbitrary epsilon to infinity yields infinity. Therefore the set $[a,\infty)$ is closed.
But, if the space you're in considers $(-\infty,-\infty)$ open, then you have a topology leading the definitions. Indeed, infinity will be considered a limit point and then $[a,\infty)$ is actually open! | 2022-08-19 05:06:40 | {"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": 0.8713712692260742, "perplexity": 168.46025906050448}, "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/1659882573623.4/warc/CC-MAIN-20220819035957-20220819065957-00313.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/evaluate-2-x-1-2-d-2-y-d-x-2-2-2-x-1-d-y-d-x-12-y-6-x-linear-differential-equation-constant-coefficient-complementary-function_57720 | Evaluate ( 2 X + 1 ) 2 D 2 Y D X 2 − 2 ( 2 X + 1 ) D Y D X − 12 Y = 6 X - Applied Mathematics 2
Sum
Evaluate (2x+1)^2(d^2y)/(dx^2)-2(2x+1)(dy)/(dx)-12y=6x
Solution
(2x+1)^2(d^2y)/(dx^2)-2(2x+1)(dy)/(dx)-12y=6x
Put (2x+1)=e^z => x=(e^x-1)/2
(dz)/(dx)=2/(2x+1) but(dy)/(dx)=(dy)/(dx)(dz)/(dx)=2(dy)/(dx)=2/(2x+1)"Dy" "where" "D"=d/(dz)
therefore(2x+1)(dy)/(dx)=2"Dy"
therefore(2x+1)^2(d^2y)/(dx^2)=2^2"D(D-1)y"
From (1),
4D(D-1)y-4Dy-12y=6((e^x-1)/2)
(4D^2-8D-12)y=3(e^z-1)
For complementary solution ,
(4D^2-8D-12)=0
∴D = -1,3
thereforey_c=c_1e^(-z)+c_2 e^(3z)
For particular integral ,
y_p=1/(f(D))X
y_p=1/(4D^2-8D-12)(3(e^z-1))
therefore y_p=3/4 1/(D^2-2D-3)(e^z-1) put D = a = 1 and D = a = 0
thereforey_p=3/4(1/3-e^z/4)
The general solution of given differential eqn is ,
thereforey_g=y_c+y_p=c_1e^(-z)+c_2e^(3z)+3/4(1/3-e^z/4)
Resubstituting 𝒛 ,
therefore y_g=c_1(2x+1)^(-1)+c_2(2x+1)^3+3/4(1/3-(2x+1)/4)
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
Is there an error in this question or solution? | 2021-03-02 07:46:06 | {"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": 0.6790145039558411, "perplexity": 4634.857678715816}, "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/1614178363782.40/warc/CC-MAIN-20210302065019-20210302095019-00106.warc.gz"} |
https://hal.inria.fr/inria-00538469 | # Integrating IoT and IoS with a Component-Based approach
1 TRISKELL - Reliable and efficient component based software engineering
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
Abstract : There is a growing interest in leveraging Service Oriented Architectures (SOA) in domains such as home automation, automotive, mobile phones or e-Health. With the basic idea (supported in e.g. OSGi) that components provide services, it makes it possible to smoothly integrate the Internet of Things (IoT) with the Internet of Services (IoS). The paradigm of the IoS indeed offers interesting capabilities in terms of dynamicity and interoperability. However in domains that involve things'' (e.g. appliances), there is still a strong need for loose coupling and a proper separation between types and instances that are well-known in Component-Based approaches but that typical SOA fail to provide. This paper presents how we can still get the best of both worlds by augmenting SOA with a Component-Based approach. We illustrate our approach with a case study from the domain of home automation.
Document type :
Conference papers
Cited literature [16 references]
https://hal.inria.fr/inria-00538469
Contributor : Didier Vojtisek <>
Submitted on : Monday, November 22, 2010 - 3:34:59 PM
Last modification on : Friday, July 10, 2020 - 4:20:08 PM
Document(s) archivé(s) le : Friday, October 26, 2012 - 4:21:35 PM
### File
Nain10a.pdf
Files produced by the author(s)
### Identifiers
• HAL Id : inria-00538469, version 1
### Citation
Grégory Nain, François Fouquet, Brice Morin, Olivier Barais, Jean-Marc Jézéquel. Integrating IoT and IoS with a Component-Based approach. Procedings of the 36th EUROMICRO Conference on Software Engineering and Advanced Applications (SEAA 2010), Date-Added = 2010-06-07 10:37:26 +0200, Date-Modified = 2010-07-23 09:56:36 +0200, 2010, Lille, France, France. ⟨inria-00538469⟩
Record views | 2020-07-11 01:54:46 | {"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": 0.3271048069000244, "perplexity": 9645.915851376461}, "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-2020-29/segments/1593655919952.68/warc/CC-MAIN-20200711001811-20200711031811-00492.warc.gz"} |
http://openstudy.com/updates/4daf8ffbd6938b0b1617ab4d | ## anonymous 5 years ago I know that similar functions can have very different antiderivatives. For example, even though int((x)/((x^4)+1)) dx looks very similar to int(((x^4)+1)/(x)), their solutions are very different. How can I prove that I'm right by solving both?
1. amistre64
you wanna indefinite integral both of them right?
2. anonymous
Yes.
3. amistre64
the last one just split the fraction: x^4/x +1/x and integrate
4. amistre64
$\int\limits_{} \frac{x^4}{x} + \frac{1}{x} dx$
5. anonymous
Right...
6. amistre64
the second one we can try subsituting u=x^4+1
7. amistre64
by second I mean first :)
8. anonymous
I gotcha :) I figured as much...first and second...same thing. right? haha!
9. anonymous
so, on the first one you end up with int(1/u)du?
10. anonymous
$\int\limits 1/u$
11. amistre64
$\int\limits_{} \frac{1}{4u(u-1)^{1/2}}$this is what I get :)
12. amistre64
..du
13. amistre64
put the u under the radial and take out the 1/4
14. amistre64
sounds good, looks bad; decompose the fraction :)
15. amistre64
$\frac{1}{4}\int\limits_{}\frac{1}{u}du - \int\limits_{} \frac{1}{(u+1)^{1/2}}du$
16. amistre64
umm... what did I do lol...hold up on that
17. amistre64
yeah, thats right, I couldnt recall where I got (u-1)^1/2 from
18. anonymous
Yeah, I was about to say that you were correct. So, would you mind walking me through how to solve the second one?
19. amistre64
and by second do you mean first?
20. anonymous
Haha! yes, the first one we talked about anyway.
21. amistre64
ok.... whenever we have 2 numbers above a single term; we can split that up to its baser seqments: x^4 + 1 x^4 1 ------- = ---- + ---- right? x x x
22. amistre64
like denominators simply merge and you work the tops; well, they can be split up just as easily
23. anonymous
yes.
24. amistre64
we get: x^3 + 1/x which integrates to: x^4/4 + ln(x) + C
25. anonymous
Oh! I understand! Thank you so much. Now, do we need to add a plus c to the second one we did?
26. amistre64
whenever we have a inegral with no limits attached to it; we have to include the "constant of integration"...so yes. +C is just a place holder till we get more information about it to anchor it to the graph.
27. amistre64
was the decomposition one right? or do we know?
28. anonymous
I don't think we know, unless we can infer that it is from the original problem I gave.
29. amistre64
let me redo that one on paper to see if I got it right....
30. amistre64
running into problems with it ...
31. amistre64
i might just be over thinking it, let me try trig substitution...
32. anonymous
I didn't really notice anything wrong, but I could just be trying to do it wrong. Who knows.
33. amistre64
when I decomped it I allowed the square root to be a negative number... so I had to go another route...
34. anonymous
OH wow.
35. amistre64
if I did it right...gonna have to try to derive it down; $\frac{\tan^{-1}(x^2)}{2}$
36. amistre64
I think thats it :)$Dx (\frac{\tan^{-1}(x^2)}{2}) \rightarrow \frac{2x}{2[(x^2)^2+1]}$
37. amistre64
which = x/(x^4 +1) :)
38. amistre64
$\frac{\tan^{-1}(x^2)}{2} + C$
39. anonymous
Okay...I see I see...Thank you so much for all of your effort and help. :)
40. amistre64
youre welcome :) | 2016-10-23 06:14:08 | {"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": 0.8287723660469055, "perplexity": 3515.195828478092}, "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-2016-44/segments/1476988719155.26/warc/CC-MAIN-20161020183839-00103-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://eliaszwang.com/paper-reviews/compositional-video-prediction/ | # Compositional Video Prediction
Ye et al., 2019
Source: Ye et al., 2019
## Summary
• Method for solving pixel-level future prediction given a single image
• Decompose scene into distinct entities that undergo motion and possibly interact
• Generates realistic predictions for stacked objects and human activities in gyms
• Links: [ website ] [ pdf ]
## Background
• Given a single image, humans can easily understand the scene and make predictions
• Many other works also model relationships between objects, but do some combination of:
• Use only simple visual stimuli
• Use state based input
• Rely on sequence of inputs
• Don’t make predictions in pixel space
## Methods
• Given starting frame $f^0$ and the location of $N$ entities $\{b^0_n\}^N_{n=1}$, predict $T$ future frames $f^1, f^2, \dots, f^T$
• Use entity predictor $\mathcal{P}$ for per-entity representations $\{x^t_n\}^N_{n=1}$
• $\{x^{t+1}_n\} \equiv \mathcal{P}(\{x^t_n\}, z_t)$
• $\{x^t_n\}^N_{n=1} \equiv \{(b^t_n, a^t_n)\}^N_{n=1}$
• $b^t_n$ is the predicted location
• $a^t_n$ is the predicted implicit features for entity appearance
• $a^0_n$ obtained using ResNet-18 on cropped region from $f^0$
• Interaction between entities modeled with graph neural network, edges are predetermined
• Use frame decoder $\mathcal{D}$ to infer pixels
• $f^t \equiv \mathcal{D}(\{x^t_n\},f^0)$
• Warp normalized spatial representation to image coordinates with predicted location for each entity
• Use soft masking for each entity to account for possible occlusions
• Add these masked features to features from initial frame $f^0$
• Single random latent variable $u$ to capture multi-modality of prediction task
• Yields per-timestep latent variables $z_t$, which are correlated accross time, via a learned LSTM
• Loss consists of three terms:
• Prediction: $l_1$ loss on decoded frame from predicted features and $l_2$ loss on predicted location
• Encoder: information bottleneck on latent variable distribution
• Decoder: $l_1$ loss on decoded frame from features extracted from the same frame (auto-encoding loss)
## Results
• Datasets:
• ShapeStacks: synthetic dataset of stacked objects that fall, different block shapes/colors and configurations
• Penn Action: real video dataset of people playing indoor/outdoor sports, annotated with joint locations
• Use gym activities subset: less camera motion, more similar backgrounds
• Metrics:
• Average MSE for entity locations
• Learned Perceptual Image Patch Similarity (LPIPS) for generated frames
• Due to stochasticity from random variable $u$, use best scores from 100 samples
• Baselines:
• No-Factor: only predicts at the level of frames
• No-Edge: no interaction among entities
• Pose Knows: also predicts poses as intermediate representation, predicts location but not appearance, no interactions
• Proposed method consistenly performs better than the simple baselines on ShapeStacks, but No-Edge is a fairly close second
• Adding adversarial loss, from Pose Knows, greatly improves performance of proposed method on Penn Action
• Qualitative results on Penn Action are generally not that convincing for any model
## Conclusion
• Relies on supervision of entity locations and graph structure
• Inclusion of latent variable to model multi-modality is interesting
• Better metrics for evaluating diversity and accuracy would be useful
• Poor performance on real world dataset suggests there is still a lot of room for improvement | 2023-03-30 04:25:43 | {"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": 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": 0.2684297263622284, "perplexity": 9549.06575354655}, "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/1679296949097.61/warc/CC-MAIN-20230330035241-20230330065241-00254.warc.gz"} |
http://openstudy.com/updates/55895336e4b09ad72e960e85 | ## anonymous one year ago How may permutations of the word “spell” are there? There are____________
1. anonymous
can anyone help????????
2. mathmate
The number of permutations with repeats is $$\Large \frac{N!}{n_1!n_2!...n_k!}$$ where N=total number of items k=number of types of items n1,n2...nk=number of items of each type. For unique items, put n=1 Note that $$\sum_1^k n_k = N$$ Example: How many permutations can we form with the word parallel? there are 2a's, 1e, 3l's, 1p, 1r for a total of 8 letters. The permutation is therefore $$\Large \frac{8!}{2!1!3!1!1!}=\frac{40320}{2\times1\times6\times1\times1 }=3360$$ See also following link if you need more explanations: http://www.mathwarehouse.com/probability/permutations-repeated-items.php
3. anonymous
oh dang hold up | 2017-01-17 13:28:21 | {"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": 0.4964871406555176, "perplexity": 3355.31625464307}, "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-2017-04/segments/1484560279915.8/warc/CC-MAIN-20170116095119-00207-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.neetprep.com/question/60330-One-mole-helium-adiabatically-expanded-its-initial-state-PiViTi-its-final-state-PfVfTf-decrease-internal-energy-associatedwith-expansion-equal-CVTiTf-CPTiTf-CPCVTiTf-CPCVTiTf/55-Physics--Thermodynamics/687-Thermodynamics | # NEET Physics Thermodynamics Questions Solved
One mole of helium is adiabatically expanded from its initial state $\left({P}_{i},{V}_{i},{T}_{i}\right)$ to its final state $\left({P}_{f},{V}_{f},{T}_{f}\right)$. The decrease in the internal energy associated with this expansion is equal to
(1) ${C}_{V}\left({T}_{i}-{T}_{f}\right)$
(2) ${C}_{P}\left({T}_{i}-{T}_{f}\right)$
(3) $\frac{1}{2}\left({C}_{P}+{C}_{V}\right)\left(Ti-{T}_{f}\right)$
(4) $\left({C}_{P}-{C}_{V}\right)\left({T}_{i}-{T}_{f}\right)$
(1) $\Delta U=\mu {C}_{V}\Delta T=1×{C}_{V}\left({T}_{f}-{T}_{i}\right)=-\text{\hspace{0.17em}}{C}_{V}\left({T}_{i}-{T}_{f}\right)$
⇒ |ΔU| = CV (TiTf)
Difficulty Level:
• 52%
• 13%
• 30%
• 6%
Crack NEET with Online Course - Free Trial (Offer Valid Till August 25, 2019) | 2019-08-23 08:53:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "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": 0.9095247387886047, "perplexity": 3563.4428031366638}, "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/1566027318243.40/warc/CC-MAIN-20190823083811-20190823105811-00145.warc.gz"} |
https://kar.kent.ac.uk/14488/ | # Towards a Theory of Tracing for Functional Programs based on Graph Rewriting
Chitil, Olaf, Luo, Yong (2006) Towards a Theory of Tracing for Functional Programs based on Graph Rewriting. In: Mackie, Ian, ed. Draft Proceedings of the 3rd International Workshop on Term Graph Rewriting, Termgraph 2006. . Termgraph
PDF | 2020-02-26 11:05:51 | {"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": 0.8143215775489807, "perplexity": 8447.970883079322}, "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/1581875146341.16/warc/CC-MAIN-20200226084902-20200226114902-00311.warc.gz"} |
https://publications.mfo.de/handle/mfo/19/browse?type=msc&value=28 | Now showing items 1-3 of 3
• #### Hölder-Differentiability of Gibbs Distribution Functions
[OWP-2007-13] (Mathematisches Forschungsinstitut Oberwolfach, 2007-03-29)
In this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil’s staircases) supported on limit sets of finitely generated conformal iterated ...
• #### Mesh Ratios for Best-Packing and Limits of Minimal Energy Configurations
[OWP-2013-13] (Mathematisches Forschungsinstitut Oberwolfach, 2013-06-10)
For $N$-point best-packing configurations $\omega_N$ on a compact metric space $(A, \rho)$, we obtain estimates for the mesh-separation ratio $\gamma(\rho_N , A)$, which is the quotient of the covering radius of $\omega_N$ ...
• #### Sharp constants in the classical weak form of the John-Nirenberg inequality
[OWP-2013-07] (Mathematisches Forschungsinstitut Oberwolfach, 2013-03-14)
The sharp constants in the classical John-Nirenberg inequality are found by using Bellman function approach. | 2020-07-12 23:16:59 | {"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": 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": 0.8448757529258728, "perplexity": 2238.7551388222623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "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-2020-29/segments/1593657140337.79/warc/CC-MAIN-20200712211314-20200713001314-00495.warc.gz"} |
https://indico.cern.ch/event/961948/timetable/?view=standard | # 4th Project MEFT Workshop
Europe/Lisbon
https://videoconf-colibri.zoom.us/j/84518800070?pwd=OE80Q0RwclYrYnk1aTFjRGFnbWlMZz09 (Zoom)
### https://videoconf-colibri.zoom.us/j/84518800070?pwd=OE80Q0RwclYrYnk1aTFjRGFnbWlMZz09
#### Zoom
(password requests by email)
, ,
Description
# Scientific committee:
• Ilídio Lopes
• Patrícia Gonçalves
• Pedro Ribeiro
Participants
• Alberto Nicolicea
• André Cordeiro
• André Pereira
• Beatriz Pereira
• Bruno Marques
• Carlo Alfisi
• Clara Severino
• Cláudia Espinha
• Daniel Neacsu
• David Fordham
• Diogo Ivo
• Diogo Ribeiro
• Duarte Esteves
• Eduardo Ferreira
• Filipa Baltazar
• Filipe Cruz
• Filipe Cruz
• Francisca Madeira
• Francisco Albergaria
• Francisco Vazão
• Gonçalo Vaz
• Guilherme Brites
• Ilidio Lopes
• Joao Gomes
• José Bastos
• João Bravo
• João Duarte
• João Lourenço
• Luís Veloso
• Maria Faria
• Miguel Martins
• Nuno Fernandes
• Pedro Costa
• Pedro Piçarra
• Ricardo Florentino
• Ricardo Miguel
• Rodolfo Simões
• Rodrigo Câmara
• Rodrigo Santo
• Tiago Martins
• Tomás Cabrito
• Vladlen Galetsky
• Óscar Amaro
Contact
• Thursday, 28 January
• 09:15 09:30
Opening Session
Convener: Mr Ilídio Lopes (CENTRA - IST)
• 09:30 09:40
Flavour Anomalies 10m
The flavour anomalies are discrepancies observed between the experimental data and the Standard Model (SM) predictions and their detection forms the strongest evidence for the existence of New Physics (NP) in current collider data. These anomalies are revealed in quark-level transitions, such as the $b \rightarrow s~l^+l^-$ transitions, which are highly suppressed in the SM and can only occur via loop diagrams. NP could be manifest by the introduction of new exotic particles, such as a heavier gauge boson Z' or leptoquarks, which could allow the existence of these decays at tree level.
A particularly sensitive realization of the aforementioned transitions is the $B^0 \rightarrow K^{*0} \mu^+ \mu^-$ decay. Its angular analysis allows one to measure several parameters which are sensitive to different NP sources. This process is at the center of the detected flavour anomalies : several tensions with the SM were found during Run 1 by the LHCb, ATLAS and Belle Collaborations. Despite the large statistics collected during Run 1, the results were dominated by statistical uncertainties. The analysis of the data collected during Run 2 will allow to increase the precision of the measurements given the much higher luminosities achieved and to confirm or falsify the observed discrepancies.
The work of this thesis will be concerned with the study of this decay using data collected by the CMS detector during the full Run 2. The main goal will be to measure the decay branching fraction. The work has the potential to significantly contribute to the CMS exploration of the alluring flavour anomalies.
Speaker: Maria Faria (LIP Laboratorio de Instrumentacao e Fisica Experimental de Part)
• 09:40 09:50
Evaluation of the potential of a gamma-ray observatory to detect astrophysical neutrinos 10m
The discovery of high-energy astrophysical neutrinos has the potential to open a new window to study violent phenomena in our Universe, such as gamma-ray bursts, and to pose stringent tests to fundamental interactions. However, due to their low interaction cross-section, few experiments in the world can detect these particles.
The Southern Wide-field Gamma-ray Observatory (SWGO) project aims to monitor the Southern sky in very-high-energy gamma rays. This detector is planned to cover a huge area with detection units based on the water-Cherenkov detection technology, thus being favorable for the detection of high-energy neutrinos.
While promising, the idea to use a gamma-ray observatory to detect high-energy neutrinos has to be better assessed via simulations and considerations regarding the cross section and flux of astrophysical neutrinos. The main goal is to evaluate the validity of using this detector setup to perform these measurements and to determine the sensitivity of SWGO to a flux of astrophysical neutrinos as a function of their energy.
Speaker: Pedro Costa (Instituto Superior Técnico)
• 09:50 10:00
End-to-end simulation of satellite-based quantum key distribution 10m
Recent advances in quantum computing and number theory have put a threat on RSA protocols and other modern public-key cryptosystems, challenging the overall fragility of the classical channels. In contrast, quantum key distribution (QKD) offers to restore security and confidentiality of the information even with eavesdropping, through the basic principles of the quantum world. However, there are still a number of problems to be addressed in this field, mainly the trade-off between security, distance, and secret key rates.
To address this situation, this work will be namely focusing on a space to ground QKD simulator between a satellite (Quantsat) and a ground station, going all the way from hardware-in-loop testing to a mission concept creation, allowing to validate future space missions and experiments to be proposed under this field.
Speaker: Vladlen Galetsky (IST)
• 10:00 10:10
Novel Optimization Strategies for Clinical FLASH Proton Therapy 10m
Over decades, the number of patients diagnosed with cancer has been increasing, with expectations for the numbers to continue to rise in the future. The use of ionizing radiation for cancer treatment - radiotherapy - has become quite important as around 50% of all cancer patients have an indication for it. Treatments with radiotherapy are associated with side effects, arising from unavoidable damage to healthy tissue, which research on the field has been trying to reduce.
A new way of achieving reduced healthy tissue toxicity has been identified by biological studies, through an effect demonstrated by cells when irradiated with a high dose, for a very short time, using a very high dose-rate - the FLASH effect. Combined with precision irradiation techniques, namely proton therapy, the potential for substantially improved plans is great.
As the FLASH biological mechanism is still not understood, it's difficult to evaluate and compare different FLASH-compatible plans and so different metrics have been suggested. This project aims at building a framework for evaluation and comparison of FLASH-compatible proton therapy treatment plans, with focus on implementing strategies for optimization of metrics under a clinical treatment planning software. Evaluation is to be performed on stereotactic lung treatment plans.
Speaker: Rodrigo Jose Santo
• 10:10 10:20
Measurement of the features of the muon number distribution using the MARTA engineering array 10m
Extensive air showers offer a unique opportunity to study high energy hadronic interactions, tune up-to-date hadronic interaction models and determine the origin and acceleration mechanisms of ultra-high-energy cosmic rays, through the analysis of shower observables and shower reconstruction. We recently showed that the muon number distribution in showers with low muonic content has a feature that can be used to constrain the production cross-section of neutral pions emerging from the first proton-Air interaction. However, the detection of muons is not easily disentangled from the detection of electromagnetic particles in current cosmic ray experiments. The goal of this presentation is then to propose the measurement of the mentioned feature using the MARTA engineering array and assess to which precision this measurement can be achieved, complementing the theoretical work we previously published.
Speaker: Miguel Martins (Instituto Superior Técnico)
• 10:20 10:30
Nonlinear optics with ultrashort mid-infrared laser pulses 10m
In the past two decades, there has been a growing interest and investment in the study of ultrafast optics in the mid-infrared (MIR 2-12 µm) region. Mainly because most gaseous and biomolecules have their fundamental vibrational absorptions within this range, leaving distinctive spectral fingerprints of key importance for industrial, medical, and scientific applications.
The first generation of high-energy and high-efficiency laser sources in the MIR region has only a few years of existence and Instituto Superior Técnico (IST) has recently installed one of these new state-of-the-art laser sources. This thesis aims to explore this new laser system, particularly its characterization and the development of the first series of experiments. These will consist of SuperContinuum Generation (SCG) and High-Harmonic Generation (HHG) and will be performed at the Laboratory for Intense Lasers (L2I) in IST. A numerical simulation will then support the experimental results.
By the end of the thesis, it is expected to present this work at an international conference and co-author a paper in an international scientific journal.
Speaker: Gonçalo Vaz (GoLP/IPFN Instituto Superior Técnico, Universidade de Lisboa)
• 10:30 10:40
Scalar Mixing in New Physics Models 10m
The discovery of the Higgs boson in 2012 was an important achievement in particle physics. This scalar particle is essential in the Standard Model to explain the mass of the other particles. However, there is nothing in the theory that restricts the scalar sector of the Standard Model to have only one particle. Therefore, theoretical physicists are trying to understand what are the consequences of adding more particles to this part of the Standard Model and if this extensions are in agreement with the experimental data.
A formalism was developed by Grimus and Neufeld to work with a general model where an arbitrary number of scalar singlets and doublets are added to the scalar sector of the Standard Model. In my thesis I will first extend this formalism to work also with models with scalar triplets and, on a second stage, I will compute some physical observables using this extended formalism.
Speaker: Francisco Albergaria (IST)
• 10:40 10:50
Neutrino masses and the origin of matter through leptogenesis 10m
During his presentation on “The Theory of Electrons and Positrons”, Paul Dirac described how quantum mechanics and relativity made possible the prediction of the positron. After pointing out the apparent symmetry between positive and negative charge, he hinted that the universe could consist of equal amounts of matter and anti-matter but, for some unknown reason, human experience is confined almost entirely to matter. Notwithstanding, the symmetry between particles and antiparticles is firmly established in collider physics, which naturally poses the question of why the observed universe is composed nearly exclusively of matter, in contrast to little or no primordial antimatter. Despite its remarkable success in describing many of the inner workings of Nature at its most fundamental level, the Standard Model struggles to explain the existence of a biased Universe. A compelling possibility is that the baryon asymmetry of the Universe is generated dinamically, a scenario that is known as baryogenesis, which implies the non-conservation of the baryon number. In the past thirty to forty years, several mechanisms for baryogenesis have been put forth: GUT baryogenesis, electroweak baryogenesis, Affleck-Dine mechanism, spontaneous baryogenesis. Nonetheless, the most compelling one is the mechanism of baryogenesis via leptogenesis, first proposed by Fukugita and Yanagida, whose simplest and theoretically best motivated realization is within the seesaw mechanism of neutrino masses.
The objective of this work shall be to analyze the viability of leptogenesis, considering a model based on modular symmetries, by means of which we will determine the BAU and neutrino parameters, followed by a phenomenological analysis.
Speaker: Ricardo Miguel
• 10:50 11:00
Coffee Break
Coffee Break
• 11:00 11:10
Towards the Space-time Picture of a QCD Parton Shower 10m
Ultra-relativistic heavy ion collisions, such as those in the Large Hadron Collider (LHC) or the Relativistic Heavy-Ion Collider (RHIC), have unlocked an extensive scientific program. Namely, the production of Quark-Gluon Plasma (QGP) offers many opportunities, from the study of early universe dynamics, to the phase diagram of Quantum Chromodynamics (QCD), and the study of QCD in extreme conditions.
Given the rapid evolution of the QGP, on the yoctosecond scale, its properties are accessed through the products of heavy ion collisions. The successive parton showers can be explored by the clustering of the final state hadrons into jets. This allows for the study of jet quenching in the QGP, which can be used to access the medium evolution.
As seen in recent developments, jet clustering algorithms can be used to access the space-time structure of the parton showers, unlocking an experimental treatment of the medium evolution. However, theoretical descriptions rest on the use of coordinate space while event generators, used for phenomenological studies, reflect the momentum space evolution of parton showers.
The aim of this work is therefore to develop a simple Monte Carlo Event Generator in coordinate space, paving the way to a full treatment of QCD showers and their modification by the presence of the QGP.
Speaker: André Cordeiro (Universidade de Lisboa, Instituto Superior Técnico)
• 11:10 11:20
Mini magnetospheres in the laboratory 10m
Ion-scale magnetospheres have been observed around comets, weakly magnetized asteroids, and localized regions on the Moon. These mini magnetospheres provide a unique environment to study kinetic-scale plasma physics, in particular in the collisionless regime. To study these systems, the space and astrophysical plasma community have produced, in the last years, multiple experiments in laboratory, as for example, the ones performed on the Large Plasma Device facility (LAPD), at UCLA.
For the MSc Thesis, we will build analytical and numerical models with collisionless particle-in-cell (PIC) simulations of laboratory-produced magnetospheres, to describe the coupling of laser-produced and magnetized plasmas and determine the properties of their interaction with magnetic obstacles. This work will not only contribute to interpret results from recent experiments, but also to support the design of future experimental studies.
Speaker: Filipe Cruz
• 11:20 11:30
Neutrino interaction with solid state 2D plasma 10m
The neutrino is one of the most 'shy' particle that nature created, it's almost massless and has no electric charge. That leads to a really low interaction between neutrino and matter, nevertheless they are important in many scenarios and studying them we can test our models. Nowadays, the detection of neutrino is based on the scattering mediated by W boson that leads to a charged lepton who is revealed by measuring its Cherenkov radiation. My work is to study the interaction between neutrino and electrons plasma with the goal to increase the sensibility of the detection.
Speaker: carlo Alfisi
• 11:30 11:40
Electron-positron production in extreme fields 10m
Electron-positron pair production is one of the most important problems in extreme plasma physics. Thanks to the recent advances in laser technology, these can now be created in the lab using the most intense lasers in the world. If all other parameters are equal, more pairs are expected with a higher laser intensity. However, if the total energy of the laser pulse is fixed, there will be a trade-off between the size of the effective interaction volume and the peak intensity. This work will extend analytical scaling laws for pair production in realistic laser-electron beam scattering, previously derived assuming an ideal plane-wave description; it will also propose optimal and innovative solutions for upcoming experiments, as well as investigate applications of quantum algorithms in extreme plasma physics.
Speaker: Óscar Amaro (Instituto Superior Técnico)
• 11:40 11:50
Cr-doped Ga2O3 for radiation detection 10m
In the field of radiotherapy, one of the major challenges is the accurate in-vivo measurement of the supplied radiation dose. In this context, chromium-doped gallium oxide is an interesting material due to its attractive electrical and optical properties. On the one hand, it is known to be a radiation hard, highly transparent wide bandgap semiconductor with fast scintillation. On the other hand, when irradiated with energetic ion beams, this semiconductor displays a red luminescence assigned to intraionic transitions of chromium at oxidation state 3+ within the first biological window that spans the range of wavelengths from 700 to 950 nm. Moreover, the yield of this red emission is enhanced by the defects that are created during the irradiation. Thus, this material presents a great potential for complementary systems of electric and optical dosimetry. The main purpose of this work is to understand the defect creation mechanisms and their role in the optical activation of the chromium ions, via ion beam-induced luminescence and thermoluminescence measurements, with the goal of developing an optical dosimeter.
Speaker: Duarte Esteves (IST)
• 11:50 12:00
Signatures of Quantum Chaos in Many Body Systems 10m
Quantum Chaos studies how quantum chaotic dynamics emerges from a classical chaotic system when the action can no longer be considered much larger than ℏ. In a seminal work, Peres considered a quantity, now called Loschmidt echo, as a measure of sensibility and reversibility of quantum evolution. For quantum chaotic dynamics the decay of the Loschmidt echo with time can be related to the Lyapunov exponent of the underlying classical system. This relation has mostly been explored for systems containing a few degrees of freedom, typically a single particle, where the semi-classical limit is well defined. However, much less is known about the many-body case, where many degrees of freedom strongly interact. In this project, we propose to study signatures of Quantum Chaos in systems containing a few degrees of freedom with a simple semi-classical analog obtained by taking the thermodynamic limit. Perhaps the simplest examples of this class are collective spin models. However, such systems correspond to a single classical degree of freedom and thus lack a classical chaotic regime. A simple generalization which does exhibit chaos is the two coupled collective spins, i.e. SU(2)×SU(2) and for this reason this system is the central subject of this work. The connection between integrable and chaotic behaviour in a mixed dynamics frame is also studied for the spectral statistics of the system.
Speaker: Rodrigo Câmara (IST)
• 12:00 13:30
Lunch 1h 30m
• 13:30 13:40
Use of MHD Activity for Disruption Prediction in Tokamaks 10m
Tokamak plasma disruptions are a threat in achieving a stable controlled nuclear fusion device, thus being very important to consider the development and interconnection between disruption alarm systems and mitigation control systems. In absence of a cohesive theoretical framework based on MHD theory that contains all possible activiy that can precede a disruption, several studies have been applied to tackle this issue. The development of mode locking has been identified as one of the most predictive features. In this work, we intend to go beyond the locked mode to identify other MHD modes that may be relevant for disruption prediction, using Machine Learning and Deep Learning based models and techniques. Additional MHD based features will be relevant to correlate with other phenomena.
Summing up, this work intends to offer a more detailed insight into the processes that cause disruptions in tokamaks, as well as a robust disruption prediction model that can be analysed and potentially be implemented in current and future tokamaks, helping to solve one of the great challenges that are imposed in achieving a stable controlled nuclear fusion device.
Speaker: Tiago Martins (Instituto Superior Técnico)
• 13:40 13:50
The First Sample of Extinction Laws for Supernova Cosmology 10m
Type Ia supernovae are a set of important cosmological objects that can be used as distance indicators. This is because their luminosity can be calibrated by applying some empirical corrections, which then allows us to compute their distance to us. One of these corrections is based on a color-luminosity relation, which is in part due to the effect interstellar dust has on the emitted light. This correction is usually obtained from a fit to a large population of supernovae and is thus assumed to be universal. However, it has been shown that dust properties can vary in the Universe and thus, by assuming a universal dust effect, we are committing a gross generalization, the impact of which is still unknown. The objective of this work is therefore to obtain the individual dust contributions for each element in a group of well known supernovae, which can be done by looking at photometric data for their host galaxies. We can then use these values to obtain new distance calibrations, allowing us to evaluate the impact of assuming a universal dust effect.
Speaker: João Duarte
• 13:50 14:00
New Ideas for physics beyond the Standard Model 10m
Even though it is one of the most successful theories in all of physics, the Standard Model (SM) of particle physics cannot be a final theory. Some experimental results, ranging from the detection of B meson decays that should involve flavour changing neutral currents, to the detection of neutrino oscillations, seem to point to the existence of new physics beyond the SM. The introduction of very heavy vector-like quarks and the addition of right-handed neutrinos in the framework of Majorana neutrinos are two promising proposals that seem to be able to accommodate some the experimental results.
With these extensions in mind, in the electroweak sector, this work aims to find new symmetries or ansatzes that could constrain the form of the fermion mass matrices, reducing the free parameters of the theory while generating the observed experimental results. The study of the possible implications of these promising extensions on CP violation will be another focus of this work.
Speaker: José Bastos
• 14:00 14:10
Probing the nature of dark matter using stars 10m
In the last decades, the fields of cosmology, theoretical astrophysics and particle physics have come across one of the most enduring problems in physics of modern times: the search for the origin and nature of dark matter particles. Numerous studies have compared and combined in a self-consistent way the most powerful cosmic probes: the cosmic microwave background, galaxy redshift surveys, galaxy cluster number counts, type Ia supernovae, and galaxy peculiar velocities. All the studies have led cosmologists to conclude that we live in a flat accelerating Universe, dominated by cold dark matter and by dark energy. Although dark energy is a relatively new problem in cosmology, the dark matter problem has been around for quite some time, without any plausible solution so far. This project proposes using stars as a new method to study the properties of dark matter, complementing that way the international efforts to solve the dark matter problem.
Speaker: David Fordham (Instituto Superior Tecnico)
• 14:10 14:20
A Monte Carlo based study of the FLASH effect in radiotherapy with protons 10m
According to the World Health Organization, cancer is the second leading cause of death globally, right after cardiovascular diseases. Thus, there is a great interest in the continuous study of this disease, not only aiming at an earlier diagnosis but at more effective treatment options.
This project focuses on radiotherapy, a treatment option commonly considered for tumour treatment in which the tumour cells are targeted with ionizing radiation, damaging the cancer cells and eventually leading to their death. Due to its ionizing effects, radiotherapy may also induce serious toxicity effects in the patients and drastically affect their quality of life. In this context, Flash radiotherapy has shown promising results since multiple pre-clinical studies have shown that the delivery of radiation at significantly greater dose rates than the ones conventionally considered, may lead to a reduction in the toxicity effects induced in the surrounding healthy cells, while maintaining an effective tumour control.
The present work will provide a first overview of my Master’s dissertation topic, which will focus on the implementation of a multi-beam treatment planning for Flash therapy using protons.
Speaker: Filipa Baltazar (IST)
• 14:20 14:30
A Computational Model for Radiotherapy Studies with Proton Mini-Beams 10m
Radiotherapy is a cornerstone of both curative and palliative cancer care. It is estimated that half of all cancer patients will receive radiotherapy during the course of their treatment. However, radiotherapy is severely limited by radiation-induced toxicities. Irradiation of noncancerous “normal” tissues during the course of therapeutic radiation can result in a range of side effects including self- limited acute toxicities, mild chronic symptoms, or severe organ dysfunction. These side effects of the used radiation treatments are the reason that the research for new types of radiation treatments continue to be investigated.
Mini-beam radiotherapy is a new type of radiotherapy that has been presenting very good results in the reduction of the effects of radiation in healthy tissues. This new type of treatment, studied with for both X-ray and proton therapy, uses a combination of spatial fractionation of the dose and millimetric filed sizes. In mini-beam radiotherapy the tumor is both radiated with very high doses and low doses. The main goal of this work is to develop a new computational model for mini-beam radiotherapy and compare the results with previous studies, to help understand how it is possible that parts of the tumor that receive almost no dose show sterilization of the cancer cells and lower ability to multiply and spread, what ultimate leads the tumor to shrink.
Speaker: Cláudia Espinha (IST)
• 14:30 14:40
Polarization patterns in the sky and their influence in astronomical observations 10m
The perfect conditions of an astronomical observation assume a sky free of light contamination. Moonlight sky polarization can also be a source of systematic errors in polarimetric studies and must be considered.
Sun's radiation, reflected by the moon, passes through Earth's atmosphere and the light interacts with her. That leads to polarization patterns. A model to characterize the observed moonlight polarization includes the localization of the observatory, the light wavelength and the composition and density of the atmosphere. Such a model for the moonlight polarization will help plan observations and correct the background moonlight polarization.
In this talk, some of the concepts and objectives in this model's construction will be summarized and explained.
Speaker: Beatriz Pereira
• 14:40 14:50
An analysis of the Portuguese energy storage strategy based on the Choquet multiple criteria preference aggregation model 10m
With the increase of renewable energy generation, and their problems related to output instability, storage systems must be implemented in parallel to account for this effect. For this reason, a model to rank the various available options is developed for the several sectors of the energy storage market, as well as a methodologically similar model for the purpose of strategic energy public policy, with the government as the main decision-maker. Beyond a critical review of the results, a robustness analysis will be performed, in order to ensure that the obtained results are credible and valid, serving as the foundation for future decisions.
Speaker: André Pereira (Instituto Superior Técnico)
• 14:50 15:00
Characterization of ultrashort mid-infrared laser pulses using frequency resolved optical gating 10m
The past decade has witnessed the emergence of ultrashort laser systems in the mid-infrared range (2 − 10 µm). This has brought several new challenges to this field, beginning with the pulse determination. Indeed, to measure an event in time, a shorter one is needed to compare it with, but these ultrashort pulses are the shortest events ever created, measuring at most 100 fs. Pulse retrieval techniques in the near infrared have already been explored for a while now and diagnostic equipment is presently commercially available. This is, however, still not the case for lasers in mid-infrared, mainly due to the lack of market.
The Institute for Plasmas and Nuclear Fusion (IPFN) has recently installed a novel 3 µm laser and it is the goal of this thesis to determine the full temporal characterization of the laser pulses of this new system using the frequency resolved optical gating technique. The work will be conducted in the Laboratory for Intense Lasers (L2I) at Instituto Superior Técnico (IST).
Speaker: Luís Veloso
• 15:00 15:10
Coffee Break
• 15:10 15:20
Magnetoresistive sensors for industrial positioning applications 10m
Almost every industry of the present day requires accurate solutions for metrology and positioning. There are several techniques that can be used, using different devices working under different physical principles. Each having their own strengths and weaknesses, they aim, however, towards the same goal: accurate and reliable positioning with high resolution. The suitability of each device will vary on the requirements of the task (type of environment, minimum resolution, among others), but this also means versatility is of some value as the device can be used in a widespread range of situations. Magnetoresistive sensors combined with magnetic scales to form a magnetic encoder, provide very accurate positioning systems with high resolution. The focus of the work will be optimizing AMR sensors for positioning applications. These sensors offer solutions with low power consumption and low prices.
Speaker: Guilherme Brites (Instituto Superior Técnico)
• 15:20 15:30
Energy transfer pathways in CO Plasma 10m
The main goal of this work is to study 𝐶𝑂 plasmas, and their energy transfer mechanisms. As we will see, these mechanisms are fundamental for the study/understanding of CO2 conversion, which is one of the missing pieces to efficiently produce fuels or other useful chemical components from greenhouse gases. I will state the importance of this topic in more detail, while addressing the advantages of using plasma technology.
Speaker: Rodolfo Simões
• 15:30 15:40
Superradiance in Binaries 10m
One of the most astounding predictions of General Relativity is the existence of Black Holes. They are known to be possible places to probe the existence and nature of Dark Matter. Black Hole binaries are also expected to serve as detectors of this matter, but the lack of solutions to Einstein's field equations makes it hard to study how in fact these systems interact with this matter. We must then find alternative ways of studying this systems. By means of the fluid dynamics - GR connection, we aim at developing toy models that might give us some insight into the behaviour of such complex systems.
Speaker: Diogo Ribeiro
• 15:40 15:50
Some Theorectical Aspects of Multi-Higgs Models 10m
The Standard Model (SM) of Particle Physics describes the fundamental forces in nature. It accounts for all observed subatomic particles and even predicted the existence of the Higgs boson, which was observed for the first time in 2012. However it also has its shortcomings, such as the inability to explain why there are three generations of fermions, the values of fermion masses, including neutrino masses, the baryonic asymmetry in the universe, or the existence of dark matter (DM). The development of Multi-Higgs models intends to address these problems. It consists essentially in an extension of the scalar sector of the Standard Model by N scalars. In this thesis, we study the Dark Matter problem. We start by studying the theoretical aspects of extended scalar sectors. We will then use those results to develop models for DM consistent with the existent experimental constraints.
Speaker: Francisco Vazão
• Friday, 29 January
• 09:50 10:00
Multi-Higgs Doublet Models 10m
In recent years, symmetric models with 3 Higgs doublets (3HDMs) have been studied in order to address shortcomings in the Standard Model. This thesis focuses on the mass spectra predicted by some of these 3HDMs, in cases where the symmetries are softly-broken.
Speaker: Diogo Ivo
• 10:00 10:10
Cryptanalysis for trusted nodes in quantum key distribution 10m
One of the major challenges of quantum key distribution (QKD) is the limited distance at which the communicating parties (Alice and Bob) can be. To mitigate this effect, trusted nodes are established, where the key is reconstructed and resent to more distant locations. But even though these nodes are trusted, they are still open to certain types of attacks, namely to the so-called "side-channel attacks", which can be exploited by quantum hackers.
The goal of this thesis is to calculate the limit of information which can be disclosed to an eavesdropper (Eve) in these trusted nodes while maintaining the key renewal perfectly secure. Moreover, we shall determine the impact of the number of trusted nodes on the key generation rate, assuming an upper limit of information disclosed by each node. We will also consider concrete QKD implementations at Instituto de Telecomunicações, based on optical fiber and on free space.
Speaker: João Bravo (Instituto Superior Técnico, ULisboa)
• 10:10 10:20
Quasi-disorder Effects in Topological Systems 10m
The search and study of topological properties of matter has proved fruitful in recent years in research in materials science and condensed matter physics. Superconductors have long been a focus of interest due to their promising applications. Superconductors with intrinsic topological properties, in particular, have recently attracted theoretical and experimental interest due to phenomena associated with the appearance of surface/edge Majorana modes. One of the question that arises is how it is possible to disturb the exotic phases that have been observed in these materials with non-trivial characteristics, and what further effects may arise from perturbing topological systems.
The main goal of this work is to study quantum topological systems, in particular topological superconductors, and how topological phases and modes are affected in the presence of quasi-disorder.
Speaker: Francisca Madeira
• 10:20 10:30
Family Symmetries and the Flavour Problem 10m
Why are there three families of quarks and leptons and the mass hierarchies and mixing matrices so different for these two types of particles? And why does the gauge sector have only a few parameters while the flavour sector has a much larger set of external parameters? These are still some unanswered questions by the Standard Model, the current model of particle physics. Their solution might be in the introduction of discrete family symmetries. In my master thesis, I will use multiple modular symmetries to construct a high energy theory, which is then broken to a low energy model with a single modular symmetry. This scheme allows multiple moduli fields to acquire different VEV's, leading to the realisation of different mass textures in the charged lepton and neutrino sectors. It is then possible to obtain a realistic mixing matrix and mass hierarchies for the leptons using a much smaller set of free parameters.
Speaker: João Lourenço (Instituto Superior Técnico)
• 10:30 10:40
Confronting MultiHiggs models with experiment 10m
The Higgs particle was predicted in 1964 and discovered at CERN on July 2012, earning Higgs and Englert the 2013 Physics Nobel Prize. This is a spin zero particle (scalar), necessary to give masses to the all other massive particles in the Standard Model of Electroweak interactions. But, there is no fundamental reason why there should be only one such scalar. In this project, one wishes to confront Multi-Higgs models with current experimental data, possibly including models with extended gauge sectors.
Speaker: Ricardo Florentino (Instituto Superior Técnico)
• 10:40 10:50
From Light-Front Wave Functions to Parton Distribution Functions 10m
Parton Distribution Functions have been a fundamental part in the calculation of experimental quantities that involve hadrons. Current understanding comes from experimental fits to data and other techniques such as Lattice QCD. We propose an alternative way of calculating the Parton Distribution Functions directly from the theory via Light-Front Wave Functions, integrated directly from the Bethe-Salpeter Wave Function. The mathematical details and prescriptions are developed for a simple scalar model to be later applied to QCD.
Speaker: Eduardo Ferreira (LIP; IST)
• 10:50 11:00
Coffee Break
Coffee Break
• 11:00 11:10
Development of GPU-Accelerated Trigger Algorithms for the ATLAS Experiment at the LHC 10m
The LHC is the highest energy particle accelerator ever built. The gigantic ATLAS experiment records proton and ion collisions produced by the LHC to study the most fundamental matter particles and the forces between them. A major upgrade, expected for the years 2025-26, will increase the LHC collision rate up to a factor 7 with respect to the nominal values, to allow acquiring a huge amount of data and pushing the limits of our understanding of Nature.
The online event selection system (trigger) is a crucial part of the experiment. It analyses in real time, the 40 MHz event rate, selecting only the potentially interesting collisions for later analysis. After the LHC upgrade, the estimated increase in collision rate, and consequently event size, lead to much longer event reconstruction times, that are not matched by the slower expected growth in computing power at fixed cost. This implies a change in paradigm, increasing parallelism in computer architecture, using concurrency and multithreading and/or hardware accelerators, such as GPUs or FPGAs for handling suitable algorithmic code.
The first ATLAS Trigger GPU prototype was implemented and evaluated in 2015-16 [1]. The LIP Portuguese team was responsible for the calorimeter reconstruction algorithms. The results obtained showed the potential gain but also the limitations of the architecture and implementation done.
The objective of this Master thesis project is to contribute to the development, optimisation and performance studies of the second calorimeter reconstruction Trigger GPU prototype.
The development will be done within the new concurrent ATLAS reconstruction framework AthenaMT, using CUDA and C++ programming languages, in collaboration with researches from CERN and other European institutions involved in this effort.
References:
[1] P. Conde Muíño on behalf of the ATLAS Collaboration, “Multi-threaded algorithms for GPGPU in the ATLAS High Level Trigger”, J. Phys.: Conf. Ser. 898 (2017) 032003.
Speaker: Nuno Fernandes
• 11:10 11:20
Development of a Magnetic Camera for Barcode and QR Magnetic Identification Tag Readout 10m
Bar-codes allow the storage of information along a 1D vector, while QR Codes allow the storage along a 2D Matrix, containing a much higher information density.
Both are very used in industry and trade and normally employ optical reading systems, given that nowadays it is also possible to do the reading using the Smartphone câmara.
We seek to develop a magnetic câmara constiting of an array of magnetic sensors that can do the reading (line-by-line or 2D) of Bar-Codes and QR Codes printed in magnetic ink.
While the optical reading systems are more efficient and already represent a standard industry, the magnetic systems allow encoding and transmission of secret information, having possible applications in industry, business and security systems.
Speaker: Alberto Nicolicea
• 11:20 11:30
Uptake and depth distribution in cells of metal-based complexes for therapeutic applications 10m
TBA
Speaker: Bruno Marques (IST)
• 11:30 11:45
Closing session
Convener: Mr Ilídio Lopes (CENTRA - IST) | 2021-05-09 08:48:39 | {"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": 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": 0.4294110834598541, "perplexity": 1393.110709348594}, "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-2021-21/segments/1620243988961.17/warc/CC-MAIN-20210509062621-20210509092621-00563.warc.gz"} |
http://programmers.stackexchange.com/questions/17355/what-are-some-famous-one-liner-or-two-liner-programs-and-equations/17361 | # What are some famous one-liner or two-liner programs and equations? [closed]
I'm experimenting with a new platform and I'm trying to write a program that deals with strings that are no longer than 60 characters and I'd like to populate the data store with some famous or well-known small chunks of code and equations, since programming and math goes along with the theme of my software. The code can be in any language and the equations from any discipline of mathematics, just so long as they're less than a total of 60 characters in length. I suspect people are gonna break out some brainfuck for this one.
For example,
#include<stdio.h>
int main(){printf ("Hi World\n");return 0;}
60 characters exactly!
Thanks so much for your wisdom!
-
## locked by ChrisF♦May 13 '13 at 19:46
This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here. This question and its answers are frozen and cannot be changed. More info: help center.
## closed as not constructive by gnat, MichaelT, GlenH7, ChrisF♦May 13 '13 at 19:46
As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.
Why was brainfk censored? Can't we be adults and not tell everyone what they can and cannot read? In this context brainfk is not an obscenity. – ChaosPandion Nov 6 '10 at 21:28
I suspect this question will be closed. Try improve it to be more constructive. See: blog.stackoverflow.com/2010/09/good-subjective-bad-subjective – bigown Nov 6 '10 at 23:41
@bigown: This is a good subjective one and is constructive. It's no different than asking for famous quotes. In fact, it's better, because it's asking for famous code/equation "quotes." :-) – Macneil Nov 7 '10 at 0:30
@Macneil:I think the same, but the question is poor, it can be improved. – bigown Nov 7 '10 at 4:22
@bigown: honestly, I can't really see how this question could be any more constructive. Not to doubt you or so, but very genuily asked, could you suggest an improvement to @BeachRunnerJoe? I actually very much enjoyed the answers and learned a lot from them. I'd love to see this question reopen. – Joris Meys Nov 7 '10 at 20:31
The classic C string copy routine is known by fewer and fewer people theses days:
while (*d++ = *s++);
-
yes, very famous...to the veterans! – BeachRunnerJoe Nov 6 '10 at 19:27
While I understand it has "historical" value it's terrible terrible code, so the fact that it's falling in disuse is a good thing =) – Andreas Bonini Nov 6 '10 at 19:34
A C veteran would recognize the pattern immediately. It's idiomatic C. – Barry Brown Nov 6 '10 at 19:47
Always thought this was incredibly cool. – Maulrus Nov 7 '10 at 1:00
I must say, I agree with @Kop. In just a few chars, it shows signifficant flaws of its standard lib and its semantics. One of the most absurd things is strings being 0-terminated instead of length-prefixed (which is safer and makes determining the length of a string O(1)). The second thing is that C doesn't have actual boolean values (which fixes the if (alarm = red) launchNukes();-trap). Dijkstra would consider this code more than harmful. I do agree it is imperative for a C programmer to at least understand this code, but I think it's more important for him to know how to do it better. – back2dos Nov 7 '10 at 13:12
not one line, but I present The World's Last C Bug:
if (status = 1)
LaunchNukes();
-
That's one of those "Oh sh*t!" errors. – the Tin Man Nov 11 '10 at 1:49
it's LaunchNukes(); – hasen Nov 12 '10 at 21:24
if that has been written as: if(GetRadarInfo()=1){...}, we wouldn't get this bug because it doesn't compile. So don't always introduce intermediate variable. – tactoth Jan 27 '11 at 2:45
I see Conway's Game of Life in APL floating around a lot:
An extra bonus is that this will make sure you're handling unicode correctly.
-
ha! that's the first thing I thought of when I saw your code, nice! – BeachRunnerJoe Nov 6 '10 at 19:24
Wow, that's impressive! – FinnNk Nov 6 '10 at 22:43
Explanation: youtube.com/watch?v=a9xAKttWgP4 – J.F. Sebastian Nov 7 '10 at 21:28
And I thought Perl looked like line noise. – the Tin Man Nov 8 '10 at 20:17
@Greg, just wait, APL uses more than the roman and greek alphabets because there weren't enough letters and symbols already; backspace (more properly called "overstrike") is also used because some characters need to be typed on top of other characters. One such was a divide symbol on top of a square, which represented matrix inversion (if unary operator, or multiplication by the inverted matrix if it was used as a binary operator). – Tangurena Nov 10 '10 at 21:32
A modified version of a famous Perl one-liner:
/^.?$|^(..+?)\1+$/
This regular expression matches strings whose length is prime.
The original version is:
/^1?$|^(11+?)\1+$/
which matches strings consisting of a prime number of 1s.
-
Quicksort:
qsort [] = []
qsort (x:xs) = qsort (filter (< x) xs) ++ [x] ++ qsort (filter (>= x) xs)
If the list is empty, the sorted result is the empty list.
If the list starts with the element x, and the rest of the list is xs, then the sorted result is list consisting of the sorted list consisting of all elements in xs less than x concatenated with the element x concatenated with the sorted list of all elements in xs larger than x.
(or in other words - divide in two piles, all less than x and all larger than x, sort them both and create a list with the less-than pile, the element x, and the larger-than pile).
Beats the understandability of the C version quite easily.
-
This is Standard ML? Or Haskell? – Barry Brown Nov 6 '10 at 21:02
Haskell. I like the mindset of the language. – user1249 Nov 7 '10 at 1:00
I like the partitioning alternative qsort (x:xs) = qsort lesser ++ equal ++ qsort greater where (lesser,equal,greater) = part x xs ([],[x],[]) – Kendall Hopkins Nov 8 '10 at 23:40
Is there a version of this that uses a random pivot instead of the head of the list? That would make it closer to C.A.R. Hoare's original. – Macneil Nov 11 '10 at 2:07
Hoare says "The item chosen [as pivot element]... should always be that which occupies the highest-addressed locations of the segment which is to be partitioned. If it is feared that this will have a harmfully non-random result, a randomly chosen item should initially be placed in the highest-addressed locations". So to be true to Hoare, we should work with the last element, not the first. – user1249 Nov 11 '10 at 10:16
1. The Ackerman function. The implementation of the Ackermann-Péter version should fit into 60 chars :)
2. This lovely hexadecimal constant: 0x5f3759df. It is the heart of the most WTFing code I've ever seen: the fast inverse square root.
3. The famous XOR swap.
4. question = /(bb|[^b]{2})/
-
+1 for inverse square root – Macneil Nov 6 '10 at 23:40
@Macneil Argh! I was just thinking of that one. – Mark C Nov 7 '10 at 4:02
When I first figured out the bash forkbomb, I thought it was really sweet.
:(){ :|:& };:
-
Wow, that's just evil! – Macneil Nov 11 '10 at 2:05
Look at all the smilies! You could call this "The Smiley bomb!" – Mark C Nov 11 '10 at 16:28
print "hello world\n";
and its derivations seems to be popular. :-)
-
+1: easily the most 'famous' - deserving or not. – Steve Evers Dec 5 '10 at 8:44
Because you mention equations, this one belongs on your list:
e^{i\pi}+1=0
(Wolfram Alpha rendering: )
-
Yes it does! Good ol' Euler, another good one! – BeachRunnerJoe Nov 6 '10 at 19:30
I remember this as e^{i/pi} = i^2 – Josh K Nov 8 '10 at 23:44
@Josh K: That's because i² == -1, so you can balance the equation by subtracting one from both sides, removing the +1 and changing the =0 to -1 or – Daenyth Nov 9 '10 at 1:08
How to detect even numbers:
x % 2 == 0
-
Or !(x%2) in sane languages. – Christian Mann Nov 7 '10 at 4:11
Or !(x & 1) in languages without optimizing compiler. – J.F. Sebastian Nov 7 '10 at 21:34
@Christian, numbers should not be booleans - too easy to make a mistake. – user1249 Dec 3 '10 at 10:37
import this in Python.
EDIT as comments cannot contain line breaks: For those without a Python interpreter handy, this is the output
Beautiful is better than ugly.
Explicit is better than implicit.
Simple is better than complex.
Complex is better than complicated.
Flat is better than nested.
Sparse is better than dense.
Special cases aren't special enough to break the rules.
Although practicality beats purity.
Errors should never pass silently.
Unless explicitly silenced.
In the face of ambiguity, refuse the temptation to guess.
There should be one-- and preferably only one --obvious way to do it.
Although that way may not be obvious at first unless you're Dutch.
Now is better than never.
Although never is often better than *right* now.
If the implementation is hard to explain, it's a bad idea.
If the implementation is easy to explain, it may be a good idea.
Namespaces are one honking great idea -- let's do more of those!
-
I'm a Python beginner. What would this achieve? – Richard Nov 12 '10 at 21:13
@Richard: Try writing this in the Python interactive interpreter :). – MAK Nov 13 '10 at 5:52
This brightened up my Sunday afternoon :) – Richard Nov 14 '10 at 12:39
@Richard Serious question: If you run this, does it give you a stack overflow? – Mark C Nov 19 '10 at 1:19
Not quite 2 lines but I would say this is quite famous:
void swap(float* x, float* y)
{
float t;
t = *x;
*x = *y;
*y = t;
}
Actually some languages can describe it in one line. Lua comes to mind but there are more.
x, y = y, x
-
definitely famous! – BeachRunnerJoe Nov 6 '10 at 19:17
with ints: a ^= b ^= a ^= b; – JulioC Nov 6 '10 at 21:44
I'm just curious how is this implemented? does it create a temporary table (y, x), then assign x the 1st element and y the 2nd element? – tactoth Nov 8 '10 at 7:58
Also I'm wondering how often do people swap values in real life programing. – tactoth Nov 8 '10 at 7:58
@tactoth - Swapping is commonly used for implementing strongly exception safe assignment in C++. – Kaz Dragon Nov 8 '10 at 9:30
My favorite lambda calculus example is the Y combinator:
Y = λf.(λx.f (x x)) (λx.f (x x))
-
From an exercise in K&R, here is a function that will return how many bits are set in the number given. At 58 characters:
int bits(int n){int b=0;while(n){n=n&(n-1);b++;}return b;}
It takes time proportional to the number of bits set. The "ah ha" part here is that
n = n & (n - 1)
Removes the rightmost set bit from n.
-
Awesome, nice K&R reference! – BeachRunnerJoe Nov 6 '10 at 19:59
### Recursive Pascal's Triangle in One Line (Haskell)
r n=take(n+1)$iterate(\a->zipWith(+)(0:a)$a++[0])[1]
Fifty-two characters, add spaces to taste. Courtesy of "Ephemient" in the comment here.
I thought this was a better example than the cryptic but brief solutions in J and K (though I'm no Haskell user, yet).
-
### Unix Roulette (DANGER!)
[ $[$RANDOM % 6 ] == 0 ] && rm -rf /* || echo Click #Roulette
(That is 62 characters long, so you can remove the comment (would it work that way?) or some non-essential spaces.)
-
Please mark this as dangerous. – Chinmay Kanchi Dec 3 '10 at 11:34
I use zsh and it doesn't work unless s/==/-eq/ :-) – Artem Ice Aug 7 '12 at 9:27
fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
-
Why not fibs = 0 : scanl (+) 0 fibs? – FUZxxl Nov 27 '11 at 22:18
DO 10 I=1.3
This is one of the most expensive bugs in history. This Fortran statement assigns the float value of 1.3 to the variable named DO10I.
The correct code - the header of the loop repeating statements until the statement labeled 10 and the loop variable I accepting values 1, 2, 3:
DO 10 I=1,3
-
Why is it an expensive bug? – Barry Brown Nov 10 '10 at 21:06
This bug was in a subroutine that calculated orbital trajectories for a 1961 Mercury space flight. However, it was caught and fixed before launch, and was therefore not a costly bug. There was a similar bug on a Mariner mission that did cause failure of the mission, though. (source: Expert C Programming, pages 31-32.) – Darel Nov 22 '10 at 20:23
void send(short *to, short *from, int count)
{
int n = (count +7 ) / 8;
switch (count % 8) {
case 0: do { *to = *from++;
case 7: *to = *from++;
case 6: *to = *from++;
case 5: *to = *from++;
case 4: *to = *from++;
case 3: *to = *from++;
case 2: *to = *from++;
case 1: *to = *from++;
} while(--n > 0);
}
}
Tom Duff unrolled a memory-mapped port write into one of the most bizarre C constructs the world has seen.
-
It doesn't fit into 60 characters, but it def is cool. I remember getting chills seeing his name scroll past in the credits to some Pixar movie. – Macneil Nov 12 '10 at 4:48
Anything to do with Hello World comes to mind. You could go with different variations if you plan on storing multiple languages.
For something more non-trivial, there's Fibbonacci.
-
Fibbonacci, nice one! Here's the code... if (k < 2) return k;else return fib(k-1) + fib(k-2); – BeachRunnerJoe Nov 6 '10 at 19:23
@BeachRunnerJoe: You might want to combine that with the conditional operator ;) – back2dos Nov 6 '10 at 19:41
yes indeed! return (k < 2) ? k : fib(k-1) + fib(k-2); – BeachRunnerJoe Nov 6 '10 at 19:45
val (minors, adults) = people.partition(_.age < 18)
The above line of Scala code partitions people (a list of Persons) into two lists based on their respective ages.
It takes the following much of code to do the same thing in Java:
List<Person> minors = new ArrayList<Person>();
for(Person p : people) {
if(p.age < 18) {
} else {
}
}
-
Swapping the values of two variables without using a third variable. This is one of the first things in programming that I was told and thought "Hmm... that's cool"
int a,b;
b=a-b;
a=a-b;
b=a+b;
-
I know you can do this using XORs, but this was my bit of nostalgia for today :) – Jonathon Dec 3 '10 at 10:51
XOR has no problem with overflow. Does this? – Job Sep 3 '11 at 21:04
Black magic from John Carmack
float Q_rsqrt( float number )
{
long i;
float x2, y;
const float threehalfs = 1.5F;
x2 = number * 0.5F;
y = number;
i = * ( long * ) &y; // evil floating point bit level hacking
i = 0x5f3759df - ( i >> 1 ); // what the ****?
y = * ( float * ) &i;
y = y * ( threehalfs - ( x2 * y * y ) ); // 1st iteration
// y = y * ( threehalfs - ( x2 * y * y ) ); // 2nd iteration, this can be removed
return y;
}
-
The largest number that can be represented by 8 Byte (Python)
print '\n'.join("%i Byte = %i Bit = largest number: %i" % (j, j*8, 256**j-1) for j in (1 << i for i in xrange(8)))
-
1. Conditional operator :
minVal = (a < b) ? a : b;
2. Switch case
3. for-each loop [Java]
-
Ohh, the ternary operator is a good one! thanks – BeachRunnerJoe Nov 6 '10 at 19:18
You're welcome :-) – Chankey Pathak Nov 6 '10 at 19:26
Actually, conditional operator is the correct name. An operator is ternary if it takes three arguments. – back2dos Nov 6 '10 at 19:46
@back2dos - Indeed, both C# and JavaScript call this the conditional operator. – ChaosPandion Nov 6 '10 at 19:49
@back2dos - I think this is our problem - I would refer to the apple as "the fruit" in that situation, but I think we're arguing grammar, not programming language syntax, and you are correct that ?: is the conditional operator ;) – grkvlt Nov 20 '10 at 20:29
This Quine from the Jargon File in C:
char*f="char*f=%c%s%c;main(){printf(f,34,f,34,10);}%c";main(){printf(f,34,f,34,10);}
There is also a LISP version there, too, but you can find many others floating around, in pretty much any language you could imaging...
-
euler's identity which links the most beautiful numbers in math universe: 1, 0, e, i and π : e^i(π) + 1 = 0
-
I had a good one and I wrote it down in the margin.
-
Nice one Fermat – Richard Nov 12 '10 at 21:15
Thanks for noticing! – Tim Nov 12 '10 at 21:44
int gcd(int a, int b)
{
while(b>0)
{
int t = a%b;
a=b;
b=t;
}
return a;
}
Probably not famous, but one of my favorites. To most it's not immediately apparent why it works.
-
This is a bit over 60 characters but it really depends on variable naming (so I'm including it!) | 2016-06-27 02:46:50 | {"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": 0.46360865235328674, "perplexity": 3402.2872150358467}, "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-2016-26/segments/1466783395620.56/warc/CC-MAIN-20160624154955-00036-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://jdh.hamkins.org/being-hod-of-a-set-is-invariant-throughout-the-generic-multiverse/?shared=email&msg=fail | # Being HOD-of-a-set is invariant throughout the generic multiverse
$\newcommand\HOD{\text{HOD}}$The axiom $V=\HOD$, introduced by Gödel, asserts that every set is ordinal definable. This axiom has a subtler foundational aspect than might at first be expected. The reason is that the general concept of “object $x$ is definable using parameter $p$” is not in general first-order expressible in set theory; it is of course a second-order property, which makes sense only relative to a truth predicate, and by Tarski’s theorem, we can have no first-order definable truth predicate. Thus, the phrase “definable using ordinal parameters” is not directly meaningful in the first-order language of set theory without further qualification or explanation. Fortunately, however, it is a remarkable fact that when we allow definitions to use arbitrary ordinal parameters, as we do with $\HOD$, then we can in fact make such qualifications in such a way that the axiom becomes first-order expressible in set theory. Specifically, we say officially that $V=\HOD$ holds, if for every set $x$, there is an ordinal $\theta$ with $x\in V_\theta$, for which which $x$ is definable by some formula $\psi(x)$ in the structure $\langle V_\theta,{\in}\rangle$ using ordinal parameters. Since $V_\theta$ is a set, we may freely make reference to first-order truth in $V_\theta$ without requiring any truth predicate in $V$. Certainly any such $x$ as this is also ordinal-definable in $V$, since we may use $\theta$ and the Gödel-code of $\psi$ also as parameters, and note that $x$ is the unique object such that it is in $V_\theta$ and satisfies $\psi$ in $V_\theta$. (Note that inside an $\omega$-nonstandard model of set theory, we may really need to use $\psi$ as a parameter, since it may be nonstandard, and $x$ may not be definable in $V_\theta$ using a meta-theoretically standard natural number; but fortunately, the Gödel code of a formula is an integer, which is still an ordinal, and this issue is the key to the issue.) Conversely, if $x$ is definable in $V$ using formula $\varphi(x,\vec\alpha)$ with ordinal parameters $\vec\alpha$, then it follows by the reflection theorem that $x$ is defined by $\varphi(x,\vec\alpha)$ inside some $V_\theta$. So this formulation of $V=HOD$ is expressible and exactly captures the desired second-order property that every set is ordinal-definable.
Consider next the axiom $V=\HOD(b)$, asserting that every set is definable from ordinal parameters and parameter $b$. Officially, as before, $V=\HOD(b)$ asserts that for every $x$, there is an ordinal $\theta$, formula $\psi$ and ordinals $\vec \alpha<\theta$, such that $x$ is the unique object in $V_\theta$ for which $\langle V_\theta,{\in}\rangle\models\psi(x,\vec\alpha,b)$, and the reflection argument shows again that this way of defining the axiom exactly captures the intended idea.
The axiom I actually want to focus on is $\exists b\,\left( V=\HOD(b)\right)$, asserting that the universe is $\HOD$ of a set. (I assume ZFC in the background theory.) It turns out that this axiom is constant throughout the generic multiverse.
Theorem. The assertion $\exists b\, (V=\HOD(b))$ is forcing invariant.
• If it holds in $V$, then it continues to hold in every set forcing extension of $V$.
• If it holds in $V$, then it holds in every ground of $V$.
Thus, the truth of this axiom is invariant throughout the generic multiverse.
Proof. Suppose that $\text{ZFC}+V=\HOD(b)$, and $V[G]$ is a forcing extension of $V$ by generic filter $G\subset\mathbb{P}\in V$. By the ground-model definability theorem, it follows that $V$ is definable in $V[G]$ from parameter $P(\mathbb{P})^V$. Thus, using this parameter, as well as $b$ and additional ordinal parameters, we can define in $V[G]$ any particular object in $V$. Since this includes all the $\mathbb{P}$-names used to form $V[G]$, it follows that $V[G]=\HOD(b,P(\mathbb{P})^V,G)$, and so $V[G]$ is $\HOD$ of a set, as desired.
Conversely, suppose that $W$ is a ground of $V$, so that $V=W[G]$ for some $W$-generic filter $G\subset\mathbb{P}\in W$, and $V=\HOD(b)$ for some set $b$. Let $\dot b$ be a name for which $\dot b_G=b$. Every object $x\in W$ is definable in $W[G]$ from $b$ and ordinal parameters $\vec\alpha$, so there is some formula $\psi$ for which $x$ is unique such that $\psi(x,b,\vec\alpha)$. Thus, there is some condition $p\in\mathbb{P}$ such that $x$ is unique such that $p\Vdash\psi(\check x,\dot b,\check{\vec\alpha})$. If $\langle p_\beta\mid\beta<|\mathbb{P}|\rangle$ is a fixed enumeration of $\mathbb{P}$ in $W$, then $p=p_\beta$ for some ordinal $\beta$, and we may therefore define $x$ in $W$ using ordinal parameters, along with $\dot b$ and the fixed enumeration of $\mathbb{P}$. So $W$ thinks the universe is $\HOD$ of a set, as desired.
Since the generic multiverse is obtained by iteratively moving to forcing extensions to grounds, and each such movement preserves the axiom, it follows that $\exists b\, (V=\HOD(b))$ is constant throughout the generic multiverse. QED
Theorem. If $V=\HOD(b)$, then there is a forcing extension $V[G]$ in which $V=\HOD$ holds.
Proof. We are working in ZFC. Suppose that $V=\HOD(b)$. We may assume $b$ is a set of ordinals, since such sets can code any given set. Consider the following forcing iteration: first add a Cohen real $c$, and then perform forcing $G$ that codes $c$, $P(\omega)^V$ and $b$ into the GCH pattern at uncountable cardinals, and then perform self-encoding forcing $H$ above that coding, coding also $G$ (see my paper on Set-theoretic geology for further details on self-encoding forcing). In the final model $V[c][G][H]$, therefore, the objects $c$, $b$, $P(\omega)^V$, $G$ and $H$ are all definable without parameters. Since $V\subset V[c][G][H]$ has a closure point at $\omega$, it satisfies the $\omega_1$-approximation and cover properties, and therefore the class $V$ is definable in $V[c][G][H]$ using $P(\omega)^V$ as a parameter. Since this parameter is itself definable without parameters, it follows that $V$ is parameter-free definable in $V[c][G][H]$. Since $b$ is also definable there, it follows that every element of $\HOD(b)^V=V$ is ordinal-definable in $V[c][G][H]$. And since $c$, $G$ and $H$ are also definable without parameters, we have $V[c][G][H]\models V=\HOD$, as desired. QED
Corollary. The following are equivalent.
1. The universe is $\HOD$ of a set: $\exists b\, (V=\HOD(b))$.
2. Somewhere in the generic multiverse, the universe is $\HOD$ of a set.
3. Somewhere in the generic multiverse, the axiom $V=\HOD$ holds.
4. The axiom $V=\HOD$ is forceable.
Proof. This is an immediate consequence of the previous theorems. $1\to 4\to 3\to 2\to 1$. QED
Corollary. The axiom $V=\HOD$, if true, even if true anywhere in the generic multiverse, is a switch.
Proof. A switch is a statement such that both it and its negation are necessarily possible by forcing; that is, in every set forcing extension, one can force the statement to be true and also force it to be false. We can always force $V=\HOD$ to fail, simply by adding a Cohen real. If $V=\HOD$ is true, then by the first theorem, every forcing extension has $V=\HOD(b)$ for some $b$, in which case $V=\HOD$ remains forceable, by the second theorem. QED
## 2 thoughts on “Being HOD-of-a-set is invariant throughout the generic multiverse”
1. The proof begins by assuming that $V\models\sf ZFC$. That stands to reason, of course. But $\mathrm{HOD}(b)$ may be a model of $\sf ZF+\lnot AC$.
In that case the theorems about ground model definability might fail.
• I was working in ZFC, so we have ZFC+V=HOD(b). So this is not an issue; but I have edited to clarify. | 2021-12-02 13:36:36 | {"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": 0.9645107388496399, "perplexity": 181.09970551649096}, "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-2021-49/segments/1637964362219.5/warc/CC-MAIN-20211202114856-20211202144856-00270.warc.gz"} |
https://www.physicsforums.com/threads/can-an-ordered-pair-have-identical-elements.897636/ | # I Can an ordered pair have identical elements?
Tags:
1. Dec 18, 2016
### Stoney Pete
Hi guys,
Here is a wacky question for you:
Suppose you have a simple recursive function f(x)=x. Given the fact that a function f(x)=y can be rewritten as a set of ordered pairs (x, y) with x from the domain of f and y from the range of f, it would seem that the function f(x)=x can be written as the set containing the ordered pair (x, x). But does such an ordered pair, with identical elements, make any sense? After all, order is everything in ordered pairs (and tuples generally), right? But how can one distinguish order when the members of the pair are the same?
I have read somewhere that the Kuratowski definition of an ordered pair (x, y) as {{x}, {x, y}} allows ordered pairs with identical elements, namely as follows: (x, x)= {{x}, {x, x}} = {{x}, {x}} = {{x}}. But how does having the set {{x}} tell us anything about order?
What are your thoughts on this? And do you know of any literature dealing with this issue? Thanks for your answers!
Stoney
2. Dec 18, 2016
### jambaugh
If you eliminate ordered pairs with identical elements you couldn't define coordinates for the points on the line y=x. An ordered pair is specifically *not* a set so yes there's nothing odd about the pair having equal elements. Constructionists are fond of defining all mathematical structures in terms of sets to allow everything to have a common axiomatic footing. But there's no need to worry about such foundations when using mathematics. An ordered pair is an ordered pair. (a,b) = (x,y) if and only if a=x and b=y. Nothing more need be said except in specific applications.
3. Dec 18, 2016
### Stoney Pete
I am kinda interested in axiomatic foundations, so for me the set-theoretic construction of ordered n-tuples is important...
When you say an ordered pair is specifically not a set, then that is in a sense wrong, since an ordered pair can be defined in set-theoretic terms (e.g. the Kuratowski definition mentioned in my first post).
And when you mention the linear function x=y you are missing my point. That linear function indeed gives a set of ordered pairs {(0,0), (1,1), (-1,-1),..., (n, n), (-n,-n)} where the elements in each pair are in a sense identical, but at the same time they specify different geometric values (the first one on the x axis, the second on the y axis). I mean a function where the input is in all respects identical to the output.
Also, I am not specifically talking about numerical functions but about functions in a more abstract logical manner as mappings from one set to another, no matter what is in the set. For example, causation can be seen as function mapping a set of causes to a set of effects. Now take the old philosophical idea of self-causation, where a thing (e.g. God) causes its own existence. You then have a function where input and output are identical. It is for such cases that I wonder whether the notion of ordered pairs still makes sense. I guess this is not so much mathematics but concerns rather formal logic of metaphysics.... Nevertheless, it specifies a clear question in set theory and the theory of functions: can the elements of an ordered pair (or any n-tuple for that matter) be identical in all respects?
4. Dec 18, 2016
### Stoney Pete
For a more mathematical example of what I have in mind, consider the function f:N→N where domain and range are the same set N. This function is a set including ordered pairs such as (1, 1), (2, 2) etc. How can we speak of ordered pairs in such cases, where there really is just one element mentioned twice?
5. Dec 18, 2016
### jambaugh
I was not mentioning a function I was mentioning a geometric object, a line, and its coordinate representation, a set of ordered pairs.
I am not sure what is causing you confusion. Remember that the ordered pair is a collection, and not the objects in the collection. If you prefer you may think of the "slots" in the pair as representing a pair of variables. You can have distinct variables x and y that may happen to have equal values (x = 3 and y= 3 at the same time).
I would avoid thinking in terms of functions explicitly as the ordered pair concept is more primitive (as in gets defined first).
Another point: in your comment you seemed to imply you were thinking of the order in the ordered pair as having to do with some intrinsic ordering of the elements. This is not the case. The pair ordering is not a property of the entries, the entries are properties of the pair object. (one is the property of "first value", the other is the value of the "second" property.)
A pair is just a list of length 2 (with the natural generalization of triples, quadruples, ... n-tuples.) The ordering is simply a matter of the pair NOT being a two element set.
Where you ask:
The answer is that we can mention the same object twice, "potato, potato". And a pair is simply the mathematical semantics of making two references to an object or objects. The two references may refer to the same object or different objects because there's no proscription denying that possibility in the definition of the pair.
Its just like the words we use in alphabetical written language. There's no problem allowing letters to repeat or occur more than once in a word. Think of an ordered pair as a "two letter" mathematical "word". mm?
Once you understand this intent in the definition of an ordered pair then you can choose your favorite way to axiomatically construct it from other objects, be they sets, or categories, or functions, or groups.
6. Dec 19, 2016
### Stephen Tashi
Pity you! $\$ But I see your point - you aren't in doubt about the intuitive meaning of "ordered pair".
However, you aren't asking a precise question. Your question asks how a particular set "tells us anything about order". This implies you have a definition of "order" in mind that the set should tell us something about. So we don't have a precise question until you offer a definition of what it means to "tell something about order".
If we take for granted that the natural numbers are defined or that ordinal numbers are defined, you could ask how the Kuratowski definition answers some question involving those mathematical concepts - i.e. Does the Kuratowski definition tell us if an element is the "first" element of an ordered pair ? However, what definition of order are we using to create the definitions of the natural numbers and ordinal numbers? - e.g. if we ask a question about a "first" thing, what definition of "first" are we using in our question?
The Kuratowski definition you quoted doesn't mention the terms "first member of the ordered pair " and "second member of the ordered pair", so it's fair to say the Kuratowski definition tells us nothing about the meaning of those terms. The game of axiomatics is to begin with certain mathematical concepts and to use those concepts to define more mathematical concepts. (It's a game that can be played in more than one way.) We want our precisely constructed creations to match our intuitive ("Platonic") notions of familiar mathematical objects. So, from the point of view of axiomatics the relevant question is not whether the Kuratowski definition contains within it the definitions of "first" and "second". The relevant question is whether we can use the Kuratowski definition and other already-defined concepts to construct a definition of "first member " and "second member" that matches our intuitive idea of those concepts.
Can we do that? ( I don't know what Kuratowski did, but I think we can accomplish that task.)
7. Dec 19, 2016
### FactChecker
The mapping:
(x,y) ↔ {{x}, {x,y}} if x ≠ y
(x,x) ↔ {{x}}
Seems well defined and usable for defining ordered pairs. I don't immediately see why one would want to do that, but that might just show my lack of imagination.
8. Dec 19, 2016
### Stephen Tashi
From an axiomatic point of view, we need the definition of "ordered pair" before we can construct the usual definition for a "mapping" as a "set of ordered pairs such that ...".
9. Dec 19, 2016
### FactChecker
Right. I guess I should have said it is an association between two alternative (hopefully equivalent) definitions.
10. Dec 19, 2016
### Stephen Tashi
An interesting technicality in Kuratowski definition of ordered pair is the question of how much human perception of notation is allowed to play a role. If we phrase the definition in the form:
"The ordered pair (a,b) is defined to be the set {{a},{a,b}}"
then we have assumed human perception distinguishes "(a,b)" from "(b,a)" and thus we assume there is a perceived property of order ( left-to-right) that is utilized in making the definition, but not explicitly explained by the definition itself.
From the purely axiomatic point of view, the definition of an ordered pair would be better written in the form like:
"P is an ordered pair" means that ....
so that no undefined notion of "first" or "leftmost" would be assumed.
I wonder how Kuratowski wrote the definition in his original papers.
11. Dec 22, 2016
### Logical Dog
yes it can. :)
in a function, the ordered pair definition usually means that the element on the left is of the domain, and that on the right is the co domain.
that is why you see usually a function as an ordered pair defined as
N x R or similar, it is said to be a binary relation
the cartesian plane R^2 is an example of an ordered pairing where same elements exist,
functions as you said, and even relations
also as far as I know an ordered pair is an element of a set, and thus, set properties and operations on it apply
the number of elements inside the ordered pair also reflect the number of sets which have been combined IN ORDER to produce it.
So {(a, b , c, d) | a in X, b in Y, c in Z, d in Q}.
the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b.
ordered pairs are also used accodring to book of proof by R HAMMACK. to define relations such as < > =.
So the = relation on the cartesian plane is the set of all ordered pairs where (a,a). google reflexive relation.
Last edited: Dec 22, 2016
12. Dec 22, 2016
### FactChecker
You are correct. Kuratowski's definition is valid. It's just that, as you said, {{a}, {a,b}} = {{a}, {b,a}} represents (a,b) and {{a}, {a,a}} = {{a}} represents (a,a).
13. Dec 22, 2016
### Logical Dog
sorry, I meant the ordered pair (a,b) that is an element of some larger set of ordered pairs, meaning the set of ordered pairs is a set, and (a,b) its element. BUT thats what you understood.
14. Dec 22, 2016
### Logical Dog
IF YOU are having trouble composing ordered pairs, it is easier to LIST out all elements horizontally, and LIST out the other sets elements Vertically, and combine them....ALSO for a function the domain is always the horizontal axis, the co domain the vertical.
{1, 2, 3} x { a, b , 1}
-----1 2 3
a
b
1
edit: HOLD on..I dont understand how (x, x)= {{x}, {x, x}} = {{x}, {x}} = {{x}}..??
for the set {{x}, {x, x}} you get this ordered pair (assuming the set is "mutliplied" by itself):
(x,x), (x,x,x) and other ugly stuff? but i agree that {x} x {x} = (x,x)
are you taking the power set of (x, x)? bUT (X,x) IS NOT a set. it is a list that is part of a set containing other lists.
power set of {(x,x)} = {{}, {(x,x)}}
Last edited: Dec 22, 2016
15. Dec 23, 2016
### FactChecker
Sorry, in post 12, I was thinking that you were the OP. See the discussion in post 1. This is just his original definition followed by the basic set property that repeated elements can be ignored.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook
Have something to add?
Draft saved Draft deleted | 2017-02-27 07:28:43 | {"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": 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": 0.7305675745010376, "perplexity": 625.3045714427076}, "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-2017-09/segments/1487501172649.58/warc/CC-MAIN-20170219104612-00530-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://web2.0calc.com/questions/just-a-converson | +0
# Just a converson!
0
37
2
Question- what is $$7.2\overline{111}$$ In radical form?
This is a question cuz I solved for something in dicimal form instead of radical.
Thx
off-topic
Apr 26, 2020
#1
+21017
+1
"Simplest radical form" means to take an answer like sqrt(40) and rewrite is as 2·sqrt(10).
7.21111.... is the common fraction 649 / 90 ...
What was the problem?
Apr 26, 2020 | 2020-06-06 02:36:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9749175906181335, "perplexity": 12101.16019730302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348509264.96/warc/CC-MAIN-20200606000537-20200606030537-00245.warc.gz"} |
https://www.tutorialspoint.com/scikit_learn/scikit_learn_lasso.htm | # Scikit Learn - LASSO
## LASSO (Least Absolute Shrinkage and Selection Operator)
LASSO is the regularisation technique that performs L1 regularisation. It modifies the loss function by adding the penalty (shrinkage quantity) equivalent to the summation of the absolute value of coefficients.
$$\displaystyle\sum\limits_{j=1}^m\left(Y_{i}-W_{0}-\displaystyle\sum\limits_{i=1}^nW_{i}X_{ji} \right)^{2}+\alpha\displaystyle\sum\limits_{i=1}^n| W_i|=loss_{-}function+\alpha\displaystyle\sum\limits_{i=1}^n|W_i|$$
sklearn.linear_model. Lasso is a linear model, with an added regularisation term, used to estimate sparse coefficients.
## Parameters
Followings table consist the parameters used by Lasso module −
Sr.No Parameter & Description
1
alpha − float, optional, default = 1.0
Alpha, the constant that multiplies the L1 term, is the tuning parameter that decides how much we want to penalize the model. The default value is 1.0.
2
fit_intercept − Boolean, optional. Default=True
This parameter specifies that a constant (bias or intercept) should be added to the decision function. No intercept will be used in calculation, if it will set to false.
3
tol − float, optional
This parameter represents the tolerance for the optimization. The tol value and updates would be compared and if found updates smaller than tol, the optimization checks the dual gap for optimality and continues until it is smaller than tol.
4
normalize − Boolean, optional, default = False
If this parameter is set to True, the regressor X will be normalized before regression. The normalization will be done by subtracting the mean and dividing it by L2 norm. If fit_intercept = False, this parameter will be ignored.
5
copy_X − Boolean, optional, default = True
By default, it is true which means X will be copied. But if it is set to false, X may be overwritten.
6
max_iter − int, optional
As name suggest, it represents the maximum number of iterations taken for conjugate gradient solvers.
7
precompute − True|False|array-like, default=False
With this parameter we can decide whether to use a precomputed Gram matrix to speed up the calculation or not.
8
warm_start − bool, optional, default = false
With this parameter set to True, we can reuse the solution of the previous call to fit as initialization. If we choose default i.e. false, it will erase the previous solution.
9
random_state − int, RandomState instance or None, optional, default = none
This parameter represents the seed of the pseudo random number generated which is used while shuffling the data. Followings are the options −
• int − In this case, random_state is the seed used by random number generator.
• RandomState instance − In this case, random_state is the random number generator.
• None − In this case, the random number generator is the RandonState instance used by np.random.
10
selection − str, default=‘cyclic’
• Cyclic − The default value is cyclic which means the features will be looping over sequentially by default.
• Random − If we set the selection to random, a random coefficient will be updated every iteration.
## Attributes
Followings table consist the attributes used by Lasso module −
Sr.No Attributes & Description
1
coef_ − array, shape(n_features,) or (n_target, n_features)
This attribute provides the weight vectors.
2
Intercept_ − float | array, shape = (n_targets)
It represents the independent term in decision function.
3
n_iter_ − int or array-like, shape (n_targets)
It gives the number of iterations run by the coordinate descent solver to reach the specified tolerance.
### Implementation Example
Following Python script uses Lasso model which further uses coordinate descent as the algorithm to fit the coefficients −
from sklearn import linear_model
Lreg = linear_model.Lasso(alpha = 0.5)
Lreg.fit([[0,0], [1, 1], [2, 2]], [0, 1, 2])
### Output
Lasso(alpha = 0.5, copy_X = True, fit_intercept = True, max_iter = 1000,
normalize = False, positive = False, precompute = False, random_state = None,
selection = 'cyclic', tol = 0.0001, warm_start = False)
### Example
Now, once fitted, the model can predict new values as follows −
Lreg.predict([[0,1]])
### Output
array([0.75])
### Example
For the above example, we can get the weight vector with the help of following python script −
Lreg.coef_
### Output
array([0.25, 0. ])
### Example
Similarly, we can get the value of intercept with the help of following python script −
Lreg.intercept_
### Output
0.75
### Example
We can get the total number of iterations to get the specified tolerance with the help of following python script −
Lreg.n_iter_
### Output
2
We can change the values of parameters to get the desired output from the model. | 2021-10-16 23:50:10 | {"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": 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": 0.30630624294281006, "perplexity": 2915.914286626665}, "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/1634323585045.2/warc/CC-MAIN-20211016231019-20211017021019-00483.warc.gz"} |
http://ipe7.sourceforge.net/manual/manual_38.html | 8.4 Using Truetype fonts
## 8.4 Using Truetype fonts
To make PDF presentations that are as "fancy" as the PowerPoint presentations of competing speakers one needs to use fancy fonts. It's not hard to find nice fonts, but they are mostly in Truetype (TTF) format. This section explains how to use TTF fonts in Ipe.
Ipe relies on Pdflatex to translate the text source representation into a string of PDF operators and font subsets, that can then be used to generate Postscript, PDF, and to display the text on the screen. Ipe can therefore use any font that Pdflatex can handle, and to use a TTF font we just have to add it to Pdflatex's font reportoire.
I've made a webpage explaining the steps necessary to add a TTF font to Pdftex's font repertoire, using the lhandw.ttf font as an example. Let's assume that you have performed these steps, and that you can access the font when running Pdflatex normally (not from Ipe).
We are then ready to try the font from within Ipe. Let's first assume you only want to use the new font in a few places in your Ipe document. You should define a command analogous to `\textrm` to switch to the new font. Open the Document properties dialog in the Edit menu, and add this line to the Latex preamble:
```\DeclareTextFontCommand{\textlh}
{\fontencoding{T1}\fontfamily{lhandw}\selectfont}
```
You can now use `\textlh` inside Ipe text objects to typeset in Lucida-Handwriting.
Finally, let's make a multi-page presentation typeset wholly using Lucida-Handwriting. This declaration in the Latex preamble will change the document fonts:
```\renewcommand{\encodingdefault}{T1}
\renewcommand{\rmdefault}{lhandw}
\renewcommand{\sfdefault}{phv}
\renewcommand{\ttdefault}{pcr}
```
Note that this switches all text fonts to TTF or Postscript fonts. This is necessary, as we use the `T1` encoding (an 8-bit encoding) for Lucida-Handwriting. Keeping Computer-Modern as the font for `\textsf` or `\texttt` would cause LaTeX to load the `T1` version of Computer-Modern. These are bitmapped "Type3" fonts, which Ipe cannot handle. | 2013-05-23 12:01:31 | {"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": 0.9632043838500977, "perplexity": 3166.083852155007}, "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/1368703298047/warc/CC-MAIN-20130516112138-00044-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=2006_AMC_10B_Problems/Problem_6&diff=next&oldid=6723 | # Difference between revisions of "2006 AMC 10B Problems/Problem 6"
## Problem
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $\frac{2}{\pi}$, as shown. What is the perimeter of this region?
$\mathrm{(A) \ } \frac{4}{\pi}\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } \frac{8}{\pi}\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } \frac{16}{\pi}$
## Solution
Since the side of the square is the diameter of the semicircle, the radius of the semicircle is $\frac{1}{2}\cdot\frac{2}{\pi}=\frac{1}{\pi}$
Since the length of one of the semicircular arcs is half the circumference of the corresponding circle, the length of one arc is $\frac{1}{2}\cdot2\cdot\pi\cdot\frac{1}{\pi}=1$
Since the desired perimeter is made up of four of these arcs, the perimeter is $4\cdot1=4\Rightarrow D$ | 2021-03-02 05:49:05 | {"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": 0.8460594415664673, "perplexity": 231.06602419184728}, "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-2021-10/segments/1614178363217.42/warc/CC-MAIN-20210302034236-20210302064236-00534.warc.gz"} |
https://kb.osu.edu/dspace/handle/1811/18487 | # HIGH RESOLUTION INFRARED SPECTRA OF THE $C=O$ STRETCH FUNDAMENTAL AND OVERTONE BANDS OF ACETALDEHYDE
Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/18487
Files Size Format View
1993-MH-03.jpg 62.80Kb JPEG image
Title: HIGH RESOLUTION INFRARED SPECTRA OF THE $C=O$ STRETCH FUNDAMENTAL AND OVERTONE BANDS OF ACETALDEHYDE Creators: Andrews, Anne M.; Tretyakov, M. Yu.; Belov, S. P.; Fraser, G. T.; Pate, Brooks H. Issue Date: 1993 Publisher: Ohio State University Abstract: The $C=O$ stretching fundamental $(\nu_{4})$ of acetaldehyde $(CH_{3}CHO)$ at $1745 cm^{-1}$ has been measured with a tunable infrared diode laser spectrometer coupled to a slit-jet nozzle source. The spectrum does not appear to be severely fractionated and assignments are currently being attempted. The overtone $2\nu_{4}$ was recorded between $3470 cm^{-1}$ and $3490 cm^{-1}$ with a color-center laser coupled to the electric-resonance optothermal spectrometer. Preliminary double-resonance assignments have been made and minimal fractionation from IVR has been observed. Description: Author Institution: Molecular Physics Division, National Institute of Standards and Technology; Microwave Spectroscopy Laboratory, Institute of Applied Physics, Russian Academy of Science URI: http://hdl.handle.net/1811/18487 Other Identifiers: 1993-MH-3 | 2015-08-03 19:18:14 | {"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": 0.3419954776763916, "perplexity": 9796.585993135086}, "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-2015-32/segments/1438042990112.92/warc/CC-MAIN-20150728002310-00002-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.vasp.at/wiki/index.php?title=IALGO&diff=9847&oldid=9825 | All requests for technical support from the VASP group must be addressed to: vasp.materialphysik@univie.ac.at
# Difference between revisions of "IALGO"
IALGO = -1 | 2-4 | 5-8 | 15-18 | 28 | 38 | 44-48 | 53-58
Default: IALGO = 8 for VASP.4.4 and older = 38 else (if ALGO is not set)
Description: IALGO selects the algorithm used to optimize the orbitals.
Mind: We strongly urge the users to select the algorithms via ALGO. Algorithms other than those available via ALGO are subject to instabilities.
Optimize each band iteratively using a conjugate gradient algorithm. Subspace-diagonalization before conjugate gradient algorithm. The conjugate gradient algorithm is used to optimize the eigenvalue of each band.
• IALGO=5 steepest descent
• IALGO=7 preconditioned steepest descent
IALGO=8 (VASP-releases older than VASP.4.5) is always fastest, IALGO=5-7 are only implemented for test purposes.
Please mind, that IALGO=8 is not supported as of VASP.4.5, since M. Teter, Corning and M. Payne hold a patent on this algorithm.
Subspace-diagonalization after iterative refinement of the eigenvectors using the conjugate gradient algorithm. These switches are retained for compatibility reasons only and should not be used any longer. Generally IALGO=5-8 is preferable. Sub-switches as above.
Subspace-diagonalization before conjugate gradient algorithm. No explicit orthonormalization of the gradients to the trial orbitals is done. This setting saves time, but does fail in most cases (mainly included for test purposes). Try IALGO=4X (RMM-DIIS) instead.
## The blocked-Davidson scheme
• IALGO=38: Blocked-Davidson algorithm (ALGO=N).
Kosugi algorithm (special blocked-Davidson iteration scheme). This algorithm is the default in VASP.4.6 and VASP.5.X. It optimizes a subset of NSIM bands simultaneously. The optimized bands are kept orthogonal to all other bands. If problems are encountered with the algorithm, try to decrease NSIM. Such problems are encountered, if linear dependencies develop in the search space. By reducing NSIM the rank of the search space is decreased.
## RMM-DIIS
• IALGO=44-48: Residual minimization method direct inversion in the iterative subspace (ALGO= F)
The RMM-DIIS algorithm reduces the number of orthonormalization steps (${\displaystyle O(N^{3})}$) considerably and is therefore much faster than IALGO=8 and IALGO=38, at least for large systems and for workstations with a small memory band width. For optimal performance, we recommend to use this switch together with LREAL=Auto). The algorithm works in a blocked mode in which several bands are optimized at the same time. This can improve the performance even further on systems with a low memory band width (default is presently NSIM=4).
The following sub-switches exist:
• IALGO=44 steepest descent eigenvalue minimization
• IALGO=46 residuum-minimization + preconditioning
• IALGO=48 preconditioned residuum-minimization (ALGO=F)
IALGO=48 is usually most reliable (IALGO=44 and 46 are mainly for test purposes).
For IALGO=4X, a subspace-diagonalization is performed before the residual vector minimization, and a Gram-Schmidt orthogonalization is employed after the RMM-DIIS step. In the RMM-DIIS step, each band is optimized individually (without the orthogonality constraint); a maximum of NDAV iterative steps per band are performed for each band. The default is NDAV=4, and we we recommend to leave this value unchanged.
Please mind, that the RMM-DIIS algorithm can fail in rare cases, whereas IALGO=38 did not fail for any system tested up to date. Therefore, if you have problems with IALGO=48 try first to switch to IALGO=38.
However, in some cases the performance gains due to IALGO=48 are so significant that IALGO=38 might not be a feasible option. In the following we try to explain what to do if IALGO=48 does not work reliably:
In general two major problems can be encountered when using IALGO=48: First, the optimization of unoccupied bands might fail for molecular dynamics and relaxations. This is because our implementation of the RMM-DIIS algorithm treats unoccupied bands more "sloppy" then occupied bands during MD's. The problem can be solved rather easily by specifying WEIMIN=0 in the INCAR file. In that case all bands are treated accurately.
The other major problem (which occurs also for static calculations) is the initialization of the orbitals. Because the RMM-DIIS algorithm tends to find eigenvectors which are close the the initial set of trial vectors there is no guarantee to converge to the correct ground state! This situation is usually very easy to recognize; whenever one eigenvector is missing in the final solution, the convergence becomes slow at the end (mind, that it is possible that one state with a small fractional occupancy above the Fermi-level is missing). If you suspect that this is the case switch to ICHARG=12 (i.e. no update of charge and Hamiltonian) and try to calculate the orbitals with high accuracy (${\displaystyle 10^{-6}}$). If the convergence is fairly slow or stucks at some precision, the RMM-DIIS algorithm has problems with the initial set of orbitals (as a rule of thumb not more than 12 electronic iterations should be required to determine the orbital for the default precision for ICHARG=12). The first thing to do in that case is to increase the number of bands (NBANDS) in the INCAR file. This is usually the simplest and most efficient fix, but it does not work in all cases. This solution is also undesirable for MD's and long relaxations because it increases the computational demand somewhat. A simple alternative - which worked in all tested cases - is to use IALGO=38 (Davidson) for a few non selfconsistent iterations and to switch then to the RMM-DIIS algorithm. This setup is automatically selected when ALGO= Fast is specified in the INCAR file (IALGO must not specified in the INCAR file in this case).
The final option is somewhat complicated and requires an understanding of how the initialization algorithm of the RMM-DIIS algorithm works: after the random initialization of the orbitals, the initial orbitals for the RMM-DIIS algorithm are determined during a non selfconsistent steepest descent phase (the number of steepest descent sweeps is given by NELMDL, default is NELMDL=-12 for RMM-DIIS). During this initial phase in each sweep, one steepest descent step per orbital is performed between each sub space rotation. This "automatic" simple steepest descent approach during the delay is faced with a rather ill-conditioned minimization problem and can fail to produce reasonable trial orbitals for the RMM-DIIS algorithm. In this case the quantity in the column "rms" will not decrease during the initial phase (12 steps), and you must improve the conditioning of the problem by setting the ENINI parameter in the INCAR file. ENINI controls the cutoff during the initial (steepest descent) phase for IALGO=48. Default for ENINI is ENINI= ENCUT. If convergence problems are observed, start with a slightly smaller ENINI; reduce ENINI in steps of 20%, till the norm of the residual vector (column "rms") decreases continuously during the first 12 steps.
A final note concerns the mixing: IALGO=48 dislikes too abrupt mixing. Since the RMM-DIIS algorithm always stays in the space spanned by the initial orbitals, and too strong mixing (large AMIX, small BMIX) might require to change the Hilbert space, the initial mixing must not be too strong for IALGO=48. Try to reduce AMIX and increase BMIX if you suspect such a situation. Increasing NBANDS also helps in this situation.
## Direct optimization
• IALGO=53-58: Treat total free energy as variational quantity and minimize the functional completely selfconsistently.[1][2][3]
These algorithms have been carefully optimized and should be selected for Hartree-Fock type as well as META-gradient functionals. The present version is rather stable and robust even for metallic systems.
Important sub-switches:
• IALGO=53 damped MD with damping term automatically determined by the given time-step (ALGO=D).
• IALGO=54 damped MD (velocity quenched or quickmin)
• IALGO=58 preconditioned conjugated gradient (ALGO=A)
Furthermore, LSUBROT determines whether the subspace rotation matrix (rotation matrix in the space spanned by the occupied and unoccupied orbitals) is optimized. The current default is LSUBROT=.FALSE. This allows for efficient groundstate calculations for insulators. When hybrid functionals are used, LSUBROT=.TRUE. can be tried for small gap semiconductors and metals. This algorithm performs standard SCF steps during the direct optimization steps in order to determine an optimal rotation matrix between occupied and unoccupied orbitals. For hybrid functionals, LSUBROT=.TRUE. is generally faster, however, in rare cases, it can lead to instabilities.[4].
The preconditioned conjugate gradient (IALGO= 58, ALGO= A) algorithm is recommended for insulators. The best stability is usually obtained if the number of bands equals half the number of electrons (non spin polarized case). In this case, the algorithm is fairly robust and fool proof and might even outperform the mixing algorithm.
For small gap systems and for metals, it is however usually required (metals) or desirable (semiconductors) to use a larger value for NBANDS. In this case, we recommend to use the damped MD algorithm (IALGO=53, ALGO=Damped) instead of the conjugate gradient one.
The stability of the all bands simultaneously algorithms depends strongly on the setting of TIME. For the conjugate gradient case, TIME controls the step size in the trial step, which is required in order to perform a line minimization of the energy along the gradient (or conjugated gradient). Too small steps make the line minimization less accurate, whereas too large steps can cause instabilities. The step size is usually automatically scaled by the actual step size minimizing the total energy along the gradient (values can range from 1.0 for insulators to 0.01 for metals with a large density of states at the Fermi-level).
For the damped MD algorithm (IALGO=53, ALGO=Damped), a sensible TIME step is even more important. In this case TIME is not automatically adjusted, and the user is entirely responsible to chose an appropriate value. Too small time-steps slow the convergence significantly, whereas too large values will always lead to divergence. It is sensible to optimize this value, in particular, if many different configurations are considered for a particular system. It is recommended to start with a small step size TIME, and to increase TIME by a factor 1.2 until the calculations diverge. The largest stable step TIME should then be used for all calculations.
The final algorithm IALGO=54 also uses a damped molecular dynamics algorithm and quenches the velocities to zero if they are antiparallel to the present forces (quick-min). It is usually not as efficient as IALGO=53, but it is also less sensitive to the TIME parameter.
Mind: it is very important to set an appropriate TIME for these algorithms. Furthermore, it might be expedient to set NELM to 1 or 2 for molecular dynamics simulations or relaxations in vasp.6. See the corresponding section in the documentation of NELM. If the ions move by a very large distance during relaxations, even NELM=3 can be tried (in particular for HF type Hamiltonians).
## Miscellaneous
• IALGO=-1: Performance test.
VASP does not perform an actual calculation, only some important parts of the program will be executed and the timing for each part is printed out at the end.
• IALGO=2: Orbitals and one-electron energies are kept fixed.
One electron occupancies and electronic density of states (DOS) are, however, recalculated. This option is only useful if a pre-converged WAVECAR file is read. The option allows to run selected post-processing tasks, such as local DOS, or the interface code to Wannier90.
• IALGO=3: Orbitals are kept fixed.
One-electron energies, band structure energies, and the electronic density of states (DOS) are, as well as, the total energy are recalculated for the present Hamiltonian (the one electron occupancies are kept fixed, however). This option is only useful if a pre-converged WAVECAR file is read. The option also allows to run selected post-processing tasks, such as local DOS, or the interface code to Wannier90.
• IALGO=4: Orbitals are updated by applying a sub-space rotation.
The Hamiltonian is evaluated in the space spanned by the orbitals (read from WAVECAR), and one diagonalization in this space is performed. No optimization outside the subspace spanned by the orbitals is performed. Note: if NBANDS is larger or equal to the total number of plane waves, the resulting one-electron orbitals are exact.
• IALGO=90: Exact Diagonalization. This flag selects an exact diagonalization of the one-electron Hamiltonian. This requires a fairly large amount of memory, and should be selected with caution. Specifically, we recommend to select this algorithm to prepare the WAVECAR for RPA or GW calculations, where many unoccupied orbitals are calculated (more than 30-50 % of the states spanned by the full plane wave basis). To speed up the calculations, we recommend to perform a routine groundstate calculation before calculating the unoccupied states. | 2020-01-18 11:16:48 | {"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": 0.7235260605812073, "perplexity": 1700.5130892389516}, "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/1579250592565.2/warc/CC-MAIN-20200118110141-20200118134141-00557.warc.gz"} |
https://www.calculus-online.com/exercise/3471 | calculus online - Free exercises and solutions to help you succeed!
# Global Extremum – Domain of a curve with absolute value – Exercise 3471
Exercise
Find the maximum value and the minimum value of the function
$$z(x,y)=x^2-xy+y^2$$
In the domain
$$D=\{ (x,y): |x|+|y|\leq 1\}$$
Final Answer
$$\max_D z(0,\pm 1)=\max_D z(\pm 1,0) =1$$
$$\min_D z(0,0) =0$$
Solution
Coming soon…
Share with Friends | 2019-11-15 05:25:54 | {"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": 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": 4, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25647053122520447, "perplexity": 1944.6593514119484}, "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-2019-47/segments/1573496668585.12/warc/CC-MAIN-20191115042541-20191115070541-00549.warc.gz"} |
http://www.r-bloggers.com/exploratory-data-analysis-2-ways-of-plotting-empirical-cumulative-distribution-functions-in-r/ | # Exploratory Data Analysis: 2 Ways of Plotting Empirical Cumulative Distribution Functions in R
(This article was first published on The Chemical Statistician » R programming, and kindly contributed to R-bloggers)
#### Introduction
Continuing my recent series on exploratory data analysis (EDA), and following up on the last post on the conceptual foundations of empirical cumulative distribution functions (CDFs), this post shows how to plot them in R. (Previous posts in this series on EDA include descriptive statistics, box plots, kernel density estimation, and violin plots.)
I will plot empirical CDFs in 2 ways:
1. using the built-in ecdf() and plot() functions in R
2. calculating and plotting the cumulative probabilities against the ordered data
Continuing from the previous posts in this series on EDA, I will use the “Ozone” data from the built-in “airquality” data set in R. Recall that this data set has missing values, and, just as before, this problem needs to be addressed when constructing plots of the empirical CDFs.
Recall the plot of the empirical CDF of random standard normal numbers in my earlier post on the conceptual foundations of empirical CDFs. That plot will be compared to the plots of the empirical CDFs of the ozone data to check if they came from a normal distribution.
#### Method #1: Using the ecdf() and plot() functions
I know of 2 ways to plot the empirical CDF in R. The first way is to use the ecdf() function to generate the values of the empirical CDF and to use the plot() function to plot it. (The plot.ecdf() function combines these 2 steps and directly generates the plot.)
First, let’s get the data and the sample size; note the need to count the number of non-missing values in the “ozone” data vector for the sample size.
### get data and calculate key summary statistics
# extract "Ozone" data vector for New York
ozone = airquality\$Ozone
# calculate the number of non-missing values in "ozone"
n = sum(!is.na(ozone))
Now, let’s use the ecdf() function to obtain the empirical CDF values. You can see what the output looks like below.
# obtain empirical CDF values
ozone.ecdf = ecdf(ozone)
> ozone.ecdf
Empirical CDF
Call: ecdf(ozone)
x[1:67] = 1, 4, 6, ..., 135, 168
Finally, use the plot() function to plot the empirical CDF.
• Note that only one argument – the object created by ecdf() – is needed.
• Also note my use of the mtext() and the expression() functions to add the desired “F-hat-of-x” label. For some strange reason, the same expression used in the ylab option in the plot() function does not show the “hat”. I’m very glad that mtext() shows the “hat”!
• The ylab option in plot() is set as ‘ ‘ to purposefully show nothing. If the ylab option is not specified, $F_n(x)$ will be shown, but this does not have the hat. (Yes, I am doing a lot of work just to add a “hat” to the “F”, but now you get to learn some more R!)
• Notice that “[n]‘ is used to write “n” as a subscript.
### plotting the empirical cumulative distribution function using the ecdf() and plot() functions
# print a PNG image to a desired folder
plot(ozone.ecdf, xlab = 'Sample Quantiles of Ozone', ylab = '', main = 'Empirical Cumluative Distribution\nOzone Pollution in New York')
# the "line" option is used to set the position of the label
# the "side" option specifies the left side
mtext(text = expression(hat(F)[n](x)), side = 2, line = 2.5)
dev.off()
# you can create the plot directly with just the plot.ecdf() function, but this doesn't produce any empirical CDF values
#### Method #2: Plotting the Cumulative Probabilities Against the Ordered Data
There is another way of plotting the empirical CDF that mirrors its definition. It uses R functions to
• calculate the cumulative probabilities
• order the data
• plot the cumulative probabilities against the ordered data.
This method does not use any function specifically created for empirical CDFs; it combines several functions that are more rudimentary in R.
• It plots the empirical CDF as a series of “steps” using the option type = ‘s’ in the plot() function.
• Notice that the vector (1:n)/n is the vector of the cumulative probabilities that are assigned to the data.
• I have also added some vertical and horizontal lines that mark the 3rd quartile; this gives the intution that the CDF increases quickly and that most of the probabilities are already assigned with the small values of the data.
• In case you’re wondering how I got the 3rd quartile, I used the summary() function on the output of the fivenum() function as applied to the ozone data.
summary(fivenum(ozone))
> summary(fivenum(ozone))
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.0 18.0 31.5 56.4 63.5 168.0
### empirical cumulative distribution function using sort() and plot()
# ordering the ozone data
ozone.ordered = sort(ozone)
# plot the possible values of probability (0 to 1) against the ordered ozone data (sample quantiles of ozone)
# notice the option type = 's' for plotting the step functions
plot(ozone.ordered, (1:n)/n, type = 's', ylim = c(0, 1), xlab = 'Sample Quantiles of Ozone', ylab = '', main = 'Empirical Cumluative Distribution\nOzone Pollution in New York')
# mark the 3rd quartile
abline(v = 62.5, h = 0.75)
legend(65, 0.7, '3rd Quartile = 63.5', box.lwd = 0)
# add the label on the y-axis
mtext(text = expression(hat(F)[n](x)), side = 2, line = 2.5)
dev.off()
#### Did the Ozone Data Come from a Normal Distribution?
Recall the empirical CDF plot of the random standard normal numbers from my last post on the conceptual foundations of empirical CDFs.
Comparing this above plot to the plots of the empirical CDFs of the ozone data, it is clear that the latter do not have the “S” shape of the normal CDF. Thus, the ozone data likely did not come from a normal distribution.
Filed under: Applied Statistics, Descriptive Statistics, Plots, R programming Tagged: abline(), airquality, cdf, cumulative distribution function, data, data analysis, ecdf(), empirical cdf, empirical cumulative distribution function, expression(), goodness of fit, legend(), missing data, missing values, mtext(), normal distribution, ozone, plot, plot.ecdf(), plots, plotting, quantile, quantiles, quartile, quartiles, R, R programming, standard normal distribution, statistics, subscript | 2014-09-16 21:34:43 | {"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": 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": 0.47230711579322815, "perplexity": 3050.9084113334516}, "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-2014-41/segments/1410657119965.46/warc/CC-MAIN-20140914011159-00015-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://www.chemicalforums.com/index.php?topic=98025.msg345632 | August 09, 2020, 11:10:18 AM
Forum Rules: Read This Before Posting
### Topic: Aftermath of a wildfire-coins in plastic! (Read 3879 times)
0 Members and 1 Guest are viewing this topic.
#### stickle
• Very New Member
• Posts: 2
• Mole Snacks: +0/-0
##### Aftermath of a wildfire-coins in plastic!
« on: December 09, 2018, 08:10:08 PM »
Everyone,
My in-laws lost their home in the recent Camp Fire in Northern California. In picking through the wreckage we discovered two related problems that we could use your knowledge to solve.
They had silver coins in a "fireproof" safe. The lining of the safe was black plastic. The plastic melted and encased the coins. Is there a solvent that will remove the plastic?
They had some other coins stored in ziplock bags. These melted together with the plastic from the coin tubes. Is there a solvent that will remove this plastic?
Is there a completely different way to go? The black plastic gets soft at around 300 degrees F. but just smears.
Your wisdom would mean a great deal to a couple of elderly fire refugees.
Thank you.
S
#### wildfyr
• Global Moderator
• Sr. Member
• Posts: 1512
• Mole Snacks: +159/-9
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #1 on: December 09, 2018, 09:23:01 PM »
I would try a good soak in paint thinner, aka toluene first.
If that doesn't work, then next I would try dichloromethane which is bought as paint Stripper in some places.
Be careful with both and DO NOT work with in enclosed spaces.
#### Enthalpy
• Chemist
• Sr. Member
• Posts: 3523
• Mole Snacks: +291/-57
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #2 on: December 10, 2018, 05:18:02 PM »
Dissolving a polymer thicker than a paint layer tends to take very long. Nice if it works, but often discouraging. Ultrasound, like in a jewellery cleaner, often helps.
==========
An alternative might be an acetylene+oxygen torch. The flame burns the plastic away quickly, more so with an excess of oxygen. It may melt the coins too, depending on how it's done, but the silver in the coins has often the same value as the coins.
Molten silver is just more difficult to sell than a silver coin. Such a coin is basically a stamping meant to guarantee a silver amount and purity.
If you try the torch:
- Blow pure oxygen once the plastic burns. The flame will be less hot than with acetylene.
- Do that outside! Nearly all burning plastics are toxic for real.
- Maybe air suffices.
==========
Possibly a strong ultraviolet source would degrade the plastic at interesting pace, so it can be broken and brushed away, but I doubt it. Sunlight takes its time, especially if the plastic is black, thick, and resides between the coins. Some sources, like arc lamps, are stronger.
==========
As most chemical methods get slower with the plastic thickness, it wouldn't be bad to first separate the coins with a chisel, hammer (and vice), leaving just thin plastic on them.
After that, just a barbecue fire followed by brushing may suffice. De-oxidising the tarnished silver is standard practice.
==========
Hey chemists, would there be some compound (A bleach on steroids? Ozone? Percarbonate?) that oxidises the plastic away? That should be faster than a dissolution.
Or one that makes the plastic fragile (again an oxidizer?) so it can be hammered and brushed away?
Would a strong base be worth trying? Breaking the polymer chemically must be faster than dissolving it.
« Last Edit: December 10, 2018, 05:29:46 PM by Enthalpy »
#### wildfyr
• Global Moderator
• Sr. Member
• Posts: 1512
• Mole Snacks: +159/-9
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #3 on: December 10, 2018, 06:57:08 PM »
Pirahna solution? Highly dangerous but would fit the bill.
We don't know what polymer it is, and without FTIR or some manufacturer info we probably never will. If its polyester then KOH would be great. If its polypropylene then its useless.
#### Corribus
• Chemist
• Sr. Member
• Posts: 2907
• Mole Snacks: +451/-22
• Gender:
• A lover of spectroscopy and chocolate.
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #4 on: December 11, 2018, 11:59:13 AM »
It's all blind speculation without knowing better what kind of polymer it is. Polypropylene and polyethylene have much different stability to, e.g., UV light. I have no idea what a safe liner would be made from but plastic bags are almost certainly polyethylene, which is unfortunately very difficult to dissolve in anything, at least not without destroying the coin underneath as well. There's a caterpillar that can eat polyethyelene: https://www.nature.com/articles/d41586-017-00593-y
Honestly I think enthalpy's suggestion of controlled burning is probably the best bet. Silver's melting point is about 960 deg. C and manycommon organic polymers decompose at around 400-600 deg. C. If you could control the furnace temperature to be safely below the silver melting point but above the polymer decomposition temperature, you may be able to burn away the polymer while leaving the coins intact. Then clean whatever reside remains with a strong solvent. I wouldn't expect any kind of mint condition coin left over at the end though.
This does seem to defeat the purpose of a fireproof safe.
What men are poets who can speak of Jupiter if he were like a man, but if he is an immense spinning sphere of methane and ammonia must be silent? - Richard P. Feynman
#### Borek
• Mr. pH
• Deity Member
• Posts: 25879
• Mole Snacks: +1692/-401
• Gender:
• I am known to be occasionally wrong.
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #5 on: December 11, 2018, 01:55:25 PM »
A lot depends on how fast you want to get rid of plastic. I agree with Corribus that polyethylene is easier burnt than dissolved, at the same time many years ago I have learned something interesting. I made a serious mistake - I left a polyethylene bottle near our fireplace. It melted, leaving a nasty stain on the grout. I have removed mechanically as much as I could, and a year later there were no traces left - apparently elevated temperature (in a place which can be typically safely touched with a bare hand) was enough to slowly decompose the polyethyelene. It can be impractical to keep something at - say - 120 deg F for several months, but it can work.
ChemBuddy chemical calculators - stoichiometry, pH, concentration, buffer preparation, titrations.info, pH-meter.info
#### stickle
• Very New Member
• Posts: 2
• Mole Snacks: +0/-0
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #6 on: December 12, 2018, 01:47:42 PM »
Thanks everyone for thinking about his problem and giving me some directions to explore.
Tried to post pics...not familiar with the process.
So far...
problem-A large mass of coins (40 lbs.) encased in a monolith of melted plastic. I used dry ice for an hour to make the plastic brittle then bashed it with a 2X4 and broke the monolith up, (is there a technical name for this process?). Most coins still stuck together in units of 2s to 10.
I wanted to start with simple and safe and proceed from there.
Coins melted with clear plastic...
Soaked the worst burnt coins in white vinegar for a week. Nothing.
Soaked others in Acetone. Nothing.
Currently soaking another batch in a stripping product that contains toluene, I see some progress but very slow.
Black plastic coins- In addition to what was done above, I put some coins that were stuck together in an "air fryer" (convection oven kinda thing) that has a temp control. at around 300 F. I could get them apart, but the plastic is still on the face and sides of the coin itself.
So I have time if time is the key variable.
Maybe if I combine the processes...flame/acetone/heat something will happen faster...
I will keep you posted!
#### wildfyr
• Global Moderator
• Sr. Member
• Posts: 1512
• Mole Snacks: +159/-9
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #7 on: December 12, 2018, 02:02:18 PM »
If you can continuously stir the stripping product bath that can speed things up quite a lot. Diffusion is slow.
#### Corribus
• Chemist
• Sr. Member
• Posts: 2907
• Mole Snacks: +451/-22
• Gender:
• A lover of spectroscopy and chocolate.
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #8 on: December 12, 2018, 02:04:28 PM »
300 °F is only 150 °C. This is too low to see polymer decomposition on a reasonable timescale. You may have more luck with like a 500 °F oven. My outdoor ceramic grill with a pizza stone will get to about 750-800 °F if I really blast it with charcoal. If you don't have something like this, you might look around for someone with a kiln for firing clays and see if you can talk them into letting you try to bake your coins at low (for them) temperature. I'd shoot for something like 350 °C (650-675 °F) as a first try. This is where most organics will start to carbonize in a reasonable amount of time. Certainly, I would only try one coin at first to make sure you don't damage the coin, and make sure it cools before you start to clean the residue off because even if you're below the silver melt, it could be soft enough to damage by abrasion or bending. If you do it right, any carbon residue should come right off.
*Actually if you have any chunks of the plastic without the coin embedded, that makes a great 'spot test' material to see if you can get it to decompose.
What men are poets who can speak of Jupiter if he were like a man, but if he is an immense spinning sphere of methane and ammonia must be silent? - Richard P. Feynman
#### Enthalpy
• Chemist
• Sr. Member
• Posts: 3523
• Mole Snacks: +291/-57
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #9 on: December 13, 2018, 03:04:04 PM »
An electric oven with pyrolysis cleaning? Just, don't use the oven for cooking any more after that. Throw it away. And please don't sniff the gases coming out of the oven. Use the oven where an internal fire makes no damage.
Though personally, I'd go the flame method. Hold each coin in the flame with coal tongs. After that, brush the dirt away as possible and clean chemically.
Cleaning apparatus exist for jewelry with ultrasound. It could speed up some chemical actions, including dissolution.
#### Enthalpy
• Chemist
• Sr. Member
• Posts: 3523
• Mole Snacks: +291/-57
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #10 on: December 17, 2018, 09:08:37 AM »
To hold the coins in a hot location, a kitchen sieve of stainless steel outperforms coal tongs. Mind any plastic at the handle.
A heat gun, often used to strip paint, would provide air at a controlled temperature.
https://en.wikipedia.org/wiki/Heat_gun
However, the plastic on the coins will ignite. You might build a small oven with a few loose bricks and blow the hot air in it.
My preferred version presently: melt and dissolve the plastic in a hot liquid. Among transparent liquids, cooking oil (maize, peanut) is a good choice, mineral oil is more a fire hazard, molten paraffin may dissolve some plastics better and is more transparent. Just do it in a pan (no reuse for food!) as for cooking, rely on the odour and fumes to estimate the temperature. The usual oil cooking temperature should be adequate for most plastics.
#### P
• Full Member
• Posts: 639
• Mole Snacks: +64/-15
• Gender:
• I am what I am
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #11 on: January 17, 2019, 05:46:03 AM »
Thanks everyone for thinking about his problem and giving me some directions to explore.
Tried to post pics...not familiar with the process.
So far...
problem-A large mass of coins (40 lbs.) encased in a monolith of melted plastic. I used dry ice for an hour to make the plastic brittle then bashed it with a 2X4 and broke the monolith up, (is there a technical name for this process?). Most coins still stuck together in units of 2s to 10.
I wanted to start with simple and safe and proceed from there.
Coins melted with clear plastic...
Soaked the worst burnt coins in white vinegar for a week. Nothing.
Soaked others in Acetone. Nothing.
Currently soaking another batch in a stripping product that contains toluene, I see some progress but very slow.
Black plastic coins- In addition to what was done above, I put some coins that were stuck together in an "air fryer" (convection oven kinda thing) that has a temp control. at around 300 F. I could get them apart, but the plastic is still on the face and sides of the coin itself.
So I have time if time is the key variable.
Maybe if I combine the processes...flame/acetone/heat something will happen faster...
I will keep you posted!
Do you want to sell the mess? I collect coins and I would find it fun to remove them from the mess. I'll give you the spot price for the silver plus postage plus a small commission. I'll enjoy removing them and cataloguing them and keeping some and selling some on. It is one of my hobbies. :-) PM me if you want to sell them to me. :-)
Regards,
P.
If the toluene is working, albeit slowly, keep it in there. Maybe score up the polymer a bit before soaking to increase surface area. Leave it for a few weeks and it will soften at least and you could then maybe hack the polymer off easier by a mechanical method. Better yet - send them to me! :-)
Tonight I’m going to party like it’s on sale for $19.99! - Apu Nahasapeemapetilon #### P • Full Member • Posts: 639 • Mole Snacks: +64/-15 • Gender: • I am what I am ##### Re: Aftermath of a wildfire-coins in plastic! « Reply #12 on: April 16, 2019, 06:17:33 AM » Do you want to sell the mess? That's a 'No' then is it? I used dry ice for an hour to make the plastic brittle then bashed it with a 2X4 and broke the monolith up You risk scratching the coins - which won't really be a problem unless they are mint/UNC/proofs or in very good condition. I used dry ice for an hour to make the plastic brittle then bashed it with a 2X4 and broke the monolith up, (is there a technical name for this process?). Smashing. I do wonder though - if you are going to sell the coins afterwards then you might want to just sell the block. Tonight I’m going to party like it’s on sale for$19.99!
- Apu Nahasapeemapetilon
#### P
• Full Member
• Posts: 639
• Mole Snacks: +64/-15
• Gender:
• I am what I am
##### Re: Aftermath of a wildfire-coins in plastic!
« Reply #13 on: April 16, 2019, 06:18:43 AM »
Also - I don't think fire is a good idea. You can get unwanted coloured toning on the coins when they take fire/heat damage.
Tonight I’m going to party like it’s on sale for \$19.99!
- Apu Nahasapeemapetilon | 2020-08-09 15:10:18 | {"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": 0.4487484395503998, "perplexity": 5849.434741945361}, "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/1596439738555.33/warc/CC-MAIN-20200809132747-20200809162747-00214.warc.gz"} |
http://mathhelpforum.com/calculus/57604-families-curves-what-b.html | # Thread: Families of curves... what is b?
1. ## Families of curves... what is b?
Find a formula for a curve of the form for with a local maximum at and points of inflection at and .
2. Hello, purplegirl1818!
$\displaystyle \text{Find a formula for a curve of the form: }\:y \:=\:e^{-\frac{(x-a)^2}{b}}\;\text{ for }b > 0$
with a local maximum at $\displaystyle x = 3$ and points of inflection at $\displaystyle x = -1\text{ and }x = 7.$
Max/min points: .$\displaystyle y' \;=\;\left(e^{-\frac{(x-a)^2}{b}}\right)\cdot\left(\text{-}2\,\frac{x-a}{b}\right) \;=\;0 \quad\Rightarrow\quad x \:=\:a$
Since there is a maximum at $\displaystyle x = 3$, then: $\displaystyle \boxed{a\:=\:3}$
$\displaystyle \text{The function is: }\;y \:=\:e^{-\frac{(x-3)^2}{b}}$
$\displaystyle \text{The derivative is: }\;y' \;=\;-\frac{2}{b}\cdot(x-3)\cdot e^{-\frac{(x-3)^2}{b}}$
$\displaystyle \text{The second derivative is: }\;y'' \;=\;-\frac{2}{b}\bigg[(x-3)e^{-\frac{x-3)^2}{b}}\cdot\frac{-2(x-3)}{b} + 1\cdot e^{-\frac{x-3)^2}{b}}\bigg]$
. . $\displaystyle y'' \;=\;-\frac{2}{b}\,e^{-\frac{x-3)^2}{b}}\left[\frac{-2(x-3)^2}{b} + 1\right] \;=\;-\frac{2}{b}\,e^{-\frac{(x-3)^2}{b}}\,\left[\frac{-2(x-3)^2 + b}{b}\right]$
Inflection points: .$\displaystyle -2(x-3)^2 + b \:=\:0 \quad\Rightarrow\quad 2(x-3)^2 \:=\:b \quad\Rightarrow\quad (x-3)^2 \:=\:\frac{b}{2}$
. . . . $\displaystyle x-3 \:=\:\pm\sqrt{\frac{b}{2}} \quad\Rightarrow\quad x \:=\:3\pm\sqrt{\frac{b}{2}}$
Since the inflection points are at $\displaystyle x = 7\text{ and }x = -1$
. . we have: .$\displaystyle \begin{Bmatrix}3 + \sqrt{\frac{b}{2}} \:=\:7\\ \\[-3mm] 3 - \sqrt{\frac{b}{2}} \:=\:\text{-}1 \end{Bmatrix}\quad\Rightarrow\quad \boxed{b \:=\:32}$
$\displaystyle \text{Therefore, the function is: }\;y \;=\;e^{-\frac{(x-3)^2}{32}}$
3. thank you so much, you are my hero lol | 2018-04-20 07:37:51 | {"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": 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": 0.9799174070358276, "perplexity": 1503.8403620196127}, "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-2018-17/segments/1524125937161.15/warc/CC-MAIN-20180420061851-20180420081851-00285.warc.gz"} |
https://brainsanswers.co.uk/mathematics/arrange-the-circles-represented-by-9261584 | , 20.03.2018 18:18 issagirl05
# Arrange the circles (represented by their equations in general form) in ascending order of their radius lengths. tiles x2 + y2 − 2x + 2y − 1 = 0 x2 + y2 − 4x + 4y − 10 = 0 x2 + y2 − 8x − 6y − 20 = 0 4x2 + 4y2 + 16x + 24y − 40 = 0 5x2 + 5y2 − 20x + 30y + 40 = 0 2x2 + 2y2 − 28x − 32y − 8 = 0 x2 + y2 + 12x − 2y − 9 = 0 sequence
### Another question on Mathematics
Mathematics, 05.02.2019 01:34
Will give brainliest! 2017 amc 12a problem 24 quadrilateral $$abcd$$ is inscribed in circle $$o$$ and has side lengths $$ab=3, bc=2, cd=6$$, and $$da=8$$. let $$x$$ and $$y$$ be points on $$\overline{bd}$$ such that $$\frac{dx}{bd} = \frac{1}{4}$$ and $$\frac{by}{bd} = \frac{11}{36}$$. let $$e$$ be the intersection of line $$ax$$ and the line through $$y$$ parallel to $$\overline{ad}$$. let $$f$$ be the intersection of line $$cx$$ and the line through $$e$$ parallel to $$\overline{ac}$$. let $$g$$ be the point on circle $$o$$ other than $$c$$ that lies on line $$cx$$. what is $$xf\cdot xg$$? show all work!
Mathematics, 05.02.2019 00:22
Elijah and his sister went to the movies. they had $34 altogether and spent$9.50 per ticket. elijah and his sister bought the same snacks. write and solve an inequality fotlr the amount that each person spent on snacks. interpret the solition
An internet company charges $8.95 per month for the first 3 months that it hosts your web site. then the company charges$11.95 per month for web hosting. how much money, in dollars, will the company charge for 8 months of web hosting? | 2020-10-01 19:46:51 | {"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": 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": 0.48034238815307617, "perplexity": 359.09315774665833}, "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-2020-40/segments/1600402131986.91/warc/CC-MAIN-20201001174918-20201001204918-00426.warc.gz"} |
https://www.chemeurope.com/en/encyclopedia/Tritium.html | My watch list
my.chemeurope.com
# Tritium
Tritium
Tritium
General
Name, symbol tritium, triton,3H
Neutrons 2
Protons 1
Nuclide data
Natural abundance trace
Half-life 12.32 years
Decay products 3He
Isotope mass 3.0160492 u
Spin 1/2+
Excess energy 14949.794± 0.001 keV
Binding energy 8481.821± 0.004 keV
Decay mode Decay energy
Beta emission 0.018590 MeV
Tritium (symbol T or ³H) is a radioactive isotope of hydrogen. The nucleus of tritium (sometimes called a triton) contains one proton and two neutrons, whereas the nucleus of protium (the most abundant hydrogen isotope) contains no neutrons.
## Decay
Tritium is radioactive with a half-life of 12.32 years. It decays into helium-3 by the reaction
${}^3_1\hbox{H}\to{}^3_2\hbox{He}^++\hbox{e}^-+\overline{\nu}_{\hbox{e}}$
releasing 18.6 keV of energy. The electron has an average kinetic energy of 5.7 keV, while the remaining energy is carried off by the nearly undetectable electron antineutrino. The low-energy beta radiation from tritium cannot penetrate human skin, so tritium is only dangerous if inhaled or ingested. Its low energy also creates difficulty detecting tritium labelled compounds except by using liquid scintillation counting.
## Production
Tritium occurs naturally due to cosmic rays interacting with atmospheric gases. In the most important reaction for natural tritium production, a fast neutron (> 4MeV [1]) interacts with atmospheric nitrogen:
${}^{14}_7\hbox{N}+{}^1\hbox{n}\to{}^{12}_6\hbox{C}+{}^3_1\hbox{H}$
Because of tritium's relatively short half-life, however, tritium produced in this manner does not accumulate over geological timescales, and its natural abundance is negligible.
Tritium is produced in nuclear reactors by neutron activation of lithium-6. This is possible with neutrons of any energy, and is an exothermic reaction yielding 4.8 MeV, which is more than one-quarter of the energy that fusion of the produced triton with a deuteron can later produce.
${}^6_3\hbox{Li}+{}^1\hbox{n}\to{}^4_2\hbox{He}(2.05 MeV)+{}^3_1\hbox{H}(2.75 Mev)$
High-energy neutrons can also produce tritium from lithium-7 in an endothermic reaction, consuming 2.822 MeV. This was discovered when the 1954 Castle Bravo nuclear test produced an unexpectedly high yield.[2]
${}^7_3\hbox{Li}+{}^1\hbox{n}\to{}^4_2\hbox{He}+{}^3_1\hbox{H}+{}^1\hbox{n}$
High-energy neutrons irradiating boron-10 will also occasionally produce tritium. [3]The more common result of boron-10 neutron capture is 7Li and a single alpha particle.[4]
${}^{10}_5\hbox{B}+{}^1\hbox{n}\to{}^4_2\hbox{He}+{}^4_2\hbox{He}+{}^3_1\hbox{H}$
The reactions requiring high neutron energies are not attractive production methods.
Tritium's decay product helium-3 has a very large cross section for the (n,p) reaction with thermal neutrons and is rapidly converted back to tritium in a nuclear reactor.
${}^3_2\hbox{He}+{}^1\hbox{n}\to{}^1_1\hbox{H}+{}^3_1\hbox{H}$
Tritium is occasionally a direct product of nuclear fission, with a yield of about 0.01% (one per 10000 fissions).[5][6] This means that tritium release or recovery needs to be considered in nuclear reprocessing even in ordinary spent nuclear fuel where tritium production was not a goal.
Tritium is also produced in heavy water-moderated reactors when deuterium captures a neutron. This reaction has a very small cross section (which is why heavy water is such a good neutron moderator) and relatively little tritium is produced; nevertheless, cleaning tritium from the moderator may be desirable after several years to reduce the risk of escape to the environment.
According to IEER's 1996 report about the United States Department of Energy, only 225 kg of tritium has been produced in the US since 1955. Since it is continuously decaying into helium-3, the stockpile was estimated as approximately 75 kg at the time of the report.[7]
Tritium for American nuclear weapons was produced in special heavy water reactors at the Savannah River Site until their shutdown in 1988; with the Strategic Arms Reduction Treaty after the end of the Cold War, existing supplies were sufficient for the new, smaller number of nuclear weapons for some time. Production was resumed with irradiation of lithium-containing rods (replacing the usual boron-containing control rods) at the commercial Watts Bar Nuclear Generating Station in 2003-2005 followed by extraction of tritium from the rods at the new Tritium Extraction Facility at SRS starting in November 2006.[8]
## Properties
Tritium has an atomic mass of 3.0160492. It is a gas (T2 or ³H2) at standard temperature and pressure. Tritium combines with oxygen to form a liquid called tritiated water T2O or partially tritiated THO.
Tritium figures prominently in studies of nuclear fusion due to its favorable reaction cross section and the high energy yield of 17.6 MeV for its reaction with deuterium:
${}^3_1\hbox{H}+{}^2_1\hbox{D}\to{}^4_2\hbox{He}+\hbox{n}$
All atomic nuclei, being composed of protons and neutrons, repel one another because of their positive charge. However, if the atoms have a high enough temperature and pressure (as is the case in the core of the Sun, for example), then their random motions can overcome such electrical repulsion (called the Coulomb force), and they can come close enough for the strong nuclear force to take effect, fusing them into heavier atoms. Since tritium has the same charge as ordinary hydrogen, it experiences the same electrostatic repulsive force (see Coulomb's law). However, due to tritium's supply of neutrons which are carried into reactions and feel the attractive strong force once delivered, tritium can more easily fuse with other light atoms. The same is also true, albeit to a lesser extent, of deuterium, and that is why brown dwarfs (so-called failed stars) cannot burn hydrogen, but do indeed burn deuterium.
Before the onset of atmospheric nuclear weapons tests, the global equilibrium tritium inventory was estimated at about 80 megacuries (MCi).
Like hydrogen, tritium is difficult to confine; rubber, plastic, and some kinds of steel are all somewhat permeable. This has raised concerns that if tritium is used in quantity, in particular for fusion reactors, it may contribute to radioactive contamination, although its short half-life should prevent any significant accumulation in the atmosphere.
Atmospheric nuclear testing (prior to the Partial Test Ban Treaty) proved unexpectedly useful to oceanographers, as the sharp spike in surface tritium levels could be used over the years to measure the rate at which the lower and upper ocean levels mixed.
## Regulatory limits
The legal limits for tritium in drinking water can vary. The U.S. limit is calculated to yield a dose of 4 mrem (or 40 microsieverts in SI units) per year.
• United States 740 Bq/L or 20,000 pCi/L (Safe Drinking Water Act)
• World Health Organization 1,000 Bq/L.
## Usage
### Self-powered lighting
The emitted electrons from small amounts of tritium cause phosphors to glow so as to make self-powered lighting devices called trasers which are now used in watches and exit signs. It is also used in certain countries to make glowing keychains, and compasses. In recent years, the same process has been used to make self-illuminating gun sights for firearms. These take the place of radium, which can cause bone cancer. These uses of radium have been banned in most countries for decades.
The aforementioned IEER report claims that the commercial demand for tritium is 400 grams per year.
### Nuclear weapons
Tritium is widely used in nuclear weapons for boosting a fission bomb or the fission primary of a thermonuclear weapon. Before detonation, a few grams of tritium-deuterium gas are injected into the hollow "pit" of fissile plutonium or uranium. The early stages of the fission chain reaction supply enough heat and compression to start DT fusion, then both fission and fusion proceed in parallel, the fission assisting the fusion by continuing heating and compression, and the fusion assisting the fission with highly energetic (14.1 MeV) neutrons. As the fission fuel depletes and also explodes outward, it falls below the density needed to stay critical by itself, but the fusion neutrons make the fission process progress faster and continue longer than it would without boosting. Increased yield comes overwhelmingly from the increase in fission; the energy released by the fusion itself is much smaller because the amount of fusion fuel is much smaller.
Besides increased yield (for the same amount of fission fuel with vs. without boosting) and the possibility of variable yield (by varying the amount of fusion fuel), possibly even more important advantages are allowing the weapon (or primary of a weapon) to have a smaller amount of fissile material (eliminating the risk of predetonation by nearby nuclear explosions) and more relaxed requirements for implosion, allowing a smaller implosion system.
Because the tritium in the warhead is continuously decaying, it is necessary to replenish it periodically. The estimated quantity needed is 4 grams per warhead.[9] To maintain constant inventory, 0.22 grams per warhead per year must be produced.
As tritium quickly decays and is difficult to contain, the much larger secondary charge of a thermonuclear weapon instead uses lithium deuteride as its fusion fuel; during detonation, neutrons split lithium-6 into helium-4 and tritium; the tritium then fuses with deuterium, producing more neutrons. As this process requires a higher temperature for ignition, and produces fewer and less energetic neutrons (only D-D fusion and 7Li splitting are net neutron producers), LiD is not used for boosting, only for secondaries.
For more details on this topic, see nuclear weapon design.
### Controlled nuclear fusion
Tritium is an important fuel for controlled nuclear fusion in both magnetic confinement and inertial confinement fusion reactor designs. The experimental fusion reactor ITER and the National Ignition Facility (NIF) will use Deuterium-Tritium (D-T) fuel. The D-T reaction is favored since it has the largest fusion cross-section (~ 5 barns peak) and reaches this maximum cross-section at the lowest energy (~65 keV center-of-mass) of any potential fusion fuel.
## History
Tritium was first predicted in the late 1920s by Walter Russell, using his "spiral" periodic table[citation needed], then produced in 1934 from deuterium, another isotope of hydrogen, by Ernest Rutherford, working with Mark Oliphant and Paul Harteck. Rutherford was unable to isolate the tritium, a job that was left to Luis Alvarez and Robert Cornog, who correctly deduced that the substance was radioactive. Willard F. Libby discovered that tritium could be used for dating water, and therefore wine.
Tritium was seen in Spider-man 2 when "Doc Ock" used it to (unsuccessfully) sustain a nuclear fusion reaction.
## References
1. ^ An Evaluation of the Neutron and Gamma-ray Production Cross Sections for Nitrgoen, Los Alamos Scientific Laboratory
2. ^ http://www.ieer.org/reports/tritium.html#(11)
3. ^ http://meetings.lle.rochester.edu/Tritium/documents/3.ppt
4. ^ http://nuclearweaponarchive.org/Nwfaq/Nfaq12.html
5. ^ Tritium (Hydrogen-3), Human Health Fact Sheet, Argonne National Laboratory, August 2005
6. ^ Serot, O.; Wagemans, C.; Heyse, J. (2005). "New Results on Helium and Tritium Gas Production From Ternary Fission". INTERNATIONAL CONFERENCE ON NUCLEAR DATA FOR SCIENCE AND TECHNOLOGY. AIP Conference Proceedings 769: 857-860.
7. ^ Tritium: The environmental, health, budgetary, and strategic effects of the Department of Energy's decision to produce tritium, Hisham Zerriffi January, 1996
8. ^ http://www.srs.gov/general/news/factsheets/tef.pdf
9. ^ http://www.ieer.org/reports/tritium.html
Hydrogen-2 Isotopes of Hydrogen Hydrogen-4 Produced from: Hydrogen-4 Decay chain Decays to: Helium-3
be-x-old:Трыт | 2020-02-18 10:48:04 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 7, "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": 0.6740410923957825, "perplexity": 3390.532087411724}, "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/1581875143646.38/warc/CC-MAIN-20200218085715-20200218115715-00432.warc.gz"} |
https://solvedlib.com/n/the-number-of-species-s-of-plants-in-guyana-in-an-area-of,13539294 | The number of species $S$ of plants in Guyana in an area of $A$ hectares can be estimated using the
Question:
The number of species $S$ of plants in Guyana in an area of $A$ hectares can be estimated using the formula $S=88.63 \sqrt[4]{A}$ The Kaieteur National Park in Guyana has an area of $63,000$ hectares. How many species of plants are in the park? Data: Hans ter Steege, "A Perspective on Guyana and its Plant Richness," as found on www.bio.uu.nl
Similar Solved Questions
6. Sketch the curve with the given polar equation by first sketching the graph of r as a function of 0 in Cartesian coordinates 4cos 20
6. Sketch the curve with the given polar equation by first sketching the graph of r as a function of 0 in Cartesian coordinates 4cos 20...
PomOavedA given partial fraction2XCx+D x2 + 1(x-1)(x+4)(x2+1) X-a X+4can be evaluated as:8/857/3513/856/23
pom Oaved A given partial fraction 2X Cx+D x2 + 1 (x-1)(x+4)(x2+1) X-a X+4 can be evaluated as: 8/85 7/35 13/85 6/23...
4) Problems on uncertainty relations. Notation: for any operator O let a-AO) a) Show that the gen...
4) Problems on uncertainty relations. Notation: for any operator O let a-AO) a) Show that the generalized uncertainty principle for the operator and H σ//σ,2 |I What can you deduce about the value of in a stationary state? b) The time dependent Schroedinger equation allows the identific...
Which of the following is a reservoir of transmission for viral upper respiratory infections? Infected animalsInfected humansArthropod vectorInfected birds
Which of the following is a reservoir of transmission for viral upper respiratory infections? Infected animals Infected humans Arthropod vector Infected birds...
1. Given the function; f (x) 16(x3/2).b. Find the quadratic approximating polynomial centered at a
1. Given the function; f (x) 16(x3/2). b. Find the quadratic approximating polynomial centered at a...
How do you simplify and find the excluded values of (x^2 + 5x + 4) / (x^2 - 16)?
How do you simplify and find the excluded values of (x^2 + 5x + 4) / (x^2 - 16)?... | 2023-03-24 12:41:41 | {"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": 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": 0.812004566192627, "perplexity": 1916.2513199433329}, "config": {"markdown_headings": false, "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-2023-14/segments/1679296945282.33/warc/CC-MAIN-20230324113500-20230324143500-00370.warc.gz"} |
https://scriptinghelpers.org/questions/45197/detect-if-a-player-press-a-key-when-the-tool-is-equipped | 1
# Detect if a player press a key when the tool is equipped?
Hello I would like to make a skill for a game only work when the tool is equipped so I wrote the following script for example and it's not working. Any ideas on how to fix it ?
local uis = game:GetService('UserInputService')
local plr = game.Players.LocalPlayer
script.Disabled = false
script.Parent.Equipped:connect(function()
script.Disabled = false
uis.InputBegan:Connect(function(input)
if input.KeyCode == Enum.KeyCode.E then
print('The tool was equipped')
end
end)
end)
script.Parent.Unequipped:connect(function()
script.Disabled = true
uis.InputBegan:Connect(function(input)
if input.KeyCode == Enum.KeyCode.E then
print('The tool was unequipped')
end
end)
end)
1
Why at the beginning do you set the script's Disabled to false? Is it disabled in the first place? Bluemonkey132 194 — 5y
0
Goulstem 8144
5 years ago
Edited 5 years ago
Okay firstly.. you can't mess with the script's Disabled property.
I suggest making an a variable true when you equip, and simply checking it upon key press.
local uis = game:GetService('UserInputService')
local plr = game.Players.LocalPlayer
local eq = false --Equipped variable
script.Parent.Equipped:Connect(function() --'connect' is deprecated
eq = true; --set equipped variable to true
script.Parent.Unequipped:Wait(); --wait for them to unequip
eq = false; --set equipped variable to false
end)
uis.InputBegan:Connect(function(input,process)
if not process then --Make sure they're not chatting
if input.KeyCode == Enum.KeyCode.E then
--check equipped val
print(eq and "The tool is equipped" or "The tool is unequipped");
end
end
end)
0
Thanks ScriptAbyss 10 — 5y | 2022-12-05 15:53:54 | {"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": 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": 0.1969439834356308, "perplexity": 13289.515132088394}, "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/1669446711017.45/warc/CC-MAIN-20221205132617-20221205162617-00428.warc.gz"} |
https://tex.stackexchange.com/questions/475463/animate-an-airplane-in-beamer/475482 | # Animate an airplane in Beamer
For my (beamer) presentation (relating to airplanes) I want an animation of an airplane. I was hoping there would be a way for the airplane to move across the screen (or any other motion will do as well ). For instance, see 09:45 in this video moving airplane in powerpoint The airplane can look like this , feel free to include any image of airplane.
This is all I have right now (a simple image of airplane in beamer template)
\documentclass[ignorenonframetext]{beamer}
\usepackage{fontawesome}
\mode<presentation>
{\usetheme{Singapore}
\setbeamercovered{transparent}
}
\usepackage[english]{babel}
\title{Beamer Example}
\author{Author}
\subject{Presentation Programs}
\institute[ University]{
Department of XZ\\
University}
\begin{document}
\section{Outline}
\frame[label=exampleframe]{
\frametitle{Example}
\faPlane
}
\end{document}
Example using Fontawesome plane (click on the image to see animation):
Update: The Beamer part, to fulfill the request completely.
To become independent from Adobe products, the whole presentation can be made in SVG format (Click to start presentation, F11 for full-screen, navigate with PgUp and PgDown; a Blink-based browser [Chromium, Chrome, Opera] may be needed for the background gradient to be rendered correctly):
Standalone animation LaTeX input:
Update: example extended (softer take-off and landing) and improved (node placing along path, as shown by user @Hafid).
\documentclass{standalone} % animated PDF
%\documentclass[dvisvgm,preview]{standalone} % animated SVG: latex + dvisvgm --font-format=woff --bbox=preview --zoom=-1
%\documentclass[export]{standalone} % multipage PDF
\usepackage{fontawesome}
\usepackage{tikz,animate}
\ExplSyntaxOn
\let\fpEval\fp_eval:n % expandable flt-point calculation with L3
\ExplSyntaxOff
\begin{document}
\begin{animateinline}[
autoplay,controls,
begin={\begin{tikzpicture}[scale=0.85]
\coordinate (a) at (110:15); \coordinate (b) at (70:15);
\node [anchor=north east] at (a) {A}; \node [anchor=north west] at (b) {B};
\useasboundingbox (a) node [anchor=north east] {A} arc [start angle=110,end angle=70,radius=15] (b) node [anchor=north west] {B};},
end={\end{tikzpicture}}
]{24}
\multiframe{161}{iPos=0+1}{%
\path[draw] (a) arc [start angle=110,end angle=70,radius=15] (b) node [pos=\fpEval{0.5*(1-cosd(180*\iPos/160))},sloped,rotate=-45]{\faPlane};
}
\newframe
\multiframe{19}{iAng=45+10}{%
\path[draw] (a) arc [start angle=110,end angle=70,radius=15] (b) node [pos=1,sloped,rotate=-\iAng]{\faPlane};
}
\newframe*
\multiframe{161}{iPos=0+1}{%
\path[draw] (a) arc [start angle=110,end angle=70,radius=15] (b) node [pos=\fpEval{0.5*(1+cosd(180*\iPos/160))},sloped,rotate=135]{\faPlane};
}
\newframe
\multiframe{19}{iAng=135+-10}{%
\path[draw] (a) arc [start angle=110,end angle=70,radius=15] (b) node [pos=0,sloped,rotate=\iAng]{\faPlane};
}
\end{animateinline}
\end{document}
Presentation (beamer-class ) LaTeX input. Compile with
latex presentation.tex
latex presentation.tex
dvisvgm --font-format=woff --bbox=papersize --zoom=-1 -p1,- presentation
\documentclass[dvisvgm,hypertex,aspectratio=169]{beamer}
\usetheme{Singapore}
\usepackage{fontawesome}
\usepackage{tikz,animate}
\ExplSyntaxOn
\let\fpEval\fp_eval:n % expandable flt-point calculation with L3
\ExplSyntaxOff
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PageDown, PageUp key event handling; navigation symbols
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[totpages]{zref}
\usepackage{atbegshi}
\usepackage{fontawesome}
\AtBeginShipout{%
\special{dvisvgm:raw
<defs>
<script type="text/javascript">
<![CDATA[
if(e.key=='PageDown'){
\ifnum\thepage<\ztotpages
document.location.replace('\jobname-\the\numexpr\thepage+1\relax.svg');%
\fi
}else if(e.key=='PageUp'){
\ifnum\thepage>1
document.location.replace('\jobname-\the\numexpr\thepage-1\relax.svg');%
\fi%
}
});
]]>
</script>
</defs>
}%
}%
\AtBeginShipoutUpperLeftForeground{%
\raisebox{-\dimexpr\height+0.5ex\relax}[0pt][0pt]{\makebox[\paperwidth][r]{%
\normalsize\color{structure!40!}%
\ifnum\thepage>1%
\href{\jobname-\the\numexpr\thepage-1\relax.svg}{\faArrowLeft}%
\else%
\textcolor{lightgray}{\faArrowLeft}%
\fi\hspace{0.5ex}%
\ifnum\thepage<\ztotpages%
\href{\jobname-\the\numexpr\thepage+1\relax.svg}{\faArrowRight}%
\else%
\textcolor{lightgray}{\faArrowRight}%
\fi%
\hspace{0.5ex}%
}}%
}%
}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{Fasten Seat Belts}
\subtitle{Use a Web browser and press \framebox{F11}}
\author{AlexG}
\date{\today}
\begin{document}
\frame{\titlepage}
\begin{frame}{Animation}
\begin{center}
\begin{animateinline}[
autoplay,controls,
begin={\begin{tikzpicture}[scale=0.95]
\coordinate (a) at (110:15); \coordinate (b) at (70:15);
\node [anchor=north east] at (a) {A}; \node [anchor=north west] at (b) {B};
\useasboundingbox (a) node [anchor=north east] {A} arc [start angle=110,end angle=70,radius=15] (b) node [anchor=north west] {B};},
end={\end{tikzpicture}}
]{24}
\multiframe{161}{iPos=0+1}{%
\path[draw] (a) arc [start angle=110,end angle=70,radius=15] (b) node [pos=\fpEval{0.5*(1-cosd(180*\iPos/160))},sloped,rotate=-45]{\faPlane};
}
\newframe
\multiframe{19}{iAng=45+10}{%
\path[draw] (a) arc [start angle=110,end angle=70,radius=15] (b) node [pos=1,sloped,rotate=-\iAng]{\faPlane};
}
\newframe*
\multiframe{161}{iPos=0+1}{%
\path[draw] (a) arc [start angle=110,end angle=70,radius=15] (b) node [pos=\fpEval{0.5*(1+cosd(180*\iPos/160))},sloped,rotate=135]{\faPlane};
}
\newframe
\multiframe{19}{iAng=135+-10}{%
\path[draw] (a) arc [start angle=110,end angle=70,radius=15] (b) node [pos=0,sloped,rotate=\iAng]{\faPlane};
}
\end{animateinline}
\end{center}
\end{frame}
\end{document}
• You need Acrobat Reader as PDF viewer. – AlexG Feb 18 at 11:30
• Thanks @AlexG. I'll have to install it then. Wont any other debian friendly PDF Viewer do? – GermanShepherd Feb 18 at 12:22
• @GermanShepherd there are a few pdf viewers that have good emulation of adobe internals on windows each adobe collaborator may have some good features the better ones are bluebeam and foxit and the lightest contender are the tracker products most of those 3 have products that should handle this form of animation well – user170109 Feb 18 at 12:23
• Firefox, Chromium. In the case of SVG output. (As animation in my answer.) – AlexG Feb 18 at 12:24
• @AlexG, Firefox does render the image animation. I 'll still need Acrobat/ some other PDF Viewer for my presentation I believe. I want the plane to fly when the slide is loaded. – GermanShepherd Feb 18 at 12:31
With decorations.markings you can transport the plane along any path and it will always be rotated to be a tangent of the path (without you having to do that manually).
\documentclass[ignorenonframetext]{beamer}
\usepackage{fontawesome}
\mode<presentation>
{\usetheme{Singapore}
\setbeamercovered{transparent}
}
\usepackage[english]{babel}
\usepackage{tikz}
\usetikzlibrary{decorations.markings,calc}
\title{Beamer Example}
\author{Author}
\subject{Presentation Programs}
\institute[ University]{
Department of XZ\\
University}
\newcount\myangle
\begin{document}
\section{Outline}
\begin{frame}[t]
\frametitle{Example}
\transduration{4}
\animate<2-21>
\animatevalue<2-21>{\myangle}{0}{19}
\begin{tikzpicture}
\tikzset{pics/.cd,
plane/.style={code={\fill (-0.6,0.2) -- (-0.5,0) -- (-0.6,-0.2)
-- (-0.4,-0.2) -- (-0.3,-0.1)-- (-0.1,-0.15) -- (-0.2,-0.5) -- (00.05,-0.5)
-- (0.15,-0.2) to[out=0,in=-90] (0.5,0) to[out=90,in=180] (0.15,0.2)
-- (00.05,0.5) -- (-0.2,0.5) -- (-0.1,0.15) -- (-0.3,0.1) -- (-0.4,0.2); }}}
\path[use as bounding box] (-5.5,-4.5) rectangle (2.5,3.5);
\draw[postaction={decorate,decoration={markings,
mark=at position \myangle/20 with {\path let \p1=($(current bounding box.east)-(current bounding box.west)$),
\n1={-atan2(\y1,\x1)} in pic[rotate=\n1]{plane};}}}] (-5,0) to (2,0) arc(90:-180:2)
--++(0,5);
\end{tikzpicture}
\end{frame}
\end{document}
This uses the beamer built-in animation facilities (as in Hafid's answer), but can be combined with \animateinline (see Raaja's answer and AlexG's answer).
The animated gif was created via
convert -density 300 -delay 34 -loop 0 -alpha remove multipage.pdf animated.gif
as explained in this great answer.
Or a 3D like version where the plane flies out of the beamer plane. (Before giving the presentation, please contact the organizers for a safety briefing. ;-)
\documentclass[ignorenonframetext]{beamer}
\mode<presentation>
{\usetheme{Singapore}
\setbeamercovered{transparent}
}
\usepackage[english]{babel}
\usepackage{tikz}
\usetikzlibrary{decorations.markings,calc}
\title{Beamer Example}
\author{Author}
\subject{Presentation Programs}
\institute[ University]{
Department of XZ\\
University}
\newcount\mydist
\begin{document}
\section{Outline}
\begin{frame}[t]
\frametitle{Example}
\transduration{4}
\animate<2-22>
\animatevalue<2-22>{\mydist}{0}{20}
\begin{tikzpicture}
\tikzset{pics/.cd,
plane/.style={code={\fill (-0.6,0.2) -- (-0.5,0) -- (-0.6,-0.2)
-- (-0.4,-0.2) -- (-0.3,-0.1)-- (-0.1,-0.15) -- (-0.2,-0.5) -- (00.05,-0.5)
-- (0.15,-0.2) to[out=0,in=-90] (0.5,0) to[out=90,in=180] (0.15,0.2)
-- (00.05,0.5) -- (-0.2,0.5) -- (-0.1,0.15) -- (-0.3,0.1) -- (-0.4,0.2); }}}
\path[use as bounding box] (-5.25,-4.5) rectangle (2.25,3.5);
\draw[postaction={decorate,decoration={markings,
mark=at position \mydist/20 with {\path let \p1=($(current bounding box.east)-(current bounding box.west)$),
\n1={-atan2(\y1,\x1)} in (0,0)
pic[rotate=\n1,scale={0.3+0.7*sin(9*\mydist)},gray!20]{plane}
(${0.01+0.04*sin(9*\mydist)}*($(current bounding
box.north east)-(current bounding box.south west)$)$)
pic[rotate=\n1,scale={0.3+0.7*sin(9*\mydist)}]{plane};}}}] (-5,0) to (2,0) arc(90:-180:2)
--++(0,5);
\end{tikzpicture}
\end{frame}
\end{document}
• neat as usual and only niggle is the shift of focus at start :-) However my question is out of interest what steps did you use to convert to gif ? – user170109 Feb 18 at 16:31
• @KJO I added that information (and also removed the initial kick;-). – user121799 Feb 18 at 16:36
• +1 more and more if I could – user170109 Feb 18 at 16:40
• Certainly the best solution here. +1 – AlexG Feb 18 at 16:43
• @HafidBoukhoulda "The animated gif was created via convert -density 300 -delay 34 -loop 0 -alpha remove multipage.pdf animated.gif as explained in this great answer". – user121799 Feb 18 at 17:09
Another solution using \animate command provided by the beamer package
\documentclass[ignorenonframetext]{beamer}
\usepackage{fontawesome}
\usepackage{tikz}
\mode<presentation>
{\usetheme{Singapore}
\setbeamercovered{transparent}
}
\usepackage[english]{babel}
\title{Beamer Example}
\author{Author}
\subject{Presentation Programs}
\institute[ University]{
Department of XZ\\
University}
\begin{document}
\section{Outline}
\frame[label=exampleframe]{
\frametitle{Example}
See the plane flying
\newcount\p
\animate<2-10>
\animatevalue<2-10>{\p}{0}{100}
\begin{tikzpicture}
\path(0,0)rectangle(0.75\paperwidth,-0.75\paperheight);
\path[draw](0,0)..controls +(30:2) and +(40:2)..+(4,-1) node [pos=\p/100,sloped,rotate=-45,allow upside down]{\faPlane};
\end{tikzpicture}
}
\end{document}
• It was this method of positioning a node along the path I was looking for. Thanks, +1! – AlexG Feb 19 at 7:53
• @AlexG Very pleased to know that my answer is helpful to you – Hafid Boukhoulda Feb 19 at 8:31
A starting point for your pursuit:
\documentclass[ignorenonframetext]{beamer}
\usepackage{fontawesome}
\mode<presentation>
{\usetheme{Singapore}
\setbeamercovered{transparent}
}
\usepackage[english]{babel}
\title{Beamer Example}
\author{Author}
\subject{Presentation Programs}
\institute[ University]{
Department of XZ\\
University}
%% you need these
\usepackage{tikz}
\usetikzlibrary{positioning, arrows}
\usepackage{animate}
\begin{document}
\section{Outline}
\frame[label=exampleframe]{
\frametitle{Example}
\faPlane
}
\begin{frame}[c]
\begin{center}
\pgfmathtruncatemacro\N{10}
\begin{animateinline}[autoplay]{5}
\multiframe{6}{iPosition=0+1}{
\begin{tikzpicture}
\node[circle,draw=black] (t1) at (0,0) {};
\node (tx) at (\iPosition,0) {\rotatebox{-45}{\faPlane}};
\draw[-] (t1.center) -- (tx.center);
\node[circle,draw=black] (t2) at (5,0) {};
\end{tikzpicture}
}
\end{animateinline}
\end{center}
\end{frame}
\end{document}
PS With @marmot's suggestion:
• Thank you @Raaja. This seems good to me.. I 'll try to play around with this. – GermanShepherd Feb 18 at 12:23
• @Raaja a dirty way for make a .gif is using a screen capture software like Apowersoft – vi pa Feb 18 at 12:32
• @GermanShepherd You are welcome! – Raaja Feb 18 at 12:39
• @Raaja Very nice and good. You airplane is very fast. :-) – Sebastiano Feb 19 at 10:13
• @Sebastiano Thank you. I'll slide you a secret, my TiKzplane possess time-warping capabilities ;-). – Raaja Feb 19 at 10:17
Slightly modified from https://github.com/samcarter/Extravanganza2018/blob/master/paulo/MaryDuck/MaryDuck.tex by @PauloCereda
\documentclass{beamer}
\usepackage{tikzducks}
\usetikzlibrary{calc,decorations.markings}
\setbeamertemplate{background}{%
\begin{tikzpicture}[remember picture,overlay,
decoration={markings, mark=at position \thepage/\insertdocumentendpage with {
\begin{scope}[xscale=-1]
\duck
\fill[orange] (0.7331,0.5229) .. controls (1.8688,-0.6326) and (2.2337,0.0383) .. (1.2819,0.7331) -- cycle;
\fill[brown] (1.3848,1.6771) .. controls (1.2665,2.2823) and (0.5559,2.2697) .. (0.4000,1.6455) .. controls (0.5711,1.6714) and (0.8503,1.6562) .. (0.9926,1.6247) .. controls (0.9703,1.4641) and (1.0307,1.0718) .. (1.1444,1.0104) .. controls (1.3485,0.9002) and (1.4461,1.4498) .. (1.3848,1.6771) -- cycle;
\fill[gray] (0.9153,1.4857) -- (0.9472,1.6278) -- (1.3926,1.5288) -- (1.3840,1.4228) -- cycle;
\fill[gray] (0.6484,1.6773) -- (0.6601,1.7155) -- (0.7558,1.6863) -- (0.7441,1.6480) -- cycle;
\draw[gray,fill=black] (0.83,1.57) circle (0.135);
\draw[gray,fill=black] (0.54,1.65) circle (0.12);
\end{scope}
}}
]
\path[postaction=decorate]
($(current page.north west)+(-1,0)$) to[out=-30,in=90]
(current page.center) to[out=-90,in=180,looseness=6,distance=4cm]
(current page.center) to[out=0,in=160]
(current page.south east);
\end{tikzpicture}%
}
\begin{document}
\begin{frame}
\pause[50]
\end{frame}
\end{document}
• Do ducks need artificial wings? – AlexG Mar 7 at 17:56
• Also, downward loopings are dangerous! – AlexG Mar 7 at 17:58
• I have always been especially fond of a plane duck! – R. Schumacher Mar 7 at 19:03
• @AlexG If you are a duck on a duck plane you can pull off downward loopings, after all you don't need a parachute in case something goes wrong but can just fly yourself :) – user36296 Mar 8 at 11:58
• @R.Schumacher duck plane: i.imgur.com/HzsQI5i.jpg :) – user36296 Mar 8 at 12:00 | 2019-09-22 20:20:40 | {"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": 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": 0.6168773174285889, "perplexity": 11187.55030958737}, "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-2019-39/segments/1568514575674.3/warc/CC-MAIN-20190922201055-20190922223055-00276.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-a-combined-approach-4th-edition/chapter-1-section-1-3-exponents-order-of-operations-and-variable-expressions-exercise-set-page-28/59 | ## Algebra: A Combined Approach (4th Edition)
Evaluate each expression when y = 3: $5y^{2}$ $5(3)^{2}$ 3*3 = 9 9*5 = 45 | 2018-08-16 15:00:44 | {"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": 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": 0.5874997973442078, "perplexity": 2245.940705421876}, "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-2018-34/segments/1534221211000.35/warc/CC-MAIN-20180816132758-20180816152758-00216.warc.gz"} |
https://www.gradesaver.com/textbooks/math/prealgebra/prealgebra-7th-edition/chapter-7-section-7-2-solving-percent-problems-with-equations-practice-page-479/3 | # Chapter 7 - Section 7.2 - Solving Percent Problems with Equations - Practice: 3
X = 90% $\times$ 0.045
#### Work Step by Step
What number means X is means = 90% of 0.045 means 90% $\times$ 0.045 X = 90% $\times$ 0.045
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-07-18 20:54:13 | {"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": 0.6739291548728943, "perplexity": 2470.087355950627}, "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-2018-30/segments/1531676590329.25/warc/CC-MAIN-20180718193656-20180718213656-00515.warc.gz"} |
http://mymathforum.com/probability-statistics/57919-expected-value.html | My Math Forum Expected Value
Probability and Statistics Basic Probability and Statistics Math Forum
June 9th, 2015, 07:39 AM #1 Newbie Joined: May 2015 From: Portland, OR. USA Posts: 8 Thanks: 0 Expected Value Here are two Expected Value problems. Can somebody please check my answers? 1. In American roulette, the wheel has the 38 numbers, 00, 0, 1, 2, ..., 34, 35, and 36, marked on equally spaced slots. If a player bets \$1 on a number and wins, then the player keeps the dollar and receives an additional \$35. Otherwise, the dollar is lost. 36 * (1/38) + -1 * (37/38) ≈ -.03 (The book says -.05. I don't understand what I did wrong) 2. A charity organization is selling \$5 raffle tickets as part of a fund-raising program. The first prize is a trip to Mexico valued at \$3450, and the second prize is a weekend spa package valued at \$750. The remaining 20 prizes are \$25 gas cards. The number of tickets sold is 6000. 3450 * (1/6000) + 750 * (1/6000) + 25 * (20/6000) + /5 (5987/6000) ≈ -4.2 Last edited by greg1313; June 10th, 2015 at 11:30 AM.
June 10th, 2015, 11:20 AM #2
Senior Member
Joined: Feb 2010
Posts: 711
Thanks: 147
Quote:
Originally Posted by statssav Here are two Expected Value problems. Can somebody please check my answers? 1. In American roulette, the wheel has the 38 numbers, 00, 0, 1, 2, ..., 34, 35, and 36, marked on equally spaced slots. If a player bets \$1 on a number and wins, then the player keeps the dollar and receives an additional \$35. Otherwise, the dollar is lost. 36 * (1/38) + -1 * (37/38) ≈ -.03 (The book says -.05. I don't understand what I did wrong)
If the question is "what are your expected winnings" then the answer is
$\displaystyle 35 \cdot \frac{1}{38} + (-1) \cdot \frac{37}{38} = \frac{-2}{38}$.
,
## a charity organization is selling $5 raffle tickets as part , # a charity organization is selling$5 raffle
Click on a term to search for related topics.
Thread Tools Display Modes Linear Mode
Similar Threads Thread Thread Starter Forum Replies Last Post jbergin Probability and Statistics 1 January 2nd, 2015 08:56 AM jbergin Advanced Statistics 0 December 30th, 2014 09:29 AM jbergin Probability and Statistics 1 December 23rd, 2014 08:14 AM kristeen0221 Advanced Statistics 0 February 16th, 2013 11:39 AM pskye Algebra 0 April 4th, 2011 04:50 PM
Contact - Home - Forums - Cryptocurrency Forum - Top | 2019-09-18 09:26:08 | {"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": 0.5379267930984497, "perplexity": 2874.6328900650997}, "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/1568514573264.27/warc/CC-MAIN-20190918085827-20190918111827-00507.warc.gz"} |
https://kr.mathworks.com/help/econ/using-the-kalman-filter-to-estimate-and-forecast-the-diebold-li-model.html | # Apply State-Space Methodology to Analyze Diebold-Li Yield Curve Model
This example shows how to use state-space models (SSM) and the Kalman filter to analyze the Diebold-Li yields-only and yields-macro models [2] of monthly yield-curve time series derived from U.S. Treasury bills and bonds. The analysis includes model estimation, simulation, smoothing, forecasting, and dynamic behavior characterization by applying Econometrics Toolbox™ SSM functionality. The example compares SSM estimation performance to the performance of more traditional econometric estimation techniques.
After the financial crisis of 2008, solvency regulations placed greater emphasis on market valuation and accounting of liabilities. As a result, many financial firms, especially insurance companies and pension funds, write annuity contracts and incur long-term liabilities that call for sophisticated approaches to model and forecast yield curves.
Because the value of long-term liabilities greatly increases with low interest rates, the probability of very low yields must be modeled accurately. The Kalman filter, with its ability to incorporate time-varying coefficients and infer unobserved factors driving the evolution of observed yields, is often appropriate for the estimating of yield curve model parameters and then simulating and forecasting yields, which are integral to insurance and pension analysis.
In this example, you build, fit, and analyze a yield curve model by using Econometrics Toolbox SSM functionality and this workflow:
1. Represent the Diebold-Li yields-only model in parametric state-space model form, as supported by Econometrics Toolbox SSM functionality.
2. For the yields-only model, reproduce the in-sample estimation results published in [2], and compare the results obtained to those of the two-step approach as published in [1].
3. For the yields-only model, compute minimum mean square error (MMSE) forecasts and show the Monte Carlo simulation capabilities of the SSM functionality.
4. Estimate the yields-macro Diebold-Li SSM, which integrates the financial and macroeconomic factors.
5. Compute impulse response functions (IRF) and forecast error variance decompositions (FEVD) in the state-space framework, which characterize the yields-to-macro and macro-to-yields links.
6. Estimate the Diebold-Li SSM augmented with exogenous macroeconomic variables.
### Diebold-Li Yield Curve Model
The Diebold-Li model is a variant of the Nelson-Siegel model [3], reparameterized from the original formulation to contain yields only. For observation date $t$ and time to maturity $\tau$, the Diebold-Li model of yield ${y}_{t}\left(\tau \right)$ is
${y}_{t}\left(\tau \right)={L}_{t}+{S}_{t}\left(\frac{1-{e}^{-\lambda \tau }}{\lambda \tau }\right)+{C}_{t}\left(\frac{1-{e}^{-\lambda \tau }}{\lambda \tau }-{e}^{-\lambda \tau }\right),$
where:
• ${L}_{t}$ is the long-term factor (level).
• ${S}_{t}$ is the short-term factor (slope).
• ${C}_{t}$ is the medium-term factor (curvature).
• $\lambda$ determines the maturity at which the loading on the curvature is maximized and governs the exponential decay rate of the model.
### State-Space Formulation of Diebold-Li Model
#### Econometrics Toolbox SSM Form
The ssm object of the Econometrics Toolbox enables you to specify a linear problem in state-space representation. To perform the following operations on the SSM, pass the you ssm object that represents it to the appropriate function.
For state vector ${x}_{t}$ and observation (response) vector ${y}_{t}$, the parametric form of the Econometrics Toolbox SSM is has the following linear state-space representation:
$\begin{array}{l}{x}_{t}={A}_{t}{x}_{t-1}+{B}_{t}{u}_{t}\\ {y}_{t}={C}_{t}{x}_{t}+{D}_{t}{\epsilon }_{t},\end{array}$
where the vectors ${u}_{t}$ and ${ϵ}_{t}$ are uncorrelated, unit-variance, white noise processes. In the SSM, the first equation is the state equation and the second is the observation equation. The model parameters ${A}_{t}$, ${B}_{t}$, ${C}_{t}$, and ${D}_{t}$ are the state-transition, state-disturbance-loading, measurement-sensitivity, and observation-innovation coefficient matrices, respectively.
Although the SSM functions accommodate time-varying (dynamic) parameters, parameters whose values and dimensions can change with time, in the Diebold-Li model, the parameters are time invariant (static).
#### Diebold-Li Yields-Only SSM
The level, slope, and curvature factors of the Diebold-Li model follow a vector autoregressive process of first order (VAR(1)), which forms a state-space system. Diebold, Rudebusch, and Aruoba [2] compose the state vector with the level, slope, and curvature factors. The resulting state transition equation, which governs the dynamics of the state vector, is
$\left[\begin{array}{c}{L}_{t}-{\mu }_{L}\\ {S}_{t}-{\mu }_{S}\\ {C}_{t}-{\mu }_{C}\end{array}\right]=\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right]\left[\begin{array}{c}{L}_{t-1}-{\mu }_{L}\\ {S}_{t-1}-{\mu }_{S}\\ {C}_{t-1}-{\mu }_{C}\end{array}\right]+\left[\begin{array}{c}{\eta }_{Lt}\\ {\eta }_{St}\\ {\eta }_{Ct}\end{array}\right],$
where ${\mu }_{\mathit{j}}$, $\mathit{j}\in \left\{\mathit{L},\mathit{S},\mathit{C}\right\}$, is the mean of factor $\mathit{j}$. The corresponding observation (measurement) equation is
$\left[\begin{array}{c}{y}_{{\tau }_{1}t}\\ {y}_{{\tau }_{2}t}\\ ⋮\\ {y}_{{\tau }_{N}t}\end{array}\right]=\left[\begin{array}{ccc}1& \frac{1-{e}^{-\lambda {\tau }_{1}}}{\lambda {\tau }_{1}}& \frac{1-{e}^{-\lambda {\tau }_{1}}}{\lambda {\tau }_{1}}-{e}^{-\lambda {\tau }_{1}}\\ 1& \phantom{\rule{0.2777777777777778em}{0ex}}\frac{1-{e}^{-\lambda {\tau }_{2}}}{\lambda {\tau }_{2}}& \frac{1-{e}^{-\lambda {\tau }_{2}}}{\lambda {\tau }_{2}}-{e}^{-\lambda {\tau }_{2}}\\ ⋮& ⋮& ⋮\\ 1& \frac{1-{e}^{-\lambda {\tau }_{N}}}{\lambda {\tau }_{N}}& \frac{1-{e}^{-\lambda {\tau }_{N}}}{\lambda {\tau }_{N}}-{e}^{-\lambda {\tau }_{N}}\end{array}\right]\left[\begin{array}{c}{L}_{t}\\ {S}_{t}\\ {C}_{t}\end{array}\right]+\left[\begin{array}{c}{w}_{{\tau }_{1}t}\\ {w}_{{\tau }_{2}t}\\ ⋮\\ {w}_{{\tau }_{N}t}\end{array}\right].$
The Diebold-Li model has the following matrix representation for the 3-D vector of mean-adjusted factors ${f}_{t}$ and observed yields ${y}_{t}$:
$\begin{array}{c}\left({f}_{t}-\mu \right)=A\left({f}_{t-1}-\mu \right)+{\eta }_{t}\\ {y}_{t}=\Lambda {f}_{t}+{w}_{t}.\end{array}$
The Diebold-Li model imposes the following assumptions on the state-equation factor disturbances ${\eta }_{t}$ and the observation-equation innovations (deviations of observed yields at various maturities) ${w}_{t}$:
• ${\eta }_{t}$ and ${w}_{t}$ are orthogonal, Gaussian, white noise processes, symbolically
$\left[\begin{array}{c}{\eta }_{t}\\ {w}_{t}\end{array}\right]\phantom{\rule{0.5em}{0ex}}\sim WN\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}Q& 0\\ 0& H\end{array}\right]\right).$
• Disturbances ${\eta }_{t}$ are contemporaneously correlated, which implies that their covariance matrix $Q$ is nondiagonal.
• Innovations in ${w}_{t}$ are uncorrelated, which implies the covariance matrix $H$ is diagonal.
Define the latent states ${\mathit{x}}_{\mathit{t}}$ as the mean-adjusted factors
${x}_{t}={f}_{t}-\mu ,$
and define the intercept-adjusted (deflated) yields $\stackrel{\sim }{\text{\hspace{0.17em}}{\mathit{y}}_{\mathit{t}}}$ as
${\underset{}{\overset{\sim }{y}}}_{t}={y}_{t}-\Lambda \mu .$
Substitute ${\mathit{x}}_{\mathit{t}}$ and $\stackrel{\sim }{\text{\hspace{0.17em}}{\mathit{y}}_{\mathit{t}}}$ into the preceding equations, and the resulting Diebold-Li state-space system is
$\begin{array}{l}{x}_{t}=A{x}_{t-1}+{\eta }_{t}\\ {\underset{}{\overset{\sim }{y}}}_{t}=\Lambda {x}_{t}+{w}_{t}.\end{array}$
Compare the Diebold-Li state-space system to the formulation that the Econometrics Toolbox SSM functionality supports, which is
$\begin{array}{l}{x}_{t}={A}_{t}{x}_{t-1}+{B}_{t}{u}_{t}\\ {y}_{t}={C}_{t}{x}_{t}+{D}_{t}{\epsilon }_{t}.\end{array}$
The state-transition coefficient matrix $A$ is the same in both formulations, and Diebold-Li model matrix $\Lambda$ is the same as the measurement-sensitivity coefficient matrix $C$ in the SSM formulation.
The relationship between the disturbance and innovation processes is more subtle. Because ${\eta }_{\mathit{t}}=\mathit{B}{\mathit{u}}_{\mathit{t}}$ and ${\mathit{w}}_{\mathit{t}}=\mathit{D}{\epsilon }_{\mathit{t}}$, the covariances of the random variables must be equal. As a result of the linear transformation property of Gaussian random vectors, the relationship between the covariances of the disturbances and innovations is
$\begin{array}{l}Q=B{B}^{\prime }\\ H=D{D}^{\prime }.\end{array}$
To prepare the Diebold-Li model for Econometrics Toolbox SSM functionality, use the ssm function to specify the SSM; ssm returns an ssm model object representing the model. The ssm function enables you to create a model containing known or unknown parameters; an ssm object containing unknown parameters is a template for estimation. You can either explicitly set the parameters, or you can specify a custom function that implicitly defines the SSM.
To create an SSM explicitly, you must specify all coefficient matrices $\mathit{A}$, $\mathit{B}$, $\mathit{C}$, and $\mathit{D}$. To indicate the presence and placement of unknown parameter values, specify NaN values. Each NaN entry corresponds to a unique parameter to estimate. This approach to model creation is convenient when each parameter affects and is uniquely associated with a single element of a coefficient matrix.
To create an SSM implicitly, you must specify a custom parameter-to-matrix mapping function that maps an input parameter vector to model parameters $\mathit{A}$, $\mathit{B}$, $\mathit{C}$, and $\mathit{D}$. The function content defines the model formulation. This approach to model creation is useful for complex models or for imposing parameter constraints.
An ssm object does not store nonzero offsets of state variables, or any parameters associated with regression components, in the observation equation. To estimate coefficients of a regression component, you must deflate the observations ${y}_{t}$. Similarly, other ssm functions expect deflated, or preprocessed, observations to account for any offsets or a regression component in the observation equation.
Because the Diebold-Li model has the following characteristics that are impossible to specify explicitly, this example creates an ssm object implicitly.
• Each factor in the Diebold-Li model includes a nonzero offset (mean), which represents a regression component.
• The model imposes a symmetry constraint on the covariance matrix $Q=B{B}^{\prime }$ and a diagonality constraint of the covariance matrix $H=D{D}^{\prime }$.
• The model includes a decay rate parameter $\lambda$.
The preceding state-space formulation is not unique. For example, you can include the factor offsets as states in the state equation instead of observation deflators. The advantage of observation deflation is that the dimensionality of the state vector directly corresponds to the 3-D yields-only factor model of Diebold, Rudebusch, and Aruoba [2]. The disadvantage is that the estimation is performed on the deflated yields, and therefore you must account for the adjustment by deflating, and then inflating, the yields when you pass the estimated model to other ssm functions.
The yield data consists of 29 years of monthly, unsmoothed Fama-Bliss U.S. Treasury zero-coupon yield measurements, as analyzed and discussed in [1] and [2]. The time series in the data represents maturities of 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, and 120 months. The yields are expressed in percent and recorded at the end of each month, beginning January 1972 and ending December 2000, for a total of 348 monthly curves of 17 maturities each. For example, the time stamp 31-Jan-1972 corresponds to the beginning of February 1972. You can access the entire unsmoothed Fama-Bliss yield curve data set, a subset of which is analyzed in [1] and [2], at https://www.sas.upenn.edu/~fdiebold/papers/paper49/FBFITTED.txt.
This example uses the entire Diebold-Li data set Data_DieboldLi.mat to reproduce the estimation results published in [2], and it compares the results of the two-step and SSM approaches. Alternatively, you can analyze the forecast accuracy of the models by partitioning the data into in-sample and out-of-sample periods, and then fitting the models to the former set and assessing forecast performance of the estimated models using the latter set. For more details on assessing forecast accuracy of Diebold-Li models, see Tables 4 through 6 in [1].
Load the Diebold-Li data set, and then extract the yield series.
load Data_DieboldLi maturities = maturities(:); % Cast to a column vector Yields = DataTable{:,1:17}; % In-sample yield series for estimation dates = DataTable.Time; % Date stamps
### Estimate Yields-Only Diebold-Li Model Using Two-Step Method
Diebold and Li [1] estimate the parameters of their yield curve model by using a two-step approach:
1. Fix $\lambda$, and then, for each monthly yield curve, estimate the level, slope, and curvature parameters. The result is a 3-D time series of estimates of the unobserved level, slope, and curvature factors.
2. Fit a first-order autoregressive model to the time series of factors derived in the first step.
By fixing $\lambda$, the estimation procedure is ordinary least squares (OLS). Otherwise, the estimation procedure is nonlinear least squares. The Nelson-Siegel framework sets $\lambda$ = 0.0609 [3], which implies that loading on the curvature (medium-term factor) is maximized at 30 months.
Because the yield curve is parameterized as a function of the factors, forecasting the yield curve is equivalent to forecasting the underlying factors, and then evaluating the Diebold-Li model as a function of the factor forecasts.
The first step equates the three factors (level, slope, and curvature) to the regression coefficients obtained by OLS, and it accumulates a 3-D time series of estimated factors by repeating the OLS fit for each observed yield curve.
For each month (row), perform the first step by using OLS to fit the following linear model to the yield curve series.
${\mathit{y}}_{\mathit{j}}=\mathit{L}+\frac{1-{\mathit{e}}^{\lambda {\tau }_{\mathit{j}}}}{\lambda {\tau }_{\mathit{j}}}\mathit{S}+\left(\frac{1-{\mathit{e}}^{\lambda {\tau }_{\mathit{j}}}}{\lambda {\tau }_{\mathit{j}}}-{\mathit{e}}^{\lambda {\tau }_{\mathit{j}}}\right)\mathit{C}$, $\mathit{j}=\left\{3,6,9,...,120\right\}$.
Store the regression coefficients and residuals of the linear model fit.
lambda0 = 0.0609; X = [ones(size(maturities)) (1-exp(-lambda0*maturities))./(lambda0*maturities) ... ((1-exp(-lambda0*maturities))./(lambda0*maturities)-exp(-lambda0*maturities))]; Beta = zeros(size(Yields,1),3); Residuals = zeros(size(Yields,1),numel(maturities)); for i = 1:size(Yields,1) EstMdlOLS = fitlm(X,Yields(i,:)','Intercept',false); Beta(i,:) = EstMdlOLS.Coefficients.Estimate'; Residuals(i,:) = EstMdlOLS.Residuals.Raw'; end
Beta contains the 3-D time series of estimated factors.
Fit a first-order autoregressive (AR) model to the time series of estimated factors. You can accomplish this task in two ways:
• Fit a univariate AR(1) model to each factor separately, as in [1].
• Fit a VAR(1) model to all 3 factors simultaneously, as in [2].
Econometrics Toolbox supports univariate and multivariate AR estimation.
Fit a VAR(1) model to the estimated factors. For consistency with the SSM formulation, which works with the mean-adjusted factors, include an additive constant to account for the mean of each factor.
MdlVAR = varm(3,1); EstMdlVAR = estimate(MdlVAR,Beta);
EstMdlVAR is a varm model object representing the estimated VAR(1) factor model.
### Yields-Only Diebold-Li SSM Estimation
Next, estimate the Diebold-Li model using the implicit approach, in which you create and specify a parameter-to-matrix mapping function.
The mapping function Example_DieboldLi.m maps a parameter vector to SSM parameter matrices, deflates the observations to account for the means of each factor, and imposes constraints on the covariance matrices. For more details, open Example_DieboldLi.m.
Create an SSM by passing, to ssm, the parameter-to-matrix mapping function Example_DieboldLi as an anonymous function with an input argument representing the parameter vector params. The additional input arguments of the mapping function specify the yield and maturity information statically, which initialize the model for estimation.
Mdl = ssm(@(params)Example_DieboldLi(params,Yields,maturities));
Mdl is an ssm model object representing the SSM expressed in Example_DieboldLi. The model is solely a template for estimation.
The maximum likelihood estimation (MLE) of an SSM via the Kalman filter is widely known to be sensitive to the initial parameter values. Therefore, this example uses the results of the two-step approach to initialize the estimation.
You must pass the initial values required by estimate as a column vector. Construct the initial-value vector by performing the following procedure:
1. Specify the initial value for the coefficient matrix $\mathit{A}$ by stacking the estimated 3-by-3 AR coefficient matrix of the VAR(1) model columnwise.
2. For the initial value of the coefficient matrix $\mathit{B}$, the 3-by-3 covariance matrix $\mathit{Q}$ is the VAR(1) model innovations covariance matrix and $Q=B{B}^{\prime }$. Therefore, the estimate of $B$ is the lower Cholesky factor of $Q$. To ensure that $Q$ is symmetric, positive definite, and allows for nonzero off-diagonal covariances, allocate the six elements associated with the lower Cholesky factor of $Q$. In other words, this specification assumes that the covariance matrix $Q$ is nondiagonal, but it reserves space for the below-diagonal elements of the lower Cholesky factor of the covariance matrix so that $Q=B{B}^{\prime }$. Arrange the initial value along and below the main diagonal by stacking the matrix columnwise.
3. Because the covariance matrix $H$ in the Diebold-Li formulation is diagonal and $H=D{D}^{\prime }$, the matrix $D$ of the SSM is diagonal. Specify the initial value of $\mathit{D}$ as the square root of the diagonal elements of the sample covariance matrix of the residuals of the VAR(1) model, one element for each of the 17 maturities in the input yield data. Stack the initial value columnwise.
4. The $C$ matrix is a fully parameterized function of the estimated decay rate parameter $\lambda$. The mapping function computes it directly using $\lambda$, so $\mathit{C}$ does not require an initial value. Set the initial value of $\lambda$ to the traditional value 0.0609; the last element of the initial parameter column vector corresponds to it.
5. For the elements of the initial parameter vector associated with the factor means, set the sample averages of the OLS regression coefficients in the first step of the two-step approach.
6. Stack all initial values in the order ${\mathit{A}}_{0}$, ${\mathit{B}}_{0}$, ${\mathit{D}}_{0}$, ${\mu }_{0}$, and $\lambda$.
A0 = EstMdlVAR.AR{1}; A0 = A0(:); Q0 = EstMdlVAR.Covariance; B0 = [sqrt(Q0(1,1)); 0; 0; sqrt(Q0(2,2)); 0; sqrt(Q0(3,3))]; H0 = cov(Residuals); D0 = sqrt(diag(H0)); mu0 = mean(Beta)'; param0 = [A0; B0; D0; mu0; lambda0];
To facilitate estimation, set optimization options. Estimate the model by passing the ssm model template Mdl, the yield data, the initial values, and the optimization options to estimate. Turn off estimation displays. Because the covariance matrix $H=D{D}^{\prime }$ is diagonal, specify the univariate treatment of a multivariate series to improve the estimation run-time performance. The Kalman filter processes the vector-valued observations one at a time.
options = optimoptions('fminunc','MaxFunEvals',25000,'algorithm','quasi-newton', ... 'TolFun',1e-8,'TolX',1e-8,'MaxIter',1000,'Display','off'); [EstMdlSSM,params] = estimate(Mdl,Yields,param0,'Display','off', ... 'options',options,'Univariate',true); lambda = params(end); % Estimated decay rate mu = params(end-3:end-1)'; % Estimated factor means
EstMdlSSM is an ssm model object representing the estimated Diebold-Li SSM. The estimation procedure applies the Kalman filter.
### Compare Estimation Results
#### Estimated Parameters
Comparing the results of the two-step estimation method and the SSM fit helps you to understand the following characteristics:
• How closely the results of the two approaches agree
• How suitable the two-step estimation results are as initial parameter values for SSM estimation
Visually compare the estimated state-transition matrix $A$ of the SSM with the AR(1) coefficient matrix obtained from the VAR model.
EstMdlSSM.A
ans = 3×3 0.9944 0.0286 -0.0221 -0.0290 0.9391 0.0396 0.0253 0.0229 0.8415
EstMdlVAR.AR{1}
ans = 3×3 0.9901 0.0250 -0.0023 -0.0281 0.9426 0.0287 0.0518 0.0125 0.7881
The estimated coefficients closely agree. The diagonal elements are nearly 1, which indicates persistent self-dynamics of each factor. The off-diagonal elements are nearly 0, indicating weak cross-factor dynamics.
Next examine the state-disturbance-loading matrix $B$. Visually compare the corresponding estimated innovations covariance matrix $Q=B{B}^{\prime }$ from both estimation methods.
EstMdlSSM.B
ans = 3×3 0.3076 0 0 -0.0453 0.6170 0 0.1421 0.0255 0.8824
QSSM = EstMdlSSM.B*EstMdlSSM.B'
QSSM = 3×3 0.0946 -0.0139 0.0437 -0.0139 0.3827 0.0093 0.0437 0.0093 0.7995
QVAR = EstMdlVAR.Covariance
QVAR = 3×3 0.1149 -0.0266 -0.0719 -0.0266 0.3943 0.0140 -0.0719 0.0140 1.2152
The estimated covariance matrices relatively agree. The estimated variances increase as the state proceeds from level to slope to curvature along the main diagonal.
Now compare the factor means from the estimation methods.
mu % SSM factor means
mu = 1×3 8.0246 -1.4423 -0.4188
mu0' % Two-step factor means
ans = 1×3 8.3454 -1.5724 0.2030
The estimated means of level and slope factors agree, but the curvature factor means differ in magnitude and sign.
#### Inferred Factors
The unobserved level, slope, and curvature factors (latent states) of the Diebold-Li model are integral to forecasting the evolution of future yield curves. In this part of the example, you examine the states inferred from each estimation method.
In the two-step estimation method, the latent states are the regression coefficients estimated in the OLS step.
In the SSM method, the smooth function implements backward smoothing of the Kalman filter algorithm: for $\mathit{t}=1,...,\mathit{T}$, the smoothed states are
$E\left({x}_{t}|{y}_{T},...,{y}_{1}\right).$
The SSM framework accounts for offset adjustments to the observed yields during estimation, as specified in the parameter-to-matrix mapping function. Specifically, the mapping function deflates the original observations, and therefore the estimate function works with the offset-adjusted yields ${\underset{}{\overset{\sim }{y}}}_{t}={y}_{t}-\Lambda \mu$ instead of the original yields ${\mathit{y}}_{\mathit{t}}$. The estimated SSM EstMdlSSM does not store data, and therefore it is agnostic of any adjustments made to the original yields. Therefore, when you call other SSM functions, such as filter or smooth, you must properly account for any offsets or a regression component associated with predictors that you include in the observation equation.
Infer the latent factors, while properly accounting for offsets, by following this procedure:
1. Deflate ${\mathit{y}}_{\mathit{t}}$ by subtracting the intercept associated with the estimated offset $C\mu =\Lambda \mu$. This action compensates for the offset adjustment that occurred during estimation.
2. Pass the estimated SSM EstMdlSSM and the deflated yields $\stackrel{\sim }{\text{\hspace{0.17em}}\mathit{y}{\text{\hspace{0.17em}}}_{\mathit{t}}}$ to smooth. The resulting smoothed state estimates correspond to the deflated yields.
3. Adjust the deflated, smoothed state estimates by adding the estimated mean $\mu$ to the factors. This action results in estimates of the unadjusted latent factors.
intercept = EstMdlSSM.C*mu'; DeflatedYields = Yields - intercept'; DeflatedStates = smooth(EstMdlSSM,DeflatedYields); EstimatedStates = DeflatedStates + mu;
Plot the individual level, slope, and curvature factors derived from the two-step estimation method and the SSM fit to compare the estimates.
plot(dates, [Beta(:,1) EstimatedStates(:,1)]) title('Level (Long-Term Factor)') ylabel('Percent') legend({'Two-step','SSM'},'Location','best')
plot(dates, [Beta(:,2) EstimatedStates(:,2)]) title('Slope (Short-Term Factor)') ylabel('Percent') legend({'Two-step','SSM'},'Location','best')
plot(dates, [Beta(:,3) EstimatedStates(:,3)]) title('Curvature (Medium-Term Factor)') ylabel('Percent') legend({'Two-step','SSM'},'Location','best')
The level and slope closely agree. The patterns that the curvature estimates form agree, but the values are slightly off.
Next display the estimated decay rate parameter $\lambda$ associated with the curvature.
lambda % SSM decay rate
lambda = 0.0778
The estimated decay rate parameter is somewhat larger than the value used by the two-step estimation method, which is 0.0609. $\lambda$ determines the maturity at which the loading on the curvature is maximized. The two-step estimation method fixes $\lambda$ at 0.0609, which reflects the decision to maximize the curvature loading at exactly 2.5 years (30 months). In contrast, the SSM estimates the maximum curvature loading to occur at just less than 2 years (23.1 months).
To see the effects of $\lambda$ on the curvature loading, plot the curvature loading with respect to maturity for each value of $\lambda$.
tau = 0:(1/12):max(maturities); % Maturity (months) decay = [lambda0 lambda]; Loading = zeros(numel(tau), 2); for i = 1:numel(tau) Loading(i,:) = ((1-exp(-decay*tau(i)))./(decay*tau(i))-exp(-decay*tau(i))); end figure plot(tau,Loading) title('Loading on Curvature (Medium-Term Factor)') xlabel('Maturity (Months)') ylabel('Curvature Loading') legend({'\lambda = 0.0609', ['\lambda = ' num2str(lambda)],},'Location','best')
The hump-shaped behavior of the curvature loading as a function of maturity reveals why the curvature is interpreted as a medium-term factor. Although differences between the two methods exist, the factors derived from each approach generally agree reasonably closely. Because the one-step SSM/Kalman filter estimation method, in which all model parameters are estimated simultaneously, is more flexible, the method is preferred over the two-step estimation method.
#### Summary of Residuals
Compare the means and standard deviations of the observation equation residuals between the two estimation methods, as in Table 2 of [2]. In the reference, the factor loadings matrix $\Lambda$ is the state measurement sensitivity matrix $C$ of the SSM formulation. Express the results in basis points (bps). This example uses custom functions for display purposes. For details, see Supporting Functions.
ResidualsSSM = Yields - EstimatedStates*EstMdlSSM.C'; Residuals2Step = Yields - Beta*X'; residualMeanSSM = 100*mean(ResidualsSSM)'; residualStdSSM = 100*std(ResidualsSSM)'; residualMean2Step = 100*mean(Residuals2Step)'; residualStd2Step = 100*std(Residuals2Step)'; compareresiduals(maturities,residualMeanSSM, ... residualStdSSM," State-Space Model ",residualMean2Step, ... residualStd2Step," Two-Step ")
------------------------------------------------- State-Space Model Two-Step ------------------- ------------------ Standard Standard Maturity Mean Deviation Mean Deviation (Months) (bps) (bps) (bps) (bps) -------- -------- --------- ------- --------- 3.0000 -12.6440 22.3639 -7.3922 14.1709 6.0000 -1.3392 5.0715 2.1914 7.2895 9.0000 0.4922 8.1084 2.7173 11.4923 12.0000 1.3059 9.8672 2.5472 11.1200 15.0000 3.7130 8.7073 4.2189 9.0558 18.0000 3.5893 7.2946 3.5515 7.6721 21.0000 3.2308 6.5112 2.7968 7.2221 24.0000 -1.3996 6.3890 -2.1168 7.0764 30.0000 -2.6479 6.0614 -3.6923 7.0129 36.0000 -3.2411 6.5915 -4.4095 7.2674 48.0000 -1.8508 9.7019 -2.9761 10.6242 60.0000 -3.2857 8.0349 -4.2314 9.0296 72.0000 1.9737 9.1370 1.2238 10.3745 84.0000 0.6935 10.3689 0.1196 9.8012 96.0000 3.4873 9.0440 3.0626 9.1220 108.0000 4.1940 13.6422 3.8936 11.7942 120.0000 -1.3074 16.4545 -1.5043 13.3544
Because the SSM yields lower residual means and standard deviations for most maturities, the SSM provides a better fit than the two-step estimation method, particularly for maturities 6 to 60 months.
### Forecast Estimated SSM
To forecast the estimated SSM EstMdlSSM, by you implement minimum mean square error (MMSE) forecasting and Monte Carlo simulation methods, which the SSM functionality supports.
Because the Diebold-Li model depends only on the estimated factors, you forecast the yield curve by forecasting each factor. Also, because the estimated SSM is based on offset yields, you must compensate for the offset adjustment when forecasting or simulating the model, as described in Inferred Factors.
#### Deterministic MMSE Forecasts
Compute MMSE forecasts by performing the following actions:
1. Pass EstMdlSSM and the deflated yields $\stackrel{\sim }{\text{\hspace{0.17em}}{\mathit{y}}_{\mathit{t}}}$ to the forecast function. forecast returns the MMSE forecasts of the deflated yields 1,2,...,12 months into the future.
2. Compute the forecasted yields by adding the estimated offset $C\mu$ to the deflated, forecasted yields.
fh = 12; % Forecast horizon (months) [ForecastedDeflatedYields,FMSE] = forecast(EstMdlSSM,fh,DeflatedYields); MMSEForecasts = ForecastedDeflatedYields + intercept';
The forecasted yield curves MMSEForecasts is a 12-by-17 matrix; each row corresponds to a period in the forecast horizon and each column corresponds to a maturity.
#### Monte Carlo Forecasts
An advantage of Monte Carlo forecast over MMSE forecasts is that you can use the large sample obtained by the Monte Carlo method to study characteristics of the forecast distribution.
To forecast the yield curves by performing Monte Carlo simulation, follow this general procedure:
1. Obtain factor estimates and their covariance matrix at period $\mathit{T}$, which is the end of the in-sample data and the last period before the forecast horizon starts, to initialize the Monte Carlo simulation. Such values ensure the simulation begins with the latest available information. The estimates and covariance matrix correspond to the deflated yields. Specify the initial values in the estimated ssm model object.
2. For all maturities, draw many sample paths of deflated yields into the forecast horizon.
3. Inflate the simulated, deflated yields.
4. For each period in the forecast horizon and maturity, compute summary statistics of the inflated yields over the simulated paths.
Obtain period $\mathit{T}$ factor estimates and their covariance matrix by passing estimated SSM EstMdlSSM and the deflated yields $\stackrel{\sim }{\text{\hspace{0.17em}}{\mathit{y}}_{\mathit{t}}}$ to smooth and return the output structure array containing all estimates and covariances by period. Extract the final element of the fields corresponding to the smoothed estimates.
[~,~,results] = smooth(EstMdlSSM,DeflatedYields); Mean0 = results(end).SmoothedStates; Cov0 = results(end).SmoothedStatesCov; Cov0 = (Cov0 + Cov0')/2; % Ensure covariance is symmetric
results is a $\mathit{T}$-by-1 structure array containing various smoothed estimates and inferences. Because the extracted state mean and covariance occur at the end of the historical data set, you can alternatively use the filter function, instead of smooth, to obtain equivalent initial state values from which to forecast.
Set the initial state mean Mean0 and covariance Cov0 properties of the SSM EstMdlSSM to the appropriate smoothed estimates.
EstMdlSSM.Mean0 = Mean0; % Initial state mean EstMdlSSM.Cov0 = Cov0; % Initial state covariance
Draw 100,000 sample paths of deflated yields from the estimated SSM into the forecast horizon. Inflate the simulated, deflated yields.
rng('default') % For reproducibility numPaths = 100000; SimulatedDeflatedYields = simulate(EstMdlSSM,fh,numPaths); SimulatedYields = SimulatedDeflatedYields + intercept';
SimulatedYields is a 12-by-17-by-100,000 numeric array:
• Each row is a period in the forecast horizon.
• Each column is a maturity.
• Each page is a random draw from the forecast distribution. In other words, each page represents the future evolution of a simulated yield curve over a 12-month forecast horizon.
Compute the sample means and standard deviations over the 100,000 draws.
MCForecasts = mean(SimulatedYields,3); MCStandardErrors = std(SimulatedYields,[],3);
MCForecasts and MCStandardErrors are the Monte Carlo simulation analogs of the MMSE forecasts MMSEForecasts and standard errors FMSE, respectively.
Visually compare the MMSE and Monte Carlo simulation forecasts and corresponding standard errors.
figure plot(maturities,[MCForecasts(fh,:)' MMSEForecasts(fh,:)']) title('12-Month-Ahead Forecasts: Monte Carlo vs. MMSE') xlabel('Maturity (Months)') ylabel('Percent') legend({'Monte Carlo','MMSE'},'Location','best')
figure plot(maturities,[MCStandardErrors(fh,:)' sqrt(FMSE(fh,:))']) title('12-Month-Ahead Forecast Standard Errors: Monte Carlo vs. MMSE') xlabel('Maturity (Months)') ylabel('Percent') legend({'Monte Carlo','MMSE'},'Location','best')
The estimates are effectively identical.
A benefit of Monte Carlo simulation is that it enables an analysis of the distribution of the yields beyond their mean and standard error. Monte Carlo simulation provides additional insight into how the distribution affects the distribution of other variables dependent on it. For example, the insurance industry commonly uses the simulation of yield curves to assess the distribution of profits and losses associated with annuities and pension contracts.
Display the distribution of the simulated 12-month yield at 1, 6, and 12 months into the future, similar to the forecasting experiment in Tables 4 through 6 of [1].
index12 = find(maturities == 12); % Page index of 12-month yield bins = 0:0.2:12; figure subplot(3,1,1) histogram(SimulatedYields(1,index12,:),bins,'Normalization','pdf') title('PDF of 12-Month Yield') xlabel('Yield 1 Month into the Future (%)') subplot(3,1,2) histogram(SimulatedYields(6,index12,:),bins,'Normalization','pdf') xlabel('Yield 6 Months into the Future (%)') ylabel('Probability') subplot(3,1,3) histogram(SimulatedYields(12,index12,:),bins,'Normalization','pdf') xlabel('Yield 12 Months into the Future (%)')
Forecasts further out in the forecast horizon are more uncertain than forecasts closer to the end of the in-sample period.
### Augment Yield Curve Model with Macro Factors
The yields-macro model extends the yields-only model by including macroeconomic and financial factors. A minimum set of variables that characterize the economic activities include manufacturing capacity utilization (CU, ${\mathit{z}}_{1,\mathit{t}}$) [7], the federal funds rate (FEDFUNDS, ${\mathit{z}}_{2,\mathit{t}}$) [5], and annual price inflation (PI, ${\mathit{z}}_{3,\mathit{t}}$) [6], which interact with the level, slope, and curvature factors in a vector autoregression. The augmented SSM is
$\begin{array}{c}\left({f}_{t}-\mu \right)=A\left({f}_{t-1}-\mu \right)+{\eta }_{t}\\ \left[\begin{array}{c}{y}_{t}\\ {z}_{t}\end{array}\right]=\left[\begin{array}{cc}\Lambda & 0\\ 0& I\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}{f}_{t}+\left[\begin{array}{c}{w}_{t}\\ 0\end{array}\right]\end{array},$
where ${f}_{t}=\left[{L}_{t},{S}_{t},{C}_{t},{z}_{1,t},{z}_{2,t},{z}_{3,t}{\right]}^{\prime }$ and ${\mathit{z}}_{\mathit{t}}={\left[{z}_{1,t},{z}_{2,\mathit{t}},{z}_{3,\mathit{t}}\right]}^{\prime }$. The dimensions of $A$ and $\mu$ increase to 6-by-6 and 6-by-1, respectively. The observation equation ${\mathit{y}}_{\mathit{t}}=\left[\begin{array}{cc}\Lambda & 0\end{array}\right]{\mathit{f}}_{\mathit{t}}+{\mathit{w}}_{\mathit{t}}$ indicates that the level, slope, and curvature factors sufficiently distill the information in the yield curve. Also, ${z}_{t}=\left(\begin{array}{c}0\phantom{\rule{0.5em}{0ex}}I\end{array}\right){f}_{t}$ indicates that the macroeconomic variables are observed without measurement error. The SSM framework accounts for missing observations, denoted by NaN values in the data, by substituting estimates derived from the Kalman filter. Like the yields-only formulation, the white noise processes ${\eta }_{t}$ and ${w}_{t}$ have the distribution
$\left[\begin{array}{c}{\eta }_{t}\\ {w}_{t}\end{array}\right]\phantom{\rule{0.5em}{0ex}}\sim WN\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}Q& 0\\ 0& H\end{array}\right]\right),$
where $Q$ is a 6-by-6 symmetric positive definite matrix, and $H$ is diagonal.
The formulation supported by the SSM functionality is
$\begin{array}{l}{x}_{t}={A}_{t}{x}_{t-1}+{B}_{t}{u}_{t}\\ {\zeta }_{t}={C}_{t}{x}_{t}+{D}_{t}{\epsilon }_{t},\end{array}$
where the states are the mean-adjusted factors, namely ${x}_{t}={f}_{t}-\mu$. The deflated observations are
${\zeta }_{t}=\left[\begin{array}{c}{y}_{t}\\ {z}_{t}\end{array}\right]-\left[\begin{array}{cc}\Lambda & 0\\ 0& I\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\mu .$
The model contains 81 unknown parameters:
• $A$ contains 36 parameters.
• $Q$ contains 21 parameters.
• $H$ contains 17 parameters.
• $\mu$ contains 6 parameters.
• $\Lambda$ contains the scalar $\lambda$.
The mapping function Example_YieldsMacro.m maps a parameter vector to SSM parameter matrices, deflates the observations to account for the means of each factor, and imposes constraints on the covariance matrices. For more details, open Example_YieldsMacro.m.
### Estimate Yields-Macro Diebold-Li SSM
The yields data set Data_DieboldLi.mat additionally contains observations of the macroeconomic series, which are available in the Federal Reserve Economic Database (FRED) [4]. The macroeconomic series in Data_DieboldLi.mat are not identical to the corresponding series in [2], but this example reproduces most of those empirical results.
Extract the macroeconomic series from the data, and determine variable dimensions.
Macro = [DataTable.CU DataTable.FEDFUNDS DataTable.PI]; numBonds = size(Yields,2); numMacro = size(Macro,2); numStates = 3 + numMacro;
The estimation of 81 parameters by maximum likelihood is computationally challenging, but it is possible with carefully specified initial values for numerical optimization.
To obtain reasonable initial values, fit a vector autoregression model (the state equation) to the estimated factors Beta and the macroeconomic series Macro. Then, extract and process the estimates from the estimated model.
MdlVAR0 = varm(numStates,1); EstMdlVAR0 = estimate(MdlVAR0,[Beta Macro]); A0 = EstMdlVAR0.AR{1}; B0 = chol(EstMdlVAR0.Covariance,'lower'); D0 = std(Residuals); mu0 = [mean(Beta) mean(Macro,'omitnan')]; p0Macro = [A0(:); nonzeros(B0); D0(:); mu0(:); lambda0];
The estimate function estimates unknown parameters of an SSM by optimizing the loglikelihood using fminunc or fmincon. At each iteration, estimate computes the loglikelihood by applying the Kalman filter. This complex optimization task can require numbers of iterations and function evaluations that are larger than their default. Also, the dimension of the observation vector is substantially larger than the state vector, and therefore the method that treats a multivariate series as univariate can improve run-time performance.
Create an optimization options object that specifies a maximum of 1000 iterations and 50,000 function evaluations for unconstrained optimization. Create an SSM by passing, to ssm, the parameter-to-matrix mapping function Example_YieldsMacro as an anonymous function with an input argument representing the parameter vector params. The additional input arguments of the mapping function specify the yields, maturity, and macroeconomic series information statically. The inputs initialize the model for estimation.
options = optimoptions('fminunc','MaxIterations',1000,'MaxFunctionEvaluations',50000); MdlMacro = ssm(@(params)Example_YieldsMacro(params,Yields,Macro,maturities));
Estimate the yields-macro Diebold-Li SSM. Turn off the estimation display and specify the optimization options and the method that treats multivariate series as univariate.
[EstMdlMacro,estParamsMacro,EstParamCovMacro,logLMacro] = estimate(MdlMacro,[Yields Macro],p0Macro, ... 'Display','off','Options',options,'Univariate',true);
Because the SSM operates on the mean-adjusted factors, you must deflate (intercept-adjust) the observations before operating on the estimated model and the observations. Get the estimated SSM coefficients and the deflated data by specifying the estimated parameters as inputs to Example_YieldsMacro.m. Display the estimated state-transition matrix $\mathit{A}$ and state-disturbance covariance matrix $\mathit{B}{\mathit{B}}^{\prime }$, which account for the macroeconomic factors.
[A,B,C,~,~,~,~,deflatedData] = Example_YieldsMacro(estParamsMacro,Yields,Macro,maturities); A
A = 6×6 0.8986 -0.0624 -0.0245 -0.0061 0.0756 0.0134 -0.4325 0.4805 0.0314 0.0312 0.3768 0.0260 0.1355 0.1182 0.8455 0.0111 -0.0918 0.0023 0.0679 -0.0257 0.0023 0.9982 -0.0655 -0.0202 -0.0064 -0.0887 0.0106 0.0461 0.9958 0.0456 -0.0348 -0.0314 -0.0150 0.0268 0.0364 0.9940
B*B'
ans = 6×6 0.0914 -0.0226 0.0466 0.0370 0.0327 0.0114 -0.0226 0.3041 0.0076 0.0772 0.2241 -0.0051 0.0466 0.0076 0.8105 0.0259 0.1690 0.0023 0.0370 0.0772 0.0259 0.3710 0.1425 0.0254 0.0327 0.2241 0.1690 0.1425 0.4494 0.0128 0.0114 -0.0051 0.0023 0.0254 0.0128 0.0475
Given the full-sample data and a fitted SSM, the Kalman smoother smooth provides state estimates. The smooth function enables you to evaluate the measurement errors of the yields. For each maturity, obtain measurement error estimates by computing the mean and standard deviation of the residuals. Visually compare the measurement error estimates of the estimated yields-only and yields-macro models.
StatesMacro = smooth(EstMdlMacro,deflatedData); ResidualsMacro = deflatedData - StatesMacro*C'; residualMeanMacro = 100*mean(ResidualsMacro,'omitnan')'; residualStdMacro = 100*std(ResidualsMacro,'omitnan')'; compareresiduals(maturities,residualMeanSSM,residualStdSSM, ... " Yields-Only Model ",residualMeanMacro(1:numBonds), ... residualStdMacro(1:numBonds),"Yields-Macro Model")
------------------------------------------------- Yields-Only Model Yields-Macro Model ------------------- ------------------ Standard Standard Maturity Mean Deviation Mean Deviation (Months) (bps) (bps) (bps) (bps) -------- -------- --------- ------- --------- 3.0000 -12.6440 22.3639 -12.5379 22.2240 6.0000 -1.3392 5.0715 -1.2658 4.8526 9.0000 0.4922 8.1084 0.5387 8.1444 12.0000 1.3059 9.8672 1.3310 9.8812 15.0000 3.7130 8.7073 3.7212 8.7603 18.0000 3.5893 7.2946 3.5846 7.3145 21.0000 3.2308 6.5112 3.2168 6.4719 24.0000 -1.3996 6.3890 -1.4201 6.3520 30.0000 -2.6479 6.0614 -2.6745 6.0864 36.0000 -3.2411 6.5915 -3.2675 6.6115 48.0000 -1.8508 9.7019 -1.8663 9.7266 60.0000 -3.2857 8.0349 -3.2855 8.0124 72.0000 1.9737 9.1370 1.9896 9.1110 84.0000 0.6935 10.3689 0.7231 10.3780 96.0000 3.4873 9.0440 3.5285 9.1650 108.0000 4.1940 13.6422 4.2447 13.7447 120.0000 -1.3074 16.4545 -1.2488 16.5814
The yields-only model fits the yield curve data well, but the yields-macro model performs better.
Because the macroeconomic variables are perfectly observed, ensure that their estimated measurement errors are close to zero, to within machine precision.
residualMeanMacro(end-numMacro+1:end)'
ans = 1×3 10-14 × -0.5399 -0.7747 -0.6235
### Characterize Yields-to-Macro and Macro-to-Yields Links
#### Impulse Response Functions of Yields-Macro Model
For an SSM, the impulse response function (IRF) measures the dynamic effects on the state and measurement equations due to an unanticipated shock to each state disturbance. In the transition equation ${x}_{t}=A{x}_{t-1}+B{u}_{t}$, the IRF is the partial derivative of ${\mathit{x}}_{\mathit{t}}$, $\mathit{t}=1,2,...$ with respect to ${u}_{1}$. The SSM IRF functionality, irf and irfplot, performs the following actions:
• Apply the state shock during period 1.
• Normalize the shock variance to one; The state-disturbance-loading matrix $B$ determines any contemporaneous effects.
• Return responses for periods $\mathit{t}=1,2,...$.
For the yields-macro model, the matrix $B$ is identified up to $Q=B{B}^{\prime }$. The IRF of the yields-macro SSM requires identification conditions, such as a recursive ordering of state variables ${L}_{t},{S}_{t},{C}_{t},{z}_{1,t},{z}_{2,t},{z}_{3,t}$ so that $B$ is a lower triangular matrix. The rationale is that yields are dated at the beginning of each month, while macroeconomic data is published with a time lag. Consequently, the yields have a contemporaneous effect on the macroeconomic variables, but not vice versa.
Use irfplot to plot the IRF of the following groups of variables:
• Yield-curve responses to yield-curve shocks
• Macro responses to macro shocks
• Macro responses to yield-curve shocks
• Yield-curve responses to macro shocks
For each plot, specify 90% confidence bounds and plot responses for the first 30 periods.
figure irfplot(MdlMacro,'Params',estParamsMacro,'EstParamCov',EstParamCovMacro, ... 'PlotU',1:3,'PlotX',1:3,'PlotY',[],'NumPeriods',30,'Confidence',0.9); sgtitle('Yield-Curve Responses to Yield-Curve Shocks')
figure irfplot(MdlMacro,'Params',estParamsMacro,'EstParamCov',EstParamCovMacro, ... 'PlotU',4:6,'PlotX',4:6,'PlotY',[],'NumPeriods',30,'Confidence',0.9); sgtitle('Macro Responses to Macro Shocks')
figure irfplot(MdlMacro,'Params',estParamsMacro,'EstParamCov',EstParamCovMacro, ... 'PlotU',1:3,'PlotX',4:6,'PlotY',[],'NumPeriods',30,'Confidence',0.9); sgtitle('Macro Responses to Yield-Curve Shocks')
figure irfplot(MdlMacro,'Params',estParamsMacro,'EstParamCov',EstParamCovMacro, ... 'PlotU',4:6,'PlotX',1:3,'PlotY',[],'NumPeriods',30,'Confidence',0.9); sgtitle('Yield-Curve Responses to Macro Shocks')
#### Variance Decompositions of Yields-Macro Model
The forecast error variance decomposition (FEVD) provides information about the relative importance of each shock in affecting the forecast error variance of a response variable. Regarding the macro and yield curve interactions, FEVD analyzes whether the macro factors are influential, compared to the idiosyncratic variation in the yield curve. In the SSM, the disturbances in the transition and measurement equations cause the forecast variance of the observations. In the presence of a nonzero, observation-innovation coefficient matrix $D$, the decomposition does not sum to one because the remaining portion is attributable to the observation noise covariance $D{D}^{\prime }$. To force the sum to one, you can account for the observation noise by rescaling the FEVD.
Compute FEVD of the 12-month yield at the forecast horizons of 1, 12, and 60 months by using the fevd function. Because fevd applies the unit shock at period 1, the forecast horizon starts at period 2. Normalize each FEVD to sum to one.
Decomposition = fevd(EstMdlMacro,'NumPeriods',61); idxyield12 = find(maturities == 12); d1 = Decomposition(1+1,:,idxyield12); d12 = Decomposition(1+12,:,idxyield12); d60 = Decomposition(1+60,:,idxyield12); d1 = d1./sum(d1); d12 = d12./sum(d12); d60 = d60./sum(d60); displayfevd([d1; d12; d60],"12-Month Yield")
Variance Decomposition, 12-Month Yield -------------------------------- Horizon L S C CU FEDFUNDS PI 1.0000 0.2977 0.4397 0.2189 0.0037 0.0400 0.0001 12.0000 0.3080 0.2302 0.1345 0.0986 0.2211 0.0076 60.0000 0.3874 0.0855 0.1258 0.2836 0.0932 0.0245
The FEVD results indicate that less than 5 percent of the variation is attributable to the macroeconomic factors at the 1-month horizon. However, the macroeconomic factors are more influential at longer horizons. At the 12-month and 60-month horizons, the macroeconomic factors account for about 30 and 40 percent of the variation, respectively.
Compute FEVD of the manufacturing capacity utilization series at the forecast horizons of 1, 12, and 60 months. Normalize each FEVD to sum to one.
cu1 = Decomposition(1+1,:,numBonds+1); cu12 = Decomposition(1+12,:,numBonds+1); cu60 = Decomposition(1+60,:,numBonds+1); displayfevd([cu1; cu12; cu60],"Capacity Utilization")
Variance Decomposition, Capacity Utilization -------------------------------- Horizon L S C CU FEDFUNDS PI 1.0000 0.0466 0.0531 0.0001 0.8989 0.0013 0.0000 12.0000 0.0913 0.0260 0.0224 0.7465 0.1084 0.0054 60.0000 0.1015 0.0605 0.0595 0.5149 0.1786 0.0849
In contrast to the FEVD of the 12-month yield, the variance decompositions of manufacturing capacity utilization show that the level, slope, and curvature factors account for a small portion of the variation at each period in the forecast horizon. The variance decompositions of the other two macroeconomic variables exhibit the same pattern.
### Augment Diebold-Li SSM with Exogenous Macroeconomic Variables
In the fitted yields-macro model, the estimated parameters in the bottom-left corner of the transition matrix $A\left(4:6,1:3\right)$ are not individually significant. This result motivates a model with a more parsimonious specification, specifically with the constraint $A\left(4:6,1:3\right)=0$ and a diagonal covariance matrix $Q$. In the constrained model, the macroeconomic variables are exogenous, which means that the macroeconomic factors impact the yield curve factors, but the link is unilateral because the yields-to-macro link is absent.
The mapping function Example_YieldsExogenous.m maps a parameter vector to SSM parameter matrices, deflates the observations to account for the means of each factor, imposes constraints on the covariance matrices, and includes exogenous variables in the state equation. For more details, open Example_YieldsExogenous.m.
Create a Diebold-Li SSM augmented with exogenous macroeconomic variables by passing, to ssm, the parameter-to-matrix mapping function Example_YieldsExogenous as an anonymous function with an input argument representing the parameter vector params. The additional input arguments of the mapping function specify the yields, maturity, and macroeconomic series information statically, which initialize the model for estimation.
MdlExogenous = ssm(@(params)Example_YieldsExogenous(params,Yields,Macro,maturities));
The constrained MdlExogenous model has 57 unknown parameters.
Estimate the Diebold-Li SSM augmented with exogenous variables. Turn off the estimation display and specify the same options and initial values as specified for the estimation of the yields-macro SSM.
Mask = true(numStates); Mask(4:end,1:3) = false; param0Exogenous = [A0(Mask); diag(B0); D0(:); mu0(:); lambda0]; [EstMdlExogenous,estParamsExogenous,~,logLExogenous] = estimate(MdlExogenous,[Yields Macro], ... param0Exogenous,'Display','off','Options',options,'Univariate',true); logLMacro
logLMacro = 2.7401e+03
logLExogenous
logLExogenous = 2.5311e+03
The maximized loglikelihood of EstMdlExogenous is necessarily lower than that of the full model EstMdlMacro, which has 81 parameters.
Get the estimated SSM coefficients and the deflated data by specifying the estimated parameters as inputs to Example_YieldsExogenous.m. Display the estimated state-transition matrix $\mathit{A}$.
[AExogenous,~,CExogenous,~,~,~,~,deflatedDataExogenous] = Example_YieldsExogenous(estParamsExogenous, ... Yields,Macro,maturities); AExogenous
AExogenous = 6×6 0.8901 -0.0741 -0.0201 -0.0052 0.0826 0.0155 -0.4488 0.4672 0.0322 0.0298 0.3888 0.0266 0.1385 0.1283 0.8378 0.0072 -0.0957 -0.0010 0 0 0 0.9785 -0.0447 -0.0259 0 0 0 0.0276 0.9597 0.0375 0 0 0 0.0289 0.0050 0.9961
For each maturity, obtain measurement error estimates by computing the mean and standard deviation of the residuals. Visually compare the measurement error estimates of the estimated yields-macro model and the Diebold-Li model augmented with exogenous variables.
StatesExogenous = smooth(EstMdlExogenous,deflatedDataExogenous); ResidualsExogenous = deflatedDataExogenous - StatesExogenous*CExogenous'; residualMeanExogenous = 100*mean(ResidualsExogenous,'omitnan')'; residualStdExogenous = 100*std(ResidualsExogenous,'omitnan')'; compareresiduals(maturities,residualMeanMacro(1:numBonds),residualStdMacro(1:numBonds), ... "Yields-Macro Model",residualMeanExogenous(1:numBonds),... residualStdExogenous(1:numBonds)," Exogenous Model")
------------------------------------------------- Yields-Macro Model Exogenous Model ------------------- ------------------ Standard Standard Maturity Mean Deviation Mean Deviation (Months) (bps) (bps) (bps) (bps) -------- -------- --------- ------- --------- 3.0000 -12.5379 22.2240 -12.5914 22.4354 6.0000 -1.2658 4.8526 -1.3164 4.8587 9.0000 0.5387 8.1444 0.4975 8.0276 12.0000 1.3310 9.8812 1.3021 9.8990 15.0000 3.7212 8.7603 3.7052 8.7212 18.0000 3.5846 7.3145 3.5806 7.2662 21.0000 3.2168 6.4719 3.2231 6.4676 24.0000 -1.4201 6.3520 -1.4056 6.3821 30.0000 -2.6745 6.0864 -2.6499 6.0769 36.0000 -3.2675 6.6115 -3.2405 6.5829 48.0000 -1.8663 9.7266 -1.8506 9.7233 60.0000 -3.2855 8.0124 -3.2913 8.0536 72.0000 1.9896 9.1110 1.9601 9.1345 84.0000 0.7231 10.3780 0.6716 10.3321 96.0000 3.5285 9.1650 3.4580 8.9382 108.0000 4.2447 13.7447 4.1581 13.6501 120.0000 -1.2488 16.5814 -1.3487 16.4289
The estimated transition matrices of the models are similar. EstMdlExogenous results in larger measurement errors of yields at some maturities, as compared to the measurement errors of EstMdlMacro. Despite this result, EstMdlExogenous fits the curve well overall.
In summary, the yields-macro and Diebold-Li model augmented with exogenous variables have significant statistical and economic differences. Despite the large difference in the fitted loglikelihoods of the models, the fitted yield curve of the SSM augmented with exogenous variables is not necessarily inferior to the yield curve of the yields-macro SSM.
### Supporting Functions
Local functions facilitate several command-line displays for this example.
function compareresiduals(maturities,residualMeanL,residualStdL,tL,residualMeanR,residualStdR,tR) header = [" -------------------------------------------------"; ... " " + tL + " " + tR;... " ------------------- ------------------"; ... " Standard Standard"; ... " Maturity Mean Deviation Mean Deviation"; ... " (Months) (bps) (bps) (bps) (bps) "; ... " -------- -------- --------- ------- ---------"]; tab = [maturities residualMeanL residualStdL residualMeanR residualStdR]; fprintf("%s\n",header) disp(tab) end function displayfevd(d,t) header = ["Variance Decomposition, " + t; ... "--------------------------------"; ... " Horizon L S C CU FEDFUNDS PI"]; tab = [[1;12;60] d]; fprintf("%s\n",header) disp(tab) end
## References
[1] Diebold, F.X., and C. Li. "Forecasting the Term Structure of Government Bond Yields." Journal of Econometrics. Vol. 130, No. 2, 2006, pp. 337–364.
[2] Diebold, F. X., G. D. Rudebusch, and B. Aruoba (2006), "The Macroeconomy and the Yield Curve: A Dynamic Latent Factor Approach." Journal of Econometrics. Vol. 131, 2006, pp. 309–338.
[3] Nelson, R. C., and A. F. Siegel. "Parsimonious Modeling of Yield Curves." Journal of Business. Vol. 60, No. 4, 1987, pp. 473–489.
[4] U.S. Federal Reserve Economic Data (FRED), Federal Reserve Bank of St. Louis, https://fred.stlouisfed.org/.
[5] Board of Governors of the Federal Reserve System (US), Effective Federal Funds Rate [FEDFUNDS], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/FEDFUNDS, March 1, 2021.
[6] U.S. Bureau of Economic Analysis, Personal Consumption Expenditures: Chain-type Price Index [PCEPI], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/PCEPI, March 1, 2021.
[7] Board of Governors of the Federal Reserve System (US), Capacity Utilization: Manufacturing (SIC) [CUMFNS], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/CUMFNS, March 1, 2021. | 2021-10-28 02:40:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 156, "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": 0.7860774397850037, "perplexity": 2176.880518317887}, "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/1634323588246.79/warc/CC-MAIN-20211028003812-20211028033812-00289.warc.gz"} |
https://ru.overleaf.com/latex/templates/introduction-to-using-latex-in-lab-reports/krvvhmtvfxyt | # Introduction to using LaTeX in lab reports
Author
Teigan O'Carroll
AbstractBasic layout for lab reports for Electromagnetism and Optics | 2020-05-27 03:46:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "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": 0.17896272242069244, "perplexity": 13674.158501357793}, "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/1590347392057.6/warc/CC-MAIN-20200527013445-20200527043445-00062.warc.gz"} |
https://www.physicsforums.com/threads/infinite-dimensional-pde.207014/ | # Infinite dimensional PDE
1. Jan 3, 2008
### jostpuur
Is there any established theory concerning infinite dimensional PDE?
2. Jan 3, 2008
### ObsessiveMathsFreak
Do you mean that the function has infinitely many variables, or that it is an infinite dimensional function of a finite number of variables?
3. Jan 3, 2008
### jostpuur
Infinitely many variables.
For example a quantum mechanical real Klein-Gordon field, if I have understood correctly, can be pretty much described by the infinite dimensional non-homogenous heat equation (the Shrodinger's equation, with certain constants and with the harmonic potential). Something like this
$$i\partial_t \Psi(t,\phi) = \sum_{k\in\mathbb{R}^3} \Big(-\alpha \partial^2_{k} + \beta |k|^2\Big)\Psi(t, \phi)$$
where
$$\Psi:\mathbb{R}\times\mathbb{R}^{\mathbb{R}^3}\to\mathbb{C}.$$
It can be solved by a separation attempt
$$\Psi(t,\phi) = \prod_{k\in\mathbb{R}^3} \Phi_k(t) \Psi_k (\phi(k)),$$
where
$$\Phi_k,\;\Psi_k:\mathbb{R}\to\mathbb{C}$$
This is total honest pseudo mathematics, motivated by physics, don't complain about it!
In fact his is a very vague example with uncountable set of variables. There could be more rigor examples with only countably many variables.
Last edited: Jan 3, 2008
4. Feb 29, 2008
### jostpuur
It could be these are supposed to be called functional differential equations, but I'm not sure. Some quick google hits were slightly confusing. | 2017-12-15 10:28:42 | {"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": 0.8829100728034973, "perplexity": 1203.4642778467553}, "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/1512948568283.66/warc/CC-MAIN-20171215095015-20171215115015-00211.warc.gz"} |
http://www.originlab.com/fileExchange/details.aspx?fid=289 | #### File Exchange > Data Analysis > Hurst Exponent
Author:
OriginLab Technical Support
2/1/2016
Last Update:
6/15/2022
107
Total Ratings:
5
File Size:
23 KB
Average Rating:
File Name:
HurstExponent.opx
File Version:
1.00
Minimum Versions:
Free
Summary:
Calculate Hurst exponent with rescaled range analysis.
Screen Shot and Video:
Description:
Purpose:
The tool can be used to:
• Calculate classical Hurst exponents (R/S) and corrected Hurst exponents (R/S-AL).
• Show a Log-Log plot of R/S statistics vs subseries length.
Installation:
Download the file HurstExponent.opx, and then drag-and-drop onto the Origin workspace. An icon will appear in the Apps Gallery window.
Operation:
1. Highlight a column of time series data in the worksheet.
2. Click the Hurst Exponent icon in the Apps Gallery window.
3. In the dialog box that opens:
1. Set Start Index and End Index to define partial series.
2. Set Minimum Subseries Length to define the minimum length of subseries when partitioning the time series. The default value is 50.
When a trend or seasonality exists in time series data, minimum subseries should exhibit the time series pattern (e.g. for seasonal data, Minimum Subseries Length should be no less than the length of the period).
3. Check the Log-Log Plot to output the plot.
4. Click the OK button to output a worksheet with results and a Log-Log plot.
Algorithm:
$$(R/S)_n = \frac{1}{d} \displaystyle \sum R_m/S_m$$
where input time series is divided into d subseries of length n, $$R_m$$and$$S_m$$are the range and standard deviation in each subseries.
$$E(R/S)_n = \begin{cases} \frac{ (n-0.5) \Gamma(\frac{n-1}{2}) }{n \sqrt{ \pi } \Gamma( \frac{n}{2} ) } \displaystyle \sum_{i=1}^{n-1} \sqrt{ \frac{n-i}{i} } &\text{for} \; n \leq 340, \\ \frac{ n-0.5 }{n \sqrt{ \frac{n \pi}{2} } } \displaystyle \sum_{i=1}^{n-1} \sqrt{ \frac{n-i}{i} } &\text{for} \; n > 340 \end{cases}$$
$$R/S-AL = (R/S)_n - E(R/S)_n + \sqrt{ \frac{ n \pi }{2}}$$
Slope of $$(R/S)_n$$ vs. n in Log-Log plot is the classic Hurst exponent, and slope of R/S-AL vs. n in Log-Log plot is the AL corrected Hurst exponent.
Reference:
1. R. Weron (2002) Estimating long range dependence: finite sample properties and confidence intervals, Physica A 312, 285-299. | 2022-06-29 03:50:02 | {"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": 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": 0.27275773882865906, "perplexity": 7001.8750030861875}, "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/1656103620968.33/warc/CC-MAIN-20220629024217-20220629054217-00369.warc.gz"} |
https://www.physicsforums.com/threads/convolution-integral-s-d-o-f-system.842417/ | # Convolution Integral (s.d.o.f. system)
1. Nov 10, 2015
### DaNud
< Mentor Note -- Poster has been reminded that they need to show their work on schoolwork questions >
Does anybody know how to solve this exercise?
Derive the response of an undamped single-degree-offreedom system to force f(t)=F_0*cos(w_n*t)*u(t) with null initial conditions, w_n=(k/m)^1/2 and u(t) being the unit step function by convolution integral. Compare the solution with the result obtained with an alternative approach.
I don't even know how to start.
Last edited by a moderator: Nov 11, 2015
2. Nov 10, 2015
### Khashishi
What have you learned about transfer functions and the convolution theorem?
3. Nov 10, 2015
### DaNud
I learned that can be applied only for linear system because I am using the superposition principle. I am summing up all the random forces of a generic F(t) at time F(delta). I don't know how can be solved in terms of mathematics.
4. Nov 10, 2015
### Khashishi
Is your undamped single-degree-of-freedom system a linear system? If yes, write down the transfer function.
5. Nov 10, 2015
### DaNud
Yes the transfer function is:
a + wn*x = F0*cos(wn*t)*u(t)
where a is the acceleration.
6. Nov 10, 2015
### Khashishi
No, that's not a transfer function. That's an equation of motion. The transfer function is the Laplace transform of the LHS.
7. Nov 10, 2015
### DaNud
I don't know how to do it. Could you please explain me?
However the equation of motion is a + wn*x = F0/m *cos(wn*t)*u(t)
8. Nov 11, 2015
### BvU
[Mentor's note: Post merged from another thread.]
Hello Nud, a belated
According to the PF rules "don't know where to start" isn't good enough. So tell us what you've got in terms of subject know-how !
At least a convolution integral should be in the "relevant equations"
Then: continuing/repeating an existing thread in another subforum is considered spamming and frowned upon ! (general PF rules)
On a positive note, some guidance:
If you don't know how to get started, perhaps you can explore the alternative approach (to the convolution approach) which is hard work...
There is a solution to the homogeneous equation $\ddot x + \omega_n^2 x = 0$. You must know that already, right ?
Next you need a particular solution, which you may well also know about ?
You have no damping, so the solution to the homogeneous equation will be there for all T > 0
--
More help -- but credit is now low -- : from the original thread I gather you have not been paying much attention when the transfer function concept was treated. My advice: do catch up ! It's very useful and makes this exercise a breeze .
--
One final lifebuoy: Read up on Laplace Transforms ! - the word convolution appears there !
--
Last edited by a moderator: Nov 11, 2015
9. Nov 11, 2015
### rude man
Don't start with the Laplace transfer function. That can come later when you're asked for an "alternative approach".
You first need to find the unit impulse response to your system. The system is defined by your 2nd order undamped diff. equation which you have correctly given in your post 7 except "a" needs to be expressed in terms of x and wn should be wn^2.
One way is to transform the system equation with the Fourier integral, find the unit impulse response h(t) from that, then convolve h(t) with the input time function using the convolution integral. That however is not sticking to the time domain; in fact it is close to what is asked for later as the "alternative approach". It's kind of cheating, but I think it's what they want you to do.
So, can you use a time domain method to solve for the impulse response? The answer is Yes. It can be done in 2 steps:
1. assume input to the quiescent x(0) = x'(0) = 0 system is f(t)/m = cU(t) for a time T. The impulse is here represented as a pulse of width T and height c, with c → ∞, T → 0 but with cT = 1. Solve the diff. eq. and find the new initial conditions x(T) and x'(T), then
2. re-solve the diff. eq. with the new initial conditions x(T) and x'(T), with zero f(t).
When you take the limit as c → ∞ and T → 0, holding cT = 1, you will get the unit impulse response to the system x(t) = h(t).
Either way of getting h(t) you now need to use the convolution integral to convolve the input f(t) with h(t).
The "alternative approach" can be the Laplace transform mentioned in post 6.
10. Nov 11, 2015
### DaNud
Thanks to everybody.
I am trying to solve with the method suggested by rude man.
I obtained
h(t)=(1/(wn*m))*(sin(wn*t))
Now I have to calculate my convolution integral
x(t)=∫(F(j)*h(t-j)dj
calculated b/w 0 and t
where j is a dummy variable.
My slides talk about the shifting procedure (t-j) but I don't understand how can be done practically. Could you please give me one more hint?
11. Nov 12, 2015
### rude man
Up to here eveything is OK. Well done!
Correct also.
The shifting procedure is better done with discrete convolutions. I suggest you carry out this integral in closed form.
Then, you're ready for the "alternative approach" which has been suggested in previous posts. | 2017-08-19 19:57:15 | {"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": 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": 0.6600136160850525, "perplexity": 1186.5193914587474}, "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/1502886105712.28/warc/CC-MAIN-20170819182059-20170819202059-00560.warc.gz"} |
https://gasstationwithoutpumps.wordpress.com/tag/pulse-monitor/page/2/ | Gas station without pumps
2016 August 20
Using 4¢ diode for log-transimpedance
In Transimpedance pulse monitor does need low-pass, I realized that Schottky diodes were not going to work well for the transimpedance amplifier, and in Using nFET body diode for log-transimpedance, I tested using the body diode of a power nFET, finding that it worked quite well over at least 7.5 decades (from 1nA to 40mA). But I wanted to see whether students could use a cheap 4¢ general-purpose diode.
I used the same setup as when testing the nFET body diodes. The results were pretty much the same whether I used a 1N914B or 1N4148 diode (they share a datasheet, but the 1N914B has somewhat better constraints on the forward voltage):
The gain (in mV/dB) is about 85% larger than using an nFET body diode.
Note that at currents over about 1mA the diode current starts to saturate, deviating from the exponential pattern.
When I tried using the 1N914B diode in the same log-transimpedance amplifier as I used for the nFET body diode, it didn’t work—I got output that looked nothing like a pulse (nor like 60Hz interference). I could recover proper behavior by putting a large (100nF) capacitor in parallel with the diode, to make a low-pass filter to remove signals above a few Hz, but that wasn’t necessary for the nFET body diode (perhaps it had enough internal capacitance to do the filtering). I could reduce the capacitor to 100pF, with 60Hz interference coming in, though not being too bad, but reducing to 10pF gave me noise again rather than the pulse signal.
I was hoping not to need that extra capacitor, because the design is already more complicated than I would like for this stage of the course, and figuring out what capacitor to use is difficult—trial and error is easier than rational design here!
I tried tracking down the big, short (less than 250µs) spikes that were corrupting the signal. The first thing I tried cleaned up the problem entirely: disconnecting the power supply from the laptop so that the USB power was coming from the laptop battery rather than the power supply . That this worked actually surprised me, since the 3.3V supply and the 1.65V Vref both had beefy bypass capacitors.
I don’t know whether the noise problems are in the microcontroller (which is providing the regulated 3.3V from the noisy USB 5V) or are coupled into the analog circuit some other way. Putting a 10µF capacitor from the USB5V to GND did not help when the power supply was connected, so perhaps the problem is radiated from the power-supply cable rather than conducted through the USB cable.
I’ve noticed problems before with noise from the laptop power supply causing problems in my analog circuits (the 90kHz interference in my ultrasound experiments), and I’ve see much bigger problems with some of the cheap Windows laptops students use. The bottom line, I guess, is that I have to tell students to run PteroDAQ from battery power, not switching-supply power, even if the power supply seems more than adequately bypassed.
2016 August 15
Using nFET body diode for log-transimpedance
Filed under: Data acquisition — gasstationwithoutpumps @ 21:20
Tags: , , ,
In Transimpedance pulse monitor does need low-pass, I realized that Schottky diodes were not going to work for the transimpedance amplifier, but I didn’t have any ordinary silicon signal diodes to test with. I’ve previously used the base-emitter junction of bipolar transistors for log amplifiers, but I decided this time to test the body diode of an nFET.
I spent a fair amount of time trying to measure the V-vs-I characteristic over a fairly wide range (though only with low currents). I ended up using several tricks:
• using several different sense resistors to measure the current
• censoring the data so that very low differential voltages across the sense resistor are not plotted
• using unity-gain buffers to provide sufficient drive for the analog-to-digital converter (essential when using large sense resistors)
• using low-pass filters after the unity-gain buffers to try to reduce 60Hz interference (not entirely successful for the largest sense resistor)
• doing fitting to estimate the effective input offset of the differential analog-to-digital input for the voltage across the sense resistor (mainly from the unity-gain buffers)
Here is the test fixture I used. (I reduced the noise a bit more on the 5.6MΩ run by increasing the low-pass filter resistors to 2.2kΩ, but that made things worse for the 120kΩ sense resistor.)
The result of all this care was one of the cleanest logarithmic response plots I’ve collected:
I have over 7 decades of data here, and the log fit is excellent over the whole range.
Measuring down to 1nA on a breadboard is not easy, as the 60Hz interference is a big problem.
I fitted the slope of the log curve by alternating between fitting the offset using the 4.7Ω data and the slope using the 120kΩ data (the 5.6MΩ data seemed a bit too noisy to me). I collected several other sets of data but the plot was too cluttered when I tried to include them, so I kept just enough to get good overlap between the ranges. I’ve got a good logarithmic fit here for about 150dB, and it looks like I could go another 20dB higher (though thermal effects might start mattering above 0.1A).
The equivalent resistance at the bottom of the current range is about 150MΩ and about 14Ω at the highest current I measured with (47mA). Because the equivalent resistance varies so much, the corner frequency of the low-pass filter made by putting a 680nF capacitor in parallel to the diode also varies a lot (17kHz@14Ω, 1.6mHz@150MΩ).
I tried using the body diode in the same minimally filtered circuit as I used for testing the Schottky diode, and got usable results in even in moderately low light:
The 60Hz noise is huge, but can be filtered out digitally.
Brighter light makes the 60Hz noise be smaller (probably because the capacitive coupling introduces a current which is a smaller fraction of the total current), but does not change the strength of the filtered signal. When I switch to very bright light (a 650 lumen bike headlamp right against the finger), then the signal gets stronger, but I had to shift the bias voltage down to 1.65V, as the DC bias on the diode got to 0.55V or more.
So the question still plaguing me: can I use a log-impedance amplifier as the second amplifier lab, given that low-pass filtering is essential?
Putting the filtering in the second and third stage is simpler than putting it in the transimpedance stage, as the corner frequency is independent of the light level then. It is sufficient to put the RC filter in just the second stage, as long as the attenuation at 60Hz is sufficient—I found that about 48dB attenuation was enough, though with just one RC element that distorts the pulse signal also, since the corner frequency has to be very low (0.24 Hz), below the 1–2Hz of the pulse. If I do low-pass filtering in both stages, I could use 3Hz cutoffs, which preserves the interesting part of the signal.
Doing two stages of low-pass filtering with 1.06Hz and 1.17Hz cutoffs provides enough suppression of the 60Hz interference that I did not need digital filtering. In medium light, I got a signal large enough to saturate the third stage (so I’d need to redesign with lower gain). With 1.6Hz and 2.6Hz cutoffs, the signal is still much larger than the 60Hz noise, and less distorted by the filters. The pulse shape is still more dependent on the filters than on the actual signal from the transimpedance amplifier, which is almost a sawtooth (hence having high-frequency components that are removed by the filter).
If the 60Hz interference is small enough that the amplifiers don’t saturate, I can eliminate it by aliasing (sampling at 60Hz, so I’m at the same place in the interference waveform on each sample). But if the 60Hz interference is too large, then the signal is clipped and aliasing can’t recover the pulse signal. So digital filtering is definitely optional here—students can get good results with just analog filters and aliasing.
Alternatively, we could look at just the transimpedance amplifier output, and use digital filtering to clean up the baseline shifts and 60Hz interference. The biggest problem is that the PteroDAQ sparkline looks like a constant—the fluctuation of a few mV is only visible once the data has been plotted on a larger scale.
2016 August 13
Transimpedance pulse monitor does need low-pass
Filed under: Circuits course,Data acquisition — gasstationwithoutpumps @ 18:16
Tags: , , ,
I’m wondering now whether I can have students do a log-pulse monitor without bandpass filtering—just high-pass to get rid of the DC signal from overall illumination. Given the new position of the lab in the course, as the second amplifier lab, I don’t really want to get too tricky with RC filtering. The “gotcha” that was a problem before is that I had to remove short glitches in the very first stage, to avoid the bandpass filter lengthening them into things that looked like pulses—I don’t want students to have to do that sort of debugging on their second amplifier lab. If I can eliminate the hardware bandpass filters, and just have them use software ones, then the lab becomes more feasible.
I was also concerned that the Schottky 1N5817 diode I had tested did not have provide gain at low currents—the low threshold voltage for Schottky diodes is a disadvantage in this application. So this morning I wired up a log-transimpedance amplifier followed by a couple of op amps as inverting amplifiers. I first tried a combined gain of 408 (which I had used before with an IR emitter as the transimpedance diode), then upped the gain to 4453. I used high-pass filters to block DC, but no low-pass filters.
The circuit was not functional without adding at least one low-pass filter (a 680nF capacitor in parallel with the diode), because the 60Hz interference saturated the amplifiers, and the smaller pulse signal was completely buried.
With the capacitor, the circuit worked fine in moderately high light, but the signal got weak in low light (due to the transimpedance amplifier having a max gain of about 35kΩ—the asymptotic equivalent resistance of the diode as current goes to 0). With just the single capacitor for filtering, the 60Hz noise was larger than the pulse signal, but a digital filter could still recover the signal:
Notch filtering does a great job of removing the 60Hz noise from this signal sampled at 360Hz.
So it looks like I do have to have students do low-pass filtering for the pulse monitor. Can I fit that into the second amplifier lab, along with the log transimpedance, or will it all get too complicated?
2016 August 12
More thoughts on log-transimpedance for pulse monitor lab
I’ve been having some more thoughts on having students do a log-transimpedance amplifier for the optical pulse monitor lab (see Pulse monitor with log-transimpedance amplifier). Previously I’ve looked at V-vs-I curves for base-emitter junctions and for the IR emitter—the silicon transistors gave me about 60mV per decade of current, and the IR emitter gave me about 105mV/decade.
I’ve been thinking of having students do the V-vs-I fitting for a simple diode. I don’t have any signal diodes at home at the moment, so I tried testing a 1N5817-TP Schottky diode (about 16¢ in 100s). I used the same setup that I used for testing power nFETs, so I could go up to a high current, but did not have good resolution at low voltages and currents.
The Schottky diode has a very similar slope to the emitter-base junctions I’ve tested in the past, but I’d really have to test down to much lower currents—we’re interested in the range 10pA to 500µA, which is buried in noise in these measurements.
I can get down to 1µA fairly easily, by eliminating the voltage dividers and just using unity-gain buffers to get low-impedance values to drive the analog-to-digital converters. I tried with four different sense resistors (470Ω, 15kΩ, 560kΩ, and 5.6MΩ) and got very consistent results. The noise levels are much lower, because the larger sense resistor and lack of voltage divider makes for much larger voltages being measured for the current-sense channel. I also used the differential ADC channel for measuring the voltage across the sense resistor, which should remove a little noise compared to taking separate measurements and subtracting them.
I have more confidence in the 60.2mV/decade and 0.37V offset from these measurements than the high-current measurements I did for the first plot.
At low currents, the diode behaves more like a 33kΩ resistor than like a logarithmic element.
The 60.2mV/decade fit seems pretty good from 10µA to 10mA, and the noisier high-current measurements suggest that it is good to 100mA. The sensitivity is less below 10µA and more above 10mA, behaving almost like a fixed 33kΩ resistor at low currents.
I can get a pretty good fit over a wide range with a three-parameter model of the equivalent resistance as a function of current: a resistor in parallel with a device that has a power-law fit for resistance as a function of current:
There is no theoretical justification for this model, but it seems to match the data better than the standard voltage-as-logarithm-of-current model, at least at low currents.
At low currents, the Schottky diode acts like a 35kΩ resistor, but at high currents, the voltage seems to be 0.409 I. This model seems to fit to better than 10% over 7 decades, which is not too bad for a 3-parameter model!
I’m wondering now whether I can have students do a log-pulse monitor without bandpass filtering—just high-pass to get rid of the DC signal from overall illumination. Given the new position of the lab in the course, as the second amplifier lab, I don’t really want to get too tricky with RC filtering. The “gotcha” that was a problem before is that I had to remove short glitches in the very first stage, to avoid the bandpass filter lengthening them into things that looked like pulses—I don’t want students to have to do that sort of debugging on their second amplifier lab. If I can eliminate the hardware bandpass filters, and just have them use software ones, then the lab becomes more feasible.
2016 June 25
Pulse monitor with log-transimpedance amplifier
Filed under: Uncategorized — gasstationwithoutpumps @ 02:04
Tags: , , , ,
I’ve been planning since the Santa Cruz Mini Maker Faire to wire up an optical pulse monitor with a log-transimpedance amplifier as the first stage, so that I could use the pulse monitor in full sun or in a dimly lit room, with a dim green LED or with a bright infrared LED. The idea is to make the output of the first stage proportional to the log of the photocurrent, rather than to the photocurrent, then use a band-pass filter to get rid of the DC component and any 60Hz fluctuation, leaving only the fluctuation due to the pulse.
This pulse signal should be independent of the overall light level but on the absorbance of the finger, because
$\log(I) = c + \log(\mbox{transmitted}) = c+ \log(\mbox{illumination}) + \log(\mbox{transmitted}/\mbox{illumination}$, for some constant $c$. If the illumination is constant or has only high-frequency components, then the bandpass filter will eliminate both $c$ and $\log(\mbox{illumination})$, leaving only the absorbance $\log(\mbox{transmitted}/\mbox{illumination})$.
I deliberately did not start working on it until I had finished my grading for the quarter, so only got it built last week, just before going to Montreal for a family reunion of my wife’s family. So I’m only now getting around to blogging about it.
To make the log-transimpedance amplifier, I need a component where the voltage is proportional to the log of the current. For this I used a diode-connected PNP transistor:
The base-to-emitter diode has a current that is exponential in the voltage, and the collector-to-emitter current is proportional to the base-to-emitter current, at least until the transistor approaches saturation (which starts around 10mA).
The A1015 PNP transistor has a voltage proportional to the log of current, with about 60mV/decade. I did not use a unity-gain buffer when measuring the voltage and current, connecting the Teensy ADC channels A10 and A11 directly to the emitter and base+collector of the transistor. Measurements at less than 5µA were difficult, because the high impedance of the sense resistor made the ADC measurements inaccurate.
I tried a pulse monitor using the A1015 PNP transistor as the log-impedance element, and it worked ok, but I can do better, I think, using an IR LED as the log-impedance element:
The WP710A10F3C IR LED has a low forward voltage, and can be used from 100nA to 30mA, given that we don’t need high accuracy on the log function. We get about 105mV/decade, so it is more sensitive than the A1015 transistor. Note: I did use a unity-gain butter for these measurements, which allowed me to get down to about 50nA—still much higher than the photocurrents I observed in very low light.
The IR LED has a wide range over which the voltage is the logarithm of the current, or $\frac{dV}{dI} \approx 241mV/I$. For 10nA, the equivalent gain is about 24MΩ, and for 1µA, the gain is about 240kΩ. For 10pA (about the smallest current I’ve observed for operating the pulse monitor in very dim light), the equivalent gain is 24GΩ.
This amplifier uses only 3 op amps: a log-transimpedance stage with an IR LED as the impedance and two bandpass inverting amplifiers.
The 330pF capacitor in parallel with the log-impedance is very important—without it I get very short glitches which the next two stages lengthen into long glitches in the passband of the filters. Making the capacitor larger reduces the glitches, but makes the corner frequency of the effective low-pass filter too low when light levels are very low, and the signal is attenuated. Any smaller, and the glitches don’t get adequately removed.
I have tested the pulse monitor over a wide range of light levels, with a DC output of the first stage from 234mV to 1.033V, corresponding to photocurrents of 11pA to 463µA, a range of 42 million (7.6 decades). At very low light levels, the signal tends to be buried in 60Hz interference, but if I ground myself, it is still usable.
In very low light, the capacitive coupling of 60Hz noise buries the signal, but the bandpass filters help recover it.
At high light levels, it is easy to get clean signals, as the 60Hz interference is swamped out by the large photocurrent.
Note that the voltage swing is almost independent of the overall light level, as it depends only on the percentage fluctuation in opacity of the finger, which depends mainly on how much pressure is applied. If you get the pressure on the finger close to the mean arterial pressure (so that the finger throbs), you can get quite a large change in opacity—I’ve computed changes of 17% in opacity.
« Previous PageNext Page » | 2021-04-10 15:05:15 | {"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": 0, "img_math": 6, "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": 0.6723819375038147, "perplexity": 1906.9077160474433}, "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/1618038057142.4/warc/CC-MAIN-20210410134715-20210410164715-00188.warc.gz"} |
https://intuitivephysics.me/faraday-derivation-part1 | # Faraday's Law of Electromagnetic Induction, derivation and misconceptions.
Written 08 May 2021 by Michael Huang
### Part 1
This is part 1 of two articles on Faraday’s Law. In the first part, I attempt to clear up some confusions about the different forms of Faraday’s Law and provide a simple derivation of Faraday’s Law. In the second part, we explore some quirks and “exceptions” of Faraday’s Law.
### Contents
#### Part 1
1. The 3 confusing forms of Faraday’s Law
2. Derivation of Faraday’s Law and motional EMF
3. Grand Conclusion and last thoughts
#### Part 2
1. Several exceptions to Faraday’s Law and why they are exceptions
2. Final Puzzle
See Maxwell’s Equations for a brief and intuitive intro to all Maxwell’s Equations and Lorentz force law.
I’ve read 3 different textbooks on electromagnetism. Each provided a different description of what it defines as Faraday’s Law.
#### Definitions
##### First Definition:
$$\mathcal{E}=-\frac{\partial \phi_{B\space{swept}}}{\partial t}$$, or in other words, the net EMF equals the negative of the amount of “magnetic field lines” swept across by the “wire” per unit time.
##### Second Definition:
$\mathcal{E}=-\frac{\partial \phi_{B\space enclosed}}{\partial t}$, or the net EMF equals the negative of the amount of magnetic flux change per unit time in some “enclosed surface”.
##### Third Definition:
This is similar to the first. $\mathcal{E_{motional}}=-\frac{\partial \phi_{B\space{swept}}}{\partial t}$, or in other words, the motional EMF equals the negative of the amount of “magnetic field” swept across by the “wire” per unit time.
##### Fourth Definition:
This is actually called Maxwell-Faraday’s Law, which is a separate thingy, but people still mix it with Faraday’s Law. $\nabla \times E=-\frac{\partial B}{\partial t}$, or in more familiar integral form $\oint\limits_{\partial \sigma}E\cdot dl=-\iint\limits_{\sigma} \frac{\partial B}{\partial t}\cdot dA$. Notice that it’s a statement regarding $B$ field, and $E$ field themselves, and has nothing to do with the shape of the wire or the velocity of the wire.
All these seem confusing. And to confuse you more, the four definitions are not equivalent. Some are applicable in more cases than others.
For example, the first definition is extremely hand-wavy (though it might be the most original to what Faraday came up with). It’s flawed in many cases. The second definition holds in general, until the concept of “enclosed surface” loses its meaning. The third definition holds unless the concept of “wire” is undefined. The fourth is again, a separate law. It’s one of the four Maxwell’s Equations. Therefore, it’s fundamental and works in all cases (where Maxwell’s electromagnetism works). I will explain all these later.
#### Motional EMF vs Induced EMF
EMF is defined as follows:
$\mathcal{E}_s = \frac{1}{q}\int\limits_s F\cdot dl$
The electromotive “force” (a bad name) a long a path $s$ is defined as the work per unit charge along that path. In other words, if you move an imaginary charged particle of charge $q$ along the path $s$, compute the work done on the charge $q$ by the electromagnetic force, then you can divide it by $q$ to get the EMF along that path.
The electromotive force (EMF) we defined here is the net EMF.
Since there’re two parts to $F$, specifically $F=F_E+F_B$, we can separate the electromagnetic force into electrostatic force and magnetic force.
$\mathcal{E}_s = \frac{1}{q}\int\limits_s F\cdot dl$ $=\frac{1}{q}\int\limits_s F_E\cdot dl + \frac{1}{q}\int\limits_s F_B\cdot dl$
Recall Lorentz force law, $F_B=qv\times B, \space F_E=qE$ (note $v,B,E$ are all vectors, so are $F_B, F_E$). We plug that in, notice the $q$ cancels (it should, because that’s the point of the $\frac{1}{q}$ in the definition)
$\mathcal{E}_s = \int\limits_s E\cdot dl + \int\limits_s (v\times B)\cdot dl$
The first part, $\int\limits_s E\cdot dl$ is called the induced EMF, the second part $\int\limits_s (v\times B)\cdot dl$ is called the motional EMF.
Apparently, the $E\cdot dl$ can only be generated (assuming it’s all due to magnetic interactions) if there’s a changing $B$ field. This is because changing $B$ field creates a curl in $E$, without it, and without any charge present (as in a circuit), $E=0$. So induced EMF is named so because it’s induced by a changing magnetic field.
On the other hand, the second half has nothing to do with a changing magnetic field. Instead, it requires motion of the path or wire, which is why it’s named motional EMF.
The first part also gives you the electrostatic voltage across path $s$. As you can see, the form of the induced EMF is close to the form of Maxwell-Faraday’s Law, so we can use Maxwell-Faraday’s Law to compute the induced EMF for a closed loop (because Maxwell-Faraday’s Law integral form only works in a closed loop).
The second part is trickier, we will derive directly from it what I called the Motional Faraday’s Law, basically the third definition of Faraday’s Law
We want to derive $\mathcal{E}=-\frac{\partial \phi_{B\space enclosed}}{\partial t}$ (our second definition). Let’s first derive the motional EMF and combine with the induced EMF to obtain net EMF.
#### Motional part of Faraday’s Law derivation
We wish to derive $\mathcal{E_{motional}}=-\frac{\partial \phi_{B\space{swept}}}{\partial t}$.
$\int\limits_s (v\times B)\cdot dl$
At first sight, it seems this must evaluate to 0, because $v\parallel dl$, right?
$v\parallel dl$ when the wire/path is stationary! But the whole point of motional EMF is that now the wire and the path are not stationary.
For example, the wire can be moving rightward while the charged particle is traveling up in the wire.
$v$ denotes the total motion relative to the magnetic field (right and up). $dl$ just points up. The same can be said of the concept of path, while the path relative to magnetic field is right and up, we are only concerned about the EMF along the upward direction (we will explore why is that in later articles), so the $dl$ points up, again.
Therefore $v=v_{along}+v_{of}$, or the net velocity = velocity along the wire or the predefined direction of the path + velocity of the wire or the predefined path at that point.
$v_{along}\parallel dl$, but $v_{of}\not\parallel dl$!
we only need to consider $v_{of}$, so $\mathcal{E_{motional}}=\int\limits_s (v_{of}\times B)\cdot dl$
Notice that $(v_{of}\times B)\cdot dl$ is a box product.
It represents the volume of the parallelepiped formed by the three vectors $v_{of}, B, dl$; so a natural question to ask is what is the physical significance of this volume? I will leave that for the readers to juggle with.
To compute the same volume, we can also swap the order of the vectors a bit.
$(v_{of}\times B)\cdot dl=(dl\times v_{of})\cdot B$
Now, this quantity is much earsier to understand. $(dl\times v_{of})$ represents the rate at which area is swept by the segment $dl$ when it moves at velocity $v_{of}$! So $dl\times v_{of}=-d\frac{dA_{of \space dl}}{dt}$. There’s a little $d$ in the front to denote that this rate of area-sweep is infinitesimally small because $dl$ is infinitesimally small. The negative sign is of little importance. It’s due to the definition of the direction of area using righthand rule.
If you prefer, we can stay away from calculus notations, and use $\Delta l \times \frac{\Delta x}{\Delta t}=\frac{\Delta l \times \Delta x}{\Delta t}=-\frac{\Delta^2 A}{\Delta t}=-\Delta \frac{\Delta A}{\Delta t}$. Again, the double $\Delta$ is to indicate the smallness of the area.
Apparently, $-\frac{dA_{of \space dl}}{dt}\cdot B=-\frac{dA_{of\space dl}\cdot B}{dt}=-\frac{d\phi_B}{dt}$, or in other words, the rate at which we sweep area times the $B$ field equals the rate at which we sweep magnetic flux.
So plugging it back in to the integral,
$\mathcal{E_{motional}}=\int\limits_s -d \frac{d\phi_B}{dt} = -\frac{d\phi_{B\space swept \space by \space s}}{dt}$
Thus we have derived the third definition of Faraday’s Law, which I like to call Motional Faraday’s Law.
One last note: The only fundamental theorem of electromagnetims we used above is the Lorentz Force Law. So, in other words, this Motional Faraday’s Law should have equal predictive power to the Lorentz Force Law. All things predictable by Motional Faraday’s Law should be predicted by Lorentz Force Law, and vice versa. In some cases, Lorentz force law is much easier (as in the “exceptions” of Motional Faraday’s Law)
#### Derivation of Loop form of Faraday’s Law
Now, we can easily derive the loop form of the Faraday’s Law from this “sweeping” form. This is included in many textbooks. The basic idea is to note that when we sweep $s$, all the $\phi_B$ we sweep across either moves from $\phi_{inside\space loop}\rightarrow \phi_{outside \space loop}$ or $\phi_{outside\space loop}\rightarrow \phi_{inside \space loop}$
So if $s$ is part of a loop, the $\frac{d\phi_{B\space swept \space by \space s}}{dt}$ just goes into $\frac{d\phi_{B\space inside\space loop\space due\space to\space change\space in\space \sigma}}{dt}$; the deduction of the sign requires some care, and so I will not bore you with that. The importance of “due to change in $\sigma$” will be evident later.
Now we arrive at:
$\mathcal{E_{motional\space in\space \sigma}}= -\frac{d\phi_{B\space inside\space loop\space due\space to\space change\space in\space \sigma}}{dt}$
Which is close to the complete Loop version of Faraday’s Law (definition 2)
To obtain the complete loop version of Faraday’s Law, we must analyze $\mathcal{E}_{induced}$, or the induced EMF.
Luckily, that’s given simply by the Maxwell-Faraday’s Law.
Recall $\mathcal{E_{induced}}=\oint\limits_{\partial \sigma}E\cdot dl$
$=-\iint\limits_{\sigma} \frac{\partial B}{\partial t}\cdot dA$
$=-\frac{\partial \phi_{B\space due \space to \space change\space in\space B}}{\partial t}$
Now we add $\mathcal{E_{motional}}, \space \mathcal{E_{induced}}$.
$\mathcal{E_{net}}=\mathcal{E_{motional}}+\mathcal{E_{induced}}$
$=-\frac{\partial\phi_{B\space inside\space loop\space due\space to\space change\space in\space \sigma}}{\partial t}-\frac{\partial \phi_{B\space due \space to \space change\space in\space B}}{\partial t}$
Notice two subtleties:
I changed the full derivative to partial derivative to make it more accurate, since other variables like location can influence $\phi_B$ as well. If you don’t know what I’m talking about, don’t let it bother you.
The two $\frac{\partial \phi_B}{\partial t}$ are mutually exclusive because they are due to different causes. In fact “change in $B$” and “change of $\sigma$” are the only possible causes of a changing flux. Change in “angle” is part of either changing $B$ or changing $\sigma$. So they add together to $\frac{\partial \phi_{B\space net}}{\partial t}$
Great, now behold the full glamour of Faraday’s Law as we have derived it:
$\mathcal{E_{net}}=-\frac{\partial \phi_{B\space net}}{\partial t}$
### Conclusion
To recap what just happened. We first explained the difference of motional EMF and induced EMF. Motional EMF is EMF created due to movement of wire/path in the magnetic field. Induced EMF is EMF created due to change of magnetic field.
Another way to think of it is, Motional EMF is due to $v\times B$ part of the force, while Induced EMF is due to $E$ part of the eletromagnetic force.
There are several definitions of Faraday’s Law. The fourth definition we gave is actually called Maxwell-Faraday’s Law. It should not be confused with Faraday’s Law, for it’s more fundamental and only deals with induced EMF.
Some versions of Faraday’s Law deal with motional EMF as in the third definition. Faraday’s Law dealing with Motional EMF can come in 2 flavors: “sweeping wire flavor” and “closed loop flavor.” They are effectually equivalent. Other versions combine the third version with Maxwell-Faraday’s Law to create a statement that handles both motional EMF and induced EMF. This version is in “closed loop flavor,” because Maxwell-Faraday’s Law is hard to manipulate when there isn’t a closed loop.
The first definition involving magnetic field line is problematic (explored in part 2), but is probably closer to what Faraday originally proposed.
Last but not least, I offered a simple proof of Faraday’s Law using Lorentz Force Law and Maxwell-Faraday’s Law. Because of this proof, we can conclude that Faraday’s Law has the same predictive power as Lorentz Force Law + Maxwell-Faraday’s Law.
See part 2 for examples of spectacular failures of Faraday’s Law (mostly provided by Feynman) and why they make sense (according to me, at least)! | 2022-09-28 05:45:25 | {"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": 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": 0.8329235315322876, "perplexity": 469.5531066624249}, "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/1664030335124.77/warc/CC-MAIN-20220928051515-20220928081515-00028.warc.gz"} |
https://www.techwhiff.com/issue/william-lloyd-garrison-published-an-abolitionist-newspaper--124437 | # William lloyd garrison published an abolitionist newspaper called
###### Question:
William lloyd garrison published an abolitionist newspaper called
### The goals an organization sets in its human resource planning process should come directly from the analysis of its labor supply and demand. True False
The goals an organization sets in its human resource planning process should come directly from the analysis of its labor supply and demand. True False...
### What are free trade agreements? a. agreements that set new quotas b. laws that create tariffs c. agreements that remove trade barriers d. laws that prohibit trade between countries
What are free trade agreements? a. agreements that set new quotas b. laws that create tariffs c. agreements that remove trade barriers d. laws that prohibit trade between countries...
### What animal did a snake come from
What animal did a snake come from...
### Why did the idea of a “chief executive” scare many citizens during the ratification period?
Why did the idea of a “chief executive” scare many citizens during the ratification period?...
### Help! Need this really fast! :(
Help! Need this really fast! :(...
### How can water be used for energy?
How can water be used for energy?...
### On a map, 1 inch equals 20 miles. Two cities are 5 inches apart on the map. What is the actual distance between the cities?
On a map, 1 inch equals 20 miles. Two cities are 5 inches apart on the map. What is the actual distance between the cities?...
### 10. Explain the relationship you see for each chemical between the temperature and the state of matter 11. Referencing your drawings, what happens to the space between particles as you go from solid to liquid to gas? 12. Referencing your drawings, what happens to the speed of the particles as you go from solid to liquid to gas?
10. Explain the relationship you see for each chemical between the temperature and the state of matter 11. Referencing your drawings, what happens to the space between particles as you go from solid to liquid to gas? 12. Referencing your drawings, what happens to the speed of the particles as you go...
### How did the practice of trench warfare lead to a stalemate
how did the practice of trench warfare lead to a stalemate...
### Why do the bones in your body KRACK!!
Why do the bones in your body KRACK!!...
### A project is expected to produce cash flows of $12,800,$15,900, and $18,000 over the next 3 years, respectively. After 3 years, the project will be discontinued. What is this project worth today at a discount rate of 12.5 percent? Select one: A.$36,808.17 B. $35,758.00 C.$34,435.74 D. $35,201.76 E.$36,582.72
A project is expected to produce cash flows of $12,800,$15,900, and $18,000 over the next 3 years, respectively. After 3 years, the project will be discontinued. What is this project worth today at a discount rate of 12.5 percent? Select one: A.$36,808.17 B. $35,758.00 C.$34,435.74 D. \$35,201.76 ...
### What is the gcf of 16 and 12 with work
What is the gcf of 16 and 12 with work...
### Si el perímetro de el siguiente paralelogramo mide 80 M cuanto mide la base cuanto mide la altura y cuanto mide el área
Si el perímetro de el siguiente paralelogramo mide 80 M cuanto mide la base cuanto mide la altura y cuanto mide el área...
### A caregiver's love, affection, and acceptance are unconditional if they're given regardless of the child's behavior or expectations the child has met. Please select the best answer from the choices provided T F
A caregiver's love, affection, and acceptance are unconditional if they're given regardless of the child's behavior or expectations the child has met. Please select the best answer from the choices provided T F...
### Which are acceptable names for the plane? Choose all answers that are correct. A. Plane AU B. Plane JAU C. Plane R D. Plane ABC
Which are acceptable names for the plane? Choose all answers that are correct. A. Plane AU B. Plane JAU C. Plane R D. Plane ABC...
### What risk come with not knowing about history
What risk come with not knowing about history...
### Ionic bonds form between two ions that have
ionic bonds form between two ions that have...
### Si mi quiz valía 40 puntos y yo saqué 17 a que nota equivale eso?
Si mi quiz valía 40 puntos y yo saqué 17 a que nota equivale eso?... | 2022-10-04 10:31:00 | {"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": 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": 0.21017959713935852, "perplexity": 7975.077208160211}, "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-2022-40/segments/1664030337490.6/warc/CC-MAIN-20221004085909-20221004115909-00039.warc.gz"} |
https://mailman.ntg.nl/pipermail/ntg-context/2019/095572.html | # [NTG-context] Bug in lmtx in display mathmode
Hans Hagen j.hagen at xs4all.nl
Mon Aug 19 08:53:30 CEST 2019
On 8/19/2019 12:20 AM, Otared Kavian wrote:
> Hi Hans,
>
> I have a long document which typesets correctly with mkiv, but I get a (cryptic for me…) fatal error message when I try to typeset it in lmtx:
>
> ! error: (nodes): trying to delete an attribute reference of a non attribute list node 4476mtx-context | fatal error: return code: 256
>
> I could finally find the origin of the error, which is exposed in the following example: if you comment out the part from \startformula ... \stopformula, the file typesets correctly.
> But with this part, one gets the above error message, the culprit being \liminfbar which is defined in the first line. Note that \liminfbar is accepted when it is inline math.
>
> Best regards: OK
>
> %%% begin bug-math-lmtx.tex
>
> \definemathcommand[liminfbar][limop] {\underline{\mfunctionlabeltext{lim}}}
>
> \starttext
> If $(u_{n})_{n}$ is a sequence of real numbers, we define its {\it limit inferior}, denoted $\liminfbar$ by setting:
> \startformula
> a_{n} := \inf\{u_{k} \; ; \; k \geq n \},
> \liminfbar_{n\to\infty} u_{n} := \sup_{n \geq 1} a_{n}.
> \stopformula
>
> \stoptext
>
> %%% end bug-math-lmtx.tex
also in yesterdays beta?
Hans
----------------------------------------------------------------- | 2019-12-14 13:47:33 | {"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": 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": 0.9805062413215637, "perplexity": 8450.30637893741}, "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-2019-51/segments/1575541157498.50/warc/CC-MAIN-20191214122253-20191214150253-00085.warc.gz"} |
https://answers.launchpad.net/moderncv/+question/178261 | # Tabular in a moderncv
Hi,
it appears that it's not possible (i.e. no border, no spaces...) to include a tabular in a moderncv document (neither tabularx or array in math mode). I guess it's because some variables are modified in the style file of moderncv.
Any hint to override this?
Best,
F
## Question information
Language:
English Edit question
Status:
Solved
For:
moderncv Edit question
Assignee:
Xavier Danaux Edit question
Solved by:
Flo__
Solved:
2011-11-16
Last query:
2011-11-16
2011-11-16
Xavier Danaux (xdanaux) said on 2011-11-16: #1
Hi Florian,
Any chance to include a simple example showing the error you get when trying to include a tabular? Moderncv uses tabulars extensively, so there should be no reason you can't use them yourself...
Kind regards,
Xavier
Flo__ (flooo) said on 2011-11-16: #2
Hi,
I don't have any compilation error, it's just that it doesn't render any tabular on the pdf.
For example, the following code :
$\begin{array}{|c|c|c|} \hline AA & BB & CC \\ \hline DD & EE & FF \end{array}$
\begin{tabular}{|c|c|c|}
\hline aa & bb & cc \\
\hline dd & ee & ff \\
\hline
\end{tabular}
Best,
F
Xavier Danaux (xdanaux) said on 2011-11-16: #3
Hi Florian,
The array and the tabular are actually rendered perfectly. If you would make the cell contents less similar, you would be able to see that the layout is perfectly rendered as requested.
You are not seeing any vertical line because you have not defined their width, hence they are rendered with a zero width...
What you are looking for, is to set \arrayrulewidth to a non-zero value. You might also want to similarly set \arraycolsep and \tabcolsep to non-zero values to create some space between the columns.
Kind regards,
Xavier
Flo__ (flooo) said on 2011-11-16: #4
Hi,
Ooups, my bad. In the usual classes, i guess these values are not set to 0 by default.
To solve my problem, I just uncommented the following line of the cls file :
% TO BE TESTED
\setlength\arraycolsep{5\p@}
\setlength\tabcolsep{6\p@}
\setlength\arrayrulewidth{.4\p@}
Thanks !
F
Dimitrios Apostolou (jimis) said on 2012-09-27: #5
Hello, can't this be fixed inside moderncv? I just spent one hour trying to find out what's wrong with my use of tabular, until I found the answer at [1]. Thanks.
Xavier Danaux (xdanaux) said on 2012-10-31: #6
I just set \arrayrulewidth to its standard value of 0.4pt by default, but won't set \arraycolsep or \tabcolsep to non-zero values as of now / yet, as moderncv uses tabulars extensively and the code would need to be adapted to take those extra spaces into account. | 2020-12-02 13:20:20 | {"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": 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": 0.7505852580070496, "perplexity": 4185.619458924894}, "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/1606141708017.73/warc/CC-MAIN-20201202113815-20201202143815-00685.warc.gz"} |
https://raweb.inria.fr/rapportsactivite/RA2021/epione/index.html | 2021
Activity report
Project-Team
EPIONE
RNSR: 201822641L
Research center
Team name:
E-Patient: Images, Data & MOdels for e-MediciNE
Domain
Digital Health, Biology and Earth
Theme
Computational Neuroscience and Medicine
Creation of the Project-Team: 2018 May 01
# Keywords
• A3.3. Data and knowledge analysis
• A3.4. Machine learning and statistics
• A4.4. Security of equipment and software
• A4.8. Privacy-enhancing technologies
• A5.2. Data visualization
• A5.3. Image processing and analysis
• A5.4. Computer vision
• A5.6. Virtual reality, augmented reality
• A5.9. Signal processing
• A6.1. Methods in mathematical modeling
• A6.2. Scientific computing, Numerical Analysis & Optimization
• A6.3. Computation-data interaction
• A8.3. Geometry, Topology
• A9. Artificial intelligence
• A9.2. Machine learning
• A9.3. Signal analysis
• A9.6. Decision support
• A9.7. AI algorithmics
• A9.9. Distributed AI, Multi-agent
• B2.2. Physiology and diseases
• B2.3. Epidemiology
• B2.4. Therapies
• B2.6. Biological and medical imaging
• B2.6.1. Brain imaging
• B2.6.2. Cardiac imaging
• B2.6.3. Biological Imaging
# 1 Team members, visitors, external collaborators
## Research Scientists
• Nicholas Ayache [Team leader, Inria, Senior Researcher, HDR]
• Irene Balelli [Inria, ISFP, from Oct 2021]
• James Benn [Inria, Starting Research Position]
• Hervé Delingette [Inria, Senior Researcher, HDR]
• Marco Lorenzi [Inria, Researcher, HDR]
• Marta Nunez Garcia [Inria, Starting Research Position, from Nov 2021]
• Xavier Pennec [Inria, Senior Researcher, HDR]
• Maxime Sermesant [Inria, Senior Researcher, HDR]
## Post-Doctoral Fellows
• Irene Balelli [Inria, until Sep 2021]
• Francisco Burgos [Université polytechnique de Catalogne - Espagne, from Feb 2021]
• Anna Calissano [Inria, from May 2021]
• Marie Deprez [Inria]
• Marta Nunez Garcia [Univ de Bordeaux, until Oct 2021]
• Dimbihery Rabenoro [Inria]
• Jesus Jairo Rodriguez Padilla [Inria, from Jul 2021]
## PhD Students
• Clement Abi Nader [Université Côte d'Azur, until Jun 2021]
• Luigi Antelmi [Univ Côte d'Azur, until Jul 2021]
• Benoit Audelan [Univ Côte d'Azur, until Jul 2021]
• Tania Marina Bacoyannis [Inria]
• Jaume Banus Cobo [Inria, until Sep 2021]
• Paul Blanc-Durand [Université Paris Descartes]
• Nicolas Cedilnik [Inria, until Feb 2021]
• Mikael Chelli [Centre hospitalier universitaire de Nice]
• Gaetan Desrues [Inria]
• Zhijie Fang [Inria, from Oct 2021]
• Yann Fraboni [Inria, Accenture Labs Sophia Antipolis, CIFRE]
• Nicolas Guigui [Inria]
• Dimitri Hamzaoui [Univ Côte d'Azur]
• Josquin Harrison [Inria]
• Etrit Haxholli [Inria]
• Florent Jousse [QuantificCare, CIFRE]
• Victoriya Kashtanova [Inria]
• Huiyu Li [Inria, from Sep 2021]
• Buntheng Ly [Univ Côte d'Azur]
• Elodie Maignant [Inria]
• Morten Pedersen [Inria]
• Santiago Smith Silva Rincon [Univ Côte d'Azur]
• Hari Sreedhar [Univ Côte d'Azur]
• Riccardo Taiello [Inria, from Oct 2021]
• Yann Thanwerdas [Univ Côte d'Azur]
• Paul Tourniaire [Inria]
• Clair Vandersteen [Centre hospitalier universitaire de Nice]
• Zihao Wang [Inria]
• Yingyu Yang [Inria]
## Interns and Apprentices
• Hava Chaptoukaev [Inria, from Apr 2021 until Aug 2021]
• Pierre Lindet [Inria, from May 2021 until Jul 2021]
• Hippolyte Mayard [Inria, from May 2021 until Sep 2021]
• Andrea Senacheribbe [Inria, until Feb 2021]
• Isabelle Strobant [Inria]
## Visiting Scientists
• Marc Olivier Gauci [Centre hospitalier universitaire de Nice, from Jun 2021 until Aug 2021]
• Pamela Moceri [Université Côte d'Azur]
• Mihaela Pop [Inria, from Sep 2021]
## External Collaborators
• Guillaume Lajoinie [University of Twente, until Jun 2021]
• Marco Milanesio [Université Côte d'Azur]
# 2 Overall objectives
## 2.1 Description
Our long-term goal is to contribute to the development of what we call the e-patient (digital patient) for e-medicine (digital medicine).
• the e-patient (or digital patient) is a set of computational models of the human body able to describe and simulate the anatomy and the physiology of the patient's organs and tissues, at various scales, for an individual or a population. The e-patient can be seen as a framework to integrate and analyze in a coherent manner the heterogeneous information measured on the patient from disparate sources: imaging, biological, clinical, sensors, ...
• e-medicine (or digital medicine) is defined as the computational tools applied to the e-patient to assist the physician and the surgeon in their medical practice, to assess the diagnosis/prognosis, and to plan, control and evaluate the therapy.
The models that govern the algorithms designed for e-patients and e-medicine come from various disciplines: computer science, mathematics, medicine, statistics, physics, biology, chemistry, etc. The parameters of those models must be adjusted to an individual or a population based on the available images, signals and data. This adjustment is called personalization and usually requires solving difficult inverse problems. The overall picture of the construction of the personalized e-patient for e-medicine was presented at the College de France through an inaugural lecture and a series of courses and seminars (fr), concluded by an international workshop.
## 2.2 Organisation
The research organization in our field is often built on a virtuous triangle. On one vertex, academic research requires multidisciplinary collaborations associating informatics and mathematics to other disciplines: medicine, biology, physics, chemistry ... On a second vertex, a clinical partnership is required to help defining pertinent questions, to get access to clinical data, and to clinically evaluate any proposed solution. On the third vertex, an industrial partnership can be introduced for the research activity itself, and also to transform any proposed solution into a validated product that can ultimately be transferred to the clinical sites for an effective use on the patients.
Keeping this triangle in mind, we choose our research directions within a virtuous circle: we look at difficult problems raised by our clinical or industrial partners, and then try to identify some classes of generic fundamental/theoretical problems associated to their resolution. We also study some fundamental/theoretical problems per se in order to produce fundamental scientific advances that can help in turn to promote new applications.
# 3 Research program
## 3.1 Introduction
Our research objectives are organized along 5 scientific axes:
1. Biomedical Image Analysis & Machine Learning
2. Imaging & Phenomics, Biostatistics
3. Computational Anatomy, Geometric Statistics
4. Computational Physiology & Image-Guided Therapy
5. Computational Cardiology & Image-Based Cardiac Interventions
For each scientific axis, we introduce the context and the long term vision of our research.
## 3.2 Biomedical Image Analysis & Machine Learning
The long-term objective of biomedical image analysis is to extract, from biomedical images, pertinent information for the construction of the e-patient and for the development of e-medicine. This relates to the development of advanced segmentation and registration of images, the extraction of image biomarkers of pathologies, the detection and classification of image abnormalities, the construction of temporal models of motion or evolution from time-series of images, etc.
In addition, the growing availability of very large databases of biomedical images, the growing power of computers and the progress of machine learning (ML) approaches have opened up new opportunities for biomedical image analysis.
This is the reason why we decided to revisit a number of biomedical image analysis problems with ML approaches, including segmentation and registration problems, automatic detection of abnormalities, prediction of a missing imaging modality, etc. Not only those ML approaches often outperform the previous state-of-the-art solutions in terms of performances (accuracy of the results, computing times), but they also tend to offer a higher flexibility like the possibility to be transferred from one problem to another one with a similar framework. However, even when successful, ML approaches tend to suffer from a lack of explanatory power, which is particularly annoying for medical applications. We also plan to work on methods that can interpret the results of the ML algorithms that we develop.
## 3.3 Imaging & Phenomics, Biostatistics
The human phenotype is associated with a multitude of heterogeneous biomarkers quantified by imaging, clinical and biological measurements, reflecting the biological and patho-physiological processes governing the human body, and essentially linked to the underlying individual genotype. In order to deepen our understanding of these complex relationships and better identify pathological traits in individuals and clinical groups, a long-term objective of e-medicine is therefore to develop the tools for the joint analysis of this heterogeneous information, termed Phenomics, within the unified modeling setting of the e-patient.
To date the most common approach to the analysis of the joint variation between the structure and function of organs represented in medical images, and the classical -omics modalities from biology, such as genomics or lipidomics, is essentially based on the massive univariate statistical testing of single candidate features out of the many available. This is for example the case of genome-wide association studies (GWAS) aimed at identifying statistically significant effects in pools consisting of up to millions of genetics variants. Such approaches have known limitations such as multiple comparison problems, leading to underpowered discoveries of significant associations, and usually explain a rather limited amount of data variance . Although more sophisticated machine learning approaches have been proposed, the reliability and generalization of multivariate methods is currently hampered by the low sample size relatively to the usually large dimension of the parameters space.
To address these issues this research axis investigates novel methods for the integration of this heterogeneous information within a parsimonious and unified multivariate modeling framework. The cornerstone of the project consists in achieving an optimal trade-off between modeling flexibility and ability to generalize on unseen data by developing statistical learning methods informed by prior information, either inspired by "mechanistic" biological processes, or accounting for specific signal properties (such as the structured information from spatio-temporal image time series). Finally, particular attention will be paid to the effective exploitation of the methods in the growing Big Data scenario, either in the meta-analysis context, or for the application in large datasets and biobanks.
Federated learning in multi-centric studies. The current research scenario is characterised by medium/small scale (typically from 50 to 1000 patients) heterogeneous datasets distributed across centres and countries. The straightforward extension of learning algorithms successfully applied to big data problems is therefore difficult, and specific strategies need to be envisioned in order to optimally exploit the available information. To address this problem, we focus on learning approaches to jointly model clinical data localized in different centres. This is an important issue emerging from recent large-scale multi-centric imaging-genetics studies in which partners can only share model parameters (e.g. regression coefficients between specific genes and imaging features), as represented for example by the ENIGMA imaging-genetics study, led by the collaborators at University of Southern California. This problem requires the development of statistical methods for federated model estimation, in order to access data hosted in different clinical institutions by simply transmitting the model parameters, that will be in turn updated by using the local available data. This approach is extended to the definition of stochastic optimization strategies in which model parameters are optimized on local datasets, and then summarized in a meta-analysis context. Finally, this project studies strategies for aggregating the information from heterogeneous datasets, accounting for missing modalities due to different study design and protocols. The developed methodology finds important applications within the context of Big Data, for the development of effective learning strategies for massive datasets in the context of medical imaging (such as with the UK biobank), and beyond.
## 3.4 Computational Anatomy, Geometric Statistics
Computational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at developing algorithms to model and analyze the biological shape of tissues and organs. The goal is not only to establish generative models of organ anatomies across diseases, populations, species or ages but also to model the organ development across time (growth or aging) and to estimate their variability and link to other functional, genetic or structural information. Computational anatomy is a key component to support computational physiology and is evidently crucial for building the e-patient and to support e-medicine.
Pivotal applications include the spatial normalization of subjects in neuroscience (mapping all the anatomies into a common reference system) and atlas to patient registration to map generic knowledge to patient-specific data. Our objectives will be to develop new efficient algorithmic methods to address the emerging challenges described below and to generate precise specific anatomical model in particular for the brain and the heart.
The objects of computational anatomy are often shapes extracted from images or images of labels (segmentation). The observed organ images can also be modeled using registration as the random diffeomorphic deformation of an unknown template (i.e. an orbit). In these cases as in many other applications, invariance properties lead us to consider that these objects belong to non-linear spaces that have a geometric structure. Thus, the mathematical foundations of computational anatomy rely on statistics on non-linear spaces.
Geometric Statistics aim at studying this abstracted problem at the theoretical level. Our goal is to advance the fundamental knowledge in this area, with potential applications to new areas outside of medical imaging. Beyond the now classical Riemannian spaces, we aim at developping the foundations of statistical estimation on affine connection spaces (e.g. Lie groups), quotient and stratified metric spaces (e.g. orbifolds and tree spaces). In addition to the curvature, one of the key problem is the introduction of singularities at the boundary of the regular strata (non-smooth and non-convex analysis).
A second objective is to develop parametric and non-parametric dimension reduction methods in non-linear space. An important issue is to estimate efficiently not only the model parameters (mean point, subspace, flag) but also their uncertainty. We also want to quantify the influence of curvature and singularities on non-asymptotic estimation theory since we always have a finite (and often too limited) number of samples. A key challenge in developing such a geometrization of statistics will not only be to unify the theory for the different geometric structures, but also to provide efficient practical algorithms to implement them.
A third objective is to learn the geometry from the data. In the high dimensional but low sample size (small data) setting which is the common situation in medical data, we believe that invariance properties are essential to reasonably interpolate and approximate. New apparently antagonistic notions like approximate invariance could be the key to this interaction between geometry and learning.
Beyond the traditional statistical survey of the anatomical shapes that is developed in computational anatomy above, we intend to explore other application fields exhibiting geometric but non-medical data. For instance, applications can be found in Brain-Computer Interfaces (BCI), tree-spaces in phylogenetics, Quantum Physics, etc.
## 3.5 Computational Physiology & Image-Guided Therapy
Computational Physiology aims at developing computational models of human organ functions, an important component of the e-patient , with applications in e-medicine and more specifically in computer-aided prevention, diagnosis, therapy planning and therapy guidance. The focus of our research is on descriptive (allowing to reproduce available observations), discriminative (allowing to separate two populations), and above all predictive models which can be personalized from patient data including medical images, biosignals, biological information and other available metadata. A key aspect of this scientific axis is therefore the coupling of biophysical models with patient data which implies that we are mostly considering models with relatively few and identifiable parameters. To this end, data assimilation methods aiming at estimating biophysical model parameters in order to reproduce available patient data are preferably developed as they potentially lead to predictive models suitable for therapy planning.
Previous research projects in computational physiology have led us to develop biomechanical models representing quasi-static small or large soft tissue deformations (e.g. liver or breast deformation after surgery), mechanical growth or atrophy models (e.g. simulating brain atrophy related to neurodegenerative diseases), heat transfer models (e.g. simulating radiofrequency ablation of tumors), and tumor growth models (e.g. brain or lung tumor growth).
To improve the data assimilation of biophysical models from patient data, a long term objective of our research will be to develop joint imaging and biophysical generative models in a probabilistic framework which simultaneously describe the appearance and function of an organ (or its pathologies) in medical images. Indeed, current approaches for the personalization of biophysical models often proceed in two separate steps. In a first stage, geometric, kinematic or/ functional features are first extracted from medical images. In a second stage, they are used by personalization methods to optimize model parameters in order to match the extracted features. In this process, subtle information present in the image which could be informative for biophysical models is often lost which may lead to limited personalization results. Instead, we propose to develop more integrative approaches where the extraction of image features would be performed jointly with the model parameter fitting. Those imaging and biophysical generative models should lead to a better understanding of the content of images, to a better personalization of model parameters and also better estimates of their uncertainty.
## 3.6 Computational Cardiology & Image-Based Cardiac Interventions
Computational Cardiology has been an active research topic within the Computational Anatomy and Computational Physiology axes of the previous Asclepios project, leading to the development of personalized computational models of the heart designed to help characterizing the cardiac function and predict the effect of some device therapies like cardiac resynchronisation or tissue ablation . This axis of research has now gained a lot of maturity and a critical mass of involved scientists to justify an individualized research axis of the new project Epione, while maintaining many constructive interactions with the 4 other research axes of the project. This will develop all the cardiovascular aspects of the e-patient for cardiac e-medicine.
The new challenges we want to address in computational cardiology are related to the introduction of new levels of modeling and to new clinical and biological applications. They also integrate the presence of new sources of measurements and the potential access to very large multimodal databases of images and measurements at various spatial and temporal scales.
# 4 Application domains
The main applications of our research are in the field of healthcare and more precisely the domain of digital medicine and biomedical data analysis. The axes of research presented above are related to many branches of medicine including cardiology, oncology, urology, neurology, otology, pneumology, radiology, surgery, dermatology, nuclear medicine. Within those branches, the applications cover the following different stages of medicine : prevention, diagnosis, prognosis, treatment.
# 5 Highlights of the year
## 5.1 Awards
• Nicholas Ayache is the laureate of the 2020 International Steven Hoogendijk Award that he received in Rotterdam in 2021. More info on this prize is available on the Wikipedia page.
• Irene Balelli was awarded with an ISFP position with the research project on Causal machine learning: from imaging to in silico trials.
• Marta Nunez Garcia was awarded with a Starting Research Position.
• Maxime Sermesant coordinated two successful EU proposals, SimCardioTest and inEurHeart.
• Benoît Audelan received the second prize of the STIC doctoral school for his PhD thesis.
• Hind Dadoun earned 2nd prize at the Pierre Laffitte PhD competition.
• Buntheng Ly received the Young Investigator Award at the 4th International Scientific Workshop of the IHU Liryc.
• Zihao Wang’s paper One-shot Learning Landmarks Detection46 was selected for the People’s Choice Award at MICCAI DALI 2021.
# 6 New software and platforms
Let us describe new/updated software.
## 6.1 New software
### 6.1.1 CardiacSegmentationPropagation
• Keywords:
3D, Segmentation, Cardiac, MRI, Deep learning
• Functional Description:
Training of a deep learning model which is used for cardiac segmentation in short-axis MRI image stacks.
• Publication:
• Authors:
Qiao Zheng, Hervé Delingette, Nicolas Duchateau, Nicholas Ayache
• Contact:
Qiao Zheng
### 6.1.2 CardiacMotionFlow
• Keywords:
3D, Deep learning, Cardiac, Classification
• Functional Description:
Creation of a deep learning model for the motion tracking of the heart, extraction of characteristic quantities of the movement and shape of the heart to classify a sequence of cine-MRI cardiac images in terms of the types of pathologies (infarcted heart, dilated , hypertrophied, abnormality of the right ventricle).
• Authors:
Qiao Zheng, Hervé Delingette, Nicholas Ayache
• Contact:
Qiao Zheng
### 6.1.3 MedINRIA
• Keywords:
Visualization, DWI, Health, Segmentation, Medical imaging
• Scientific Description:
MedInria aims at creating an easily extensible platform for the distribution of research algorithms developed at Inria for medical image processing. This project has been funded by the D2T (ADT MedInria-NT) in 2010, renewed in 2012. A fast-track ADT was awarded in 2017 to transition the software core to more recent dependencies and study the possibility of a consortium creation.The Empenn team leads this Inria national project and participates in the development of the common core architecture and features of the software as well as in the development of specific plugins for the team's algorithm.
• Functional Description:
MedInria is a free software platform dedicated to medical data visualization and processing.
• URL:
• Contact:
Olivier Commowick
• Participants:
Maxime Sermesant, Olivier Commowick, Théodore Papadopoulo
• Partners:
HARVARD Medical School, IHU - LIRYC, NIH
### 6.1.4 GP-ProgressionModel
• Name:
GP progression model
• Keywords:
Data modeling, Data visualization, Data integration, Machine learning, Biostatistics, Statistical modeling, Medical applications, Evolution, Brain, Uncertainly, Uncertainty quantification, Alzheimer's disease, Probability, Stochastic models, Stochastic process, Trajectory Modeling, Marker selection, Health, Statistic analysis, Statistics, Bayesian estimation
• Functional Description:
Disease progression modeling (DPM) of Alzheimer's disease (AD) aims at revealing long term pathological trajectories from short term clinical data. Along with the ability of providing a data-driven description of the natural evolution of the pathology, DPM has the potential of representing a valuable clinical instrument for automatic diagnosis, by explicitly describing the biomarker transition from normal to pathological stages along the disease time axis.
In this software we reformulate DPM within a probabilistic setting to quantify the diagnostic uncertainty of individual disease severity in an hypothetical clinical scenario, with respect to missing measurements, biomarkers, and follow-up information. The proposed formulation of DPM provides a statistical reference for the accurate probabilistic assessment of the pathological stage of de-novo individuals, and represents a valuable instrument for quantifying the variability and the diagnostic value of biomarkers across disease stages.
This software is based on the publication:
Probabilistic disease progression modeling to characterize diagnostic uncertainty: Application to staging and prediction in Alzheimer's disease. Marco Lorenzi, Maurizio Filippone, Daniel C. Alexander, Sebastien Ourselin Neuroimage. 2019 Apr 15,190:56-68. doi: 10.1016/j.neuroimage.2017.08.059. Epub 2017 Oct 24. HAL Id : hal-01617750 https://hal.archives-ouvertes.fr/hal-01617750/
• Release Contributions:
- New interface and output - Completely based on Pytorch
• URL:
• Publication:
• Contact:
Marco Lorenzi
• Participant:
Marco Lorenzi
### 6.1.5 Music
• Name:
Multi-modality Platform for Specific Imaging in Cardiology
• Keywords:
Medical imaging, Cardiac Electrophysiology, Computer-assisted surgery, Cardiac, Health
• Functional Description:
MUSIC is a software developed by the Asclepios research project in close collaboration with the IHU LIRYC in order to propose functionalities dedicated to cardiac interventional planning and guidance. This includes specific tools (algorithms of segmentation, registration, etc.) as well as pipelines. The software is based on the MedInria platform.
• URL:
• Contact:
Maxime Sermesant
• Participants:
Florent Collot, Mathilde Merle, Maxime Sermesant
• Partner:
IHU- Bordeau
### 6.1.6 SOFA
• Name:
Simulation Open Framework Architecture
• Keywords:
Real time, Multi-physics simulation, Medical applications
• Functional Description:
SOFA is an Open Source framework primarily targeted at real-time simulation, with an emphasis on medical simulation. It is mostly intended for the research community to help develop new algorithms, but can also be used as an efficient prototyping tool. Based on an advanced software architecture, it allows the creation of complex and evolving simulations by combining new algorithms with algorithms already included in SOFA, the modification of most parameters of the simulation (deformable behavior, surface representation, solver, constraints, collision algorithm etc.) by simply editing an XML file, the building of complex models from simpler ones using a scene-graph description, the efficient simulation of the dynamics of interacting objects using abstract equation solvers, the reuse and easy comparison of a variety of available methods.
• News of the Year:
The new version v20.06 has been released including new elements on SoftRobots + ModelOrderReduction integration, in addition to an improved architecture and lots of cleans and bugfixes.
• URL:
• Contact:
Hugo Talbot
• Participants:
Christian Duriez, François Faure, Hervé Delingette, Stephane Cotin, Hugo Talbot, Maud Marchal
• Partner:
IGG
### 6.1.7 geomstats
• Name:
Computations and statistics on manifolds with geometric structures
• Keywords:
Geometry, Statistic analysis
• Scientific Description:
Geomstats is an open-source Python package for computations and statistics on manifolds. The package is organized into two main modules: “geometry“ and “learning“.
The module geometry implements concepts in differential geometry, and the module learning implements statistics and learning algorithms for data on manifolds.
The goal is to provide an easily accessible library for learning algorithms on Riemannian manifolds.
• Functional Description:
Geomstats is a Python package that performs computations on manifolds such as hyperspheres, hyperbolic spaces, spaces of symmetric positive definite matrices and Lie groups of transformations. It provides efficient and extensively unit-tested implementations of these manifolds, together with useful Riemannian metrics and associated Exponential and Logarithm maps. The corresponding geodesic distances provide a range of intuitive choices of Machine Learning loss functions. We also give the corresponding Riemannian gradients. The operations implemented in geomstats are available with different computing backends such as numpy, tensorflow and keras. Geomstats manifold computations have are integrated into keras deep learning framework thanks to GPU-enabled implementations.
• Release Contributions:
Major update of the library with full documentation and code restructuring before publication in JMLR.
• News of the Year:
In 2021, several coding sprints were organized in order to restructure and enhance the library. Par of the PhD of Nicolas Guigui was dedicated to the mathematical structuring of quotient spaces through submersion mechanisms and to parallel transport with applications to Kendal shape spaces. A challenge was organized in conjunction with ICLR 2021.
• URL:
• Publications:
• Contact:
Nicolas Guigui
• Participants:
Nicolas Guigui, Xavier Pennec, Yann Thanwerdas, Nina Miolane
• Partner:
Stanford Department of Statistics
### 6.1.8 MC-VAE
• Name:
Multi Channel Variational Autoencoder
• Keywords:
Machine learning, Artificial intelligence, Medical applications, Dimensionality reduction, High Dimensional Data, Unsupervised learning, Heterogeneity
• Scientific Description:
Interpretable modeling of heterogeneous data channels is essential in medical applications, for example when jointly analyzing clinical scores and medical images. Variational Autoencoders (VAE) are powerful generative models that learn representations of complex data. The flexibility of VAE may come at the expense of lack of interpretability in describing the joint relationship between heterogeneous data. To tackle this problem, this software extends the variational framework of VAE to introduce sparsity of the latent representation, as well as interpretability when jointly account for latent relationships across multiple channels. In the latent space, this is achieved by constraining the variational distribution of each channel to a common target prior. Parsimonious latent representations are enforced by variational dropout. Experiments on synthetic data show that our model correctly identifies the prescribed latent dimensions and data relationships across multiple testing scenarios. When applied to imaging and clinical data, our method allows to identify the joint effect of age and pathology in describing clinical condition in a large scale clinical cohort.
• Functional Description:
This software implements the work published in the paper "Sparse Multi-Channel Variational Autoencoder for the Joint Analysis of Heterogeneous Data" presented at the conference ICML 2019 (Long Beach, California, USA).
The software extends classical variational autoencoders by identifying a joint latent code associated to heterogeneous data represented in different channels. The software is implemented in Python and is based on Pytorch. It can be applied to any kind of data arrays, and provides functions for optimisation, visualisation and writing of the modelling results.
• Release Contributions:
First release
• News of the Year:
Method presented in the International Conference on Machine Learning (ICML 2019).
• URL:
• Contact:
Luigi Antelmi
• Participants:
Luigi Antelmi, Marco Lorenzi, Nicholas Ayache
• Partner:
CoBteK
### 6.1.9 SOFA-CardiacReduction
• Keywords:
Simulation, 3D modeling, Model Order Reduction, Cardiac
• Scientific Description:
Modification of a finite element deformation model : meshless approach and frame-based description, reduction in the number of affine degrees of freedom and integration points.
• Functional Description:
This SOFA plugin is intented to build a reduced model for deformable solids (especially cardiac simulations).
• Contact:
Gaetan Desrues
• Participants:
Gaetan Desrues, Hervé Delingette, Maxime Sermesant
### 6.1.10 Fed-BioMed
• Name:
A general software framework for federated learning in healthcare
• Keywords:
Federated learning, Medical applications, Machine learning, Distributed Applications, Deep learning
• Scientific Description:
While data in healthcare is produced in quantities never imagined before, the feasibility of clinical studies is often hindered by the problem of data access and transfer, especially regarding privacy concerns. Federated learning allows privacy-preserving data analyses using decentralized optimization approaches keeping data securely decentralized. There are currently initiatives providing federated learning frameworks, which are however tailored to specific hardware and modeling approaches, and do not provide natively a deployable production-ready environment. To tackle this issue, Fed-BioMed proposes an open-source federated learning frontend framework with application in healthcare. Fed-BioMed framework is based on a general architecture accommodating for different models and optimization methods
• Functional Description:
The project is based on the development of a distributed software architecture, and the establishment of a server instance from which remote experiments are triggered on the clients sites. The software is distributed to the client's sites, allowing to run machine learning models on the local data. Model parameters are then trasmitted to the server for federated aggregation.
Fed-BioMed finds application in all projects based on the development of learning models for multi-centric studies.
• Release Contributions:
Fed-BioMed is based on Python and Pytorch. The software was recently revised (under an ADT) to adopt the libraries Pygrid and Pysyft.
• News of the Year:
The develpment is ongoing and Fed-BioMed is currently at the version 3.4. For up-to-date news refer to the official project's page: https://fedbiomed.gitlabpages.inria.fr/
• URL:
• Publication:
• Authors:
Marco Lorenzi, Santiago Smith Silva Rincon, Marc Vesin, Tristan Cabel, Irene Balelli, Andréa Senacheribbe, Carlos Zubiaga Pena, Jonathan Levy
• Contact:
Marco Lorenzi
### 6.1.11 EchoFanArea
• Name:
delineation of the border of the fan in ultrasound
• Keywords:
Ultrasound fan area, Deep learning, Statistics, Image processing
• Functional Description:
This software allows the delimitation of the acquisition cone in ultrasound imaging and the inpainting of annotations (lines, characters) inside the cone. It allows both the perfect de-identification of the images but also to standardize the content of the images. It relies on a parametric probabilistic approach to generate a training dataset with region of interest (ROI) segmentation masks. This data will then be used to train a deep U-Net network to perform the same task in a supervised manner, thus considerably reducing the calculation time of the method, one hundred and sixty times faster. These images are then processed with existing filling methods to remove annotations present within the signal area.
• URL:
• Publication:
• Authors:
Hind Dadoun, Hervé Delingette, Nicholas Ayache, Anne-Laure Rousseau
• Contact:
• Partner:
Nhance
### 6.1.12 ProMFusion
• Keywords:
Image segmentation, Data fusion
• Functional Description:
Code related to the paper "Robust Fusion of Probability Maps" by Benoît Audelan, Dimitri Hamzaoui, Sarah Montagne, Raphaële Renard-Penna and Hervé Delingette. The proposed approach allows to fuse probability maps in a robust manner with a spatial regularization of the consensus.
• URL:
• Publication:
• Contact:
Benoit Audelan
• Participants:
Benoit Audelan, Hervé Delingette, Dimitri Hamzaoui
• Name:
Disease progression modeling for clinical intervention simulation
• Keywords:
Data modeling, Clinical analysis, Clinical trial simulator, Alzheimer's disease, Pytorch, Variational Autoencoder, Ordinary differential equations
• Scientific Description:
Recent failures of clinical trials in Alzheimer’s Disease underline the critical importance of identifying optimal intervention time to maximize cognitive benefit. While several models of disease progression have been proposed, we still lack quantitative approaches simulating the effect of treatment strategies on the clinical evolution. In this work, we present a data-driven method to model dynamical relationships between imaging and clinical biomarkers. Our approach allows simulating intervention at any stage of the pathology by modulating the progression speed of the biomarkers, and by subsequently assessing the impact on disease evolution.
• Functional Description:
A machine-learning framework allowing to simulate the impact of intervention on the long-term progression of imaging and clinical biomarkers from collections of healthcare data
• Contact:
Marco Lorenzi
• Participants:
Clément Abi Nader, Marco Lorenzi, Nicholas Ayache, Philippe Robert
• Partner:
Université Côte d'Azur (UCA)
### 6.1.14 SOFA-CardiacReduction
• Keywords:
Simulation, 3D modeling, Model Order Reduction, Cardiac
• Functional Description:
This SOFA plugin is intented to build a reduced model for deformable solids (especially cardiac simulations).
• Contact:
Hervé Delingette
# 7 New results
## 7.1 Medical Image Analysis & Machine Learning
### 7.1.1 Lung Nodule Detection for Lung cancer screening
Keywords: Image Detection, Lung cancer, Deep Learning.
Participants: Benoit Audelan [Correspondent], Stéphanie Lopez, Hervé Delingette.
This work was partially funded by the French government, through the UCAJEDI “Investments in the Future” project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01
We have trained a deep learning system for lung nodule detection from low dose CT scans on the LIDC-IDRI dataset containing 888 CT scans. In this dataset, radiologists characterized nodules exclusively based on radiological criteria.
We have then tested that algorithm on 1179 patients from the National Lung Screening Trial (NLST) study with biopsy-confirmed nodule malignancy (Figure 4). We have demonstrated the ability of our system to detect malignant lesions one year prior to their diagnosis.
This work was published in the 2021 ESR congress48 and included in the thesis of Benoit Audelan56.
### 7.1.2 AI-based real-time diagnostic aid for abdominal organs in ultrasound
This work has been supported by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002.
Keywords: Ultrasound; Medical Image Analysis; Computer Vision; Deep Learning.
Participants: Hind Dadoun [Correspondent], Hervé Delingette, Anne-Laure Rousseau, Eric De Kerviler, Nicholas Ayache.
The goal of the thesis is to develop AI and machine learning methods for the automatic detection of abdominal organs from time series of ultrasound imaging and the automatic detection and diagnosis of pathologies. Ultrasound (US) images usually contain identifying information outside the ultrasound fan area and manual annotations placed by the sonographers during exams. For those images to be exploitable in a Deep Learning framework we propose an open-source tool 34 to process those images and remove all annotations as shown in Figure 5.
### 7.1.3 Machine Learning for MR-TRUS Image Fusion
This work has been supported by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002.
Keywords: Multimodal Image Registration; Image-Guided Therapy; Medical Image Analysis; Statistical Learning.
Participants: Zhijie Fang [Correspondent], Eric Gaudard, Michael Baumann, Nicholas Ayache, Hervé Delingette.
The diagnosis of prostate cancer widely relies on multiparametric MR images. The localisation of cancer lesions in those images by a specialized radiologist serves to plan targeted prostate biopsies through the fusion of MR and intra-operative transrectal ultrasound (TRUS) images (Figure 6).
The objective of this work is to address the probabilistic registration of MR-TRUS prostate images. The challenge associated with this objective is to jointly learn the spatial transformation and the MR-US similarity metrics within an efficient framework.
### 7.1.4 AI-Based Diagnosis of Prostate Cancer from Multiparametric MRI
This work has been supported by the French government, through the 3IA Côte d'Azur Investments and UCA DS4H Investments in the Future project managed by the National Research Agency (ANR) with the reference numbers ANR-19-P3IA-0002and ANR-17-EURE-0004.
Keywords: Prostate cancer; Segmentation; detection; Variability.
Participants: Dimitri Hamzaoui [Correspondent], Sarah Montagne, Raphaele Renard-Penna, Nicholas Ayache, Hervé Delingette.
The aim of our work is to detect and characterize tumorous lesions of the prostate from multiparametric MRI using deep learning-based methods. We investigated several aspects of this problem, including:
• The automatic zonal segmentation of the prostate thanks to a 3D UNet-based network and the detection of lesions using zonal information provided by it (Figure 7, left).
• The study of the variability between the segmentations from different raters 27 and between their volume measurements 20 (Figure 7).
### 7.1.5 Attention-based Multiple Instance Learning with Mixed Supervision
This work has been supported by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002
Keywords: Histopathology; Deep Learning; Multiple Instance Learning.
Participants: Paul Tourniaire [Correspondent], Marius Ilié, Paul Hofman, Nicholas Ayache, Hervé Delingette.
We have developped an Attention-based Multiple Instance learning algorithm for the classification and detection of tumors in Whole Slide Images.
More precisely, we have introduced the possibility of mixed supervision, combining tile-based and slide-based annotations in the Clustering-constrained Attention Multiple Instance Learning (CLAM) model. This has led to an improvement of both classification and localization performances. This work was published at COMPAY 2021 Workshop 45. An overview of the work done is available in Figure 8.
### 7.1.6 Deep generative learning for medical data processing, analysis and modeling: application to cochlea CT imaging
This work is funded by the Provence-Alpes-Côte-d'Azur region, the Université Côte d'Azur and Oticon Medical through CIMPLE research project.
Keywords: Generative Learning; Bayesian Learning; Stochastic Flow; Deep Learning.
Participants: Zihao Wang [Correspondent], Clair Vandersteen, Charles Raffaelli, Nicolas Guevara, François Patou, Hervé Delingette.
Our work is focused on generative learning which can help in many aspects of the processing, understanding, and modeling of CT images of the inner ear :
• First, we propose a novel Bayesian framework 30for shape constrained image segmentation based on parametric shape models (instead of parametric spatial transformations). The trade-off between the appearance and shape models is governed by an interpretable parameter : the reference length.
• We also propose a GAN-based metal artifact reduction method that relies on simulated training data and is suitable for preand postoperative images. To the best of our knowledge, our approach is the first MAR algorithm that combines the physical simulation of metal artifacts with 3D GAN networks. 31
• We also tackled the issue of automatic landmark annotation of 3D volumetric images from a single example based on a one-shot learning method 46. It involves the iterative training of a shallow convolutional neural network combined with a 3D registration algorithm in order to perform automatic organ localization and landmark matching.
• Finally, we proposed in 70, a variational auto-encoder based on Langevin dynamic flow. Tight approximate posterior distribution was achieved by incorporating the gradients information in the inference
All these works are also described in the PhD manuscript of Zihao Wang 59.
## 7.2 Imaging & Phenomics, Biostatistics
### 7.2.1 Statistical Learning of Heterogeneous Data in Large-Scale Clinical Databases
Acknowledgments: IDEX UCAJEDI MNC3 Project (ANR-15-IDEX-01); 3IA (ANR-19-P3IA-0002); the OPAL infrastructure from Université Côte d'Azur.
Keywords: Ordinary Differential Equations, Generative models, Alzheimer's Disease, Disease Progression Modelling.
Participants: Clement Abi Nader [Correspondent], Nicholas Ayache, Philippe Robert, Marco Lorenzi.
Alzheimer's disease is a neurodegenerative disorder whose dynamics still remain partially unknown. The aim of this work is to develop computational models to better understand Alzheimer's disease progression, based on the analysis of imaging and clinical data. Lately, a collaboration with the LANVIE laboratory from the Geneva University Hospitals (HUG) has allowed to test the developed model in a clinical context.
• Development of SimulAD, a statistical model for describing the evolution of Alzheimer's disease 54.
• The method relies on probabilistic generative model, coupled with a system of Ordinary Differential Equations (ODE) to model Alzheimer's disease (Figure 10).
• Recent work focuses on simulating the effect of drug intervention on the progression of Alzheimer's disease 6.
• Assessing the validity and generalization of the method, by applying SimulAD on a new cohort from the Geneva Memory Center (GMC) 7.
### 7.2.2 Combining Multi-Task Learning and Multi-Channel Variational Auto-Encoders to Exploit Datasets with Missing Observations - Application to Multi-Modal Neuroimaging Studies in Dementia
Acknowledgments: IDEX UCAJEDI MNC3 Project (ANR-15-IDEX-01); 3IA (ANR-19-P3IA-0002); the OPAL infrastructure from Université Côte d'Azur.
Keywords: Multi Task Learning, Missing Data, Multimodal Data Analysis..
Participants: Luigi Antelmi [Correspondent], Nicholas Ayache [IRCCS Centro San Giovanni di Dio, Fatebenefratelli, Brescia], Philippe Robert [IRCCS Centro San Giovanni di Dio, Fatebenefratelli, Brescia], Federica Ribaldi [IRCCS Centro San Giovanni di Dio, Fatebenefratelli, Brescia], Valentina Garibotto [Department of Medical Imaging, Geneva University Hospitals, Geneva], Giovanni B. Frisoni [Department of Psychiatry, Geneva University Hospitals, Geneva], Marco Lorenzi.
The aim of this work is to build scalable learning models for the joint analysis of heterogeneous biomedical data, to improve diagnosis, treatment, and monitoring of neurological and psychiatric diseases, in collaboration with the Institut Claude Pompidou (CHU of Nice), within the MNC3 initiative (Médecine Numérique: Cerveau, Cognition, Comportement). Data comes from collections of brain imaging, biological and clinical evaluations available in current large-scale databases such as ADNI and local clinical cohorts, where the joint analysis is affected by data missingness, represented by non-overlapping sets of modalities across subjects (e.g. imaging data, clinical scores, biological measurements) and across datasets. The main contributions and results of this work are:
• a multi-task generative latent-variable model (Fig. where the common variability across datasets stems from the estimation of a shared latent representation across heterogeneous data modalities 55;
• to consistently analyze high-dimensional and heterogeneous information with missing data (Figure 11).
### 7.2.3 Development of Genome-to-Phenome association methods using biologically inspired constraints
This work was funded by the Neuromod Institut of Université Côte d'Azur and the grant ANR PARIS-15087.
Keywords: Bayesian; Variational Dropout; Genome; Phenome; Regression; Biological constraint.
Participants: Marie Deprez, Julien Moreira, Maxime Sermesant, Marco Lorenzi.
In the study of complex and rare diseases, genomics analyses have the potential to lead to important discoveries in identifying the genetics underpinning pathological traits. Yet, such analyses are faced with important challenges posed by the large dimensionality of the data, and by the need for interpretable findings. In this paper, we propose Bayesian Genome-to-Phenome Sparse Regression (G2PSR), a novel multivariate regression method to improve the detection and interpretation of genome-to-phenome associations, using biologically inspired constraints and variable selection through sparsity 15. Through benchmark experiment, we demonstrate that G2PSR outperforms state-of-the-art methods with better accuracy at identifying phenotypic-related genes, and by associating them to multiple phenotypic features. Additionally, we observe that our framework is highly scalable with respect to the overall number of SNPs and genes to be jointly analyzed. Overall, our study shows that G2PSR is an effective tool to identify genome-to-phenome associations in a high-dimension/low-sample size regime, and can thus be employed in future challenging applications, such as imaging genetics and rare disease analysis. This project was developed following multiple steps:
• Development of the G2PSR method (Bayesian Genome-to-Phenome Sparse Regression), Figure 12.A.,
• Benchmark of G2PSR and comparison with state-of-the-art group-sparse methods on synthetic genome-to-phenome data,
• Application to real data from ADNI, Figure 12.B.C.
• Preparation of the following steps for the upgrading of the G2PSR method, such as accounting for multi-omics data types and constraints for the upcoming MITOMICS project (Inserm programme MAladies Rares).
### 7.2.4 Exploring latent dynamical models for failure prediction in time-series of high-dimensional and heterogeneous data
This work was funded by 3IA Côte d'Azur
Keywords: Anomaly Detection; Density Estimation; Extreme value Theory; Neural ODEs; Normalizing Flows.
Participants: Etrit Haxholli, Marco Lorenzi.
Normalizing flows provide a theoretical framework for normality modelling, while extreme value theory is useful in the choice of an anomaly threshold. Our contributions include:
• Under some assumptions, we give theoretical guarantees that the tail of the marginal distribution coincides with the thickest tail of distributions defined on the range of the marginalized out variables (Figure 13).
• We augment Neural ODE flows in a manner such that the transformation remains a diffeomorphism and give a closed form of its Jacobian using our continuous generalization of the chain rule. Using complimentary dual autoregressive neural networks, we model incompressible vector fields (Figure 13).
### 7.2.5 On the role of Fetal Fraction in NIPT (Non-Invasive Prenatal Testing) procedures
ANR-15-IDEX-01
Keywords: prenatal testing; fetal aneuploidies prediction; fetal fraction; bioinformatic pipeline; benchmark.
Participants: Marco Milanesio, Véronique Duboc, David Pratella, John Boudjarane, Stéphane Descombes, Véronique Paquis-Flucklinger, Silvia Bottini.
Noninvasive prenatal testing (NIPT) consists of determining fetal aneuploidies by quantifying copy number alteration from the sequencing of cell-free DNA (cfDNA) from maternal blood. Due to the presence of cfDNA of fetal origin in maternal blood, in silico approaches have been developed to accurately predict fetal aneuploidies. In this work 16 we:
• run a benchmark on several tools for NIPT;
• empirically demonstrated results heterogeneity across different tools;
• developed an integrated pipeline to assess the validity of NIPT tests (Figure 14);
• developed a novel method for accurate gender prediction in NIPT.
The code is publicly available on Github github.com/uca-msi/niptune.
## 7.3 Computational Anatomy & Geometric Statistics
### 7.3.1 Population of Networks Analysis
This work was funded by the ERC grant Nr. 786854 G-Statistics from the European Research Council under the European Union's Horizon 2020 research and innovation program.
Keywords: Network Valued Data; Graph Space; Stratified Spaces; Brain fMRI networks.
Participants: Anna Calissano [Correspondent], Xavier Pennec.
• We have been studying the variability and the Fréchet Mean of fMRI brain connectivity graphs in the Graph Space 71;
• We have been exploring possible stratified spaces as interesting embeddings for set of graphs with different number of nodes and Euclidean attributes on edges and nodes;
• We have been exploring how to define a statistical testing procedure for the analysis of a set of graphs with unlabelled nodes.
### 7.3.2 Geometric computational tools for shape comparison
This work was funded by the ERC grant Nr. 786854 G-Statistics from the European Research Council under the European Union's Horizon 2020 research and innovation program.
Keywords: Parallel Transport; Geomstats.
Participants: Nicolas Guigui [Correspondent], Elodie Maignant [UC Santa-Barbara, USA], Nina Miolane [UC Santa-Barbara, USA], Xavier Pennec.
The development of the geomstats.ai python library was pursued with the support of Inria SED-SAM team (Service d'Expérimentation et de Développement, Sophia Antipolis Méditerranée) through an ADT and the organisation of a hackathon at the SCAI in Paris. In particular, this led to theoretical developments on the computation of parallel transport on Lie groups 39 with an invariant metric, and on the space of Kendall shapes 37. We obtained improved convergence speed over state of the art methods.
The application of parallel transport as a tool to normalize deformations (Figure 15) was investigated in 38 and allowed to compare the effects of congenital heart diseases on the function of the right ventricle. An unexpected effect of the size of the ventricles on the transported motion parameters was in particular put into evidence. More details are available in the PhD mansucript of N. Guigui 58.
### 7.3.3 Computational tools for geometry in Medical Imaging
This work was funded by the ERC grant Nr. 786854 G-Statistics from the European Research Council under the European Union's Horizon 2020 research and innovation program.
Keywords: Parallel Transport.
Participants: Nicolas Guigui [Correspondent], Xavier Pennec.
Our work on Schild's and Pole ladder algorithms (Figure 16 (a)) was consolidated and published in Foundations of Computational Mathematics 19. We proved that this simple constructions based on geodesic parallelograms converged with a quadratic speed, and rate proportional to the Riemannian curvature. This work closed several years of attempts to establish the numerical accuracy of ladders methods. Illustrations on the 2-sphere and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy (Figure 16 (b)).
### 7.3.4 Geodesic squared exponential kernel for non-rigid shape registration
This work was supported by the ANRT Cifre contract 2019/0101 with QuantifiCare & Inria and by the French government through the 3IA Côte d'Azur Investments ANR-19-P3IA-0002 managed by the National Research Agency.
Keywords: Statistical shape modelling; Shape registration; Shape deformations.
Participants: Florent Jousse [Correspondent], Xavier Pennec, Hervé Delingette, Arnaud Bletterer, Matilde Gonzalez.
We addressed in 41 the problem of non-rigid registration of 3D scans, which is at the core of shape modeling techniques. Firstly, we propose a new kernel based on geodesic distances for the Gaussian Process Morphable Models (GPMMs) framework. The use of geodesic distances into the kernel makes it more adapted to the topological and geometric characteristics of the surface and leads to more realistic deformations around holes and curved areas. Since the kernel possesses hyperparameters we have optimized them for the task of face registration in the FaceWarehouse dataset. We show that the Geodesic squared exponential kernel performs significantly better than state of the art kernels for the task of face registration on all the 20 expressions of the FaceWarehouse dataset (see Figure 17). Second, we propose a modification of the loss function used in the non-rigid ICP registration algorithm, that allows to weight the correspondences according to the confidence given to them. As a use case, we show that we can make the registration more robust to outliers in the 3D scans, such as non-skin parts.
### 7.3.5 Analysis in Kendall shape spaces
Keywords: Kendall shape spaces; Parallel transport.
Participants: Elodie Maignant [Correspondent], Guigui Nicolas, Trouvé Alain, Pennec Xavier.
This works focuses on analysis in Kendall shape spaces and its applications. In 37 we propose a new implementation of the parallel transport in these spaces (Figure 18). Kendall shape spaces and their related tools have been implemented in the library geomstats, together with some visualisation modules 64.
### 7.3.6 A geometric formulation of the central limit theorem for Principal Component Analysis
This work is funded by ERC AdG G-Statistics.
Keywords: PCA; Flag manifolds; Central limit theorems on manifolds; Statistical estimation on manifolds.
Participants: Dimbihery Rabenoro, Xavier Pennec.
The goal of this project is to determine the uncertainty of the estimation of subspaces arising from dimension reduction techniques for data living in Euclidian spaces or more generally in manifolds. For that, we study geometric methods using quotient spaces to characterize the concentration of the estimated subspace or family of subspaces resulting from the PCA procedure.
### 7.3.7 The quotient-affine geometry of full-rank correlation matrices
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant G-Statistics agreement No 786854). This work has been supported by the French government, through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01 and through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002.
Keywords: Correlation matrices; SPD matricesRiemannian geometry; Quotient manifold; Quotient-affine metric.
Participants: Yann Thanwerdas [Correspondent], Xavier Pennec.
The set of full-rank correlation matrices is a manifold called the open elliptope. It is not a vector space so Euclidean tools are not adapted to compute with correlation matrices. In 44, we describe the quotient-affine geometry of the open elliptope (Riemannian metric, Levi-Civita connection, geodesics, curvature), which allows:
1. to interpolate, extrapolate, compute Fréchet means, perform PCA on correlation matrices,
2. to define new geometries on covariance matrices which decouple the scales of the variables from the correlations between the variables (cf. Figure 19).
## 7.4 Computational Cardiology & Image-Based Cardiac Interventions
### 7.4.1 Deep Probabilistic Generative Model for the Inverse Problem of Electrocardiography.
This work is funded within the ERC Project ECSTATIC with the IHU Liryc, in Bordeaux.
Keywords: Cardiac Activation; Computational Modelling; Deep Learning; Electrocardiography; Inverse Problem; Generative Model.
Participants: Tania-Marina Bacoyannis [Inria, Université Côte d'Azur, Epione team, Sophia Antipolis, France], Hubert Cochet [IHU Liryc, Bordeaux, France], Maxime Sermesant [Inria, Université Côte d'Azur, Epione team, Sophia Antipolis, France].
The aim of this project is to propose a new method to solve the inverse problem of Electrocardiography, known to be ill-posed. The core of the model we have developed1 is a variational and conditional autoencoder using deep generative neural networks (Figure 20). Our unique model is able to produce advanced maps of cardiac activation by conditioning the electrical activity of the heart with signals measured on the torso surface and cardiac geometry. This method will soon be evaluated on clinical data.
### 7.4.2 Discovering the link between cardiovascular pathologies and neurodegeneration through biophysical and statistical models of cardiac and brain images
This project is funded by Université Côte d'Azur (UCA)
Keywords: Lumped models - Biophysical simulation - Statistical learning.
Participants: Jaume Banus Cobo [Correspondent], Maxime Sermesant [Pompeu Fabra University, Barcelona, Spain], Marco Lorenzi [Pompeu Fabra University, Barcelona, Spain], Oscar Camara Rey [Pompeu Fabra University, Barcelona, Spain].
The project aims at developing a computational model of the relationship between cardiac function and brain damage from large-scale clinical databases of multimodal and multi-organ medical images (Figure 21) 57. The model is based on advanced statistical learning tools for discovering relevant imaging features related to cardiac dysfunction and brain damage 11. However, the limited cardiac data available in datasets focused on neurodegenerative diseases hinders the study of the role of cardiac function in these diseases. Hence, we developed a joint imputation framework based on variational inference and Gaussian process (GP) regression. The GP emulates the cardiac dynamics of a mechanistic model and constraints the data imputation conditioned on the available brain information.
### 7.4.3 Automated-spectral smartphone for eye fundus diagnosis
This project has received funding from the European Union's Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement No. 801342 (Tecniospring INDUSTRY) and the Government of Catalonia's Agency for Business Competitiveness (ACCIÓ).
Keywords: Eye fundus; Retina; Choroid; Multispectral Imaging; Variational Autoenconder.
Participants: Francisco Javier Burgos-Fernández [Correspondent], Buntheng Ly, Fernando Díaz-Doutón, Meritxell Vilaseca, Jaume Pujol, Maxime Sermesant.
We modified the conditional variational autoencoder developed by Ly et al. 43 for the automatic classification of eye fundus through multispectral images acquired in the visible and near-infrared range of the electromagnetic spectrum (400 nm - 1300 nm) from healthy and diseased eye fundus (Figure 22). The model 47 led to a classification accuracy of 96.43%, a loss of 0.20, a sensitivity of 92.86% and a specificity of 100.00% when discriminating between healthy and diseased fundus of the test dataset.
### 7.4.4 Re-entrant Ventricular Tachycardia Simulation from Non-invasive Data
This work is funded by the IHU Liryc, Bordeaux.
Keywords: computed tomography, cardiac electrophysiology, image-based modelling, ventricular tachycardia.
Participants: Nicolas Cedilnik [Correspondent], Pierre Jaïs [CHU Bordeaux], Frédéric Sacher [CHU Bordeaux], Hubert Cochet [IHU Liryc, Bordeaux, France], Maxime Sermesant.
This project aims at improving ventricular tachycardia ablation procedures planning with in silico electro-anatomical exploration of the arrhythmogenic substrate using cadiac imaging as a support for model personalisation.
• We conducted a large scale study of the relationship between imaging and electrophysiological features using in-vivo human data (Figure 23).
• We used our findings to parametrise a reaction diffusion model of activation wave propagation.
• We showed that this personalisation framework was able to reproduce re-entrant patterns that match intra cardiac.
### 7.4.5 3D electromechanical cardiac modelling for heart failure patients stratification and prediction of cardiac resynchronisation therapy response
This work has been supported by the French government through the National Research Agency (ANR) Investments in the Future 3IA Côte d'Azur (ANR-19-P3IA-000) and by Microport CRM funding.
Keywords: Personalisation; Digital twin; Cardiac electromechanical model; Electrophysiology; Electrocardiogram.
Participants: Gaëtan Desrues [Correspondent], Serge Cazeau, Thierry Legay, Delphine Feuerstein, Maxime Sermesant.
Patient-specific 3D electromechanical models can help in improving patient selection, therapy optimisation and interventional guidance. The aim of this project is to study cardiac desynchronisation and the resynchronisation therapy, based on personalised models of the heart 35.
• We first build an atlas of deformed heart geometries, including hypertrophic and dilated cardiomyophaties. A regularized linear regression is used to generate a geometry based on patient data (Figure 24 (a)).
• After the heterogeneous meshing and His-Purkinje network generation, parameters of the electrical propagation model (such as tissue conductivities) are optimized so that the simulated 12-lead ECG matches the patient's one (Figure 24 (b)).
• Finally, the parameters of the mechanical model (such as stiffness or contractility) are optimized using the CMA-ES algorithm. The generated contraction for a left bundle branch bloc is presented (Figure 24 (c)).
### 7.4.6 Predicting thrombosis from Left-Atrium CT-Scan
This work is funded by PARIS EU Project, in collaboration with Simula, Oslo, Norway and Hambourg University Hospital, Germany. Also in collaboration with the IHU Liryc of Bordeaux and the Pompeu Fabra University in Barcelona
Keywords: Diffeomorphic Registration; Graph Representation; Latent Variable Models.
Participants: Josquin Harrison [Inria, Université Côte d'Azur, Epione team, Sophia Antipolis, France], Hubert Cochet [IHU Liryc, Bordeaux, France], Marco Lorenzi [Inria, Université Côte d'Azur, Epione team, Sophia Antipolis, France], Oscar Camara Rey [Pompeu Fabra University, Barcelona, Spain], Xavier Pennec [Inria, Université Côte d'Azur, Epione team, Sophia Antipolis, France], Maxime Sermesant [Inria, Université Côte d'Azur, Epione team, Sophia Antipolis, France].
The objective of this project is to find image-based biomarkers in order to predict the risk of thrombus-related stroke for atrial fibrillation patients. 3D shapes of the left atrium are analysed, in combination with detailed blood flow simulations. A retrospective database of 300 patients from Bordeaux University Hospital is currently analysed. A focus on the orientations of pulmonary veins lead to a graph representation of the left atrium and a novel latent variable model 40. The complete pipeline is shown in Figure 25
### 7.4.7 EP-Net 2.0: Learning Cardiac Electrophysiology Dynamics with DL
PhD funding from 3IA Côte d'Azur
Keywords: Electrophysiology; Simulation; Deep learning.
Participants: Victoriya Kashtanova [Correspondent], Nicolas Cedilnik [LIP6, Paris], Maxime Sermesant [LIP6, Paris], Ibrahim Ayed [LIP6, Paris], Patrick Gallinari [LIP6, Paris].
We propose a deep learning approach (EP-Net 2.0) to learn the cardiac electrophysiology dynamics from data in the presence of scars in the cardiac tissue slab. We show experimentally in 42 that this model is able to reproduce the transmembrane potential dynamics in situations close to the training context and is able to generalize to new conditions including more complex scar geometries, multiple signal onsets and various conduction velocities. Figure 26 presents the EP-Net 2.0 general experimental setting.
The testing results and trained model are publicly available on Github github.com/KVict-new/EP-Net-2.0.
### 7.4.8 Scar-related Ventricular Arrhythmia Prediction using Explainable Deep Learning
Part of this work has been supported by the French Government, through the National Research Agency (ANR) 3IA Côte d'Azur (ANR-19-P3IA-0002), IHU Liryc (ANR- 10-IAHU-04), Université Côte d'Azur and Ecole Doctorale Sciences et Technologies de l'Information et de la Communication (EDSTIC).
Keywords: Ventricular Arrhythmia; Myocardium Scar; CT Imaging; Explainable Learning.
Participants: Buntheng Ly [Correspondent], Sonny Finsterbach, Marta Nuñez-Garcia, Hubert Cochet, Maxime Sermesant.
We proposed an automatic pipeline for ventricular arrhythmia (VA) prediction using the cardiac CT imaging 43. Our theory was based on the relation of the left ventricular (LV) wall thinning in the CT imaging and the myocardium scar, the substrate of the VA mechanism. The complete pipeline is summarised in Figure 27, where the 2D LV thickness map was extracted and used as input to our prediction model. Our model outperformed the LV ejection fraction, the current gold standard predictor of VA.
### 7.4.9 Personalisation of 3D cardiac electromechanical models
European Union's Horizon 2020 research and innovation programme under grant agreement No 101016496 (SimCardioTest)
Keywords: Personalisation; Cardiac electromechanical modeling; Computational Cardiology; Finite Element; Simulations.
Participants: Jairo Rodriguez-Padilla [Correspondent], Maxime Sermesant.
Personalised patient-specific electromechanical models of the heart can be a huge asset for clinicians to improve therapy selection in multiple cardiac diseases. However, personalisation of 3D models is still challenging due to the complexity of the system itself, the high number of parameters to be tuned and computational demand. In this work we:
• Use the Bestel-Clément-Sorine model to compute electromechanical activity in the heart (Figure 28, panel (A)),
• personalise the model via Covariance Matrix Adaptation Evolution Strategy (CMA-ES),
• perform a further sensitivity analysis on the impact that APD increase has in mechanical output parameters used in cardiac resyinchronisation therapy (CRT) (Figure 28, panels B, C and D).
### 7.4.10 Generalisable segmentation of 2D echocardiography
ANR-19-P3IA-0002
Keywords: segmentation; Registration; Motion tracking; Deep learning; Generalisation.
Participants: Yingyu Yang [Correspondent], Maxime Sermesant.
Ultrasound images usually suffer from low ratio of signal-to-noise, boundary ambiguity and out-of-view motion. Automatic algorithms of 2D echocardiography analysis ca nhelp improve diagnostic efficiency as well as improve the accessibility of ultrasound in less developped areas. In this work 51 we:
• proposed methods of introducing shape priors from global, regional and pixel levels for echocardiography segmentation (Figure 29)
• empirically demonstrated results of different shape priors strategies and introducing shape prior through a contour-aware loss in pixel level manifested the best result
• presented the generalisation result of proposed methods in a large unseen dataset
## 7.5 Multi-centric data and Federated Learning
### 7.5.1 Differentially private Bayesian federated learning framework for heterogeneous multi-view data assimilation
This project received financial support by the French government through the Agence Nationale de la Recherche (ANR), ref. num. ANR-19-CE45-0006.
Keywords: Federated Learning; Hierarchical Generative Model; Heterogeneity; Differential Privacy.
Participants: Irene Balelli [Correspondent], Santiago Silva, Marco Lorenzi.
In 10 we developed a novel Bayesian learning framework based on a hierarchical generative model (Figure 30 (a)) for heterogeneous multi-views data assimilation across federated datasets. The Bayesian approach used for parameter estimation allows to handle heterogeneous distribution of datasets across centers, and missing views, while providing an interpretable model of data variability and a tool for missing views imputation (Figure 30 (b)). We further investigate the coupling of our framework with differential privacy: theoretical privacy guarantees against data leakage are provided, and we show that parameter utility can be reasonably preserved (Figure 30 (c)).
• A novel Bayesian federated learning paradigm
• Hierarchical generative model for joint integration of multi-views heterogeneous decentralized data
• High quality data reconstruction even in the presence of missing views
• Formal privacy guarantees
### 7.5.2 Bias in Federated Learning
Keywords: Federated learning; Client sampling; Clustered sampling; FedAvg; Convergence rate; Convergence guarantees.
Participants: Yann Fraboni [Correspondent], Marco Lorenzi.
Federated learning (FL) enables different clients to jointly learn a global model without sharing their data. Considering a subset of clients instead of all of them for optimization is called client sampling and leads to faster FL convergence. In 63, we theoretically prove the impact of a client sampling scheme on FL convergence speed. In 49, we introduce clustered sampling, a new client sampling strategy outperforming the state-of-the-art (Figure 31).
In 50, we also investigate the feasibility of free-rider attacks on FL. We propose two strategies for free-riders to successfully retrieve the learnt model without contributing to the learning process.
### 7.5.3 Federated mixed effects modeling
H2020 / Marie Sklodowska-Curie Actions COFUND number 847579
Keywords: Multi-centric data; Harmonization; SVI; Federated learning.
Participants: Santiago Silva [Correspondent], Marco Lorenzi.
Data harmonization is crucial to obtain generalized models in machine learning as well as data privacy and governance, important factors reflected in regulations such as the CCPA/GDPR. We propose to address multi-centric data harmonization by estimating mixed-effects using stochastic variational inference, a method for uncertainty estimation that allows a direct extrapolation to federated learning for secure modeling on decentralized data (Figure 32). An extension of previous works was submitted to JMLR.
### 7.5.4 Data Structuration and Security in Large-Scale Collaborative Healthcare Data Analysis
This work has been supported by the French government, through the 3IA Côte d'Azur Investments
Keywords: Federated Learning; Biomedical Application; Privacy Preserving Machine Learning.
Participants: Riccardo Taiello [Correspondent], Marco Lorenzi [EURECOM], Melek Önen [EURECOM], Olivier Humbert.
Our work aims to study the impact of privacy-preserving methods in Federated Learning for biomedical applications, which an architecture example is shown in Figure 33 Due to the highly sensitive nature of the medical data being processed, we also plan to investigate the use or design of privacy-preserving primitives for dedicated operations performed with federated learning. We, therefore, intend to make use of advanced cryptographic tools such as homomorphic encryption (HE) or secure multi-party computation (MPC), which allow the processing of the data without the leakage of additional information.
# 8 Bilateral contracts and grants with industry
## 8.1 Bilateral contracts with industry
### 8.1.1 Quantificare
Participants: Xavier Pennec, Hervé Delingette.
The company Quantificare is funding the PhD of Florent Jousse through a CIFRE grant, on the statistical analysis of shapes, deformations and appearance of anatomical surfaces for computer-aided dermatology and plastic surgery. The primary purpose is to model complex face deformations such as natural aging, facial expressions, surgical interventions and posture motions.
### 8.1.2 Oticon Medical
Participants: Zihao Wang, Hervé Delingette.
Oticon Medical, Vallauris, France, is co-funding the PhD work of Zihao Wang which aims at developing robust medical image algorithms for cochlea image segmentation.
### 8.1.3 Accenture Labs
Participants: Marco Lorenzi, Yann Fraboni.
Accenture Labs, Sophia Antipolis, France, is funding the CIFRE PhD work of Yann Fraboni, which aims at investigating the problem of bias and fairness in federated learning applications.
### 8.1.4 Microport CRM
Participants: Maxime Sermesant, Gaetan Desrues.
Microport CRM is a cardiac impantable device company, and is funding the PhD work of Gaëtan Desrues, which aims at using cardiac electromechanical modelling for cardiac resynchronisation therapy planning.
### 8.1.5 Spin-off company inHEART
Participants: Maxime Sermesant.
inHEART2 is a spin-off of the Epione team and IHU Liryc founded in 2017. inHEART provides a service to generate detailed anatomical and structural meshes from medical images, that can be used during ablation interventions. inHEART received 2 awards, one from Aquitaine region and one i-LAB from the BPI. It raised 3.2 million euros in 2020. It currently employs 19 people.
# 9 Partnerships and cooperations
## 9.1 International initiatives
### 9.1.1 Participation in other International Programs
Participants: Xavier Pennec, Marco Lorenzi, Hervé Delingette, Maxime Sermesant, Nicholas Ayache, Irene Balelli.
#### Department of Computer Science, University of Copenhagen, DK
Xavier Pennec is collaborating with Pr. Stefan Sommer on stochastic and sub-Riemannian approached to statistics, in the framework of his ERC G-statistics and of the jointly advised PhD thesis of Morten Akhoj Pedersen.
#### University College London (UCL), London, UK
Marco Lorenzi is collaborator of the COMputational Biology in Imaging and geNEtics (COMBINE) group within the Centre for Medical Image Computing (CMIC) of UCL, and with the UCL Institute of Ophtalmology. His collaboration is on the topic of spatio-temporal analysis of medical images, with special focus on brain imaging analysis and biomarker development. He is also collaborating with the “Progression Over Neurodegenerative Disorders” (POND) group (Prof. Daniel Alexander) for developing new computational models and techniques for learning characteristic patterns of disease progression using large longitudinal clinical data sets, with special focus on dementias.
#### Imaging Genetics Center (IGC), University of Southern California (USC), CA, USA and Illinois Institute of Technology (IIT, IL, USA)
Marco Lorenzi is currently collaborator of IGC and IIT for the investigation of the complex relationship between brain atrophy and genetics in Alzheimer's disease, in particular for demonstrating the effectiveness of multivariate statistical models in providing a meaningful description of the relationship between genotype and brain phenotype.
#### Laboratory of Neuroimaging of Aging (LANVIE), Faculty of Medicine, Geneva University Hospitals (HUG)
Marco Lorenzi collaborates with the LANVIE laboratory led by Prof. Giovanni B. Frisoni. The collaboration consists in developing and translating novel approaches for disease progression modeling in neurodegenerative disorders, such as Alzheimer's disease. St Thomas' Hospital, King's College London, United Kingdom
#### Laboratory of Physics of Fluids, University of Twente, NL
Herve Delingette is collaborating with Assistant Professor Guillaume Lajoinie, on the topics of Deep Learning for ultrasound imaging in the framework of the BoostUrCareer Cofund program and the thesis of Hari Sreedhar.
#### Other International Hospitals
Collaborations with several other European hospitals have been established through the European projects VP2HF, MD PAEDIGREE, SysAFib and with BarcelonaBeta research centre for Alzheimer.
## 9.2 International research visitors
Pr Sébastien Ourselin visited our team in Nov. 2021 in the context of his international Chair at the 3IA Côte d'Azur.
### 9.2.1 Visits of international scientists
##### Sarang Joshi
• Status:
Professor
• Institution of origin:
SCI Institute, Univ. Utah, Salt-lake City
• Country:
USA
• Dates:
15-11 to 20-11-2021
• Context of the visit:
visit of the G-Statistics team.
• Mobility program/type of mobility:
invitation.
## 9.3 European initiatives
### 9.3.1 FP7 & H2020 projects
#### ERC G-Statistics
• Title:
Geometric Statistics
• Type:
ERC
• Program:
H2020
• Duration:
2018-2023
• Inria contact:
Xavier Pennec
• Coordinator:
Inria
• Summary:
G-Statistics aims at exploring the foundations of statistics on non-linear spaces with applications in the Life Siences. Invariance under gauge transformation groups provides the natural structure explaining the laws of physics. In life sciences, new mathematical tools are needed to estimate approximate invariance and establish general but approximate laws. Rephrasing Poincaré: a geometry cannot be more true than another, it may just be more convenient, and statisticians must find the most convenient one for their data. At the crossing of geometry and statistics, G-Statistics aims at grounding the mathematical foundations of geometric statistics and to exemplify their impact on selected applications in the life sciences.
So far, mainly Riemannian manifolds and negatively curved metric spaces have been studied. Other geometric structures like quotient spaces, stratified spaces or affine connection spaces naturally arise in applications. G-Statistics will explore ways to unify statistical estimation theories, explaining how the statistical estimations diverges from the Euclidean case in the presence of curvature, singularities, stratification. Beyond classical manifolds, particular emphasis will be put on flags of subspaces in manifolds as they appear to be natural mathematical object to encode hierarchically embedded approximation spaces.
In order to establish geometric statistics as an effective discipline, G-Statistics will propose new mathematical structures and characterizations of their properties. It will also implement novel generic algorithms and illustrate the impact of some of their efficient specializations on selected applications in life sciences. Surveying the manifolds of anatomical shapes and forecasting their evolution from databases of medical images is a key problem in computational anatomy requiring dimension reduction in non-linear spaces and Lie groups. By inventing radically new principled estimations methods, we aim at illustrating the power of the methodology and strengthening the “unreasonable effectiveness of mathematics” for life sciences.
#### ERC ECSTATIC
• Title:
Electrostructural Tomography – Towards Multiparametric Imaging of Cardiac Electrical Disorders
• Type:
ERC
• Program:
H2020
• Duration:
2017 - 2022
• Inria contact:
Maxime Sermesant
• Coordinator:
U. Bordeaux
• Summary:
Cardiac electrical diseases are directly responsible for sudden cardiac death, heart failure and stroke. They result from a complex interplay between myocardial electrical activation and structural heterogeneity. Current diagnostic strategy based on separate electrocardiographic and imaging assessment is unable to grasp both these aspects. Improvements in personalized diagnostics are urgently needed as existing curative or preventive therapies (catheter ablation, multisite pacing, and implantable defibrillators) cannot be offered until patients are correctly recognized.
ECSTATIC aims at achieving a major advance in the way cardiac electrical diseases are characterized and thus diagnosed and treated, through the development of a novel non-invasive modality (Electrostructural Tomography), combining magnetic resonance imaging (MRI) and non-invasive cardiac mapping (NIM) technologies.
The approach will consist of: (1) hybridising NIM and MRI technologies to enable the joint acquisition of magnetic resonance images of the heart and torso and of a large array of body surface potentials within a single environment; (2) personalising the inverse problem of electrocardiography based on MRI characteristics within the heart and torso, to enable accurate reconstruction of cardiac electrophysiological maps from body surface potentials within the 3D cardiac tissue; and (3) developing a novel disease characterisation framework based on registered non-invasive imaging and electrophysiological data, and propose novel diagnostic and prognostic markers.
This project will dramatically impact the tailored management of cardiac electrical disorders, with applications for diagnosis, risk stratification/patient selection and guidance of pacing and catheter ablation therapies. It will bridge two medical fields (cardiac electrophysiology and imaging), thereby creating a new research area and a novel semiology with the potential to modify the existing classification of cardiac electrical diseases.
#### SimCardioTest
• Title: Simulation of Cardiac Devices $&$ Drugs for in-silico Testing and Certification
• Type:
Resarch and Innovation Action
• Program:
H2020
• Duration:
2021-2024
• Inria contact:
Maxime Sermesant
• Coordinator:
Inria
• Summary:
Computer modelling and simulation have the power to increase speed and reduce costs in most product development pipelines. The EU-funded SimCardioTest project aims to implement computer modelling, simulation and artificial intelligence to design and test cardiac drugs and medical devices. Scientists will establish a platform for running in silico trials and obtaining scientific evidence based on controlled investigations. The simulation of disease conditions and cohort characteristics has the potential to overcome clinical trial limitations, such as under-representation of groups. It also reduces the size and duration of human clinical trials as well as animal testing, and offers robust, personalised information. Leveraging in silico technology in healthcare will expedite product and drug certification and offer patients the best possible care.
## 9.4 National initiatives
#### Consulting for Industry
• Nicholas Ayache has been a scientific consultant for the company Mauna Kea Technologies (Paris) until 31 Dec 2021. He joined the Scientific Advisory Board of Caranx Medical in Oct 2021.
• Marco Lorenzi was a scientific consultant for the company MyDataModels (Sophia Antipolis), and for the company Flexper (Sophia Antipolis).
• Maxime Sermesant is a scientific advisor for the company inHEART (Bordeaux).
#### Institute 3IA Côte d'Azur
• The 3IA Côte d'Azur 3ia.univ-cotedazur.eu/ is one of the four "Interdisciplinary Institutes of Artificial Intelligence" that were created in France in 2019. Its ambition is to create an innovative ecosystem that is influential at the local, national and international levels, and a focal point of excellence for research, education and the world of AI.
• Epione is heavily involved in this institute since its 5 permanents researchers (N. Ayache, H. Delingette, M. Lorenzi, M. Sermesant and X.Pennec) are chair holders in this institute, and N. Ayache serves as scientific director. H. Delingette and N. Ayache are members of its scientific committee.
#### Funded projects
• Hervé Delingette is among the main investigators of the DAICAP project (2020-2022, 300k€) selected by the Health Data Hub, the Grand Défi « Amélioration des diagnostics médicaux par l’Intelligence Artificielle », and Bpifrance in July 2020. That project aims to develop an algorithm able to produce a standardised MRI scan report to improve the early detection of prostate cancer.
• Marco Lorenzi is principal investigator of the ANR JCJC project Fed-BioMed (2020-2023, 196k€).
#### Collaboration with national hospitals
The Epione project team collaborates with the following 3 French IHU (University Hospital Institute): the IHU-Strasbourg (Pr J. Marescaux and L. Soler) on image-guided surgery, the IHU-Bordeaux (Pr M. Haïssaguere and Pr P. Jaïs) on cardiac imaging and modeling and the IHU-Pitié Salpétrière (Dr. O. Colliot and S. Durrleman) on neuroimaging.
The Epione project team is involved in the following research projects with the Assistance Publique des Hôpitaux de Paris (AP-HP) : NHANCE project on abdominal ultrasound image analysis with Dr Anne-Laure Rousseau (Hospital St-Louis), PAIMRI project on prostate cancer detection with Pr Raphaele Renard-Penna (Hospital La Pitié Salpêtrière), CLARITI project on PET-CT anomaly detection with Pr Florent Besson (Hospital Kremlin Bicêtre), PET-CT lesion detection with Dr Paul Blanc-Durand (Hospital Henri Mondor).
We also have long term collaborations with the CHU Nice and Centre Antoine Lacassagne in Nice.
## 9.5 Regional initiatives
• N. Ayache and P. Robert are principal investigators of the project MNC3 (Médecine Numérique, Cerveau, Cognition, Comportement) funded by Idex Jedi UCA (2017-2021, 450k€). M. Lorenzi (Inria) actively participates to the supervision of this project with the help of V. Manera (ICP).
• Hervé Delingette is the principal investigator of the LungMark project funded by Idex UCA JEDI (2018-2021).
• Hervé Delingette is the principal investigator of the CIMPLE project, funded by Idex UCA JEDI (2018-2021), the region PACA and Oticon Medical. The region PACA and Oticon Medical are co-funding the Phd of Zihao Wang.
• Marco Lorenzi and Hervé Delingette received funding for a PhD salary from UCA under the European Program BoostUrCareer (Marie Sklodowska-Curie agreement 847581).
• Maxime Sermesant is principal investigator of the project "The Digital Heart" and the innovation action "Digital Heart Phantom" with General Electrics, funded by Idex UCA JEDI. These projects gather the local cardiac research in academia, clinics and industry.
# 10 Dissemination
Participants: Nicholas Ayache, Irene Balelli, Hervé Delingette, Marco Lorenzi, Xavier Pennec, Maxime Sermesant.
## 10.1 Promoting scientific activities
### 10.1.1 Scientific events: organisation
#### General chair, scientific chair
• Marco Lorenzi was Area Chair of the Conferences “Conference on Computer Vision and Pattern Recognition” (CVPR 2021) and Medical Image Computing and Computer Assisted Intervention (MICCAI 2021).
• M. Sermesant was a co-chair of the MICCAI 2021 Workshop Statistical Atlases and Computational Models of the Heart (STACOM 2021), which was held virtually in September 2021.
#### Member of the organizing committees
• Nicholas Ayache was a member of the organizing committee of the 4th Conference on Medical Imaging in the era of AI, held at the Brain Institute in Paris in June 2021.
• Hervé Delingette is a member of the organizing committee of the 2021 "SophI.A. summit" a scientific event that was held virtually and face to face in Sophia Antipolis from Nov. 15th till Nov. 17th 2021.
• Marco Lorenzi is part of the organizing committee of the Winter School AI4Health, co-organised by the Health Data Hub, 3IA Côte d'Azur, PRAIRIE (Paris), and MIAI (Grenoble).
• Marco Lorenzi is organiser of the Special Session “Security and Fairness in Collaborative Healthcare Data Analysis”, presented at the International Symposium of Biomedical Imaging (ISBI) 2021.
• Marco Lorenzi is co-organiser of the Tutorial “Neurological Disease Progression Modelling”, presented at the International Symposium of Biomedical Imaging (ISBI) 2021.
• Marco Lorenzi is co-organiser of the Tutorial “Disease progression modeling with cross-sectional and longitudinal data”, presented at the International Conference Medical Image Computing and Computer Assisted Intervention (MICCAI) 2021.
### 10.1.2 Scientific events: selection
#### Member of the conference program committees
• Hervé Delingette is a Program Committee member of the 2022 AAAI conference focusing on various aspects of artificial intelligence.
• X. Pennec was a member of the program committee of the Geometric Science of Information (GSI 2021) and the ICPR W. on Manifold Learning from Euclid to Riemann (Manlearn, Milan, January 2021).
#### Reviewer
• H. Delingette was a reviewer for the Conference on Computer Vision and Pattern Recognition (CVPR'21), the International Conference on Learning Representations (ICLR 2022), the International Symposium on Biomedical Imaging (ISBI'21).
• M. Lorenzi was a reviewer for the conferences Neural Information Processing Systems (NeurIPS 2021), International Conference on Machine Learning (ICML 2021), International Conferene on Artificial Intelligence and Statistics (AISTATS 2022), Medical Image Computing and Computer Aided Intervention (MICCAI 2021), International Conference on Learning Representations (ICLR 2021), Information Processing in Medical Imaging (IPMI 2021), International Joint Conference on Artificial Intelligence (IJCAI 2021), Alzheimer's Association International Conference (AAIC 2021)
• X. Pennec was reviewer for the Information Processing in Medical Imaging (IPMI 2021) Conference.
• M. Sermesant was a reviewer for Medical Image Computing and Computer Aided Intervention (MICCAI 2021), the MICCAI workshop STACOM and the Computing in Cardiology conference.
### 10.1.3 Journal
#### Member of the editorial boards
• N. Ayache is the co-founder and the Co-Editor in Chief with J. Duncan (Professor at Yale) of Medical Image Analysis journal. This scientific journal was created in 1996 and is published by Elsevier.
• N. Ayache is a member of the editorial board of the following journal: Journal of Computer Assisted Surgery (Wiley).
• H. Delingette is a member of the editorial board of the journal Medical Image Analysis (Elsevier).
• M. Lorenzi was member of the editorial board of the journal Medical Image Analysis (Elsevier), and of Scientific Reports (Nature Publishing Group); he is also member of the Board of Statisticians of the Journal of Alzheimer’s Disease (IOS Press).
• X. Pennec is a member of the editorial board of the journal Medical Image Analysis (MedIA, Elsevier), of the International Journal of Computer Vision (IJCV, Springer), and of the Journal of Mathematical Imaging and Vision (JMIV, Springer).
• I. Strobant is editorial coordinator for Medical Image Analysis, Elsevier (since october 2001).
#### Reviewer - reviewing activities
• H. Delingette was a reviewer for the following journals: Medical Image Analysis (Elsevier), IEEE Transactions on Pattern Analysis and Machine Intelligence, Pattern Recognition.
• M. Lorenzi was a reviewer for the following journals: Journal of Alzheimer's Disease, Medical Image Analysis, IEEE Transactions on Medical Imaging, NeuroImage, International Journal of Computer Vision, Scientific Reports.
• X. Pennec was a reviewer for the following journals: PLOS One, Bernoulli, SIAM Journal on Mathematics of Data Science (SIMODS), SIAM Journal on Matrix Analysis and Applications (SIMAX), Information Geometry, Medical Image Analysis.
• M. Sermesant was a reviewer for the following journals: Journal of Machine Learning Research, Journal of the American College of Cardiology, IEEE Transactions on Medical Imaging, IEEE Transactions on Biomedical Engineering, Medical Image Analysis and Computers in Biology and Medecine.
### 10.1.4 Invited talks
• Nicholas Ayache gave the following plenary invited talks:
• AI4Health - Winter School of HDH, January 4th, 2021 (visio);
• French-German International Symposium, May 10th, 2021 (visio);
• International Steven Hoogendijk Award Lecture, Rotterdam City Hall, October 1st, 2021;
• Sofcot Forum (Othopedic surgery) in Paris, November 9th, 2021;
• International 3IA Prairie Workshop in Paris, November 10th, 2021;
• Hervé Delingette gave the following invited talks:
• University of Twente (NL) colloquium on Image analysis, February 19th, 2021;
• 5th ACE Impact Workshop, April 29th, 2021;
• EuroBiomed colloquium, on October 11th, 2021.
• Marco Lorenzi gave the following invited talks:
• keynote at the MICCAI Workshop on Distributed And Collaborative Learning 2021, October 1st, 2021;
• IEEE EMBS Public Forum on Data Science and Engineering in Healthcare, February 10th, 2021.
• X. Pennec gave the following invited talks:
• BIRS-CMO Workshop Geometry & Learning from Data, Casa Mathematica de Oaxaca, Mexico, October 24-29, 2021;
• Algebraic and Combinatorial Perspectives in Mathematical Sciences (ACPMS) European Seminar, Mai 28th, 2021;
• Manifold Learning from Euclid to Riemann, Workshop in conjunction with the 25th ICPR, Milan, Italy, January 11th, 2021.
### 10.1.5 Scientific expertise
• N. Ayache is a member of the following scientific committees:
• 2021- Scientific Committee of the French Health Data Hub;
• 2016 -: Scientific advisory committee for Région Ile de France (20 members);
• 2010 -: Scientific Advisory Boards in London (ICL,KCL).
• H. Delingette is a member of the scientific committee of the institute 3IA Côte d'Azur and was reviewer of the funding agency ANR (Agence Nationale de la Recherche, France), a reviewer of the PEPR Santé Numérique for the UDICE consortium. He was the lead participant in the French-canadian round table on AI for health care in March 25th.
• M. Lorenzi was reviewer of the funding agency ANR (Agence Nationale de la Recherche, France), and of the Fondation France Alzheimer (France).
• X. Pennec was a panel member for the Collaborative Research in Computational Neuroscience (CRCN) (joint call from NSF (US), NIH (US), ANR (FR), BMBF (DE), BSF (US-IL), NICT (JP)) and of the Inria International Chair committee.
• Nicholas Ayache is the scientific director of the 3IA Côte d'Azur since its creation in April 2019.
• Hervé Delingette is a representive of Inria at the Federation Hospitalo-Universitaire Oncoage led by the CHU Nice. He was nominated as one of the 3 scientific directors of the IDEX program UCA JEDI under the direction of the president of the Université Côte d'Azur. He is an administrator of the Groupement de Coopération Sanitaires (GCS) CARES involving the Université Côte d'Azur and the 3 local hospitals (CHU Nice, Centre Antoine Lacassagne, Fondation Lenval).
• Hervé Delingette is a member of the committee of coordination between Université Côte d'Azur and Inria which supervises the strategic agreement between the two entities.
• Marco Lorenzi is a member of the local steering committee of the technological platforms (Comités Scientifiques de Pilotage des Plateformes) in charge of Cluster, Grid, Cloud, and HPC technologies. He is supervisor representative of the BoostUrCAreer Doctoral Program of Université Côte d'Azur.
• Marco Lorenzi is part of the working group of the PEPR Santé Numerique (axis “Secure, safe, and fair machine learning for healthcare” and “MultiScale AI for SingleCell-Based Precision Medicine”).
• Xavier Pennec is co-director of the Ecole doctorale STIC of Université Côte d'Azur. He is a member of the committee of EDSTIC, of the Doctoral follow-up Committee (CSD) at Inria Sophia Antipolis, and participated to the PhD fellowship granting committees of EDSTIC, Cofund at UCA, CORDI at Inria. He is a member of the "Comité de la Recherche Biomédicale en Santé Publique (CRBSP)" of the Nice University Hospital (CHU). At University Côte d'Azur /UCA JEDI, he is a member of the executive committee of the Academy 4 (Living systems Complexity and diversity), of the Scientific committee of the Academy 2 (Complex Systems), and of the Advanced Research Program Committee. At the national level, X. Pennec is a member of the Evaluation Committee of Inria. As such, he participated in 2021 to the promotion commitees for CRHC, DR1, DR0, to the PEDR committe and to the CRCN recruitment committee at Inria Bordeaux.
• Maxime Sermesant is an elected member of the Inria Sophia Antipolis research centre committee.
## 10.2 Teaching - Supervision - Juries
### 10.2.1 Teaching
• Master: H. Delingette and X. Pennec, Introduction to Medical Image Analysis, 21h course (28.5 ETD), Master 2 MVA, ENS Saclay, France.
• Master: M. Lorenzi and I. Balelli, Bayesian Learning, 30h course, Master Data Science, Univ. Côte d'Azur, France.
• Master : H. Delingette, 1h, Challenges in AI, University Diploma for AI in Healthcare, Univ. Côte d'Azur, France.
• Master: M. Lorenzi, Model Selection and Resampling Methods, 30h course, Master Data Science, Univ. Côte d'Azur, France.
• Master: I. Balelli, Modeling of biological systems, 10h ETD, Master 2 BIM, Univ. Côte d'Azur, France.
• M. Lorenzi and I. Balelli presented a one-day hands-on session on the software Fed-BioMed to the Winter School AI4Health (HDH and 3IA), and to the Inria DFKI Summer School.
### 10.2.2 Supervision: defended PhDs
• Clément Abi Nader, Modelling and simulating the progression of Alzheimer's disease through the analysis of multi-modal neuroimages and clinical data, Université Côte d'Azur. Defended on June 15th, 2021. Directed by Nicholas Ayache and Philippe Robert, and co-supervised by Marco Lorenzi.
• Luigi Antelmi, Statistical learning on heterogeneous medical data with bayesian latent variable models : application to neuroimaging dementia studies, Université Côte d'Azur. Defended on July 7th, 2021. Directed by Nicholas Ayache and Philippe Robert, and co-supervised by Marco Lorenzi.
• Benoit Audelan, Probabilistic segmentation modelling and deep learning-based lung cancer screening, Université Côte d'Azur. Defended on July 22nd, 2021. Directed by Hervé Delingette. Received second prize from the doctoral school EDSTIC.
• Tania-Marina Bacoyannis, Cardiac Imaging and Machine Learning for Electrostructural Tomography, Université Côte d'Azur. Defended on December 17th, 2021. Co-directed by Maxime Sermesant and Hubert Cochet (IHU Liryc).
• Jaume Banus Cobo, Linking cardiovascular pathologies and neurodegeneration with a combined biophysical and statistical methodology, Université Côte d'Azur. Defended on May 26th, 2021. Directed by Maxime Sermesant and co-supervised by Marco Lorenzi.
• Nicolas Guigui, Computational methods for statistical estimation on Riemannian manifolds and application to cardiac deformations, Université Côte d'Azur. Defended on November 18, 2021. Directed by Xavier Pennec 58.
• Zihao Wang, Deep generative learning for medical data processing, analysis and modeling: application to cochlea CT imaging, Université Côte d'Azur. Defended on Sept. 17th, 2021. Directed by Hervé Delingette.
### 10.2.3 Supervision: ongoing PhDs
• Paul Blanc-Durand, Deep neural networks in Nuclear Medicine, Université Paris Descartes, co-supervised by Hervé Delingette
• Mikael Chelli, AI for Othopedic Surgery, Centre hospitalier universitaire de Nice. Started in August 2019. Co-directed by Nicholas Ayache, Hervé Delingette and Jean Chaoui (Imascap).
• Hind Dadoun, AI-Based Real Time Diagnosis of Abdominal Ultrasound, Université Côte d'Azur. Started in December 2019. Co-directed by Nicholas Ayache and Hervé Delingette.
• Gaëtan Desrues, 3D electromechanical cardiac modelling for heart failure patients stratification and prediction of cardiac resynchronisation therapy response, Université Côte d'Azur. Started in 2020. Directed by Maxime Sermesant.
• Zhijie Fang, Robust Multimodal Non rigid image registration based on machine learning, Université Côte d'Azur. Started in 2020. Co-directed by Hervé Delingette and Nicholas Ayache.
• Yann Fraboni, Bias in Federated Learning, Université Côte d'Azur. Started in 2020. Directed by Marco Lorenzi.
• Dimitri Hamzaoui, AI-Based Diagnosis of Prostate Cancer from Multiparametric MRI, Université Côte d'Azur. Started in 2020. Co-directed by Nicholas Ayache and Hervé Delingette.
• Josquin Harrison, Medical Imaging and learning for the prediction of strokes in the case of atrial fibrillation, Université Côte d'Azur. Started in 2020. Directed by Maxime Sermesant.
• Etrit Haxholli, Exploring latent dynamical models for failure prediction in time-series of high-dimensional and heterogeneous data, Université Côte d'Azur. Started in 2020. Directed by Marco Lorenzi.
• Florent Jousse, Analyse statistique de forme, de déformations et d'apparences de surfaces anatomiques pour l'aide à la dermatologie et à la chirurgie plastique. Started in 2019. Cifre fellowship with Quantificare. Co-directed by Xavier Pennec and Hervé Delingette.
• Victoriya Kashtanova, Learning Cardiac 3D Electromechanical Dynamics with PDE-based Physiological Constraints for Data-Driven Personalized Predictions in Cardiology, Université Côte d'Azur. Started in 2020. Co-directed by Maxime Sermesant and Patrick Gallinari (Sorbonne University, LIP6, Paris).
• Huiyu Li, Anonymisation and protection of medical data based on deep neural networks. Université Côte d'Azur. Started in 2020. Co-directed by Hervé Delingette and Nicholas Ayache.
• Buntheng Ly, Cardiac Image Analysis for Sudden Cardiac Death Prediction, Université Côte d'Azur. Started in 2019. Co-directed by Maxime Sermesant and Hubert Cochet (IHU Liryc).
• Elodie Maignan, Geometric learning of manifolds, Université Côte d'Azur. Started in 2020. Co-directed by Xavier Pennec and Alain Trouvé (ENS Paris-Saclay).
• Morten Pedersen, Lie group action, approximate invariance and Sub-Riemannian geometry in statistics, Université Côte d'Azur and University of Copenhagen. Started in 2020. Co-directed by Xavier Pennec and Stefan Sommer (Univ Copenhagen, DK).
• Santiago Smith Silva Rincon, Federated learning of biomedical data in large-scale networks of multicentric imaging-genetics information, Université Côte d'Azur. Started in 2020. Co-directed by Marco Lorenzi and Barbara Bardoni (INSERM)
• Hari Sreedhar, Learning-based detection and classification of thyroid nodules from ultrasound images, Université Côte d'Azur. Started in 2020. Co-directed by Hervé Delingette and Charles Raffaelli (CHU-Nice).
• Riccardo Taiello, Data Structuration and Security in Large-Scale Collaborative Healthcare Data Analysis, Université Côte d'Azur. Started in 2021. Co-supervised by Melek Önen (EURECOM) and Olivier Humbert (Centre Antoine Lacassagne).
• Yann Thanwerdas, Statistical Dimension Reduction in Non-Linear Manifolds for Brain Shape Analysis, Connectomics & Brain-Computer Interfaces, Université Côte d'Azur. Started in 2019. Directed by Xavier Pennec.
• Paul Tourniaire, AI-based selection of imaging and biological markers predictive of therapy response in lung cancer, Université Côte d'Azur. Started in 2019. Co-directed by Nicholas Ayache and Hervé Delingette.
• Yingyu Yang, Artificial Intelligence for Automatic Cardiac Function Analysis: Multimodal and Biophysical Approach with Application to a Portable Imaging Device, Université Côte d'Azur. Started in 2020. Co-directed by Maxime Sermesant and Pamela Moceri (CHU-Nice)
### 10.2.4 Juries
• H. Delingette was PhD jury member for the thesis of Z. Wang (UCA, Thesis Director), of B. Audelan (UCA, Thesis Director), of Theo Estienne (Ecole Centrale Paris, reviewer). He was a member of the medicine thesis committee of Dr A. Berembaum (Faculté de médecine Paris) and Dr Karime Wu (Faculté de médecine Paris).
• M. Lorenzi was PhD jury Member for the thesis of Vikram Venkatraghavan (VUMC, NL), Arnaud Fernandez (IPMC, UCA), Julien Audibert (EURECOM), Radia Zeghari (Institut Claude Pompidou), Rosa Candela (EURECOM), Gia-Lac Tran (EURECOM).
• X. Pennec was reviewer for the HDR of Salem Said (Univ. Bordeaux), and member of PhD jury of Nicolas Guigui (Univ. Côte d'Azur).
## 10.3 Popularization
• Hervé Delingette participated to a TV program on Azur TV on Feb. 25th dedicated to "AI in healtcare".
• Hervé Delingette and Hind Dadoun participated to the "Fête de la science" by presenting the concept of AI in healthcare to high school students at the "Maison de l'IA" in Sophia Antipolis on Oct. 5th.
# 11 Scientific production
## 11.1 Major publications
• 1 miscY.Yann Fraboni, R.Richard Vidal, L.Laetitia Kameni and M.Marco Lorenzi. Clustered Sampling: Low-Variance and Improved Representativity for Clients Selection in Federated Learning.May 2021
• 2 articleNumerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds.Foundations of Computational MathematicsJune 2021
• 3 articleJ.Julian Krebs, H.Hervé Delingette, N.Nicholas Ayache and T.Tommaso Mansi. Learning a Generative Motion Model from Image Sequences based on a Latent Motion Matrix.IEEE Transactions on Medical ImagingFebruary 2021
• 4 articleM.Maxime Sermesant, H.Hervé Delingette, H.Hubert Cochet, P.Pierre Jaïs and N.Nicholas Ayache. Applications of artificial intelligence in cardiovascular imaging.Nature Reviews CardiologyMarch 2021
• 5 articleZ.Zihao Wang, T.Thomas Demarcy, C.Clair Vandersteen, D.Dan Gnansia, C.Charles Raffaelli, N.Nicolas Guevara and H.Hervé Delingette. Bayesian Logistic Shape Model Inference: application to cochlear image segmentation.Medical Image AnalysisOctober 2021
## 11.2 Publications of the year
### International journals
• 6 articleC.Clément Abi Nader, N.Nicholas Ayache, G. B.Giovanni B Frisoni, P.Philippe Robert and M.Marco Lorenzi. Simulating the outcome of amyloid treatments in Alzheimer's Disease from multi-modal imaging and clinical data.Brain CommunicationsFebruary 2021
• 7 articleC.Clément Abi Nader, F.Federica Ribaldi, G. B.Giovanni B Frisoni, V.Valentina Garibotto, P.Philippe Robert, N.Nicholas Ayache and M.Marco Lorenzi. SimulAD: A dynamical model for personalized simulation and disease staging in Alzheimer's disease.Neurobiology of Aging2022
• 8 articleB.Benoît Audelan, D.Dimitri Hamzaoui, S.Sarah Montagne, R.Raphaële Renard-Penna and H.Hervé Delingette. Robust Bayesian fusion of continuous segmentation maps.Medical Image AnalysisMarch 2022
• 9 articleT.Tania Bacoyannis, B.Buntheng Ly, N.Nicolas Cedilnik, H.Hubert Cochet and M.Maxime Sermesant. Deep Learning Formulation of ECGI Integrating Image & Signal Information with Data-driven Regularisation.EP-Europace23Supplement_1March 2021, i55-i62
• 10 articleI.Irene Balelli, S.Santiago Silva and M.Marco Lorenzi. A Probabilistic Framework for Modeling the Variability Across Federated Datasets of Heterogeneous Multi-View Observations.Information processing in medical imaging : proceedings of the ... conference.2021
• 11 articleJ.Jaume Banus, M.Marco Lorenzi, O.Oscar Camara and M.Maxime Sermesant. Biophysics-based statistical learning: Application to heart and brain interactions.Medical Image Analysis72August 2021
• 12 articleP.Paul Blanc-Durand, S.Simon Jégou, S.Salim Kanoun, A.Alina Berriolo-Riedinger, C. M.Caroline M Bodet-Milin, F.Françoise Kraeber-Bodéré, T.Thomas Carlier, S.Steven Le Gouill, R.-O.René-Olivier Casasnovas, M.Michel Meignan and E.Emmanuel Itti. Fully automatic segmentation of diffuse large B cell lymphoma lesions on 3D FDG-PET/CT for total metabolic tumour volume prediction using a convolutional neural network.European Journal of Nuclear Medicine and Molecular Imaging2021, 1362-1370
• 13 articleA.Adrià Casamitjana, M.Marco Lorenzi, S.Sebastiano Ferraris, L.Loïc Peter, M.Marc Modat, A.Allison Stevens, B.Bruce Fischl, T.Tom Vercauteren and J. E.Juan Eugenio Iglesias. Robust joint registration of multiple stains and MRI for multimodal 3D histology reconstruction: Application to the Allen human brain atlas.Medical Image AnalysisOctober 2021
• 14 articleH.Hind Dadoun, A.-L.Anne-Laure Rousseau, E.Eric De Kerviler, J. M.Jean Michel Correas, A.-M.Anne-Marie Tissier, F.Fanny Joujou, S.Sylvain Bodard, K.Kemel Khezzane, C.Constance de Margerie-Mellon, H.Hervé Delingette and N.Nicholas Ayache. Detection, Localization, and Characterization of Focal Liver Lesions in Abdominal US with Deep Learning.Radiology: Artificial Intelligence2022
• 15 articleM.Marie Deprez, J.Julien Moreira, M.Maxime Sermesant and M.Marco Lorenzi. Decoding genetic markers of multiple phenotypic layers through biologically constrained Genome-to-Phenome Bayesian Sparse Regression.Frontiers in Molecular Medicine2021
• 16 articleV.Véronique Duboc, D.David Pratella, M.Marco Milanesio, J.John Boudjarane, S.Stéphane Descombes, V.Véronique Paquis-Flucklinger and S.Silvia Bottini. NiPTUNE: an automated pipeline for noninvasive prenatal testing in an accurate, integrative and flexible framework.Briefings in BioinformaticsSeptember 2021
• 17 articleN.Nicolas Duchateau, P.Pamela Moceri and M.Maxime Sermesant. Direction-dependent decomposition of 3D right ventricular motion: beware of approximations.Journal of The American Society of Echocardiography3422021, 201-203
• 18 articleS.Sara Garbarino and M.Marco Lorenzi. Investigating hypotheses of neurodegeneration by learning dynamical systems of protein propagation in the brain.NeuroImage235July 2021, 117980
• 19 articleNumerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds.Foundations of Computational MathematicsJune 2021
• 20 articleD.Dimitri Hamzaoui, S.Sarah Montagne, B.Benjamin Granger, A.Alexandre Allera, M.Malek Ezziane, A.Anna Luzurier, R.Raphaelle Quint, M.Mehdi Kalai, N.Nicholas Ayache, H.Hervé Delingette and R.Raphaele Renard-Penna. Prostate volume prediction on MRI: tools, accuracy and variability.European RadiologyFebruary 2022
• 21 articleD.Dimitri Hamzaoui, S.Sarah Montagne, R.Raphaele Renard-Penna, N.Nicholas Ayache and H.Hervé Delingette. Automatic Zonal Segmentation of the Prostate from 2D and 3D T2-weighted MRI and Evaluation for Clinical Use.Journal of Medical Imaging2022
• 22 articleJ.Julian Krebs, H.Hervé Delingette, N.Nicholas Ayache and T.Tommaso Mansi. Learning a Generative Motion Model from Image Sequences based on a Latent Motion Matrix.IEEE Transactions on Medical ImagingFebruary 2021
• 23 articleJ.Julian Krebs, T.Tommaso Mansi, H.Hervé Delingette, B.Bin Lou, J.Joao Lima, S.Susumu Tao, L.Luisa Ciuffo, S.Sanaz Norgard, B.Barbara Butcher, W.Wei Lee, E.Ela Chamera, T.-M.Timm-Michael Dickfeld, M.Michael Stillabower, J.Joseph Marine, R.Robert Weiss, G.Gordon Tomaselli, H.Henry Halperin, K.Katherine Wu and H.Hiroshi Ashikaga. CinE caRdiac magneTic resonAnce to predIct veNTricular arrhYthmia (CERTAINTY).Scientific Reports111November 2021
• 24 articleA.Alice Le Brigant, N.Nicolas Guigui, S.Sana Rebbah and S.Stéphane Puechmorel. Classifying histograms of medical data using information geometry of beta distributions.IFAC-PapersOnLineJune 2021
• 25 articleP.Pamela Moceri, N.Nicolas Duchateau, D.Delphine Baudouy, F.Fabien Squara, S. S.Sok Sithikun Bun, E.Emile Ferrari and M.Maxime Sermesant. Additional prognostic value of echocardiographic follow-up in pulmonary hypertension - role of 3D right ventricular area strain.European Heart Journal - Cardiovascular Imaging2021
• 26 articleP.Pamela Moceri, N.Nicolas Duchateau, B.Benjamin Sartre, D.Delphine Baudouy, F.Fabien Squara, M.Maxime Sermesant and E.Emile Ferrari. Value of 3D right ventricular function over 2D assessment in acute pulmonary embolism.Echocardiography2021
• 27 articleS.Sarah Montagne, D.Dimitri Hamzaoui, A.Alexandre Allera, M.Malek Ezziane, A.Anna Luzurier, R.Raphaelle Quint, M.Mehdi Kalai, N.Nicholas Ayache, H.Hervé Delingette and R.Raphaele Renard Penna. Challenge of prostate MRI segmentation on T2-weighted images: inter-observer variability and impact of prostate morphology.Insights into Imaging121June 2021
• 28 articleT. M.Talia M Nir, J.-P.Jean-Paul Fouche, J.Jintanat Ananworanich, B. M.Beau M Ances, J.Jasmina Boban, B. J.Bruce J Brew, J. R.Joga R Chaganti, L.Linda Chang, C. R.Christopher R K Ching, L. A.Lucette A Cysique, T.Thomas Ernst, J.Joshua Faskowitz, V.Vikash Gupta, J.Jaroslaw Harezlak, J. M.Jodi M Heaps-Woodruff, C. H.Charles H Hinkin, J.Jacqueline Hoare, J. A.John A Joska, K. J.Kalpana J Kallianpur, T.Taylor Kuhn, H. Y.Hei Y Lam, M.Meng Law, C.Christine Lebrun-Frénay, A. J.Andrew J Levine, L.Lydiane Mondot, B. K.Beau K Nakamoto, B. A.Bradford A Navia, X.Xavier Pennec, E. C.Eric C Porges, L. E.Lauren E Salminen, C. M.Cecilia M Shikuma, W.Wesley Surento, A. D.April D Thames, V.Victor Valcour, M.Matteo Vassallo, A. J.Adam J Woods, P. M.Paul M Thompson, R. A.Ronald A Cohen, R.Robert Paul, D. J.Dan J Stein and N.Neda Jahanshad. Association of Immunosuppression and Viral Load With Subcortical Brain Volume in an International Sample of People Living With HIV.JAMA Network Open41January 2021, e2031190
• 29 articleM.Maxime Sermesant, H.Hervé Delingette, H.Hubert Cochet, P.Pierre Jaïs and N.Nicholas Ayache. Applications of artificial intelligence in cardiovascular imaging.Nature Reviews CardiologyMarch 2021
• 30 articleZ.Zihao Wang, T.Thomas Demarcy, C.Clair Vandersteen, D.Dan Gnansia, C.Charles Raffaelli, N.Nicolas Guevara and H.Hervé Delingette. Bayesian Logistic Shape Model Inference: application to cochlear image segmentation.Medical Image Analysis75October 2021, 102268
• 31 articleZ.Zihao Wang, C.Clair Vandersteen, T.Thomas Demarcy, D.Dan Gnansia, C.Charles Raffaelli, N.Nicolas Guevara and H.Hervé Delingette. Inner-ear Augmented Metal Artifact Reduction with Simulation-based 3D Generative Adversarial Networks.Computerized Medical Imaging and Graphics93October 2021, 101990
• 32 articleZ.Zhifei Xu, Z.Zihao Wang, Y.Yin Sun, C.Chulsoon Hwang, H.Hervé Delingette and J.Jun Fan. Jitter Aware Economic PDN Optimization with a Genetic Algorithm.IEEE Transactions on Microwave Theory and TechniquesJune 2021
• 33 articleK.Kevin Zhou, H. N.Hoang Ngan Le, K.Khoa Luu, H.Hien Van Nguyen and N.Nicholas Ayache. Deep reinforcement learning in medical imaging: A literature review.Medical Image Analysis73October 2021, 102193
### International peer-reviewed conferences
• 34 inproceedingsH.Hind Dadoun, H.Hervé Delingette, A.-L.Anne-Laure Rousseau, E.Eric De Kerviler and N.Nicholas Ayache. Combining Bayesian and Deep Learning Methods for the Delineation of the Fan in Ultrasound Images.ISBI 2021 - 18th IEEE International Symposium on Biomedical ImagingNice, FranceApril 2021
• 35 inproceedingsG.Gaëtan Desrues, D.Delphine Feuerstein, T.Thierry Legay, S.Serge Cazeau and M.Maxime Sermesant. Personal-by-design: a 3D Electromechanical Model of the Heart Tailored for Personalisation.FIMH 2021 - 11th International Conference on Functional Imaging and Modeling of the HeartStanford, CA, United StatesJune 2021
• 36 inproceedingsM.Maxime Di Folco, N.Nicolas Guigui, P.Patrick Clarysse, P.Pamela Moceri and N.Nicolas Duchateau. Investigation of the impact of normalization on the study of interactions between myocardial shape and deformation.FIMH 2021 - 11th International Conference on Functional Imaging and Modeling of the HeartStanford, United StatesJune 2021, In press
• 37 inproceedingsN.Nicolas Guigui, E.Elodie Maignant, A.Alain Trouvé and X.Xavier Pennec. Parallel Transport on Kendall Shape Spaces.GSI 2021 - 5th conference on Geometric Science of Information12829Lecture Notes in Computer ScienceParis, FranceSpringerJuly 2021, 103-110
• 38 inproceedingsN.Nicolas Guigui, P.Pamela Moceri, M.Maxime Sermesant and X.Xavier Pennec. Cardiac Motion Modeling with Parallel Transport and Shape Splines.ISBI 2021 - 18th IEEE International Symposium on Biomedical ImagingIEEE 18th International Symposium on Biomedical Imaging (ISBI), 2021Nice, FranceIEEEApril 2021, pp. 1394-1397
• 39 inproceedingsA reduced parallel transport equation on Lie Groups with a left-invariant metric.GSI 2021 - 5th conference on Geometric Science of Information12829Lecture Notes in Computer ScienceParis, FranceSpringer, ChamJuly 2021, 119-126
• 40 inproceedingsJ.Josquin Harrison, M.Marco Lorenzi, B.Benoit Legghe, X.Xavier Iriart, H.Hubert Cochet and M.Maxime Sermesant. Phase-independent Latent Representation for Cardiac Shape Analysis.MICCAI 2021 - 24th International Conference on Medical Image Computing and Computer Assisted Intervention12906LNCS - Lecture Notes in Computer ScienceStrasbourg, FranceSeptember 2021
• 41 inproceedingsF.Florent Jousse, X.Xavier Pennec, H.Hervé Delingette and M.Matilde Gonzalez. Geodesic squared exponential kernel for non-rigid shape registration.FG 2021 - IEEE International Conference on Automatic Face and Gesture RecognitionJODHPUR, IndiaDecember 2021
• 42 inproceedingsV.Victoriya Kashtanova, I.Ibrahim Ayed, N.Nicolas Cedilnik, P.Patrick Gallinari and M.Maxime Sermesant. EP-Net 2.0: Out-of-Domain Generalisation for Deep Learning Models of Cardiac Electrophysiology.Functional Imaging and Modeling of the Heart; Functional Imaging and Modeling of the HeartFIMH 2021 - 11th International Conference on Functional Imaging and Modeling of the Heart12738Lecture Notes in Computer ScienceStanford, CA (virtual), United StatesSpringer International PublishingJune 2021, 482-492
• 43 inproceedingsB.Buntheng Ly, S.Sonny Finsterbach, M.Marta Nuñez-Garcia, H.Hubert Cochet and M.Maxime Sermesant. Scar-Related Ventricular Arrhythmia Prediction from Imaging Using Explainable Deep Learning.FIMH 2021 - 11th International Conference on Functional Imaging and Modeling of the Heart12738Lecture Notes in Computer ScienceStanford, United StatesSpringer International PublishingJune 2021, 461-470
• 44 inproceedingsY.Yann Thanwerdas and X.Xavier Pennec. Geodesics and Curvature of the Quotient-Affine Metrics on Full-Rank Correlation Matrices.GSI 2021 - 5th conference on Geometric Science of Information12829Proceedings of Geometric Science of InformationParis, FranceSpringer, ChamJuly 2021, 93-102
• 45 inproceedingsP.Paul Tourniaire, M.Marius Ilie, P.Paul Hofman, N.Nicholas Ayache and H.Hervé Delingette. Attention-based Multiple Instance Learning with Mixed Supervision on the Camelyon16 Dataset.Proceedings of Machine Learning ResearchMICCAI Workshop on Computational Pathology156Proceedings of Machine Learning ResearchStrasbourg, FrancePMLRSeptember 2021, 216-226
• 46 inproceedingsZ.Zihao Wang, C.Clair Vandersteen, C.Charles Raffaelli, N.Nicolas Guevara, F.François Patou and H.Hervé Delingette. One-shot Learning Landmarks Detection.The MICCAI workshop on Data Augmentation, Labeling, and Imperfections13003Lecture Notes in Computer Science (LNCS)strasbourg, FranceSpringerSeptember 2021, 163-172
### National peer-reviewed Conferences
• 47 inproceedingsF. J.Francisco J Burgos-Fernández, B.Buntheng Ly, F.Fernando Díaz-Doutón, M.Meritxell Vilaseca, J.Jaume Pujol and M.Maxime Sermesant. Automatic classification of multispectral eye fundus images using deep learning.Libro de Resúmenes RNO2021Online, SpainNovember 2021
### Conferences without proceedings
• 48 inproceedingsB.Benoît Audelan, L.Lopez Stéphanie, P.Pierre Fillard, Y.Yann Diascorn, B.Bernard Padovani and H.Hervé Delingette. Validation of lung nodule detection a year before diagnosis in NLST dataset based on a deep learning system.ERS 2021 - European Respiratory Society International CongressVirtual, United KingdomSeptember 2021
• 49 inproceedingsY.Yann Fraboni, R.Richard Vidal, L.Laetitia Kameni and M.Marco Lorenzi. Clustered Sampling: Low-Variance and Improved Representativity for Clients Selection in Federated Learning.International Conference on Machine Learningonline, FranceJuly 2021
• 50 inproceedingsY.Yann Fraboni, R.Richard Vidal and M.Marco Lorenzi. Free-rider Attacks on Model Aggregation in Federated Learning.AISTATS 2021 - 24th International Conference on Artificial Intelligence and StatisticsSan Diego, United StatesApril 2021
• 51 inproceedingsY.Yingyu Yang and M.Maxime Sermesant. Shape Constraints in Deep Learning for Robust 2D Echocardiography Analysis.FIMH 2021 - 11th International Conference on Functional Imaging and Modeling of the HeartStanford, United StatesJune 2021
### Scientific book chapters
• 52 inbookN.Nicholas Ayache. Foreword.Digital Anatomy - Applications of Virtual, Mixed and Augmented RealityHuman–Computer Interaction SeriesSpringer NatureMay 2021
• 53 inbookStatistical analysis of organs' shapes and deformations: the Riemannian and the affine settings in computational anatomy.Digital Anatomy - Applications of Virtual, Mixed and Augmented RealityHuman–Computer Interaction SeriesSpringer NatureMay 2021
### Doctoral dissertations and habilitation theses
• 54 thesisC.Clément Abi Nader. Modelling and simulating the progression of Alzheimer's disease through the analysis of multi-modal neuroimages and clinical data.Université Côte d'AzurJune 2021
• 55 thesisL.Luigi Antelmi. Statistical learning on heterogeneous medical data with bayesian latent variable models : application to neuroimaging dementia studies.Université Côte d'AzurJuly 2021
• 56 thesisB.Benoît Audelan. Probabilistic segmentation modelling and deep learning-based lung cancer screening.Université Côte d'AzurJuly 2021
• 57 thesisJ.Jaume Banus. Heart & Brain. Linking cardiovascular pathologies and neurodegeneration with a combined biophysical and statistical methodology.Université Côte d'AzurMay 2021
• 58 thesisComputational methods for statistical estimation on Riemannian manifolds and application to the study of the cardiac deformations.Université Côte d'AzurNovember 2021
• 59 thesisZ.Zihao Wang. Deep generative learning for medical data processing, analysis and modeling: application to cochlea ct imaging.Inria - Sophia Antipolis; Université côte d'azurSeptember 2021
### Reports & preprints
• 60 miscL.Luigi Antelmi, N.Nicholas Ayache, P.Philippe Robert, F.Federica Ribaldi, V.Valentina Garibotto, G. B.Giovanni B Frisoni and M.Marco Lorenzi. Combining Multi-Task Learning and Multi-Channel Variational Auto-Encoders to Exploit Datasets with Missing Observations -Application to Multi-Modal Neuroimaging Studies in Dementia.May 2021
• 61 miscJ.James Benn and S.Stephen Marsland. The Measurement and Analysis of Shapes.February 2022
• 62 miscQ.Quentin Clairon, C.Chloé Pasin, I.Irene Balelli, R.Rodolphe Thiébaut and M.Mélanie Prague. Parameter estimation in nonlinear mixed effect models based on ordinary differential equations: An optimal control approach.September 2021
• 63 miscY.Yann Fraboni, R.Richard Vidal, L.Laetitia Kameni and M.Marco Lorenzi. On The Impact of Client Sampling on Federated Learning Convergence.December 2021
• 64 miscN.Nina Miolane, M.Matteo Caorsi, U.Umberto Lupo, M.Marius Guerard, N.Nicolas Guigui, J.Johan Mathe, Y.Yann Cabanes, W.Wojciech Reise, T.Thomas Davies, A.António Leitão, S.Somesh Mohapatra, S.Saiteja Utpala, S.Shailja Shailja, G.Gabriele Corso, G.Guoxi Liu, F.Federico Iuricich, A.Andrei Manolache, M.Mihaela Nistor, M.Matei Bejan, A. M.Armand Mihai Nicolicioiu, B.-A.Bogdan-Alexandru Luchian, M.-S.Mihai-Sorin Stupariu, F.Florent Michel, K. D.Khanh Dao Duc, B.Bilal Abdulrahman, M.Maxim Beketov, E.Elodie Maignant, Z.Zhiyuan Liu, M.Marek Černý, M.Martin Bauw, S.Santiago Velasco-Forero, J.Jesus Angulo and Y.Yanan Long. ICLR 2021 Challenge for Computational Geometry & Topology: Design and Results.December 2021
• 65 miscT.Talia Nir, J.-P.Jean-Paul Fouche, J.Jintanat Ananworanich, B.Beau Ances, J.Jasmina Boban, B.Bruce Brew, L.Linda Chang, J.Joga Chaganti, C. R.Christopher R.K. Ching, L.Lucette Cysique, T.Thomas Ernst, J.Joshua Faskowitz, V.Vikash Gupta, J.Jaroslaw Harezlak, J.Jodi Heaps-Woodruff, C.Charles Hinkin, J.Jacqueline Hoare, J.John Joska, K.Kalpana Kallianpur, T.Taylor Kuhn, H.Hei Lam, M.Meng Law, C.Christine Lebrun-Frenay, A.Andrew Levine, L.Lydiane Mondot, B.Beau Nakamoto, B.Bradford Navia, X.Xavier Pennec, E.Eric Porges, C.Cecilia Shikuma, A.April Thames, V.Victor Valcour, M.Matteo Vassallo, A.Adam Woods, P.Paul Thompson, R.Ronald Cohen, R.Robert Paul, D.Dan Stein and N.Neda Jahanshad. Smaller limbic structures are associated with greater immunosuppression in over 1000 HIV-infected adults across five continents: Findings from the ENIGMA-HIV Working Group.January 2021
• 66 miscY.Yann Thanwerdas and X.Xavier Pennec. O(n)-invariant Riemannian metrics on SPD matrices.September 2021
• 67 miscY.Yann Thanwerdas and X.Xavier Pennec. The geometry of mixed-Euclidean metrics on symmetric positive definite matrices.November 2021
• 68 miscY.Yann Thanwerdas and X.Xavier Pennec. Theoretically and computationally convenient geometries on full-rank correlation matrices.January 2022
• 69 miscZ.Zihao Wang and H.Hervé Delingette. Attention for Image Registration (AiR): an unsupervised Transformer approach.May 2021
• 70 miscZ.Zihao Wang and H.Hervé Delingette. Quasi-Symplectic Langevin Variational Autoencoder.June 2021
### Other scientific publications
• 71 miscA.Anna Calissano, A.Aasa Feragen and S.Simone Vantini. Graph-Valued Models for Dimensionality Reduction and Regression.Online, United StatesAugust 2022
## 11.3 Other
### Patents
• 72 patentJ.Julian Krebs, H.Hiroshi Ashikaga, T.Tommaso Mansi, B.Bin Lou, K.Katherine Chih-ching Wu and H.Henry Halperin. Risk prediction for sudden cardiac death from image derived cardiac motion and structure features.US20210059612A1United StatesMarch 2021
• 73 patentJ.Julian Krebs, T.Tommaso Mansi and B.Bin Lou. Patient specific risk prediction of cardiac events from image-derived cardiac function features.US20210057104A1United StatesFebruary 2021 | 2022-07-02 08:13:28 | {"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": 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": 0.2006850391626358, "perplexity": 13530.012286727448}, "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/1656103989282.58/warc/CC-MAIN-20220702071223-20220702101223-00013.warc.gz"} |