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http://sylefeb.blogspot.com/2011/06/ | ## Wednesday, June 22, 2011
### [latex] Math in algorithms
I am using the algorithmic package to add code in my latex documents. I came across the following issue: How to include math in the code? Simple solution:
\begin{lstlisting}[mathescape]
Thanks to http://stackoverflow.com/questions/2809836/latex-math-symbols-in-listings | {"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": 0.8530155420303345, "perplexity": 3672.9387383483286}, "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-2019-13/segments/1552912203123.91/warc/CC-MAIN-20190324002035-20190324024035-00149.warc.gz"} |
https://www.parabola.unsw.edu.au/1964-1969/volume-2-1965/issue-3 | Volume 2
1965
This article is concerned with the decimal expansions of numbers of the form $\frac{a}{b}$, where $a$ and $b$ are positive integers with no common factor except $1$, and $a$ is less than $b$.
Each answer is the recurring block of digits in the "decimal" expansion of a rational number $\frac{a}{p}$, using $S$ as the base of the number system.
The theory of combinatorial configurations abounds in unsolved problems, some of which can be stated in simple non-technical terms.
The word induction is used to describe two quite different processes for reaching conclusions.
J41 In the following equation: - $$29+38+10+4+5+6+7 = 99$$ the left hand side contains every digit exactly once. Either find a similar expression (involving only $+$ signs) whose sum is $100$, or prove that it is impossible to do so.
J31 This is set again in the Problems Section.
Q.1 As for the Junior Section - see Parabola Vol. 2, No. 1. | {"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": 0.8766383528709412, "perplexity": 256.0476744173924}, "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/1614178364008.55/warc/CC-MAIN-20210302125936-20210302155936-00369.warc.gz"} |
https://tex.stackexchange.com/questions/8918/hello-world-in-pdflatex/8926#8926 | Hello World in pdfLaTeX
After many years being silent, TeX can now talk! Almost minimal code below! It should be of particular interest to Physicists and Cosmologists. Need to have Adobe Reader installed.
\documentclass{scrartcl}
\usepackage[T1]{fontenc}
\usepackage[scaled =.92]{helvet}
\setlength{\paperwidth}{5.2075in}
\setlength{\paperheight}{4.90in}
\renewcommand*{\familydefault}{phv}
\usepackage[pdftex,margin=0.5in]{geometry}
\usepackage{soul}
\usepackage{fancyhdr}
\lfoot{}\cfoot{}\rfoot{}
\pagestyle{fancy}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage[pdftex,pdfpagelayout=SinglePage,
]{hyperref}
\definecolor{background}{rgb}{0.99,0.98,0.90}
\pagecolor{background}
\setlength{\parindent}{0.0cm}
\usepackage[pdftex]{insdljs}
\begin{insDLJS}[test]{test}{JavaScript}
function Hello()
{
var cSpeaker = tts.getNthSpeakerName(0);
tts.speaker = cSpeaker;
tts.qText ("Hello, Tex Stack Exchange. Helloooo, can anybody hear me? Helloooo?");
tts.talk();
}
function HelloWorld()
{
var cSpeaker = tts.getNthSpeakerName(0);
tts.speaker = cSpeaker;
tts.qText ("Hello, Tex Stack Exchange. Helloooo, can anybody hear me?");
tts.qText("This is a new and alien TeX world.");
tts.talk();
}
function WhatsUp()
{
var cSpeaker = tts.getNthSpeakerName(0);
tts.speaker = cSpeaker;
tts.qText ("Humans are not proud of their ancestors, and rarely invite them round to dinner.");
tts.qText("What\'s up?");
tts.talk();
}
function Universe()
{
var cSpeaker = tts.getNthSpeakerName(0);
tts.speaker = cSpeaker;
tts.qText ("In the beginning the Universe was created. This has made a lot of people very angry and has been widely regarded as a bad move.In those days spirits were brave, the stakes were high, men were real men, women were real women and small furry creatures from Alpha Centauri were real small furry creatures from Alpha Centauri. There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened. I am Douglas Adams");
tts.qText("Who are you? I may be a sorry case, but I don't write jokes in base 13");
tts.talk();
}
\end{insDLJS}
%% This must be here
\OpenAction{/S/JavaScript/JS(Hello();)}
%% Short hand commands
\newcommand{\textforlabel}[2]{%
\TextField[name={#1}, value={#2}, width=9em,align=2,%
bordercolor={0.990 .980 .85},%
}
\begin{document}
\phantom{0}
\begin{center}
\vspace{2cm}
\textbf{\Huge Hello World!\\[0.2cm] Can you hear me?}
\end{center}
\vfill
\newpage
\begin{Form}
\begin{center}
%% Push button is defined here
\PushButton[name=hello,%
onclick={HelloWorld();}, bordercolor={0.650 .790 .94}%
]{Hello World!}
\\[10pt]
\PushButton[name=hello,%
onclick={WhatsUp();}, bordercolor={0.650 .790 .94}%
]{What's Up?}
\\[10pt]
\PushButton[name=hello,%
onclick={Universe();}, bordercolor={0.650 .790 .94}%
]{What's with the Universe?}
\\[2cm]
\end{center}
\end{Form}
\end{document}
On a more serious note, does (all)TeX have a place in the "new" media?
• Not that I want to spoil the fun, but what exactly is the question here? Jan 13 '11 at 23:10
• @Yiannis: There are packages to typeset music. What about adjoining to these the possibility of playing the music that is typeset? Jan 14 '11 at 12:46
• @Bruno Le Floch: TeX itself typesets text, so should it also be a text-to-speech reader? External utilities do that, as they should. I would think that score-to-music is significantly easier, since the notation is totally unambiguous and highly structured, so can't the same (or similar) utilities do this as well? Jan 14 '11 at 13:24
• @Ryan Reich: TeX is (among other things) about placing text in the most pleasing way. Couldn't similar algorithms (or the same) be used to decide the time between two words, sentences, etc, instead of the spacing that TeX usually caters for? Jan 14 '11 at 16:04
• @Bruno Le Floch: Oh, I see. That is an interesting idea for adapting the spacing algorithm. Just create a tonal "font" with chords interpreted to be words, and assign a space factor of the appropriate fraction to tones of various lengths. It would be up to the composer/author not to write "words" with tones of different lengths, though I don't see off the top of my head how to get chords in which one note hangs while the others change. Perhaps this should be implemented as box widths instead... Jan 14 '11 at 16:45
TeX is designed to solve a single problem: setting letters into words into lines into paragraphs into pages, interspersed with mathematics and with displays or inserts of various kinds. To facilitate the automation of this task in a variety of unanticipated circumstances, it provides a complete (if obtuse) programming language with many hooks into the typesetting functions with which it operates. The fact that this language is Turing-complete leads enthusiastic people to claim that TeX can do anything, but this is not true: the task in question must be capable of description in this language. By virtue of the hooks, those tasks include typesetting, but the only facilities provided for other tasks are: the \special command and the very limited input/output commands (particularly \write18). Effectively, TeX allows other kinds of programming applications only through the wholesale importation of external utilities.
I think it's a mistake to try to make TeX do more than it is designed to be easily capable of doing, and a mistake to try to insert new features into that design. For example, hyperref works because one can use \special simply to feed information to the viewer, but it would be silly to try to use some kind of Javascript program to rewrite the document in accordance with a hyperlink. The reason is that TeX thinks of a document as being a finished product, and so "runtime" operations are not even considered. This is necessary for the problem of typesetting ever to be solved. Of course, if your Javascript has no runtime effect on the document it can be embedded via \special, but since that is already possible I doubt it is the question. Whether it is a matter of practicality or of good taste...my opinion was already given in the first sentence. | {"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": 0.9013171792030334, "perplexity": 2235.6385146614543}, "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-2022-05/segments/1642320304915.53/warc/CC-MAIN-20220126041016-20220126071016-00201.warc.gz"} |
http://www.webupd8.org/2009/04/4-ways-to-create-bootable-live-usb.html | # 4 Ways to Create Bootable Live USB Drives (For Windows, Linux and Mac OS X)
Tweet
Below you'll find 4 tools along with instructions on creating bootable live USB drives under Linux (Ubuntu), Windows and Mac OS X.
## 1. Using UNetbootin (for Windows and Linux)
UNetbootin allows you to create bootable Live USB drives for a variety of Linux distributions from Windows or Linux, without requiring you to burn a CD. You can either let it download one of the many distributions supported out-of-the-box for you, or supply your own Linux .iso file if you've already downloaded one or your preferred distribution isn't on the list.
UNetbootin has built-in support for automatically downloading and loading the following distributions: Ubuntu, Debian, Fedora, PCLinuxOS, Linux Mint, Sabayon Linux, Gentoo, MEPIS, openSUSE, Zenwalk, Slax, Dreamlinux, Arch Linux, Elive, CentOS, Damn Small Linux, Mandriva, SliTaz, FaunOS, Puppy Linux, FreeBSD, gNewSense, Frugalware Linux, NetBSD but can work with others too.
UNetbootin can also be used to load various system utilities, including:
• Parted Magic, a partition manager that can resize, repair, backup, and restore partitions.
• Super Grub Disk, a boot utility that can restore and repair overwritten and misconfigured GRUB installs or directly boot various operating systems
• Backtrack, a utility used for network analysis and penetration testing.
• Ophcrack, a utility which can recover Windows passwords.
• NTPasswd, a utility which can reset Windows passwords and edit the registry.
• Gujin, a graphical bootloader that can also be used to boot various operating systems and media.
• Smart Boot Manager (SBM), which can boot off CD-ROM and floppy drives on computers with a faulty BIOS.
• FreeDOS, which can run BIOS flash and other legacy DOS utilities.
## Installation & Screenshots
1. If using Windows, run the file, select a distribution, floppy/hard disk image, or kernel/initrd to load, select a target drive (USB Drive or Hard Disk), then reboot once done.
2. If using Linux, make the file executable (using either the command chmod +x ./unetbootin-linux, or going to Properties->Permissions and checking "Execute"), then start the application, you will be prompted for your password to grant the application administrative rights, then the main dialog will appear, where you select a distribution and install target (USB Drive or Hard Disk), then reboot when prompted.
3. After rebooting, if you created a Live USB drive by selecting "USB Drive" as your install target, press the appropriate button (usually F1, F2, F12, ESC, or backspace) while your computer is starting up to get to your BIOS boot menu and select USB drive as the startup target; otherwise if there's no boot selection option, go to the BIOS setup menu and change the startup order to boot USB by default. Otherwise, if you did a "frugal install" by selecting "Hard Disk" as your install target, select the UNetbootin entry from the Windows Boot Menu as the system boots up.
To create a Live USB using UNetbootin, download an ISO file, select it under UNetbootin's "diskimage" option, and specify your target USB disk under "Drive:". After pressing OK, wait as the ISO is extracted to your USB drive; once done, you will have a bootable Linux Mint Live USB drive.
### Requirements
1 GB or larger USB drive, formatted as Fat32 (most USB drives come formatted as FAT32 by default, but if you need to format it, on Windows, go to My Computer->right click your USB drive->format, or on Linux, use GParted or another partition manager)
Supported operating systems: Windows 2000 and above OR a modern Linux distribution
Additional dependencies (Linux Only): You will need the packages syslinux and p7zip-full installed (no dependencies on Windows)
## 2. Using Win32 Image Writer (Windows only)
### Graphical Interface
3. Note the drive letter assigned to your flash media
4. Start Disk Imager
6. Remove your flash media when the operation is complete
### Command Line alternative
4. Run flashnul -p
5. Note the physical device number for the USB drive
7. Answer "yes" if the selected destination device is correct
8. Remove your USB drive when the command completes
## 3. Using usb-imagewriter (Ubuntu Only)
### Graphical Interface
2. Install the usb-imagewriter package
3. Open Applications -> Accessories -> Image Writer
5. Select the downloaded file and flash device, and click "Write to Device"
6. Remove your device when the operation is complete
### Command Line alternative
2. Open a terminal and insert your flash media
3. Look at the output of dmesg | tail -20 to determine the device node assigned to your flash media (e.g. /dev/sdb)
4. Run sudo umount /dev/device/node
6. Remove your flash media when the command completes
Update: Ubuntu now includes "Startup disk creator". You can access it via System > Administration > Startup Disk Creator and it's a very easy to use tool. Simply enter your memory stick into the USB drive, then open Startup Disk Creator and select "Format" (the USB stick needs to be formatted first), then select the ISO image you want to write on the USB memory stick and click "Make startup disk". That's it. | {"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": 0.1610850691795349, "perplexity": 17745.311877207314}, "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-2017-26/segments/1498128323842.29/warc/CC-MAIN-20170629015021-20170629035021-00451.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=222312 | # The capacitance of the two metal sphere system
by Indis Nenhrma
Tags: capacitance, metal, sphere
P: 8
1. The problem statement, all variables and given/known data
What is the capacitance of the two metal sphere system.
*The question has been attached shematically. All the variables are given in the attached file, there is no missing variable.
thx for any help.
Attached Images
two metal sphere.bmp (230.4 KB, 120 views)
Attached Files
two metal sphere.doc (23.5 KB, 84 views)
P: 8 I thought that I could solve this problem by integration. I can divide two spheres to small circular plates as parallel to each other and consider them as parallel plates. Then, i can add them up. It ain't difficult, but it requires calculus knowledge.
Sci Advisor P: 1,256 You can't do it that way. It is a very difficult problem by any method. You could try image, but a large number of image charges would be needed.
P: 2 The capacitance of the two metal sphere system You can find the electric-field due to one sphere. Then, intagrate it from r to L-r to find the potential between spheres due to one sphere. I think the spheres must have a charge +Q and -Q, thus, potentials due to each sphere are same and total potential is two times the calculated one with integration. Once you have found an equality consists of Q and V, you can find the capacitancy by writing this equality in form of Q = C*V.
P: 8
Quote by Porter Then, intagrate it from r to L-r to find the potential between spheres due to one sphere.
Why did you choose limits as r and L-r?
P: 2 Because potential diffrence inside a sphere is zero, there is no potential difference between 0 and r, you do not need the to add potential difference between these points. And, of course, electric-field inisde sphere is zero, not depending on distance, so you can not even intagrate electric-field by choosing limits involving 0 to r and L-r to L intervals.
P: 8 thx dude.
P: 455 But the surface charge density on each sphere is not uniform.
Related Discussions Engineering, Comp Sci, & Technology Homework 1 Classical Physics 1 Electrical Engineering 7 Introductory Physics Homework 5 Introductory Physics Homework 1 | {"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": 0.9066718816757202, "perplexity": 1040.6659437123046}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510272940.33/warc/CC-MAIN-20140728011752-00027-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://mathhelpforum.com/geometry/178323-points-sine-wave.html | # Thread: Points on a sine wave
1. ## Points on a sine wave
Given 3 points : p1(x1,sin(x1)) p2(x2,sin(x2)) p3(x3,sin(x3)) does the following identity hold ?
if x2 < x3 and | x1 - x2 | < | x1 - x3| then distance(p1,p2) < distance(p1,p3)
2. Consider x_1 = 0, x_2 = pi/2, x_3 = pi. | {"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": 0.9395889639854431, "perplexity": 4999.40007911811}, "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-2017-09/segments/1487501170614.88/warc/CC-MAIN-20170219104610-00355-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://jakesmathlessons.com/limits/limits-intro/ | # Limits – Intro
Limits are an important topic to understand in calculus. The reason for this is that most of the other main categories in calculus are built around limits.
## But what is a limit?
When you take the limit of a function, we are essentially asking: what $y$ value does this function approach as $x$ gets really really close to a certain value.
Now we’ll get into a few examples of what this looks like graphically. If you’d like to learn more about evaluating limits algebraically, I’d recommend starting with the 8 limit properties.
### Example 1
For example, take the function graphed below, $y = x^2$. Imagine we travel along the function, from both sides, near $x=2$. As we get closer and closer to the $x$ value of $2$, what $y$ value do we also get closer and closer to?
We get closer and closer to a $y$ value of $4$. This is a simple example, but this is the general idea behind all limits. If you need to find a limit of a function as $x$ approaches a certain value, you can figure this out quite easily with a graph of the function.
Just trace along the function from both the left and right side of that $x$ value and move in towards the specific $x$ value. Whatever $y$ value you close in on is the value of the limit of that function as $x$ approaches the given value.
In this example, we would say: “the limit of $x^2$, as $x$ approaches $2$, is $4$.” Or “the limit as $x$ approaches $2$, of $x^2$, is $4$.” In mathematical notation, that phrase would look something like this:
$$\lim_{x \to 2} x^2 = 4$$
It just so happens that the value of the limit is the same value we would get by plugging $x = 2$ into this function. What I mean by that is, $f(2) = 2^2 = 4$ is exactly what we got for the limit of $x^2$ as $x$ approaches $2$. This is not a coincidence. I will go into more detail about why this is important when we discuss continuous functions and continuity. But for now, let’s do a few harder examples.
### Example 2
I’m going to stick with a couple more examples of finding limits using graphs, because it is a skill you will need to know, and I’ve noticed that it’s something a lot of introductory calculus students have a hard time with. So if finding limits from graphs is something that confuses you, don’t worry.
Let’s try finding a little more challenging limit. Below is a graph of $f(x)$.
Using the graph in Figure 1.2, we will consider two different limits. First let’s consider the following:
$$\lim_{x \to 1}f(x)$$
Let’s look at what’s going on in this graph at and around $x = 1$. It looks like the function was supposed to go through the point $(1,\ 2)$, but that point got taken out. Instead, there is a hole there, and the graph of $f(x)$ includes the point $(1,\ 4)$. Therefore, we can say $f(1) = 4$. All this means is that this function returns a value of $4$, or $y = 4$, when $x = 1$. However, this has no impact at the limit we are considering.
Limits aren’t impacted by what happens at a specific point, only by what is happening around that point. As a result, knowing that $f(1)=4$ doesn’t help us to find
$$\lim_{x \to 1}f(x).$$
#### So what do we need to look at?
To find this limit, we need to look at what’s happening around $x = 1$, and ignore what’s happening at $x = 1$. As we get closer and closer to $x = 1$, from the left and the right, what $y$ value do we get close to? Imagine traveling along this function, perhaps starting at $x = 2$, and moving to the left. As you pass $x = 1.5$, then $x = 1.25$, then $x = 1.125$, and so on, we get closer and closer to the hole I mentioned earlier. We get closer and closer to a $y$ value of $2$.
The same thing happens if we start on that function at $x = 0$ and move to the right, toward $x = 1$. From both sides we get closer and closer to $y = 2$.
It is important to realize that we never actually get to $y = 2$. This is because we never get to $x = 1$, just infinitely close to it. As I said before, when solving for a limit, it doesn’t matter what happens at the point it’s asking about. It only matters what happens as you get infinitely close. This is the reason why, in this example, $f(1) = 4$, but
$$\lim_{x \to 1}f(x) = 2.$$
The other important thing to point out in this example is that we approached $x = 1$ from both sides and they both led us closer and closer to the same $y$ value of $2$. If both sides didn’t give us the same value, this limit would not exist. This is illustrated in greater detail in my lesson about one-sided limits.
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If you would like to continue reading about limits, check out my limits page. There’s plenty more lessons and example problems to read there. If your questions are not answered there, I’d love your feedback. Email me at jakesmathlessons@gmail.com with any unanswered questions you have and I’ll make sure to answer your question! | {"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": 112, "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": 0.7038897275924683, "perplexity": 139.95710575540983}, "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/1679296949644.27/warc/CC-MAIN-20230331144941-20230331174941-00636.warc.gz"} |
https://biomechanical.asmedigitalcollection.asme.org/IDETC-CIE/proceedings-abstract/IDETC-CIE2018/51791/V004T08A012/275003 | The modal decomposition of tapping mode atomic force microscopy microcantilevers in air and liquid environment was experimentally investigated to identify their complex responses. In experiment, the flexible microcantilevers and a flat HOPG sample were used. The responses of the microcantilevers were obtained to extract the linearized modes and orthogonal values using the methods for the proper orthogonal decomposition and the smooth orthogonal decomposition. The influence of the tapping setpoints and the hydrodynamic damping forces were investigated with the multi-mode response of microcantilevers. The results show that the first mode is dominant under normal operating conditions in tapping mode. However, at lower setpoint, the flexible microcantilever behaved uncertain modal distortion near the tip on the sample. The dynamics tapping effect and the damping between microcantilever and liquid influenced their responses.
This content is only available via PDF. | {"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": 0.9022627472877502, "perplexity": 2098.1283370561164}, "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-2022-33/segments/1659882570730.59/warc/CC-MAIN-20220807211157-20220808001157-00244.warc.gz"} |
https://en.wikipedia.org/wiki/B%C3%A9zout%27s_lemma | # Bézout's identity
(Redirected from Bézout's lemma)
Bézout's identity (also called Bézout's lemma) is a theorem in the elementary theory of numbers: let a and b be nonzero integers and let d be their greatest common divisor. Then there exist integers x and y such that
${\displaystyle ax+by=d.}$
• the greatest common divisor d is the smallest positive integer that can be written as ax + by
• every integer of the form ax + by is a multiple of the greatest common divisor d.
The integers x and y are called Bézout coefficients for (a, b); they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm. If both a and b are nonzero, the extended Euclidean algorithm produces one of the two pairs such that ${\displaystyle |x|\leq \left|{\frac {b}{d}}\right|}$ and ${\displaystyle |y|\leq \left|{\frac {a}{d}}\right|}$ (equality may occur only if one of a and b is a multiple of the other).
Many theorems of elementary theory of numbers, such as Euclid's lemma or Chinese remainder theorem, result from Bézout's identity.
A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that results from Bézout's identity is thus true in all these domains.
## Structure of solutions
When one pair of Bézout coefficients (x, y) has been computed (e.g., using extended Euclidean algorithm), all pairs can be represented in the form
${\displaystyle \left(x+k{\frac {b}{\gcd(a,b)}},\ y-k{\frac {a}{\gcd(a,b)}}\right),}$
where k is an arbitrary integer and the fractions simplify to integers.
Among these pairs of Bézout coefficients, exactly two of them satisfy
${\displaystyle |x|<\left|{\frac {b}{\gcd(a,b)}}\right|\quad {\text{and}}\quad |y|<\left|{\frac {a}{\gcd(a,b)}}\right|.}$
This relies on a property of Euclidean division: given two integers c and d, if d does not divide c, there is exactly one pair (q,r) such that c = dq + r and 0 < r < |d|, and another one such that c = dq + r and 0 < -r < |d|.
The Extended Euclidean algorithm always produces one of these two minimal pairs.
### Example
Let a = 12 and b = 42, gcd (12, 42) = 6. Then we have the following Bézout's identities, with the Bézout coefficients written in red for the minimal pairs and in blue for the other ones.
{\displaystyle {\begin{aligned}\vdots \\12&\times \color {blue}{-10}&+\;\;42&\times \color {blue}{3}&=6\\12&\times \color {red}{-3}&+\;\;42&\times \color {red}{1}&=6\\12&\times \color {red}{4}&+\;\;42&\times \color {red}{-1}&=6\\12&\times \color {blue}{11}&+\;\;42&\times \color {blue}{-3}&=6\\12&\times \color {blue}{18}&+\;\;42&\times \color {blue}{-5}&=6\\\vdots \end{aligned}}}
## Proof
Bézout's lemma is a consequence of the defining property of Euclidean division, namely: that dividing a positive integer a by a positive integer b yields a remainder greater than or equal to zero and strictly less than b. For given positive integers a and b there is a smallest positive integer d = as + bt among all those of the form ax + by with x and y integers. Now the remainder yielded by dividing either a or b by d is also of the form ax + by since it is obtained by subtracting a multiple of d = as + bt from a or b; so the remainder must be greater than or equal to zero and strictly smaller than d. This leaves 0 as only possibility for such a remainder, so d divides both a and b exactly.
If c is a common divisor of a and b, then c also divides d = as + bt. Since c divides d, c must be less than or equal to d, thus d is the greatest common divisor of a and b; the proof is complete.
This proof does not provide a method for computing Bézout's coefficients. However, Bézout's lemma is also a corollary of the proof of the Extended Euclidean algorithm and this algorithm does provide an efficient method of computing these coefficients. This algorithm and the associated proof may also be extended to any Euclidean domain.
## Generalizations
### For three or more integers
Bézout's identity can be extended to more than two integers: if
${\displaystyle \gcd(a_{1},a_{2},\ldots ,a_{n})=d}$
then there are integers ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ such that
${\displaystyle d=a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}}$
has the following properties:
• d is the smallest positive integer of this form
• every number of this form is a multiple of d
### For polynomials
Bézout's identity works for univariate polynomials over a field exactly in the same ways as for integers. In particular the Bézout's coefficients and the greatest common divisor may be computed with the Extended Euclidean algorithm.
As the common roots of two polynomials are the roots of their greatest common divisor, Bézout's identity and fundamental theorem of algebra imply the following result:
For univariate polynomials f and g with coefficients in a field, there exist polynomials a and b such that af + bg = 1 if and only if f and g have no common root in any algebraically closed field (commonly the field of complex numbers).
The generalization of this result to any number of polynomials and indeterminates is Hilbert's Nullstellensatz.
### For principal ideal domains
As noted in the introduction, Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain (PID). That is, if R is a PID, and a and b are elements of R, and d is a greatest common divisor of a and b, then there are elements x and y in R such that ax + by = d. The reason: the ideal Ra+Rb is principal and indeed is equal to Rd.
An integral domain in which Bézout's identity holds is called a Bézout domain.
## History
French mathematician Étienne Bézout (1730–1783) proved this identity for polynomials.[1] However, this statement for integers can be found already in the work of another French mathematician, Claude Gaspard Bachet de Méziriac (1581–1638).[2][3][4] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 9, "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": 0.9388786554336548, "perplexity": 247.96604010816387}, "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-2016-36/segments/1471982296571.15/warc/CC-MAIN-20160823195816-00228-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://www.dfranke.us/posts/2014-10-14-how-poodle-happened.html | ## How POODLE Happened
Bodo Möller, Thai Duong, and Krzysztof Kotowicz have just broken the internet again with POODLE[20], a new and devastating attack against SSL. POODLE, an acronym for Padding Oracle On Downgraded Legacy Encryption, permits a man-in-the-middle attacker to rapidly decrypt any browser session which utilizes SSL v3.0 — or, as is generally the case, any session which can be coerced into utilizing it. POODLE is a death blow to this version of the protocol; it can only reasonably be fixed by disabling SSL v3.0 altogether.
This post is meant to be a “simple as possible, but no simpler” explanation of POODLE. I’ve tried to make it accessible to as many readers as possible and yet still go into full and accurate technical detail and provide complete citations. However, as the title implies, I have a second goal, which is to explain not merely how POODLE works, but the historical mistakes which allow it to work: mistakes that are still with us even though we’ve known better for over a decade.
### A brief history of SSL
POODLE is the latest in a long line of line of similar attacks against known weaknesses in SSL’s use of cipher block chaining (CBC). This explanation of it, therefore, will begin with a bit of a history lesson. I’ve focused this timeline on vulnerabilities that are relevant to CBC; it is nowhere near being a complete list of past problems with SSL. I’ve left off the Bleichenbacher attacks on RSA[6, 14], compression ratio attacks like CRIME[27] and BREACH[24], and implementation bugs like Heartbleed[16].
• 1994: A team including Taher Elgamal at Netscape Communications designs SSL (Secure Socket Layer) v1.0; this version is never released publicly[22].
• 1995: SSL v2.0 is released as a feature of Netscape Navigator[22].
• 1996: Paul Kocher et. al., also at Netscape, develop SSL v3.0[15], a nearly-complete redesign of the protocol, addressing several serious vulnerabilities in SSL v2.0. SSL v3.0 is the first version of the protocol to authenticate handshake messages, thus in theory preventing attackers from triggering downgrades to earlier protocol versions.
• 1999: The IETF publishes RFC 2246[9], standardizing TLS (Transport Layer Security) v1.0, closely based on SSL v3.0.
• 2001: Serge Vaudenay notes a vulnerability in the padding schemes used for cipher block chaining in SSL v3.0 and TLS v1.0, but his attack is at the time believed to be impractical under normal circumstances[30].
• 2002: Wei Dai publishes an attack against SSH’s practice of using predictable initialization vectors for cipher block chaining[8]; Bodo Möller observes that SSL v3.0 and TLS v1.0 have the same flaw but does not leverage it into an attack[18].
• 2003: Brice Canvel, Alain Hiltgen, Serge Vaudenay, and Martin Vuagnoux combine Vaudenay’s padding attack with a timing attack to produce practical results against OpenSSL[7].
• 2006: Gregory Bard publishes a “challenging but feasible” way of exploiting the predictable-IV flaw observed by Dai and Möller to extract small amounts of information from an encrypted SSL session[3].
• 2006: With the release of Internet Explorer 7, all major browsers ship with TLS 1.0 enabled by default and SSL 2.0 disabled by default[31].
• 2006: RFC 4346[10] defines TLS v1.1, fixing the predictable-IV flaw and providing implementation advice for mitigating the Vaudenay attack. It is slow to be adopted.
• 2008: RFC 5246[11] defines TLS v1.2, with support for AEAD cipher modes. Again, adoption is slow.
• 2011: Thai Duong and Juliano Rizzo sucessfully use the Vaudenay attack to exploit ASP.NET[12].
• 2011: Thai Duong and Juliano Rizzo publish the BEAST attack[13], building on Gregory Bard’s work to provide a much more powerful attack against TLS v1.0 and SSL v3.0. Some servers switch to using RC4, which, being a stream cipher, is not vulnerable to BEAST or any other variation of the Vaudenay attack.
• 2012–2014: Motivated in part by BEAST, browser support for TLS v1.1 and v1.2 finally improves[31].
• 2013: Nadhem AlFardan and Kenny Paterson publish Lucky13[1], a new twist on the Canvel-Hiltgen-Vaudenay-Vuagnoux timing attack of 2003. It impacts CBC mode in all versions of SSL.
• 2013: Nadhem AlFardan, Daniel Bernstein, Kenny Paterson, Bertram Poettering and Jacob Schuldt publish a practical attack against long-known weaknesses in RC4[2]. As a cure for BEAST, RC4 can now be considered worse than the disease.
• 2014: Bodo Möller, Thai Duong, and Krzysztof Kotowicz publish POODLE[20].
### The downgrade dance, or why the ’90s just won’t stop calling
You might gather by reading this timeline that nobody who has been paying attention is really shocked hear about another attack against SSL’s handling of cipher block chaining. It’s been a thorn in our side for at least 13 years and has jabbed us again and again.
Some readers are probably wondering why we still care about SSL v3.0 vulnerabilities. TLS v1.0 was published 15 years ago, and support for it has practically been universal since the demise of Internet Explorer 6 (which supports TLS v1.0 but shipped with it disabled by default). The handshake authentication introduced in SSL v3.0 is supposed to prevent downgrade attacks, and in theory it should. In the overwhelming majority of cases where both client and server support TLS v1.0 or greater, we shouldn’t have to care about anything that’s wrong with SSL v3.0 — right?
Unfortunately, no — that’s wrong. The problem stems from browser vendors’ desire to be able to cope with buggy servers and middleboxes which advertise a protocol version that they can’t actually support. To work around such broken behavior, when an SSL handshake fails most browsers (all but Opera[5]) will fall back to an earlier protocol version and retry. This browser behavior, called the “downgrade dance”, makes it trivially vulnerable to downgrade attacks. All an attacker has to do is temporarily cut the client’s connection a few times until the browser’s fallback behavior is triggered.
### A primer on SSL bulk encryption
SSL is a big, complex protocol, and stepping through the whole thing would make a much longer post than this one. Fortunately, most of that complexity lives in the initial steps of the protocol, the handshake phase. The handshake phase is the part of the protocol which enables two previously-unfamiliar parties (a client and a server) to agree on a shared secret key for communication, in addition to some non-secret, auxiliary parameters such as which cipher algorithms they’ll be using. The attacks we’re concerned with in this article don’t involve the handshake: they attack the bulk encryption phase, where the communicating parties are already exchanging application-level messages such as HTTP requests. So, for the remainder of this discussion, you can assume that the client and server already have a key shared between the two of them and nobody else.
That shared key is actually several shared keys. Each side has one key that it uses for encryption (preventing attackers from reading the contents of the communication), and one that it uses for authentication (preventing attackers from modifying the contents of the comunication without being immediately detected). Client-to-server communication uses one pair of keys and server-to-client communication uses another, bringing the total to four.
The modern consensus (e.g. [28]) among cryptographers is that right way to produce an authenticated, encrypted block of data is either to use a dedicated AEAD construct such as AES-GCM, or to encrypt-then-MAC: first encrypt the plaintext, producing ciphertext, then compute a message authentication code (MAC) over the ciphertext. However, when SSL was designed in mid-90s, this consensus was not yet established, and SSL does things in a different order: MAC-then-encrypt. It first computes a MAC over the plaintext, then encrypts the plaintext together with the MAC. While this construction isn’t inherently broken per se, it’s fraught with danger and it directly enabled Vaudenay’s attack and POODLE.
Most algorithms supported by SSL v3.0 – TLS v1.1 (in practice, all but RC4) are based on block ciphers. A block cipher is a cryptographic primitive which takes as input a key of a fixed size (typically 128–256 bits, but much shorter for some legacy ciphers), and a plaintext of fixed size (typically 64–128 bits), and produces a ciphertext of equal length to the plaintext. A corresponding decryption primitive, given the ciphertext and the same key, returns the original plaintext.
Because raw block ciphers can only operate on fixed-sized inputs, a bit of additional machinery, called a block cipher mode of operation is needed for encrypting longer messages. The mode of operation used in SSL is called cipher block chaining, or CBC, and it works as follows:
1. Take the first block of plaintext and XOR it with an extra parameter called the initialization vector, or IV.
2. Encrypt the result of step 1, producing a ciphertext. Output it.
3. Take the next block of input and XOR it with the previous ciphertext.
4. Encrypt the result of step 3, producing another ciphertext. Output it.
5. Repeat steps 3–4 until the input is exhausted
CBC allows you to use a block cipher to encrypt a message of effectively-unbounded length, but the length of the input stills needs to be a multiple of the block size. If this isn’t already the case, the input needs to be padded with some extra bytes in order to make it so.
The whole structure used as input for encryption looks like this ([15] §5.2.3.2):
To put that into English, this means the structure consists of four fields concatenated together:
1. The plaintext.
2. A MAC computed over the plaintext, also covering some auxiliary data such as the sequence number of the message.
3. As many padding bytes as are needed to make the message fit the block size.
SSL v3.0 doesn’t specify what goes into the padding bytes; they’re often random. TLS is more strict about them: each byte of padding must take on the same value as the length byte at the end. So, if you have three bytes of padding plus the one length byte, then the last four bytes of your message will be 0x03 0x03 0x03 0x03. This detail turns out to thwart some attacks, including POODLE.
In both SSL v3.0 and all versions of TLS, the initialization vector used to encrypt the first message of the bulk encryption phase is determined during the handshake. TLS v1.1 changes how the initialization vectors for subsequent messages are chosen. In v1.1 and beyond, a fresh initialization vector is transmitted explicitly with every message. In v1.0 and prior, the last block of ciphertext of each message is used as the initialization vector for the next. The old approach is insecure and is what enabled BEAST, but the difference has no relevance to POODLE.
The notion of an oracle is a central concept of cryptology. The term is borrowed from theoretical computer science. In CS theory, an oracle is a “black box” which, given a string, returns an instantaneous yes-or-no answer as to whether that string satisfies some particular logical predicate. CS theorists commonly study how some particular oracle would augment the capability of a Turing machine, by enabling it to solve new problems or to solve old ones more efficiently.
The sorts of oracles that interest CS theorists are usually physically unrealizable, but the sort that interest cryptologists are everywhere: they’re often built by accident. A cryptologic oracle is usually manifested as a network service which has access to some secret piece of information, such as a cryptographic key. When it interacts with a (perhaps hostile) network client, it somehow leaks some bit of information which could not otherwise be obtained without direct access to the secret.
The general idea behind a padding oracle is that when an SSL server (or client) receives and decrypts some ciphertext, its behavior afterward reveals information about the padding bytes of the plaintext. If the padding is valid, it continues with the protocol; if it is invalid, it aborts with an error.
### The Vaudenay attack
Suppose an SSL client sends the following message to a server, using AES-128 (block size = 16 bytes) for encryption and HMAC-SHA-1 (tag size = 20 bytes) for authentication:
Before encryption, a hex dump of the message would look like this:
I’ve broken the lines at 16-byte intervals, so every line corresponds to one AES block. The 20 bytes following the body of the message, starting with bc and ending with 5b, are the MAC tag. After the end of the MAC tag, we’re 15 bytes into a 16-byte block, so we append no padding, and then a padding-length byte of 00.
So that you can follow along and try this yourself, I’m using a all-null keys and IVs. In practice, of course, these would be replaced by real keys, and the attacker wouldn’t know them.
Here’s the above message after encryption:
Let’s see what happens if we start flipping bits in this ciphertext. See that 47 byte at the end of the first block? Let’s change it to 46 and then see what happens after decryption:
The first block of plaintext has been entirely corrupted, just like you’d probably expect. But what happened to the second block? The “V” at the end has changed to a “W”, yet the rest of it is intact. To understand why, recall how CBC works. Before encryption, each plaintext block is XORed with the previous ciphertext block. Therefore, CBC decryption needs to work by first decrypting each ciphertext block and then XORing the result with the previous ciphertext block. Therefore, flipping a bit in a ciphertext block will cause the corresponding bit in the next plaintext block to get flipped, without impacting the rest of the block or any succeeding block.
Now let’s try another experiment. I’ll take the second ciphertext block, the one that contains the password, and copy it to the end of the message, where the padding belongs. So my ciphertext now looks like this:
If you decrypt this you get:
The last byte, where the padding-length belongs, is now 82. Why 82? The plaintext byte from that block that I copied to the end was originally 56, or “V”, the last character of the password. But now it’s preceeded by a ciphertext block whole final byte is 93 rather than 47 like it was originally. So the plaintext ends up as 56 $$\oplus$$ 93 $$\oplus$$ 47 $$=$$ 82.
Have you ever seen the Hollywood trope where the hero is trying to break the door code to get into the evil overlord’s lair, so he plugs a gizmo into the control panel which slowly locks in one digit at a time until the whole code is broken? That’s a bit like how the Vaudenay attack actually works.
Let’s suppose that I as an attacker make a lucky guess that the user’s password ends in “V”. Then I could form the following ciphertext:
This ciphertext is the same as the last two lines of the one above, except that I’ve replaced the 93 at the end of the first block with 11, which is 93 $$\oplus$$ 82. Now if I decrypt this, I get plaintext ending in 00, which is the correct padding-length byte! If I can get the server to reveal this fact to me, then I can confirm that my “V” guess was correct.
This is the basis of the Vaudenay padding-oracle attack. An attacker who can get the server to reveal whether a ciphertext decrypts to something with valid padding or not, can then guess the contents of any block of plaintext one character at a time, and get confirmation when the guess is correct.
Vaudenay discovered this attack in 2001, but didn’t immediately recognize the full extent of its implications. He wasn’t convinced that it was actually possible for an attacker to tell valid from invalid padding based on the server’s behavior. Even if the attacker successfully generates valid padding in a tampered message, the tampering will still be detected because MAC verification will still fail (except, not necessarily — we’ll reconsider this when we talk about POODLE). TLS v1.0 generates a different error message for bad padding than it does for a bad MAC, but that error message is encrypted so it’s not obvious how an attacker can tell the difference. Vaudenay suggested getting the error message out of an unprotected log file, but this isn’t very plausible. Later, in 2003, Vaudenay co-authored a paper which uses a timing attack to tell the difference between the two errors, and in 2013 the Lucky13 attack resurrected this idea.
Vaudenay also originally believed that the fact that TLS treats all padding errors as fatal, shutting the connection and discarding the session key, meant that the full attack wasn’t possible: that the attacker got to take one guess at one byte and nothing more. POODLE, using ideas already foreshadowed by BEAST, shows that in the browser context, this isn’t necessarily so.
### POODLE
Recall that SSL v3.0 treats its padding differently than TLS does. In TLS, every padding byte is determined: it must take the same value as the padding-length. In SSL v3.0, the padding is random: it can be anything. Therefore, in SSL v3.0, since there’s no such thing as an “invalid” padding byte, it may be impossible to determine whether it has been tampered with. This is especially true when the padding fills an entire block. I’ll add an extra space to the plaintext from my previous example, so that it pushes into the next block and results in an entire block of padding:
Here’s the ciphertext:
I’ve padded this message the way that TLS v1.0 would, but if we’re speaking SSL v3.0 then all but the last of those bytes would be ignored: you could fill anything at all in there and the message would still be accepted. That means I can modify the final ciphertext block any way I want, and since the final byte is the only one that matters, there’s a 1-in-256 chance that the message will be accepted: even the MAC will still be valid! That means, in particular that I can do this:
I took the block containing the password and copied it into the final block, much like before. 255 times out of 256, this will result in an error and the session will abort. One time in 256, though, it will just contiune along as normal — and then I can do the simple XOR math which tells me what the last character of the password must have been.
In the context of web browser, if I have a man-in-the-middle position on the victim’s network and the secret I’m trying to steal is inside an HTTPS-only cookie, it’s easy for me to force the client to keeping resending the same message until this attack succeeds. All I have to do is wait for the victim to visit any plain-HTTP site, and insert an invisible iframe into it which runs some Javascript. The Javascript will keep making requests to site whose cookie I’m trying to steal, and I’ll keep tampering with each request as it occurs. Each failed attempt will result in the connection dropping and then being renegotiated with new key material, so each attempt has an independent 1-in-256 chance of succeeding. Once I’ve successfully determined one byte of the secret cookie, I then increase the length of the URL being requested by one, so that the next unknown byte is now positioned at the end of a block. I also adjust the length of something after the cookie, such as the POST body, so that there is still a full block of padding at the end. I repeat my attack in this fashion until I’ve decrypted the entire cookie.
### The workaround
Within the confines of SSL v3.0, POODLE cannot be fixed. However, the downgrade dance which enables it can be. For this purpose, Bodo Möller and Adam Langley, of Google, have introduced a proposal called TLS_FALLBACK_SCSV[19].
SCSV stands for “signaling cipher suite value”, and it’s essentially a hack which allows TLS clients to indicate to servers that they support some extension to TLS, while ensuring that servers that don’t understand the extension will simply ignore it. TLS already provides an extension mechanism which is supposed to satisfy this purpose, but a lot of TLS servers choke if they receive an extension they don’t recognize. SCSV works by appending a special, bogus value to the list of ciphersuites that the client advertises support for; it turns out that TLS servers more reliably ignore unrecognized ciphersuites than unrecognized extensions. The SCSV mechanism has previously and successfully been used to advertise support for the workaround developed for Marsh Ray’s renegotiation attack[25].
The TLS_FALLBACK_SCSV proposal is that when a client is retrying a TLS connection with an earlier protocol version as part of the downgrade dance, it will signal that it is doing so by including TLS_FALLBACK_SCSV in the cipher list. Legacy servers won’t recognize what it means, so they’ll proceed as normal and allow the downgrade to occur. However, newly-patched servers, which ought to handle recent protocol versions properly, should recognize TLS_FALLBACK_SCSV and refuse the connection. The idea is that a well-behaved server should never trigger the downgrade dance, so therefore if it occurs it must be a result of adversarial interference, and the right thing to do is kill the connection rather than allow the attack to proceed.
I admit that this workaround leaves me a bit puzzled. Sooner rather than later, there are going to be buggy servers out there which understand TLS_FALLBACK_SCSV, yet choke when the client offers TLS v1.2 (or, soon, v1.3). The result will be that clients which support TLS_FALLBACK_SCSV and TLS v1.2 will be unable to connect to them. This is the same outcome as if the browser simply avoided the downgrade dance altogether. If this outcome is acceptable — and it should be — wouldn’t disabling the downgrade dance be a cleaner way to go about it?
Anyway, the Googlers behind the TLS_FALLBACK_SCSV proposal have a lot more collective experience interfacing with broken SSL implementations than I do; they probably have their reasons for doing it this way. It’s not a standard yet, and I’m sure that at some point in the IETF process this argument will be raised and they’ll answer to it.
### The fix
The only correct way to fix POODLE is to disable SSL v3.0 altogether.
I think that last sentence will be mostly uncontroversial. Now, though, I am going to step onto my soapbox and say: disabling SSL v3.0 does not go far enough. It is time to aggressively deprecate as many old versions of TLS as possible. POODLE is not a one-off. It exploits a known mistake that has bitten us before. Many more similar mistakes still exist in TLS v1.0, and some time very soon one of them is going to bite us again.
Every revision of TLS contains fixes for dangerous errors committed by earlier versions. TLS v1.0 dictates the format of padding, preventing POODLE. v1.1 gets rid of IV-chaining, preventing BEAST. v1.2 introduces support for AEAD ciphersuites, providing an alternative to the dangerous MAC-then-encrypt construct. TLS v1.3 will eliminate the RSA handshake protocol[29], which lacks forward secrecy.
Currently, browser support for TLS versions beyond v1.0 is not deployed widely enough to make it practical for most servers to disable v1.0 or anything after it. This must be fixed. For browser vendors, this means backporting TLS v1.2 support to older branches. For website operators, it means displayping nag screens encouraging users to upgrade their browser. For enterprise IT administrators, it means updating your desktop image and fixing or replacing legacy applications that rely on old browser versions.
It’s time to put the cryptographic mistakes of the ’90s behind us.
### References
[1]: Nadhem J. AlFardan and Kenneth G. Paterson, 2013. “Lucky Thirteen: Breaking the TLS and DTLS Record Protocols”. <http://www.isg.rhul.ac.uk/tls/TLStiming.pdf>
[2]: Nadhem J. AlFardan, Daniel J. Bernstein, Kenneth G. Paterson, Bertram Poettering, and Jacob C.N. Schuldt, 2013. “On the security of RC4 in TLS and WPA”. <http://www.isg.rhul.ac.uk/tls/RC4biases.pdf>
[3]: Gregory V. Bard, 2006. “A Challenging but Feasible Blockwise-Adaptive Chosen-Plaintext Attack on SSL”. <https://eprint.iacr.org/2006/136.pdf>
[4]: Mihir Bellare and Chanathip Namprempre, 2000. “Authenticated Encryption: Relations abong notions and analysis of the generic composition paradigm”. <ftp://ftp.iks-jena.de/pub/mitarb/lutz/crypt/symmetric/Bellare-Namprempre%3AAuthenticated_Encryption%3AAnalysis_of_Composition_Paradigm.pdf>
[6]: Daniel Bleichenbacher, 1998. “Chosen Ciphertext Attacks Against Protocols Based on the RSA Encryption Standard PKCS #1”. <http://archiv.infsec.ethz.ch/education/fs08/secsem/Bleichenbacher98.pdf>
[7]: Brice Canvel, Alain Hiltgen, Serge Vaudenay, and Martin Vuagnoux, 2003. “Password Interception in a SSL/TLS Channel”. <http://canvel.free.fr/crypto/pdf/CHVV03.pdf>
[8]: Wei Dai, 2002. “An attack against SSH2 protocol”. <http://www.weidai.com/ssh2-attack.txt>
[9]: Tim Dierks and Philip Karlton, 1999. “RFC 2246: The TLS Protocol Version 1.0”. <https://tools.ietf.org/html/rfc2246>
[10]: Tim Dierks and Eric Rescorla, 2006. “RFC 4246: The Transport Layer Security (TLS) Protocol Version 1.1”. <https://tools.ietf.org/html/rfc4346>
[11]: Tim Dierks and Eric Rescorla, 2008. “RFC 5246: The Transport Layer Security (TLS) Protocol Version 1.2”. <https://tools.ietf.org/html/rfc5246>
[12]: Thai Duong and Juliano Rizzo, 2011. “Cryptography in the Web: The Case of Cryptographic Design Flaws in ASP.NET”. <http://www.ieee-security.org/TC/SP2011/PAPERS/2011/paper030.pdf>
[13]: Thai Duong and Juliano Rizzo, 2011. “Here Come the $$\oplus$$ Ninjas”. <http://www.hpcc.ecs.soton.ac.uk/~dan/talks/bullrun/Beast.pdf>
[14]: Hal Finney, 2006. “Bleichenbacher’s RSA signature forgery based on implmentation error”. <http://www.imc.org/ietf-openpgp/mail-archive/msg06063.html>
[15]: Alan Freier, Philip Karlton, and Paul Kocher, 2011. “RFC 6101: The Secure Sockets Layer (SSL) Protocol Version 3.0”. <https://tools.ietf.org/html/rfc6101>
[16]: Marko Laakso, 2014. “Heartbleed Bug”. <http://heartbleed.com>
[17]: Moxie Marlinspike, 2011. “The Cryptographic Doom Principle”. <http://www.thoughtcrime.org/blog/the-cryptographic-doom-principle/>
[18]: Bodo Möller, 2002. “Security of CBC Ciphersuites in SSL/TLS: Problems and Countermeasures”. <https://www.openssl.org/~bodo/tls-cbc.txt>
[19]: Bodo Möller and Adam Langley, 2014. “TLS Fallback Signaling Cipher Suite Value (SCSV) for Preventing Protocol Downgrade Attacks (work in progress)”. <https://tools.ietf.org/html/draft-bmoeller-tls-downgrade-scsv-02>
[20]: Bodo Möller, Thai Duong, and Krzysztof Kotowicz, 2014. “This POODLE Bites: Exploiting the SSL 3.0 Fallback”. <https://www.openssl.org/~bodo/ssl-poodle.pdf>
[22]: Rolf Oppliger, 2009. “SSL and TLS: Theory and Practice”. <http://books.google.com/books?id=dR2G0oPufe0C&pg=PA68>
[23]: Colin Percival, 2009. “Encrypt then MAC”. <http://www.daemonology.net/blog/2009-06-24-encrypt-then-mac.html>
[24]: Angelo Prado, Neal Harris, and Yoel Gluck, 2013. “BREACH ATTACK”. <http://breachattack.com>
[25]: Marsh Ray and Steve Dispensa, 2009. “Renegotiating TLS”. <https://kryptera.se/Renegotiating%20TLS.pdf>
[26]: Eric Rescorla, Marsh Ray, and Steve Dispensa, 2010. “Transport Layer Security (TLS) Renegotiation Indication Extension”. <https://tools.ietf.org/html/rfc5746>
[27]: Juliano Rizzo and Thai Duong, 2012. “The CRIME Attack”. <https://docs.google.com/presentation/d/11eBmGiHbYcHR9gL5nDyZChu_-lCa2GizeuOfaLU2HOU>
[28]: Philip Rogaway, 2002. “Authenticated-Encryption with Associated-Data”. <http://seclab.cs.ucdavis.edu/papers/ad.pdf>
[29]: Joseph Salowey, 2014. “Re: [TLS] Confirming Consensus on removing RSA key Transport from TLS 1.3”. <https://www.ietf.org/mail-archive/web/tls/current/msg12266.html>
[30]: Serge Vaudenay, 2002. “Security Flaws Induced by CBC Padding; Applications to SSL, IPSEC, WTLS…”. <https://www.iacr.org/archive/eurocrypt2002/23320530/cbc02_e02d.pdf>
[31]: Wikimedia Foundation, 2014. “Transport Layer Security: Web browsers”. <https://en.wikipedia.org/w/index.php?title=Transport_Layer_Security&oldid=629101662#Web_browsers> | {"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": 0.37562593817710876, "perplexity": 4092.6706391681087}, "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-2016-36/segments/1471982947760.88/warc/CC-MAIN-20160823200907-00058-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://www.amansmathsblogs.com/inverse-trigonometric-functions-question-and-answer-set-1/ | Monday, June 24, 2019
Home > Question and Answer > Inverse Trigonometric Functions Question And Answer Set 1
# Inverse Trigonometric Functions Question and Answer Set 1
Hi students, welcome to Amans Maths Blogs (AMB). On this post, you will get the Inverse Trigonometric Functions Question and Answer Set 1. It will help you to practice the questions on the topics of inverse trigonometric functions based questions with answers.
Inverse Trigonometric Functions Question and Answer Set 1: Ques No 1
The value of tan-1(sin(-Pi/2)) is
Options:
A. Pi/2
B. Pi/4
C. -Pi/2
D. -Pi/4
Inverse Trigonometry Functions Question and Answer Set 1: Ques No 2
The value of tan-1(x/y) – tan-1( (x – y) / (x + y) )is
Options:
A. Pi/2
B. Pi/3
C. Pi/4
D. Pi/6
Inverse Trigonometric Functions Question and Answer Set 1: Ques No 3
The value of x in the equation 3cot(cos-1x) = 2 is
Options:
A. 2/3
B. 2/Root(13)
C. 1/Root(13)
D. 1/3
Inverse Trigonometric Functions Question and Answer Set 1: Ques No 4
The value of x (x > 0) in the inverse trigonometric equation tan-1( (1 – x) / (1 + x) ) = (1/2) tan-1(x) is
Options:
A. 2/3
B. 2/Root(3)
C. 1/Root(3)
D. 1/3
Inverse Trigonometric Functions Question and Answer Set 1: Ques No 5
The value of sin(sin-1(3/5) + sin-1(8/17)) is
Options:
A. 91/85
B. 77/85
C. 11/85
D. 26/85
Inverse Trigonometry Functions Question and Answer Set 1: Ques No 6
The value of sin-1(3/5) – sin-1(8/17) is
Options:
A. cos-1(84/85)
B. sin-1(64/85)
C. sin-1(91/85)
D. cos-1(67/85)
Inverse Trigonometric Functions Question and Answer Set 1: Ques No 7
If tan-1 [(x – 1) / (x + 1)] + tan-1 [(x + 1) / (x + 2)] = Pi/4, then the value of x (x > 0) is
Options:
A. 1/Root(3)
B. 1/2Root(2)
C. 1/Root(2)
D. 1/2Root(3)
Inverse Trigonometric Functions Question and Answer Set 1: Ques No 8
If tan-1 [2x] + tan-1 [3x] = Pi/4, then the value of x is
Options:
A. 1/2
B. -1/3
C. -1
D. 1/6
Inverse Trigonometry Functions Question and Answer Set 1: Ques No 8
The simplified form of the trigonometric expression tan-1 , where (x != 0) is
Options:
A. (1/2)tan-1x
B. Pi – tan-1x
C. tan-1x
D. 2tan-1x | {"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": 0.9371654987335205, "perplexity": 8896.434844740319}, "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/1560627999298.86/warc/CC-MAIN-20190624084256-20190624110256-00246.warc.gz"} |
https://math.msu.edu/seminars/TalkView.aspx?talk=29421 | ## Seminar in Cluster algebras
• Jiuzu Hong, University of North Carolina at Chapel Hill
• BD Schubert varieties of parahoric group schemes and global Demazure modules of twisted current algebras
• 09/26/2022
• 12:30 PM - 1:30 PM
• C304 Wells Hall
• Linhui Shen (shenlin1@msu.edu)
It is well-known that there is a duality between affine Demazure modules and the spaces of sections of line bundles on Schubert varieties in affine Grassmannians. This should be regarded as a local theory. In this talk, I will explain an algebraic theory of global Demazure modules of twisted current algebras. Moreover, these modules are dual to the spaces of sections of line bundles on Beilinson-Drinfeld Schubert varieties of certain parahoric groups schemes, where the factorizations of global Demazure modules are compatible with the factorizations of line bundles. This generalizes the work of Dumanski-Feigin-Finkelberg in the untwisted setting. In order to establish this duality in the twisted case, following the works of Zhu, we prove the flatness of BD Schubert varieties, and establish factorizable and equivariant structures on the rigidified line bundles over BD Grassmannians of these parahoric group schemes. This work is joint with Huanhuan Yu.
## Contact
Department of Mathematics
Michigan State University | {"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": 0.8297567963600159, "perplexity": 1022.905611570473}, "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/1674764500056.55/warc/CC-MAIN-20230203122526-20230203152526-00210.warc.gz"} |
https://everything.explained.today/Josephson_effect/ | # Josephson effect explained
The Josephson effect is the phenomenon of supercurrent, a current that flows continuously without any voltage applied, across a device known as a Josephson junction (JJ), which consists of two or more superconductors coupled by a weak link. The weak link can consist of a thin insulating barrier (known as a superconductor–insulator–superconductor junction, or S-I-S), a short section of non-superconducting metal (S-N-S), or a physical constriction that weakens the superconductivity at the point of contact (S-c-S).
The Josephson effect is an example of a macroscopic quantum phenomenon. It is named after the British physicist Brian David Josephson, who predicted in 1962 the mathematical relationships for the current and voltage across the weak link.[1] [2] The DC Josephson effect had been seen in experiments prior to 1962,[3] but had been attributed to "super-shorts" or breaches in the insulating barrier leading to the direct conduction of electrons between the superconductors. The first paper to claim the discovery of Josephson's effect, and to make the requisite experimental checks, was that of Philip Anderson and John Rowell.[4] These authors were awarded patents on the effects that were never enforced, but never challenged.
Before Josephson's prediction, it was only known that normal (i.e. non-superconducting) electrons can flow through an insulating barrier, by means of quantum tunneling. Josephson was the first to predict the tunneling of superconducting Cooper pairs. For this work, Josephson received the Nobel Prize in Physics in 1973.[5] Josephson junctions have important applications in quantum-mechanical circuits, such as SQUIDs, superconducting qubits, and RSFQ digital electronics. The NIST standard for one volt is achieved by an array of 20,208 Josephson junctions in series.[6]
## Applications
Types of Josephson junction include the φ Josephson junction (of which π Josephson junction is a special example), long Josephson junction, and superconducting tunnel junction. A "Dayem bridge" is a thin-film variant of the Josephson junction in which the weak link consists of a superconducting wire with dimensions on the scale of a few micrometres or less.[7] [8] The Josephson junction count of a device is used as a benchmark for its complexity. The Josephson effect has found wide usage, for example in the following areas.
SQUIDs, or superconducting quantum interference devices, are very sensitive magnetometers that operate via the Josephson effect. They are widely used in science and engineering.
In precision metrology, the Josephson effect provides an exactly reproducible conversion between frequency and voltage. Since the frequency is already defined precisely and practically by the caesium standard, the Josephson effect is used, for most practical purposes, to give the standard representation of a volt, the Josephson voltage standard.
Single-electron transistors are often constructed of superconducting materials, allowing use to be made of the Josephson effect to achieve novel effects. The resulting device is called a "superconducting single-electron transistor".[9]
The Josephson effect is also used for the most precise measurements of elementary charge in terms of the Josephson constant and von Klitzing constant which is related to the quantum Hall effect.
\scriptstyle 1 2e
h
that carries the digital information: the absence of switching is equivalent to 0, while one switching event carries a 1.
Josephson junctions are integral in superconducting quantum computing as qubits such as in a flux qubit or others schemes where the phase and charge act as the conjugate variables.[10]
Superconducting tunnel junction detectors (STJs) may become a viable replacement for CCDs (charge-coupled devices) for use in astronomy and astrophysics in a few years. These devices are effective across a wide spectrum from ultraviolet to infrared, and also in x-rays. The technology has been tried out on the William Herschel Telescope in the SCAM instrument.
Quiterons and similar superconducting switching devices.
Josephson effect has also been observed in superfluid helium quantum interference devices (SHeQUIDs), the superfluid helium analog of a dc-SQUID.[11]
## The Josephson equations
\psiA=\sqrt{n
i\phiA A}e
, and superconductor B
\psiB=\sqrt{n
i\phiB B}e
, which can be interpreted as the wave functions of Cooper pairs in the two superconductors. If the electric potential difference across the junction is
V
, then the energy difference between the two superconductors is
2eV
, since each Cooper pair has twice the charge of one electron. The Schrödinger equation for this two-state quantum system is therefore:[12]
i\hbar \partial \partialt
\begin{pmatrix}
i\phiA \sqrt{n A}e
\\
i\phiB \sqrt{n B}e
\end{pmatrix} = \begin{pmatrix} eV&K\\ K&-eV \end{pmatrix} \begin{pmatrix}
i\phiA \sqrt{n A}e
\\
i\phiB \sqrt{n B}e
\end{pmatrix},
where the constant
K
is a characteristic of the junction. To solve the above equation, first calculate the time derivative of the order parameter in superconductor A:
\partial \partialt
i\phiA (\sqrt{n )= A}e
• \sqrt{n
i\phiA A}e
+\sqrt{nA}(i
• \phi
A
i\phiA e )=(
• \sqrt{n
A}+i\sqrt{nA}
• \phi
A)
i\phiA e
,
and therefore the Schrödinger equation gives:
• (\sqrt{n
A}+i\sqrt{nA}
• \phi
A)
i\phiA e =
1 i\hbar
i\phiA (eV\sqrt{n A}e
i\phiB +K\sqrt{n B}e
).
The phase difference of Ginzburg-Landau order parameters across the junction is called the Josephson phase:
\varphi=\phiB-\phiA
.
The Schrödinger equation can therefore be rewritten as:
• \sqrt{n
A}+i\sqrt{nA}
• \phi
A = 1 i\hbar
(eV\sqrt{nA}+K\sqrt{n
i\varphi B}e
),
and its complex conjugate equation is:
• \sqrt{n
A}-i\sqrt{nA}
• \phi
A = 1 -i\hbar
(eV\sqrt{nA}+K\sqrt{n
-i\varphi B}e
).
Add the two conjugate equations together to eliminate
• \phi
A
:
• 2\sqrt{n
A}= 1 i\hbar
i\varphi (K\sqrt{n B}e
-i\varphi -K\sqrt{n )= B}e
K\sqrt{nB
} \cdot 2\sin \varphi.
Since
• \sqrt{n
A}=
• n A
2\sqrt{nA
}, we have:
• n
A= 2K\sqrt{nAnB
}\sin \varphi.
Now, subtract the two conjugate equations to eliminate
• \sqrt{n
A}
:
2i\sqrt{nA}
• \phi
A = 1 i\hbar
(2eV\sqrt{nA}+K\sqrt{n
i\varphi B}e
-i\varphi +K\sqrt{n B}e
),
which gives:
• \phi
(eV+K\sqrt{
A =- 1 \hbar
nB nA
}\cos \varphi).
Similarly, for superconductor B we can derive that:
• n
B= 2K\sqrt{nAnB
}\sin \varphi, \, \dot \phi_B=\frac(eV-K\sqrt\cos \varphi).
Noting that the evolution of Josephson phase is
• \varphi=\phi
• B-\phi
A
and the time derivative of charge carrier density
• n
A
is proportional to current
I
, the above solution yields the Josephson equations:[13]
I(t)=Ic\sin(\varphi(t))
(1st Josephson relation, or weak-link current-phase relation)
\partial\varphi \partialt
=
2eV(t) \hbar
(2nd Josephson relation, or superconducting phase evolution equation)
where
V(t)
and
I(t)
are the voltage across and the current through the Josephson junction, and
Ic
is a parameter of the junction named the critical current. The critical current of the Josephson junction depends on the properties of the superconductors, and can also be affected by environmental factors like temperature and externally applied magnetic field.
The Josephson constant is defined as:
K
J= 2e h
,
and its inverse is the magnetic flux quantum:
\Phi
0= h 2e
=2\pi
\hbar 2e
.
The superconducting phase evolution equation can be reexpressed as:
\partial\varphi \partialt
=2\pi[KJV(t)]=
2\pi \Phi0
V(t).
If we define:
\Phi=\Phi
0 \varphi 2\pi
,
then the voltage across the junction is:
V= \Phi0 2\pi
\partial\varphi = \partialt
d\Phi dt
,
which is very similar to Faraday's law of induction. But note that this voltage does not come from magnetic energy, since there is no magnetic field in the superconductors; Instead, this voltage comes from the kinetic energy of the carriers (i.e. the Cooper pairs). This phenomenon is also known as kinetic inductance.
## Three main effects
There are three main effects predicted by Josephson that follow directly from the Josephson equations:
### The DC Josephson effect
The DC Josephson effect is a direct current crossing the insulator in the absence of any external electromagnetic field, owing to tunneling. This DC Josephson current is proportional to the sine of the Josephson phase (phase difference across the insulator, which stays constant over time), and may take values between
-Ic
and
Ic
.
### The AC Josephson effect
With a fixed voltage
VDC
across the junction, the phase will vary linearly with time and the current will be a sinusoidal AC (Alternating Current) with amplitude
Ic
and frequency
KJVDC
. This means a Josephson junction can act as a perfect voltage-to-frequency converter.
### The inverse AC Josephson effect
\omega
can induce quantized DC voltages[14] across the Josephson junction, in which case the Josephson phase takes the form
\varphi(t)=\varphi0+n\omegat+a\sin(\omegat)
, and the voltage and current across the junction will be:
V(t)=
\hbar 2e
\omega(n+a\cos(\omegat)),andI(t)=Ic
infty \sum m=-infty
Jm(a)\sin(\varphi0+(n+m)\omegat),
The DC components are:
VDC=n
\hbar 2e
\omega,andIDC=IcJ-n(a)\sin\varphi0.
This means a Josephson junction can act like a perfect frequency-to-voltage converter,[15] which is the theoretical basis for the Josephson voltage standard.
## Josephson inductance
When the current and Josephson phase varies over time, the voltage drop across the junction will also vary accordingly; As shown in derivation below, the Josephson relations determine that this behavior can be modeled by a kinetic inductance named Josephson Inductance.[16]
Rewrite the Josephson relations as:
\begin{align} \partialI \partial\varphi
&=
I
c\cos\varphi,\\ \partial\varphi \partialt
&=
2\pi \Phi0
V. \end{align}
Now, apply the chain rule to calculate the time derivative of the current:
\partialI \partialt
=
\partialI \partial\varphi
\partial\varphi \partialt
=I
c\cos\varphi ⋅ 2\pi \Phi0
V,
Rearrange the above result in the form of the current–voltage characteristic of an inductor:
V=
\Phi0 2\piIc\cos\varphi
\partialI =L(\varphi) \partialt
\partialI \partialt
.
This gives the expression for the kinetic inductance as a function of the Josephson Phase:
L(\varphi)=
\Phi0 2\piIc\cos\varphi
=
LJ \cos\varphi
.
Here,
L
J=L(0)= \Phi0 2\piIc
is a characteristic parameter of the Josephson junction, named the Josephson Inductance.
Note that although the kinetic behavior of the Josephson junction is similar to that of an inductor, there is no associated magnetic field. This behaviour is derived from the kinetic energy of the charge carriers, instead of the energy in a magnetic field.
## Josephson energy
Based on the similarity of the Josephson junction to a non-linear inductor, the energy stored in a Josephson junction when a supercurrent flows through it can be calculated.[17]
The supercurrent flowing through the junction is related to the Josephson phase by the current-phase relation (CPR):
I=Ic\sin\varphi.
The superconducting phase evolution equation is analogous to Faraday's law:
V=\operatorname{d}\Phi/\operatorname{d}t.
Assume that at time
t1
, the Josephson phase is
\varphi1
; At a later time
t2
, the Josephson phase evolved to
\varphi2
. The energy increase in the junction is equal to the work done on the junction:
\DeltaE=
2 \int 1
IV\operatorname{d}{t} =
2 \int 1
I\operatorname{d}\Phi =
\varphi2 \int \varphi1
Ic\sin\varphi
\operatorname{d}\left(\Phi
0 \varphi 2\pi
\right) =-
\Phi0Ic 2\pi
\Delta\cos\varphi.
This shows that the change of energy in the Josephson junction depends only on the initial and final state of the junction and not the path. Therefore the energy stored in a Josephson junction is a state function, which can be defined as:
E(\varphi)=- \Phi0Ic 2\pi
\cos\varphi=-EJ\cos\varphi .
Here
EJ=|E(0)|=
\Phi0Ic 2\pi
is a characteristic parameter of the Josephson junction, named the Josephson Energy. It is related to the Josephson Inductance by
EJ=
2 L c
. An alternative but equivalent definition
E(\varphi)=EJ(1-\cos\varphi)
is also often used.
Again, note that a non-linear magnetic coil inductor accumulates potential energy in its magnetic field when a current passes through it; However, in the case of Josephson junction, no magnetic field is created by a supercurrent — the stored energy comes from the kinetic energy of the charge carriers instead.
## The RCSJ model
The Resistively Capacitance Shunted Junction (RCSJ) model,[18] [19] or simply shunted junction model, includes the effect of AC impedance of an actual Josephson junction on top of the two basic Josephson relations stated above.
As per Thévenin's theorem,[20] the AC impedance of the junction can be represented by a capacitor and a shunt resistor, both parallel[21] to the ideal Josephson Junction. The complete expression for the current drive
Iext
becomes:
Iext=CJ
\operatorname{d V}{\operatorname{d}t}
+Ic\sin\varphi+
V R
,
where the first term is displacement current with
CJ
- effective capacitance, and the third is normal current with
R
- effective resistance of the junction.
## Josephson penetration depth
The Josephson penetration depth characterizes the typical length on which an externally applied magnetic field penetrates into the long Josephson junction. It is usually denoted as
λJ
and is given by the following expression (in SI):
λ
J=\sqrt{ \Phi0 2\pi\mu0d'jc
},
where
\Phi0
is the magnetic flux quantum,
jc
is the critical supercurrent density (A/m2), and
d'
characterizes the inductance of the superconducting electrodes[22]
d'=dI1\tanh\left(
d1 2λ1
\right) 2\tanh\left(
d2 2λ2
\right),
where
dI
is the thickness of the Josephson barrier (usually insulator),
d1
and
d2
are the thicknesses of superconducting electrodes, and
λ1
and
λ2
are their London penetration depths. The Josephson penetration depth usually ranges from a few μm to several mm if the critical supercurrent density is very low.[23]
## Notes and References
1. B. D. Josephson . 1962 . Possible new effects in superconductive tunnelling . Phys. Lett. . 1 . 7 . 251–253 . 10.1016/0031-9163(62)91369-0. 1962PhL.....1..251J.
2. B. D. Josephson . 1974 . The discovery of tunnelling supercurrents . Rev. Mod. Phys. . 46 . 2 . 251–254 . 1974RvMP...46..251J . 10.1103/RevModPhys.46.251. 54748764 .
3. Web site: Josephson . Brian D. . The Discovery of Tunneling Supercurrents (Nobel Lecture) . December 12, 1973 .
4. P. W. Anderson . J. M. Rowell . 1963 . Probable Observation of the Josephson Tunnel Effect . Phys. Rev. Lett. . 10 . 6 . 230 . 1963PhRvL..10..230A . 10.1103/PhysRevLett.10.230.
5. https://www.nobelprize.org/nobel_prizes/physics/laureates/1973/ The Nobel prize in physics 1973
6. Steven Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion, 2003.
7. P. W. Anderson . A. H. Dayem . 1964 . Radio-frequency effects in superconducting thin film bridges . Phys. Rev. Lett. . 13 . 6 . 195 . 10.1103/PhysRevLett.13.195. 1964PhRvL..13..195A.
8. Web site: Dawe . Richard . SQUIDs: A Technical Report – Part 3: SQUIDs . http://rich.phekda.org . 28 October 1998 . website . 2011-04-21 . https://web.archive.org/web/20110727172927/http://rich.phekda.org/squid/technical/part3.html . 27 July 2011 . dead .
9. T. A. Fulton . P. L. Gammel . D. J. Bishop . L. N. Dunkleberger . G. J. Dolan . 1989 . Observation of Combined Josephson and Charging Effects in Small Tunnel Junction Circuits . Phys. Rev. Lett. . 63 . 12 . 1307–1310 . 1989PhRvL..63.1307F . 10.1103/PhysRevLett.63.1307 . 10040529.
10. V. Bouchiat . D. Vion . P. Joyez . D. Esteve . M. H. Devoret . 1998 . Quantum coherence with a single Cooper pair . Physica Scripta . T76 . 165 . 1998PhST...76..165B . 10.1238/Physica.Topical.076a00165. free .
11. Physics Today, Superfluid helium interferometers, Y. Sato and R. Packard, October 2012, page 31
12. Web site: The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, Section 21-9: The Josephson junction. feynmanlectures.caltech.edu. 2020-01-03.
13. Book: Physics and Applications of the Josephson Effect. Barone. A.. Paterno. G.. John Wiley & Sons. 1982. 978-0-471-01469-0. New York.
14. Langenberg. D. N.. Scalapino. D. J.. Taylor. B. N.. Eck. R. E.. 1966-04-01. Microwave-induced D.C. voltages across Josephson junctions. Physics Letters. 20. 6. 563–565. 10.1016/0031-9163(66)91114-0. 0031-9163.
15. Levinsen. M. T.. Chiao. R. Y.. Feldman. M. J.. Tucker. B. A.. 1977-12-01. An inverse ac Josephson effect voltage standard. Applied Physics Letters. 31. 11. 776–778. 10.1063/1.89520. 0003-6951.
16. cond-mat/0411174. M. Devoret. A. Wallraff. Superconducting Qubits: A Short Review. 2004. Martinis. J.
17. [Michael Tinkham]
18. McCumber. D. E.. 1968-06-01. Effect of ac Impedance on dc Voltage-Current Characteristics of Superconductor Weak-Link Junctions. Journal of Applied Physics. 39. 7. 3113–3118. 10.1063/1.1656743. 0021-8979.
19. Chakravarty. Sudip. Ingold. Gert-Ludwig. Kivelson. Steven. Zimanyi. Gergely. 1988-03-01. Quantum statistical mechanics of an array of resistively shunted Josephson junctions. Physical Review B. 37. 7. 3283–3294. 10.1103/PhysRevB.37.3283. 9944915.
20. Web site: AC Thevenin's Theorem. hyperphysics.phy-astr.gsu.edu. 2020-01-03.
21. Web site: Dynamics of RF SQUID. phelafel.technion.ac.il. 2020-01-11.
22. Weihnacht. M. Influence of Film Thickness on D. C. Josephson Current. Physica Status Solidi B. 32. 2. 169. 10.1002/pssb.19690320259. 1969PSSBR..32..169W. 1969.
23. Book: Buckel . Werner . Kleiner . Reinhold . Supraleitung . 2004 . Wiley-VCH Verlag GmbH&Co.KGaA . Tübingen . 3527403485 . 67 . 6.. | {"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": 0.9277838468551636, "perplexity": 4263.0953918675195}, "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/1637964358570.48/warc/CC-MAIN-20211128164634-20211128194634-00113.warc.gz"} |
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https://stacks.math.columbia.edu/tag/03VP | Remark 63.30.1. Here are some observations concerning this notion.
1. If modeling projective curves then we can use cohomology and we don't need factor $q^ n$.
2. The only examples I know are $\Gamma = \pi _1(X, \overline\eta )$ where $X$ is smooth, geometrically irreducible and $K(\pi , 1)$ over finite field. In this case $q = (\# k)^{\dim X}$. Modulo the proposition, we proved this for curves in this course.
3. Given the integer $q$ then the sets $S_ d$ are uniquely determined. (You can multiple $q$ by an integer $m$ and then replace $S_ d$ by $m^ d$ copies of $S_ d$ without changing the formula.)
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | {"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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9767836332321167, "perplexity": 307.0797799350542}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711394.73/warc/CC-MAIN-20221209080025-20221209110025-00740.warc.gz"} |
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Minors of a Matrix
Tool for calculating the minors of a matrix, i.e. the values of the determinants of its square sub-matrices (removing one row and one column of the starting matrix).
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# Minors of a Matrix
## Minors of NxN Matrix Calculator
### What is a matrix minor? (Definition)
The minors of a square matrix $M = m_{i, j}$ of size $n$ are the determinants of the square sub-matrices obtained by removing the row $i$ and the column $j$ from $M$.
Sometimes minors are defined by removing opposing rows and columns (ie. row $n-i$ and column $n-j$).
### How to calculate a matrix minors?
For a square matrix of order 2, finding the minors is calculating the matrix of cofactors without the coefficients.
For larger matrices like 3x3, calculate the determinants of each sub-matrix.
Example: $$M = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$
The determinant of the sub-matrix obtained by removing the first row and the first column is: $ei-fh$\$, do the same for all combinations of rows and columns.
### What is the difference between a minor and a cofactor?
For a square matrix, the minor is identical to the cofactor except for the sign (indeed, the cofactors can have a - sign depending on their position in the matrix). Minors do not take this minus sign.
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https://www.physicsforums.com/threads/finding-nearest-neighbour-equilibrium-distance.957187/ | # Finding nearest neighbour equilibrium distance
• Start date
• Tags
• #1
295
4
## Homework Statement
The energy per ion in for CsCl is nearly – (αe 2 /(4πε0)) + 8Ae -(R/ρ) , where α is the Madelung constant and A = 5.64 x 103 eV and ρ = 0.34 Å. Calculate the nearest neighbour equilibrium distance.
alpha = 2 ln 2
## The Attempt at a Solution
I think that CsCl is a simple cubic structure
I found online
For a simple cubic lattice, it is clear that the nearest neighbor distance is just the lattice parameter, a. Therefore, for a simple cubic lattice there are six (6) nearest neighbors for any given lattice point.
so then my answer would be 0.34 A ? Is this correct ?
• #2
Homework Helper
Gold Member
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2,580
The first line of the problem statement appears incomplete. I don't see any definition of ## R ## and also what do you take a derivative of to set it equal to zero? It looks like you may have a typo or two in your equation.
• #3
295
4
The first line of the problem statement appears incomplete. I don't see any definition of ## R ## and also what do you take a derivative of to set it equal to zero? It looks like you may have a typo or two in your equation.
That's the entire problem copied and pasted from the assignment.
I'm not sure what you mean by what do I take the derivative of ?
• #4
Homework Helper
Gold Member
5,325
2,580
That's the entire problem copied and pasted from the assignment.
I'm not sure what you mean by what do I take the derivative of ?
The equilibrium distance for a system is normally found as the position where the potential energy is a minimum, so that ## \frac{dV}{dR}=0 ##. ## \\ ## Consider for example a mass on a spring in a gravitational field.: ## U=\frac{1}{2}kx^2+mgx ## . Taking derivative and setting equal to zero: ## kx+mg=0 ## ==>> ## x_{equilibrium}=-\frac{mg}{k} ##, which the spring constant equation also tells you the forces are balanced there.
Last edited:
• #5
295
4
The equilibrium distance for a system is normally found as the position where the potential energy is a minimum, so that ## \frac{dV}{dR}=0 ##. ## \\ ## Consider for example a mass on a spring in a gravitational field.: ## U=\frac{1}{2}kx^2+mgx ## . Taking derivative and setting equal to zero: ## kx+mg=0 ## ==>> ## x_{equilibrium}=-\frac{mg}{k} ##, which the spring constant equation also tells you the forces are balanced there.
Ok so it is not enough to say
For a simple cubic lattice, it is clear that the nearest neighbor distance is just the lattice parameter, a.
I would have to derive the energy equation that is given to me and set =0
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# The total cost of an office dinner was shared equally by k
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The total cost of an office dinner was shared equally by k [#permalink]
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06 Oct 2008, 14:17
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The total cost of an office dinner was shared equally by k of the n employees who attended the dinner. What was the total cost of the dinner?
(1) Each of the k employees who shared the cost of the dinner paid $19. (2) If the total cost of the dinner had been shared equally by k + 1 of the n employees who attended the dinner, each of the k + 1 employees would have paid$18.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient
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06 Oct 2008, 14:25
albany09 wrote:
The total cost of an office dinner was shared equally by k of the n employees who attended the dinner. What was the total cost of the dinner?
(1) Each of the k employees who shared the cost of the dinner paid $19. (2) If the total cost of the dinner had been shared equally by k + 1 of the n employees who attended the dinner, each of the k + 1 employees would have paid$18.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient
(1) Insufficient
Cost = 19k
(2) Insufficient
Cost = 18(k+1)
(1) and (2) Sufficient
19k = 18(k + 1)
19k = 18k + 18
k = 18
Cost = 19*18
Re: math [#permalink] 06 Oct 2008, 14:25
Display posts from previous: Sort by | {"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": 0.275738000869751, "perplexity": 4439.731120384908}, "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-2016-26/segments/1466783395160.19/warc/CC-MAIN-20160624154955-00148-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/why-are-linear-equations-usually-written-down-as-matrices.851125/ | # Why are linear equations usually written down as matrices?
• Thread starter japplepie
• Start date
• #1
93
0
I've been taught that for any system of linear equations, it has a corresponding matrix.
Why do people sometimes use systems of linear equations to describe something and other times matrices?
Is it all just a way of writing things down faster or are there things you could do to matrices that you couldn't do to linear equations?
## Answers and Replies
• #2
Mark44
Mentor
35,128
6,873
I've been taught that for any system of linear equations, it has a corresponding matrix.
Why do people sometimes use systems of linear equations to describe something and other times matrices?
Is it all just a way of writing things down faster or are there things you could do to matrices that you couldn't do to linear equations?
Mostly matrices are a shorthand way of writing a system of linear equations, but there is one other advantage for certain systems : the ability to use a matrix inverse to solve the system.
For example, suppose we have this system:
2x + y = 5
x + 3y = 5
This system can be written in matrix form as:
##\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} 5 \\5 \end{bmatrix}##
Symbolically, the system is Ax = b, where A is the matrix of coefficients on the left, and b is the column vector whose entries are 5 and 5. (x is the column vector of variables x and y.)
Because I cooked this example up, I know that A has an inverse; namely ##A^{-1} = \frac 1 5 \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}##
If I apply this inverse to both sides of Ax = b, I get ##A^{-1}Ax = A^{-1}b = \frac 1 5 \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 5 \\5 \end{bmatrix}##
##= \begin{bmatrix} 2 \\1 \end{bmatrix}##
From this I see that x = 2 and y = 1. You can check that this is a solution by substituting these values in the system of equations.
Likes japplepie and suremarc
• #3
93
0
Mostly matrices are a shorthand way of writing a system of linear equations, but there is one other advantage for certain systems : the ability to use a matrix inverse to solve the system.
For example, suppose we have this system:
2x + y = 5
x + 3y = 5
This system can be written in matrix form as:
##\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} 5 \\5 \end{bmatrix}##
Symbolically, the system is Ax = b, where A is the matrix of coefficients on the left, and b is the column vector whose entries are 5 and 5. (x is the column vector of variables x and y.)
Because I cooked this example up, I know that A has an inverse; namely ##A^{-1} = \frac 1 5 \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}##
If I apply this inverse to both sides of Ax = b, I get ##A^{-1}Ax = A^{-1}b = \frac 1 5 \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 5 \\5 \end{bmatrix}##
##= \begin{bmatrix} 2 \\1 \end{bmatrix}##
From this I see that x = 2 and y = 1. You can check that this is a solution by substituting these values in the system of equations.
I see, thank you very much!
• #4
HallsofIvy
Science Advisor
Homework Helper
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Essentially, matrices allow you to write any system of linear equations as the single equation "Ax= b", the simplest form.
• #5
FactChecker
Science Advisor
Gold Member
6,369
2,514
The shorthand notation provided by the matrix is very beneficial. Keeping track of the variables that the matrix operates on often clutters up the calculations. If you compose a sequence of linear operations ( E = A * B * C * D ), you can do the matrix manipulations easily. If you try to name and keep track of all the intermediate values, it is just an unnecessary mess. ( x2 = Dx1; x3 = Cx2; x4 = Bx3; x5 = Ax4; so x5 = E x1 )
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3K | {"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": 0.9934324622154236, "perplexity": 317.95330704020245}, "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-2021-31/segments/1627046153816.3/warc/CC-MAIN-20210729043158-20210729073158-00159.warc.gz"} |
http://www.fightfinance.com/?q=72,408, | # Fight Finance
#### CoursesTagsRandomAllRecentScores
Portfolio Details Stock Expected return Standard deviation Correlation Beta Dollars invested A 0.2 0.4 0.12 0.5 40 B 0.3 0.8 1.5 80
What is the beta of the above portfolio?
You just bought a house worth $1,000,000. You financed it with an$800,000 mortgage loan and a deposit of $200,000. You estimate that: • The house has a beta of 1; • The mortgage loan has a beta of 0.2. What is the beta of the equity (the$200,000 deposit) that you have in your house?
Also, if the risk free rate is 5% pa and the market portfolio's return is 10% pa, what is the expected return on equity in your house? Ignore taxes, assume that all cash flows (interest payments and rent) were paid and received at the end of the year, and all rates are effective annual rates. | {"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": 0.4334474205970764, "perplexity": 2098.87968607167}, "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/1563195527196.68/warc/CC-MAIN-20190721185027-20190721211027-00432.warc.gz"} |
http://mathhelpforum.com/math-topics/61243-tranformations.html | 1. ## tranformations
describe the transformations on y=squarerootx for the following
1. y= squarerootx+3
2. y=squarerootx (-4)
3. y=squarerootx-2
4. y=squarerootx (+5)
2. Originally Posted by william
describe the transformations on y=squarerootx for the following
1. y= squarerootx+3
2. y=squarerootx (-4)
3. y=squarerootx-2
4. y=squarerootx (+5)
1. $y= \sqrt x + 3$
2. $y= \sqrt x (-4)$
3. $y= \sqrt x - 2$
4. $y= \sqrt x (+5)$
Are these the correct Equations? Your expressions are a bit hard to interpret.
3. Originally Posted by euclid2
1. $y= \sqrt x + 3$
2. $y= \sqrt x (-4)$
3. $y= \sqrt x - 2$
4. $y= \sqrt x (+5)$
Are these the correct Equations? Your expressions are a bit hard to interpret.
Yes, can you help me with them?
4. Originally Posted by william
describe the transformations on y=squarerootx for the following
1. y= squarerootx+3
2. y=squarerootx (-4)
3. y=squarerootx-2
4. y=squarerootx (+5) | {"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": 8, "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": 0.8646645545959473, "perplexity": 14210.985308908177}, "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-2016-50/segments/1480698541896.91/warc/CC-MAIN-20161202170901-00474-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://inquiryintoinquiry.com/2019/07/23/animated-logical-graphs-23/ | ## Animated Logical Graphs • 23
The following Table will suffice to show how the “streamer-cross” forms C.S. Peirce used in his essay on “Qualitative Logic” and Spencer Brown used in his Laws of Form, as they are extended through successive steps of controlled reflection, translate into syntactic strings and rooted cactus graphs:
$\text{Form}$ $\text{String}$ $\text{Graph}$
$\texttt{(} a \texttt{)}$
$\texttt{(} a \texttt{,} b \texttt{)}$
$\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}$
### 3 Responses to Animated Logical Graphs • 23
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"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": 6, "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": 0.2587127387523651, "perplexity": 5458.211678855673}, "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-2021-10/segments/1614178368687.25/warc/CC-MAIN-20210304082345-20210304112345-00529.warc.gz"} |
http://www.ck12.org/algebra/Excluded-Values-for-Rational-Expressions/lecture/Simplifying-Rational-Expressions/r1/ | <meta http-equiv="refresh" content="1; url=/nojavascript/">
# Excluded Values for Rational Expressions
## Excluding values that result in division by zero
%
Progress
Practice Excluded Values for Rational Expressions
Progress
%
Simplifying Rational Expressions
Learn how to simply a fraction that has polynomials in the numerator and denominator. | {"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": 0.9380685091018677, "perplexity": 5313.160455804954}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375093400.45/warc/CC-MAIN-20150627031813-00014-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/124500/sums-of-squares?sort=newest | # Sums of Squares
Every prime $p = 4k + 1$ can be uniquely expressed as sum of two squares, but for which integers $x$ is $x^2 + y^2 =$ some prime $p$? Stated differently, does the square of every positive integer appear as one of the squares in the representation of some prime $p$?
-
Even the case $x=1$, i.e. determining primes of the form $n^2+1$ is an open problem. – Keivan Karai Mar 14 '13 at 11:01
But it's known that there is at least one n such that n^2 + 1 is prime. So the answer for 1 is yes. – C. T. Jorgensen Mar 14 '13 at 11:20
I believe is has not been proved that for every $x$ there is a $y$ such that $x^2+y^2$ is prime. – Gerry Myerson Mar 14 '13 at 11:25
Nice question. I'd like to see an elaboration of Gerry's comment. – Joël Mar 14 '13 at 11:58
the computational evidence on this is remarkably regular - for p up to about 108. say x (or y) represents p if x2+y2 = p. if we then examine the primes (of form 4k+1) until all positive integers up to m have been used in a representation, then we will need to examine the first m2 primes p=4k+1. the regularity is impressive. it is also tempting to conjecture that all squares are used equally often (in some asymptotic sense). – geoffreyexoo Mar 14 '13 at 13:19
If the square of every positive integer appears as one of the squares in the representation of some prime -- that is, if for each $y$ there is an $x$ such that $x^2 + y^2$ is prime -- then it follows that there are infinitely many primes of the form $X^2 + Y^4$ (by restricting to $y$s that themselves are squares). This corollary happens to be true, but it was a breakthrough result of Friedlander and Iwaniec from about 15 years ago, so it seems unlikely that the much stronger question the OP is asking has been proven.
This is a special case of Bateman–Horn conjecture, which in this case states that for given $y\in\mathbb{N}$ the polynomial $p(x)=x^2+y^2$ assumes prime values for infinitely many $x\in\mathbb{N}$, more specifically, $$\#\{x\leq N: p(x)\text{ is prime}\}\sim\frac{1}{2}\prod_{p\nmid y}\frac{p-1-(-1)^{\frac{p-1}{2}}}{p-1}\cdot\frac{N}{\ln N},$$ thus the asymptotics should depend on $y$, but only by a multiplicative constant. | {"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": 0.9780398607254028, "perplexity": 215.2710218297569}, "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-2014-49/segments/1416931006855.76/warc/CC-MAIN-20141125155646-00101-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://www.ias.ac.in/describe/article/pmsc/121/04/0405-0416 | • Fixed Points of $IA$-Endomorphisms of a Free Metabelian Lie Algebra
• # Fulltext
https://www.ias.ac.in/article/fulltext/pmsc/121/04/0405-0416
• # Keywords
Free metabelian Lie algebra; fixed point.
• # Abstract
Let 𝐿 be a free metabelian Lie algebra of finite rank at least 2. We show the existence of non-trivial fixed points of an $IA$-endomorphism of 𝐿 and give an algorithm detecting them. In particular, we prove that the fixed point subalgebra Fix 𝜑 of an $IA$-endomorphism 𝜑 of 𝐿 is not finitely generated.
• # Author Affiliations
1. Department of Mathematics, Çukurova University, Adana, Turkey
2. Department of Mathematics, Melikşah University, Kayseri, Turkey
• # Proceedings – Mathematical Sciences
Volume 132, 2022
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019
Click here for Editorial Note on CAP Mode
© 2021-2022 Indian Academy of Sciences, Bengaluru. | {"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": 0.6437882781028748, "perplexity": 6889.506453960832}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335448.34/warc/CC-MAIN-20220930082656-20220930112656-00638.warc.gz"} |
https://uk.mathworks.com/help/comm/ref/comm.coarsefrequencycompensator-system-object.html | # comm.CoarseFrequencyCompensator
Compensate for frequency offset for PAM, PSK, or QAM
## Description
The `CoarseFrequencyCompensator` System object™ compensates for the frequency offset of received signals.
To compensate for the frequency offset of a PAM, PSK, or QAM signal:
1. Define and set up your coarse frequency compensator object. See Construction.
2. Call `step` to compensate for the frequency offset of a PAM, PSK, or QAM signal according to the properties of `comm.CoarseFrequencyCompensator`. The behavior of `step` is specific to each object in the toolbox.
Note
Starting in R2016b, instead of using the `step` method to perform the operation defined by the System object, you can call the object with arguments, as if it were a function. For example, `y = step(obj,x)` and `y = obj(x)` perform equivalent operations.
## Construction
`CFC = comm.CoarseFrequencyCompensator` creates a coarse frequency offset compensator object, `CFC`. This object uses an open-loop technique to estimate and compensate for the carrier frequency offset in a received signal.
`CFC = comm.CoarseFrequencyCompensator(Name,Value)` creates a coarse frequency offset compensator object, `CFC`, with the specified property `Name` set to the specified `Value`. You can specify additional name-value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.
## Properties
`Modulation`
Modulation type
Specify the signal modulation type as `BPSK`, `QPSK`, `OQPSK`, `8PSK`, `PAM`, or `QAM`. The default is `QAM`. This property is nontunable.
`Algorithm`
Algorithm used to estimate frequency offset
Specify the estimation algorithm as one of `FFT-based` or `Correlation-based`. The default is `FFT-based`. This property is nontunable.
The table shows the allowable combinations of the modulation type and the estimation algorithm.
ModulationFFT-Based AlgorithmCorrelation-Based Algorithm
`BPSK`, `QPSK`, `8PSK`, `PAM`
`OQPSK`, `QAM`
Use the correlation-based algorithm for HDL implementations and for other situations in which you want to avoid using an FFT.
This property appears when `Modulation` is `'BPSK'`, `'QPSK'`, `'8PSK'`, or `'PAM'`.
`FrequencyResolution`
Frequency resolution (Hz)
Specify the frequency resolution for the offset frequency estimation as a positive, real scalar of data type `double`. This property establishes the FFT length used to perform spectral analysis and must be less than the sample rate. The default is `0.001`. This property is nontunable.
`MaximumFrequencyOffset`
Maximum measurable frequency offset (Hz)
Specify the maximum measurable frequency offset as a positive, real scalar of data type `double`.
The value of this property must be less than fsamp / M, where fsamp is the sample rate and M is the modulation order. As a best practice, set `MaximumOffset` to less than r / (4M). This property applies only if `Algorithm` is `Correlation-based`. The default is `0.05`. This property is nontunable.
`SampleRate`
Sample rate (Hz)
Specify the sample rate in samples per second as a positive, real scalar of data type `double`. The default is `1`. This property is nontunable.
`SamplesPerSymbol`
Samples per symbol
Specify the number of samples per symbol, s, as a real positive finite integer scalar, such that s ≥ 2. The default value is `4`. This property is nontunable.
This property appears when `Modulation` is `'OQPSK'`.
## Methods
info Characteristic information about coarse frequency compensator step Compensate for frequency offset
Common to All System Objects
`release`
Allow System object property value changes
`reset`
Reset internal states of System object
## Examples
collapse all
Compensate for a 4 kHz frequency offset imposed on a noisy QPSK signal.
Set the example parameters.
```nSym = 2048; % Number of input symbols sps = 4; % Samples per symbol nSamp = nSym*sps; % Number of samples fs = 80000; % Sampling frequency (Hz) ```
Create a square root raised cosine transmit filter.
```txfilter = comm.RaisedCosineTransmitFilter(... 'RolloffFactor',0.2, ... 'FilterSpanInSymbols',8, ... 'OutputSamplesPerSymbol',sps); ```
Create a phase frequency offset object to introduce the 4 kHz frequency offset.
```freqOffset = comm.PhaseFrequencyOffset(... 'FrequencyOffset',-4000, ... 'SampleRate',fs); ```
Create a coarse frequency compensator object to compensate for the offset.
```freqComp = comm.CoarseFrequencyCompensator(... 'Modulation','QPSK', ... 'SampleRate',fs, ... 'FrequencyResolution',1); ```
Generate QPSK symbols, filter the modulated data, pass the signal through an AWGN channel, and apply the frequency offset.
```data = randi([0 3],nSym,1); modData = pskmod(data,4,pi/4); txSig = txfilter(modData); rxSig = awgn(txSig,20,'measured'); offsetData = freqOffset(rxSig); ```
Compensate for the frequency offset using `freqComp`. When the frequency offset is high, it is beneficial to do coarse frequency compensation prior to receive filtering because filtering suppresses energy in the useful spectrum.
```[compensatedData,estFreqOffset] = freqComp(offsetData); ```
Display the estimate of the frequency offset.
```estFreqOffset ```
```estFreqOffset = -3.9999e+03 ```
Return information about the `freqComp` object. To obtain the FFT length, you must call `freqComp` prior to calling the `info` method.
```freqCompInfo = info(freqComp) ```
```freqCompInfo = struct with fields: FFTLength: 131072 Algorithm: 'FFT-based' ```
Create a spectrum analyzer object and plot the offset and compensated spectra. Verify that the compensated signal has a center frequency at 0 Hz and that the offset signal has a center frequency at -4 kHz.
```specAnal = dsp.SpectrumAnalyzer('SampleRate',fs,'ShowLegend',true, ... 'ChannelNames',{'Offset Signal' 'Compensated Signal'}); specAnal([offsetData compensatedData]) ```
Correct for a phase and frequency offset in a noisy QAM signal using a carrier synchronizer. Then correct for the offsets using both a carrier synchronizer and a coarse frequency compensator.
Set the example parameters.
```fs = 10000; % Symbol rate (Hz) sps = 4; % Samples per symbol M = 16; % Modulation order k = log2(M); % Bits per symbol ```
Create a QAM modulator and an AWGN channel.
```channel = comm.AWGNChannel('EbNo',20,'BitsPerSymbol',k,'SamplesPerSymbol',sps); ```
Create a constellation diagram object to visualize the effects of the offset compensation techniques. Specify the constellation diagram to display only the last 4000 samples.
```constdiagram = comm.ConstellationDiagram(... 'ReferenceConstellation',qammod(0:M-1,M), ... 'SamplesPerSymbol',sps, ... 'SymbolsToDisplaySource','Property','SymbolsToDisplay',4000, ... 'XLimits',[-5 5],'YLimits',[-5 5]); ```
Introduce a frequency offset of 400 Hz and a phase offset of 30 degrees.
```phaseFreqOffset = comm.PhaseFrequencyOffset(... 'FrequencyOffset',400,... 'PhaseOffset',30,... 'SampleRate',fs); ```
Generate random data symbols and apply 16-QAM modulation.
```data = randi([0 M-1],10000,1); modSig = qammod(data,M); ```
Create a raised cosine filter object and filter the modulated signal.
```txfilter = comm.RaisedCosineTransmitFilter('OutputSamplesPerSymbol',sps, ... 'Gain',sqrt(sps)); txSig = txfilter(modSig); ```
Apply the phase and frequency offset, and then pass the signal through the AWGN channel.
```freqOffsetSig = phaseFreqOffset(txSig); rxSig = channel(freqOffsetSig); ```
Apply fine frequency correction to the signal by using the carrier synchronizer.
```fineSync = comm.CarrierSynchronizer('DampingFactor',0.7, ... 'NormalizedLoopBandwidth',0.005, ... 'SamplesPerSymbol',sps, ... 'Modulation','QAM'); rxData = fineSync(rxSig); ```
Display the constellation diagram of the last 4000 symbols.
```constdiagram(rxData) ```
Even with time to converge, the spiral nature of the plot shows that the carrier synchronizer has not yet compensated for the large frequency offset. The 400 Hz offset is 1% of the sample rate.
Repeat the process with a coarse frequency compensator inserted before the carrier synchronizer.
Create a coarse frequency compensator to reduce the frequency offset to a manageable level.
```coarseSync = comm.CoarseFrequencyCompensator('Modulation','QAM','FrequencyResolution',1,'SampleRate',fs*sps); ```
Pass the received signal to the coarse frequency compensator and then to the carrier synchronizer.
```syncCoarse = coarseSync(rxSig); rxData = fineSync(syncCoarse); ```
Plot the constellation diagram of the signal after coarse and fine frequency compensation.
```constdiagram(rxData) ```
The received data now aligns with the reference constellation.
## Algorithms
### Correlation-Based
The correlation-based estimation algorithm, which can be used to estimate the frequency offset for PSK and PAM signals, is described in [1]. To determine the frequency offset, Δf, the algorithm performs a maximum likelihood (ML) estimation of the complex-valued oscillation `exp`(j2πΔft). The observed signal, rk, is represented as
`${r}_{k}={e}^{j\left(2\pi \Delta fk{\text{T}}_{s}+\theta \right)},\text{\hspace{0.17em}}1\le k\le N\text{\hspace{0.17em}},$`
where Ts is the sampling interval, θ is an unknown random phase, and N is the number of samples. The maximum likelihood estimation of the frequency offset is equivalent to seeking the maximum of the likelihood function, Λ(Δf),
`$\Lambda \left(\Delta f\right)\approx {|\sum _{i=1}^{N}{r}_{i}{e}^{-j2\pi \Delta fi{T}_{s}}|}^{2}=\sum _{k=1}^{N}\sum _{m=1}^{N}{r}_{k}{r}_{m}^{*}{e}^{-j2\pi \Delta f{T}_{s}\left(k-m\right)}\text{\hspace{0.17em}}.$`
After simplifying, the problem is expressed as a discrete Fourier transform, weighted by a parabolic windowing function. It is expressed as
`$\mathrm{Im}\left\{\sum _{k=1}^{N-1}k\left(N-k\right)R\left(k\right){e}^{j2\pi \Delta \stackrel{^}{f}{T}_{s}}\right\}=0\text{\hspace{0.17em}},$`
where R(k) denotes the estimated autocorrelation of the sequence rk and is represented as
`$R\left(k\right)\triangleq \frac{1}{N-k}\sum _{i=k+1}^{N}{r}_{i}\text{\hspace{0.17em}}{r}_{i-k}^{*},\text{\hspace{0.17em}}0\le k\le N-1\text{\hspace{0.17em}}.$`
The term k(N–k) is the parabolic windowing function. In [1], it is shown that R(k) is a poor estimate of the autocorrelation of rk when k = 0 or when k is close to N. Consequently, the windowing function can be expressed as a rectangular sequence of 1s for k = 1, 2, ..., L, where LN – 1. The results is a modified ML estimation strategy in which
`$\mathrm{Im}\left\{\sum _{k=1}^{L}R\left(k\right){e}^{-j2\pi \Delta \stackrel{^}{f}k{T}_{s}}\right\}=0\text{\hspace{0.17em}}.$`
This results in an estimate of $\Delta \stackrel{^}{f}$ in which
`$\Delta \stackrel{^}{f}\cong \frac{{f}_{samp}}{\pi \left(L+1\right)}\mathrm{arg}\left\{\sum _{k=1}^{L}R\left(k\right)\right\}\text{\hspace{0.17em}}.$`
The sampling frequency, fsamp, is the reciprocal of Ts. The number of elements used to compute the autocorrelation sequence, L, are determined as
`$L=\mathrm{round}\left(\frac{{f}_{samp}}{{f}_{max}}\right)-1,\text{\hspace{0.17em}}$`
where fmax is the maximum expected frequency offset and `round` is the nearest integer function. The frequency offset estimate improves when L ≥ 7 and leads to the recommendation that fmaxfsamp / (4M).
### FFT-Based
FFT-based algorithms can be used to estimate the frequency offset for all modulation types. Two variations are used in comm.CoarseFrequencyCompensator.
• For `BPSK`, `QPSK`, `8PSK`, `PAM`, or `QAM` modulations the FFT-based algorithm used is described in [2]. The algorithm estimates $\Delta \stackrel{^}{f}$ by using a periodogram of the mth power of the received signal and is given as
`$\Delta \stackrel{^}{f}=\frac{{f}_{samp}}{N\cdot m}\mathrm{arg}\underset{f}{\mathrm{max}}|\sum _{k=0}^{N-1}{r}^{m}\left(k\right){e}^{-j2\pi kt/N}|,\text{ }\left(-\frac{{R}_{sym}}{2}\le f\le \frac{{R}_{sym}}{2}\right)\text{\hspace{0.17em}},$`
where m is the modulation order, r(k) is the received sequence, Rsym is the symbol rate, and N is the number of samples. The algorithm searches for a frequency that maximizes the time average of the mth power of the received signal multiplied by various frequencies in the range of [–Rsym/2, Rsym/2]. As the form of the algorithm is the definition of the discrete Fourier transform of rm(t), searching for a frequency that maximizes the time average is equivalent to searching for a peak line in the spectrum of rm(t). The number of points required by the FFT is
`$N={2}^{⌈{\mathrm{log}}_{2}\left(\frac{{f}_{samp}}{{f}_{r}}\right)⌉}\text{\hspace{0.17em}},$`
where fr is the desired frequency resolution.
• For `OQPSK` modulation the FFT-based algorithm used is described in [4]. The algorithm searches for spectral peaks at +/- 200 kHz around the symbol rate. This technique locates desired peaks in the presence of interference from spectral content around baseband frequencies due to filtering.
## References
[1] Luise, M. and R. Regiannini. “Carrier recovery in all-digital modems for burst-mode transmissions.” IEEE® Transactions on Communications. Vol. 43, No. 2, 3, 4, Feb/Mar/April, 1995, pp. 1169–1178.
[2] Wang, Y., K. Shi, and E. Serpedi. “Non-Data-Aided Feedforward Carrier Frequency Offset Estimators for QAM Constellations: A Nonlinear Least-Squares Approach.” EURASIP Journal on Applied Signal Processing. 2004:13, pp. 1993–2001.
[3] Nakagawa, T., M. Matsui, T. Kobayashi, K. Ishihara, R. Kudo, M. Mizoguchi, and Y. Miyamoto. “Non-Data-Aided Wide-Range Frequency Offset Estimator for QAM Optical Coherent Receivers.” Optical Fiber Communication Conference and Exposition (OFC/NFOEC), 2011 and the National Fiber Optic Engineers Conference. March 2011, pp. 1–3.
[4] Olds, Jonathan. "Designing an OQPSK demodulator". | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 11, "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": 0.8842774033546448, "perplexity": 1600.698130603648}, "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/1631780057303.94/warc/CC-MAIN-20210922011746-20210922041746-00318.warc.gz"} |
https://arxiv.org/abs/1712.01669 | cond-mat.str-el
(what is this?)
# Title: Competition between spin-orbit coupling, magnetism, and dimerization in the honeycomb iridates: $α$-Li$_{2}$IrO$_{3}$ under pressure
Abstract: Single-crystal x-ray diffraction studies with synchrotron radiation on the honeycomb iridate $\alpha$-Li$_{2}$IrO$_{3}$ reveal a pressure-induced structural phase transition with symmetry lowering from monoclinic to triclinic at a critical pressure of $P_{c}$ = 3.8 GPa. According to the evolution of the lattice parameters with pressure, the transition mainly affects the $ab$ plane and thereby the Ir hexagon network, leading to the formation of Ir--Ir dimers. These observations are independently predicted and corroborated by our \textit{ab initio} density functional theory calculations where we find that the appearance of Ir--Ir dimers at finite pressure is a consequence of a subtle interplay between magnetism, correlation, spin-orbit coupling, and covalent bonding. Our results further suggest that at $P_{c}$ the system undergoes a magnetic collapse. Finally we provide a general picture of competing interactions for the honeycomb lattices $A_{2}$$M$O$_{3}$ with $A$= Li, Na and $M$ = Ir, Ru.
Comments: 6 pages, 2 figures Subjects: Strongly Correlated Electrons (cond-mat.str-el) Journal reference: Phys. Rev. B 97, 020104 (2018) DOI: 10.1103/PhysRevB.97.020104 Cite as: arXiv:1712.01669 [cond-mat.str-el] (or arXiv:1712.01669v1 [cond-mat.str-el] for this version)
## Submission history
From: Christine Kuntscher [view email]
[v1] Tue, 5 Dec 2017 14:40:17 GMT (420kb,A) | {"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": 0.45507797598838806, "perplexity": 4427.678728092088}, "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-2018-22/segments/1526794867977.85/warc/CC-MAIN-20180527004958-20180527024958-00227.warc.gz"} |
http://help.unc.edu/help/microsoft-key-management-services-kms/ | # Microsoft Key Management Services (KMS)
## What is KMS?
Microsoft Key Management Services (KMS) provides a way to activate volume license editions of Microsoft Windows and Microsoft Office. It is designed to allow Volume Licensing customers to automate the activation process in a way that is transparent to end users. Volume Activation applies to systems that are covered under a Volume Licensing program and is used strictly as a tool for activation and not tied to license invoicing or billing. KMS is required for the following versions of Windows and Office:
• Windows Vista
• Windows 7
• Windows 8
• Windows Server 2008
• Windows Server 2008 R2
• Windows Server 2012
• Office 2010
• Office 2013
## How KMS Works
When a supported version of Windows or Office is installed, the computer will attempt to discover the KMS Server on the network. This is accomplished by checking for a SRV record in the DNS Zone of the computer. If that record is present, the computer will contact the KMS server and automatically activate the license. KMS activations are valid for 180 days. The computer will periodically check in with the KMS server to renew the computer’s activation.
Please note: the initial KMS activation process requires network connectivity to the campus network, either through a wired, wireless, or VPN connection. Computers must also connect to the campus network at least once every 180 days to re-activate.
KMS is not a suitable activation method for computers that do not have network access or do not connect to the campus network at least once every 180 days. In these cases, there is an alternate activation method, Multiple Activation Key (MAK).
## Automatic Discovery
For automatic discovery to work, the KMS SRV record must be present in the DNS zone of the computer. For example, a computer with the DNS address computer_name.ad.unc.edu would check the ad.unc.edu zone. A computer with the DNS address computer_name.its.unc.edu would check the its.unc.edu zone.
To determine if the required record is present in a domain, run the following command using command prompt for windows:
nslookup -type=srv _vlmcs._tcp.<domain>
If the correct record is present, you should see something like this:
_vlmcs._tcp.<domain> SRV service location:
priority = 0
weight = 100
port = 1688
If this test fails, you will need to have the following DNS record added to the DNS zone:
_vlmcs._tcp.<domain>. 3600 IN SRV 0 100 1688 its-kms1.ad.unc.edu.
## Manual Windows Activation
In cases where automatic discovery doesn’t work, Windows can be manually pointed to the KMS server. To manually activate Windows, use the following steps:
1. Open a command prompt with elevation (Right click and run as administrator).
2. Run the following command to point Windows to the KMS server.cscript c:\windows\system32\slmgr.vbs -skms its-kms1.ad.unc.edu
3. Run the following command to activate Windows.cscript c:\windows\system32\slmgr.vbs -ato
## Manual Office Activation
In cases where automatic discovery doesn’t work, Office can be manually pointed to the KMS server. To manually activate Office, use the following steps:
1. Open a command prompt with elevation.
2. Navigate to the Office installation folder using the “cd” command. These are the typical folder locations:
• Office 2010 32-bit: C:\Program Files (x86)\Microsoft Office\Office14
• Office 2010 64-bit: C:\Program Files\Microsoft Office\Office14
• Office 2013 32-bit: C:\Program Files (x86)\Microsoft Office\Office15
• Office 2013 64-bit: C:\Program Files\Microsoft Office\Office15
Example: cd C:\Program Files\Microsoft Office\Office15
3. Run the following command to point Office to the KMS server.cscript ospp.vbs /sethst:its-kms1.ad.unc.edu
4. Run the following command to activate Office.cscript ospp.vbs /act | {"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": 0.9442754983901978, "perplexity": 8145.438895455747}, "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-2015-14/segments/1427131297622.30/warc/CC-MAIN-20150323172137-00213-ip-10-168-14-71.ec2.internal.warc.gz"} |
http://www.lojban.org/publications/cll.before-20160614/cll_v1.1_xhtml-section-chunks/section-miscellany.html | ## 18.21. Miscellany
A few other points:
se can be used to convert an operator as if it were a selbri, so that its arguments are exchanged. For example:
Example 18.139.
li ci se vu'u vo du li pa The-number three (inverse) minus four equals the-number one.
3 subtracted from 4 equals 1.
The other converters of selma'o SE can also be used on operators with more than two operands, and they can be compounded to create (probably unintelligible) operators as needed.
Members of selma'o NAhE are also legal on an operator to produce a scalar negation of it. The implication is that some other operator would apply to make the bridi true:
Example 18.140.
li ci na'e su'i vo du li pare The-number 3 non- plus 4 equals the-number 12.
Example 18.141.
li ci to'e vu'u re du li mu The-number 3 opposite-of- minus 2 equals the-number 5.
The sense in which plus is the opposite of minus is not a mathematical but rather a linguistic one; negated operators are defined only loosely.
la'e and lu'e can be used on operands with the usual semantics to get the referent of or a symbol for an operand. Likewise, a member of selma'o NAhE followed by bo serves to scalar-negate an operand, implying that some other operand would make the bridi true:
Example 18.142.
li re su'i re du li na'ebo mu The-number 2 plus 2 equals the-number non- 5.
2 + 2 = something other than 5.
The digits 0-9 have rafsi, and therefore can be used in making lujvo. Additionally, all the rafsi have CVC form and can stand alone or together as names:
Example 18.143.
la zel. poi gunta la tebes. pu nanmu Those-named “Seven” who attack that-named “Thebes” [past] are-men.
The Seven Against Thebes were men.
Of course, there is no guarantee that the name zel. is connected with the number rafsi: an alternative which cannot be misconstrued is:
Example 18.144.
la zemei poi gunta Those-named-the Sevensome who attack
la tebes. pu nanmu that-named Thebes [past] are-men.
Certain other members of PA also have assigned rafsi: so'a, so'e, so'i, so'o, so'u, da'a, ro, su'e, su'o, pi, and ce'i. Furthermore, although the cmavo fi'u does not have a rafsi as such, it is closely related to the gismu frinu, meaning fraction; therefore, in a context of numeric rafsi, you can use any of the rafsi for frinu to indicate a fraction slash.
A similar convention is used for the cmavo cu'o of selma'o MOI, which is closely related to cunso (probability); use a rafsi for cunso in order to create lujvo based on cu'o. The cmavo mei and moi of MOI have their own rafsi, two each in fact: mem/ mei and mom/ moi respectively.
The grammar of mekso as described so far imposes a rigid distinction between operators and operands. Some flavors of mathematics (lambda calculus, algebra of functions) blur this distinction, and Lojban must have a method of doing the same. An operator can be changed into an operand with ni'enu'a, which transforms the operator into a matching selbri and then the selbri into an operand.
To change an operand into an operator, we use the cmavo ma'o, already introduced as a means of changing a lerfu string such as fy. into an operator. In fact, ma'o can be followed by any mekso operand, using the elidable terminator te'u if necessary.
There is a potential semantic ambiguity in ma'o fy. [te'u] if fy. is already in use as a variable: it comes to mean the function whose value is always f. However, mathematicians do not normally use the same lerfu words or strings as both functions and variables, so this case should not arise in practice. | {"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": 0.683483362197876, "perplexity": 6632.105319605941}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891377.59/warc/CC-MAIN-20180122133636-20180122153636-00781.warc.gz"} |
http://electev.blogspot.com/2013/01/ | ## Saturday, January 26, 2013
### Metal detector part 2, the LM386
OK, I made some mistakes in the last post, so let's take a step back and revisit parts of it.
Last time I wrote that we wanted a low impedance in the connection between the two oscillators. I also calculated the impedance in that node at 20Hz, since that's the mixed frequency we will listen to when we look for treasure.
Wrong! The frequency we are mixing is around 400kHz, the lower frequency comes in the next node. So we need to make new impedance calculations to match the pull up/pull down network.
But there is more. I did try to connect the two oscillators with large capacitors, but suddenly the two oscillators where oscillating together. What's happened? I'm not completely sure about the mechanism, but having a small impedance at the emitter will load the emitter significantly, compared to the 22k$\Omega$ resistor. Loading the emitter in the oscillator is bad, since it would remove the small charge injection into the oscillation tank. We need the opposite, just like the original EDN article, which uses 1.2pF capacitors. I found a couple of 2.2pF capacitors, which will do. This gives us
$|Z| = \frac{1}{2 \pi 400\text{kHz} 2.2\text{pF}} = 181\text{k}\Omega$. Pull up/pull down resistors with a parallel resistance of 1.8M$\Omega$ should suffice.
Now comes the next problem. The audio amplifier we are building needs to have a high impedance input, but BJT's are usually hard to design as high very high impedance emitter followers. I asked an experienced colleague about the problem and he suggested that I look in the data sheet of the audio amplifier that EDN use, to see if it shows how the transistors in the IC connected. Great idea!
Look at the data sheet page two. At the inputs (pin 2 and 3) you can see cascaded transistors. Connecting transistors this way is called a Darlington connection and it gives a very high input impedance, ($h_{fe1}\cdot h_{fe2}\cdot R_e$), and a very high gain. This is what we will try out as the first part of our audio amplifier. But now that we are at it, we might as well see if we can understand the rest of the LM386.
The Darlington inputs are connected to a differential amplifier, the two mirrored npn's below the input stage. It amplifies the difference between the two inputs. The diff amplifiers output goes to a push-pull amplifier. This one has a transistor connected to ground (straight below the two diodes), that compensates for temperature induce gain change in the the rest of the amplifier. The lower amplifier stage has a Sziklai connected stage instead of a single transistor. It basically has the same function as the Darlington connection, it gives a higher input impedance and higher gain. If you want to know more about these circuits, I highly recommend reading chapter two of Horowitz & Hill(1989).
By the way, does anyone know what the circle with the arrow, at the top of the diagram, means? My guess would be some kind of diode. In Horowitz & Hill(1989), page 93, there is a resistor in that position.
This is it for now. I'll report back when I have tried the described circuit change and the Darlington emitter follower.
References
Horowitz, P., & Hill, W. (1989). The art of electronics. Cambridge university press.
## Wednesday, January 16, 2013
### Metal detector using Colpitt oscillators
This is a project I found through http://www.geotech1.com/. It's an old EDN article describing a very simple metal detector I thought I'd use treasure hunting with one of my daughters.
### Oscillator circuits
The basic principle is to use two oscillators. At least one of them should use an inductor as a part of the oscillator tank. The easiest design is a Colpitt's oscillator, since it only use one simple inductance together with a couple of capacitors. An LC circuit would oscillate spontaneously if it weren't for the resistance in the coil, capacitors, wires etc. To compensate for the resistance we use a transistor to give a small charge push in every oscillation. Here's an LTSpice simulation of such a circuit:
Colpitt oscillator
The output of the oscillator is taken from the emitter, and if you look at the simulated currents in the transistor, you will see some strange oscillations. These come from the base going above the collector for a short period of time. They will probably not be present in the real circuit due to internal resistance in the various elements of the circuit (this will of course be checked in reality..)
The oscillator frequencies depends on the inductance of the coil. By winding a big air coil, the frequency of the oscillator will change when the surrounding permeability changes according to this formula:
$L = \mu k$
where L is inductance, $\mu$ is permeability and k is a constant depending on the physical shape of the coil.
### Frequency mixing
Frequency is typically around 400kHz, so we need to convert it down to audio levels if we want to use a loudspeaker as indicator. This is done using a local oscillator with a fixed frequency. The two oscillator outputs are connected via capacitors, like in the schematic below. We also need to pull the out signal somewhere, not to leave output floating. The pull up/pull down resistors are chosen to have a total resistance of 75k$\Omega$ (you will see why when we design the audio amplifier). We can see the capacitors as output from one stage and the resistors as inputs to the next stage. One wants a low output impedance connected to a high input impedance to avoid loading the oscillator. The impedance of the capacitors are less than 7.5k$\Omega$ according to the formula $|Z| = \frac{1}{2 \pi f C} \approx 4000 \Omega$ at our lowest required frequency 20Hz. The capacitors and resistors make a high pass RC-filter, so we also need to check that the filter doesn't filter out sound above 20Hz. The cut-off frequency is: $f_c = \frac{1}{2 \pi R C} \approx 1$Hz.
Simplified mixer circuit
So, what will the output of this setup be? Let the capacitors be complex impedance's:
$z_1 = \frac{-j}{2 \pi f_1 C_1}$
$z_2 = \frac{-j}{2 \pi f_2 C_2}$
The current through the resistor is the sum of the current from the capacitors.
$I_{C1}+I_{C2}=I_R$
The voltage drop over R is
$V_{out} = R \cdot I_R = R (I_{C1}+I_{C2})$
The current from the capacitors will be:
$I_{C1} = \frac{V_1}{z_1} = \frac{-j\cdot V_1}{2 \pi f_1 C_1}$
$I_{C2} = \frac{-j\cdot V_2}{2 \pi f_2 C_2}$
so
$V_{out} = R \left (\frac{-j\cdot V_1}{2 \pi f_1 C_1}+\frac{-j\cdot V_2}{2 \pi f_2 C_2}\right)$
Since we are dealing with real voltages, the output voltage will be
$|V_{out}| = R \left (\frac{\sin(2 \pi f_1 t)}{2 \pi f_1 C_1}+\frac{\sin (2 \pi f_2 t)}{2 \pi f_2 C_2}\right )$
When adding two sine waves, we can use this identity:
$\sin(\alpha)+\sin(\beta) = 2 \sin\left (\frac{\alpha + \beta}{2}\right ) \cos\left (\frac{\alpha - \beta}{2}\right )$
Without actually calculating the amplitude of the resulting wave ( which we are less interested in) we can see that the resulting wave will have both the added and the subtracted frequencies of the two original waves. It will look for example like this,
20 beats per second from a 400.00kHz and a 400.02KHz sine wave.
By using a local oscillator with a frequency very close to the variable oscillator, we can generate a frequency component in the audible range that is proportional to the permeability close to the search coil.
The first step is to build the oscillators and trim them so they are close enough to each other. The second step will be building an audio amplifier to amplify the mixed signal. In the EDN article, they use a dedicated audio amplifier chip. I will build the amplifier from transistors, since that is one of the purposes of this project, to learn about transistor (BJT) electronics.
## Sunday, January 13, 2013
### Boost regulator efficiency meter
This is a project I've been working on from time to time for a while now, but now it's time to wrap it up and document it. The whole thing started with me building a couple of boost regulators for charging the cell phone (inspired by Minty Boost). I used one regulator from LT and one from TI, and I wanted to compare their energy efficiency.
Boost regulator from LT (LT1302)
Calibration was done using a known current source. By knowing the current ,$I_{in}$, and reading the ADC binary output $I_{bin}$, we can get the conversion factor (k) if we have the linear relationship $I_{in}=k \cdot I_{bin}$. By making a few measurements of this kind we can use the very smart website http://www.wolframalpha.com/ to get a linear fit with the measured data. Here's one of the calculations (actually for a voltage input, but the principle is exactly the same) using the command: linear fit {{1023,5.44},{511,2.7},{255,1.33},{127, 0.65}}. In the end, what we will calculate in matter of ADC outputs is $E=(k_{out-I} \cdot I_{out-bin} \cdot k_{out-V} \cdot V_{out-bin})/(k_{in-I} \cdot I_{out-bin} \cdot k_{in-V} \cdot V_{out-bin)}$, and hopefully we can group all k's into a nice, single, integer (0.6 in our case, not so nice...). The observant reader will notice that there is also an offset term missing in $I_{in}$, which is actually there in the Wolfram Alpha output. I haven't looked into this yet, but for now I will ignore this since I actually don't a whole lot of accuracy and the offset is pretty small. A complete error analysis of the measurement system would be interesting but time consuming, maybe I'll come back to that on a rainy day. (By the way, I used MathJax to display mathematical equations...) | {"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": 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": 0.7197679877281189, "perplexity": 749.5765147001728}, "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/1510934806768.19/warc/CC-MAIN-20171123085338-20171123105338-00632.warc.gz"} |
http://www.astro.lu.se/~danielc/ | # Research: Formation and evolution of planetary systems
I am a finishing PhD student at Lund Observatory. I am interested in the formation and early dynamical evolution of planetary systems, especially in the context of rocky planets in the habitable zone. I work with Anders Johansen and Melvyn B. Davies. My strongest skills include programming, statistics, orbital dynamics, and dust-gas dynamics in a protoplanetary disk. I conduct hydrodynamic simulations with the Pencil Code and N-body simulations with MERCURY. I am currently looking for a postdoctoral position where I can continue to work on planet formation, protoplanetary disks, and/or planetary habitability.
In addition to my astronomy research, I have worked as a software developer in the private sector, I teach MATLAB, and I have taught the Masters course in statistics at Lund Observatory. I am interested in programming languages, robust statistics, and high-performance computing.
## Survival of habitable planets in unstable planetary systems
Published: Carrera et al, 2016, MNRAS, 463, 3226 (also as arXiv:1605.01325)
Many observed giant planets lie on eccentric orbits. These orbits may be the result of strong scatterings with other giant planets at a time when the giant planets were dynamically unstable. In this work I investigate whether life in a habitable planet could survive this period of dynamical instability. In many cases, a habitable planet is ejected from the system or experiences a physical collision with another planet. In a few cases, the planet survives but acquires a new orbit that renders the planet uninhabitable.
We conducted 1,200 N-body simulations with MERCURY in which we modeled the orbital evolution of rocky planets and giant planets in various configurations. We measure the resilience of habitable planets as a function of the observed, present-day masses and orbits of the giant planets. We find that the survival rate of habitable planets depends strongly on the giant planet architecture; equal-mass giant planets are far more destructive than hierarchical systems where the giant planets have unequal masses. Our final result is shown in the figure below.
Click to Expand
We say that a habitable planet is resiliently habitable if it can avoid ejections, collisions, or orbital changes that would destroy life on the planet. The left plot shows the probability that a rocky planet in the habitable zone is resiliently habitable (colour scale) as a function of the present-day semimajor-axis and eccentricity of the observed giant planet. As a general rule, giant planets with eccentricity higher than 0.4 most likely came from a planet system where the giant planets had similar masses and experienced very strong scatterings. In these systems, rocky planets in the habitable zone are almost universally destroyed, mostly through ejection. If the present-day eccentricity is low, that indicates a dynamically quient past, that would have been conductive to the survival of habitable planets. The three white lines correspond to resilience probabilities of 25%, 50%, and 75%. The plot on the right shows the same three lines, now marked in red. The blue dots are the currently known exoplanets with a mass of at least 0.3 Jupiter masses and a stellar mass between 0.95 and 1.05 solar masses.
## How to form planetesimals from mm-sized chondrules and chondrule aggregates
Published: Carrera et al, 2015, A&A, 579, A43 (also as arXiv:1501.05314)
The size distribution of asteroids and Kuiper belt objects in the solar system is difficult to reconcile with a bottom-up formation scenario due to the observed scarcity of objects smaller than $$\sim$$100 km in size. Instead, planetesimals appear to form top-down, with large 100-1000 km bodies forming from the rapid gravitational collapse of dense clumps of small solid particles. We investigated the conditions under which solid particles can form dense clumps in a protoplanetary disk. We used the Pencil Code to model the interaction between solid particles and gas in a protoplanetary disk for a range of particle sizes. We found that particles down to millimeter sizes can form dense particle clouds through the streaming instability. We made a map of the range of conditions (particle size vs concentration) needed to form dense particle clumps.
Click to Expand
Spacetime diagrams for select runs showing the solid surface density $$\Sigma_{\rm solid}$$ (shown by color) as a function of the radial coordinate $$x$$ and simulation time. The right hand axis shows the mean solid concentration $$Z = \langle \Sigma_{\rm solid} \rangle / \langle \Sigma_{\rm total} \rangle$$. Three of the runs form visible filaments.
Click to Expand
Final results. The plot shows the range of particle sizes and concentration where particle clumps form (Green; Streaming regime). The particle concentration is $$Z = \langle \Sigma_{\rm solid} \rangle / \langle \Sigma_{\rm total} \rangle$$. The particle size is measured in Stokes number.
Finally, we estimated the distribution of collision speeds between mm-sized particles. We calculated the rate of sticking collisions and obtain a robust upper limit on the particle growth timescale of ~105 years. This means that mm-sized chondrule aggregates can grow on a timescale much smaller than the disk accretion timescale ( $$\sim$$106-107 years). | {"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": 0.8963266611099243, "perplexity": 1391.108810647118}, "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-2017-13/segments/1490218190236.99/warc/CC-MAIN-20170322212950-00080-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://hackage.haskell.org/package/hecc-0.4 | # hecc: Elliptic Curve Cryptography for Haskell
[ bsd3, codec, cryptography, deprecated, library ] [ Propose Tags ]
Deprecated. in favor of eccrypto
Pure math & algorithms for Elliptic Curve Cryptography in Haskell
[Skip to Readme]
Versions 0.1, 0.2, 0.3, 0.3.1, 0.3.2, 0.3.3, 0.4, 0.4.0.1, 0.4.1.0, 0.4.1.1 base (==4.*), cereal, crypto-api, hF2 [details] BSD-3-Clause (c) Marcel Fourné, 2009-2013 Marcel Fourné Marcel Fourné (hecc@bitrot.dyndns.org) Cryptography, Codec by MarcelFourne at Sat Jan 12 10:10:42 UTC 2013 NixOS:0.4.1.1 3712 total (19 in the last 30 days) (no votes yet) [estimated by rule of succession] λ λ λ Docs not available All reported builds failed as of 2016-12-22 Hackage Matrix CI
## Modules
• Codec
• Crypto
• ECC
• Codec.Crypto.ECC.Base
• Codec.Crypto.ECC.StandardCurves
## Downloads
#### Maintainer's Corner
For package maintainers and hackage trustees
## Readme for hecc-0.4
[back to package description]
ECC
---
RSA just doesn't cut it anymore for fast public-key crypto. Keys are large for reasonable security making it quite slow...
Enter elliptic curves: smaller numbers are necessary and everything is faster. Maybe this library is not for embedded system usage, but now people can experiment with ECC for those use-cases where some form of RSA would be chosen otherwise.
Hecc.Base
-----------
This is the Haskell-Elliptic-Curve-Cryptography-library, or maybe more appropriately atm it is only the basic math for many ECC-algorithms the user of this library may wish to implement.
As an example the EC-variant of the Diffie-Hellman key-exchange is included which shows how the values can be computed with this library (a better variant will follow, this is just an example).
Also included is a basic speed-benchmarking-file (point multiplication) for example on the NIST Curve P-256 (the author wants some usage results and performance-numbers... so...).
The API
-------
...is not stable right now. If anybody wants to use the library in its current state for serious cryptographic uses, then by all means contact the author!
The Code began as a prototyped script and has since been polished, but this is best-effort work in progress.
pmul
----
Point multiplication can be done by this library using one of two algorithms: double-and-add and mongomery ladder. The latter is the default and is intended to be mostly resistant to timing attacks, the former may be faster, depending on the number it multiplies by.
Timing Attack Resistance
------------------------
The point multiplication uses the montgomery ladder algorithm which should be timing attack resistant, but when mul by a number in binary form 1000..0 the operation gets strangely fast (us instead of ms) and 1000..0001 it is strangely slow (1.5 times), which hints to something fishy going on. More research will follow, but sidechannel-resistance is not totally out-of-focus.
Testing has given me the idea that the following-zeroes-case massively benefits from branch-prediction and the trailing-one-case throws it totally off (will have to check that on other CPUs). "More natural" numbers are safer (tested), but I wouldn't dare to say that the matter is resolved.
P.S.: 2^N-1 does not show the cache-problem, only long rows of zeroes.
Plan
----
Some algorithms using these primitives will likely follow (ECDH, ECDSA, OpenPGP; also: better versions of the primitives).
Speed is a good goal, may have some more improvements in the future.
A testing suite is on the list.
Hyperelliptic Curves... maybe.
Motivation
----------
This is a side-project from which other people may benefit. Due to time-constraints, I can't work as much on it as I would like. If you use/like it or want to make some criticism heard, please write me an email. | {"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": 0.22588422894477844, "perplexity": 5632.6272773987685}, "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-2018-22/segments/1526794869732.36/warc/CC-MAIN-20180527170428-20180527190428-00149.warc.gz"} |
https://mathoverflow.net/questions/381848/do-schlichtings-and-balmers-definitions-of-higher-witt-groups-of-a-scheme-agre | # Do Schlichting's and Balmer's definitions of higher Witt groups of a scheme agree when 2 is inverted?
My question is whether the construction of higher Witt groups of a scheme in Schlichting's Hermitian K-theory of Exact Categories agrees with the definition in Balmer's chapter in the Handbook of K-theory when 2 is inverted (i.e. the schemes are over $$\mathbb{Z}[\frac{1}{2}]$$, although I'd also be happy to know whether this is true when we tensor the Witt groups with $$\mathbb{Z}[\frac{1}{2}]$$). See below for both definitions of higher Witt groups. When 2 is not inverted, they only necessarily agree on the 0-th Witt group.
More background if necessary
Schlichting defines higher Witt groups of an exact category with duality $$(\mathcal{E}, *, \eta)$$ as the homotopy groups of the geometric realization of a Hermitian Q construction, which is roughly as follows
Given an exact category $$\mathcal{E}$$ with duality, define the category $$Q^h\mathcal{E}$$ as the category where
• objects are symmetric spaces in $$\mathcal{E}$$, i.e. pairs $$(X,\varphi)$$ of $$X\in \text{ob }\mathcal{E}$$ and an isomorphism $$\varphi: X\mapsto X^{*}$$ such that $$\varphi^*\eta_X = \varphi$$
• morphisms from $$(X, \varphi)$$ to $$(Y, \psi)$$ are equivalence classes of diagrams $$X \xleftarrow{p}U\xrightarrow{i}Y$$ where $$p$$ is an admissible epimorphism, $$i$$ an admissible monomorphism, and the restrictions of the symmetric forms on $$X$$ and $$Y$$ to $$U$$ agree.
• composition is pullback
(For a more precise statement, see Definition 4.1 in p.12 of Schlichting)
The $$0$$-th Witt group $$\pi_0(|Q^h\mathcal{E}|)$$ turns out to be the usual $$W_0(\mathcal{E}) = \{\text{isoclasses of symmetric spaces}\}/\{\text{metabolics}\}$$ (metabolics are defined in Schlichting, p. 6, Def 2.5, the 0th Witt group is defined in Schlichting p. 7, first paragraph of 2.2 and also in Balmer p. 7 Definition 1.1.27)
We can take Witt groups of a scheme $$X$$ by using the exact category of vector bundles over $$X$$ with the usual duality.
On the other hand, Balmer defines higher Witt groups of a triangulated category $$\mathcal{K}$$ with duality by taking $$W_0(\mathcal{K}) = \{\text{isoclasses of symmetric spaces}\}/\{\text{metabolics}\}$$ as above, and taking $$W_n(\mathcal{K}) = W_0(T^n\mathcal{K})$$ where $$T^n$$ denotes "applying the shift functor $$n$$ times" in the triangulated category, which can change the duality (see Balmer Def 1.4.1 and Def 1.4.4 for details). Now we can take Balmer Witt groups of a scheme $$X$$ by using the category of perfect complexes over $$X$$.
These definitions certainly agree on $$W_0$$, but they don't agree in the higher Witt groups (as can be seen in Remark 4.2 of Schlichting), and I want to know whether they do agree when $$2$$ is inverted as I said above. Thanks!
No, the definition in Schlichting's first paper are not the "correct" definition of higher Witt groups (in any case they are not the analogue of Balmer's Witt groups), rather they are some shifted higher Grothendieck-Witt groups. He provided later a different definition that does coincide with Balmer's (but which is defined only when 2 is invertible).
For brevity let me refer to the following three papers by Marco Schlichting:
In [Sch3] definition of the Grothendieck-Witt spectrum is provided when 2 is invertible. From this in [Prop. 7.2, Sch3] he deduces a model for Balmer's higher Witt groups as homotopy groups of the L-theory spectrum.
However, by using Proposition 6 in [Sch2] we see that the space $$|Q^h\mathcal{E}|$$ used in [Sch1] to define the Witt groups is not the 0-th space of the underlying spectrum. Rather it is the first space of the shifted Grothendieck-Witt spectrum $$\operatorname{GW}^{[-1]}(\mathcal{E})$$. Therefore we have $$\pi_i |Q^h\mathcal{E}|\cong \pi_{i-1}\operatorname{GW}^{[-1]}(\mathcal{E})$$
In particular its homotopy groups are not 4-periodic, and in general are not Witt groups.
• Maybe I should add that I do not know any definition of higher Witt groups that do not pass through some version of chain complexes, especially when 2 is not invertible (otherwise you can cheat via periodicity and use simple formulas in low degrees via formations, which are basically length one chain complexes) In fact if you know one I'd be rather interested in it... Jan 21, 2021 at 23:10
• Thank you Denis, this is perfect! Jan 21, 2021 at 23:10
• Unfortunately, I'm new to this subject so I don't know either... :( Jan 21, 2021 at 23:13 | {"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": 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": 35, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9701996445655823, "perplexity": 277.04237512060865}, "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/1664030335004.95/warc/CC-MAIN-20220927100008-20220927130008-00070.warc.gz"} |
http://cs.brown.edu/research/pubs/techreports/reports/CS-89-45.html | # Tech Report CS-89-45
## Parallel Transitive Closure and Point Location in Planar Structures
### Abstract:
We present parallel algorithms for several graph and geometric problems, including transitive closure and topological sorting in planar {\em st}-graphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal $O( \log n)$ running time using $n / \log n$ processors in the EREW PRAM model, $n$ being the number of vertices.
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http://accesspediatrics.mhmedical.com/content.aspx?bookid=1345§ionid=72121589 | Chapter 15
#### High-Yield Facts
• Focused assessment with sonography in trauma (FAST) in the hypotensive child allows for rapid identification of life-threatening intra-abdominal hemorrhage. In the stable traumatized child, serial FAST improves the identification of occult intra-abdominal injuries and may prove to be a useful screening tool aimed at reducing the number of computed tomography scans obtained.
• Pediatric point-of-care ultrasound (P-POCUS) allows for the more accurate identification of skin and soft-tissue infections requiring incision and drainage.
• P-POCUS improves the safety and efficiency with which central venous access is obtained in children and may improve the first-time success rate and efficiency of peripheral vascular access in children with difficult peripheral vascular access.
#### Introduction
Pediatric point-of-care ultrasound (P-POCUS) is a skill that is enabling physicians to use ultrasound technology as an extension of the physical examination to more accurately, efficiently, and safely manage children with acute medical, surgical, and trauma-related conditions. In this chapter, an introduction to three common indications for P-POCUS will be briefly reviewed—abdominal and torso trauma, skin and soft-tissue infections, and vascular access.
#### Torso Trauma
The focused assessment with sonography in trauma (FAST) scan was shown to reduce time to operative care, hospital length of stay, use of computed tomography (CT) and hospital costs, as well as improved morbidity in adult trauma patients.1 In the persistently unstable child with torso trauma, FAST similarly allows for the rapid identification and management of intra-abdominal hemorrhage2. On the other hand, most children with intra-abdominal injuries do not require surgery and therefore FAST may serve a different goal. Some have studied the utility of pediatric FAST scanning in diagnosing intra-abdominal hemorrhage versus diagnosing any injury versus clinically important injuries. The difficulty is in the consistent finding that approximately 15% of patients with torso trauma, for whom trauma code activation criteria were met, and likely some of those in whom it was not, have significant occult injuries. The current gold standard is still CT scanning. When FAST is used in the stable pediatric trauma patient,3 in conjunction with other diagnostic examinations, such as physical examination,4 liver function test,5 and/or serial FAST,6 its performance at identifying clinically important intra-abdominal injuries improves significantly. Most likely, a clinical decision rule incorporating serial FAST will improve the sensitivity and specificity of algorithms such as that proposed by Holmes et al.7 thereby assuring the appropriate group of injured children have diagnostic imaging, while not missing clinically important injuries with the goal of minimizing unnecessary radiation exposure.
The FAST technique using a low-frequency curvilinear probe in pediatrics is often easier to perform compared with adults given their smaller size and more echogenic tissue. One begins by looking at the hepatorenal space (Morison's pouch) in the right upper quadrant, which is the second most dependent part of the supine abdomen and where blood from the pelvis (the most dependent part but very ...
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## Subscription Options
### AccessPediatrics Full Site: One-Year Subscription
Connect to the full suite of AccessPediatrics content and resources including 20+ textbooks such as Rudolph’s Pediatrics and The Pediatric Practice series, high-quality procedural videos, images, and animations, interactive board review, an integrated pediatric drug database, and more. | {"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": 0.2801295518875122, "perplexity": 6611.29182437267}, "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-2017-09/segments/1487501171271.48/warc/CC-MAIN-20170219104611-00373-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/27261/ifsym-document-compiles-successfully-but-without-correct-output | # ifsym: Document compiles successfully, but without correct output
I have MacOSX Leopard, TeXShop and have copied the ifsym package to /usr/local/texlive/2009/texmf-dist/doc/latex/, downloaded from ftp://ftp.dante.de/tex-archive/help/Catalogue/entries/ifsym.html and successfully installed it etc. (updated my index: sudo texhash)
I tried to compile this piece of LaTeX code:
\documentclass{article}
\usepackage[geometry]{ifsym}
\begin{document}
$\FilledTriangleRight$
\end{document}
It compiles successfully but without the correct output, it prints a small italic letter 'd', I mean just d in the PDF.
Does anyone have an idea what I should do, so that it will print a filled triangle pointing to the left? (from the ifsym package of course, no others)
-
Welcome to TeX.sx! I cleaned up your question a bit, I hope you don't mind. – doncherry Sep 2 '11 at 14:02
The commands of ifsym must not be used in math mode. If you need them in a formula, enclose them in \mbox. That's a not so nice "feature" of the package.
On an aside, you should be using tlmgr in order to install packages in TeX Live (or, on the Mac, TeX Live Utility that should be in your /Applications/TeX folder). Consider also to upgrade to TeX Live 2011 at http://tug.org/mactex
-
Don't use math mode.
\documentclass{article}
\usepackage[geometry]{ifsym}
\begin{document}
\RightDiamond \FilledTriangleRight
\end{document}
Btw: The fonts are bitmap fonts, they won't scale well in a pdf.
- | {"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": 0.9873760938644409, "perplexity": 4863.037496117422}, "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-2014-35/segments/1408500829955.75/warc/CC-MAIN-20140820021349-00299-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://arxiv.org/abs/cond-mat/0212086 | cond-mat
# Title:Crossover from Scale-Free to Spatial Networks
Abstract: In many networks such as transportation or communication networks, distance is certainly a relevant parameter. In addition, real-world examples suggest that when long-range links are existing, they usually connect to hubs-the well connected nodes. We analyze a simple model which combine both these ingredients--preferential attachment and distance selection characterized by a typical finite interaction range'. We study the crossover from the scale-free to the spatial' network as the interaction range decreases and we propose scaling forms for different quantities describing the network. In particular, when the distance effect is important (i) the connectivity distribution has a cut-off depending on the node density, (ii) the clustering coefficient is very high, and (iii) we observe a positive maximum in the degree correlation (assortativity) which numerical value is in agreement with empirical measurements. Finally, we show that if the number of nodes is fixed, the optimal network which minimizes both the total length and the diameter lies in between the scale-free and spatial networks. This phenomenon could play an important role in the formation of networks and could be an explanation for the high clustering and the positive assortativity which are non trivial features observed in many real-world examples.
Comments: 4 pages, 6 figures, final version Subjects: Disordered Systems and Neural Networks (cond-mat.dis-nn); Statistical Mechanics (cond-mat.stat-mech) Journal reference: Europhys. Lett. 63 (6) pp.915-921 (2003) DOI: 10.1209/epl/i2003-00600-6 Cite as: arXiv:cond-mat/0212086 [cond-mat.dis-nn] (or arXiv:cond-mat/0212086v3 [cond-mat.dis-nn] for this version)
## Submission history
From: Marc Barthelemy [view email]
[v1] Wed, 4 Dec 2002 09:44:02 UTC (32 KB)
[v2] Tue, 18 Feb 2003 09:53:40 UTC (26 KB)
[v3] Mon, 22 Sep 2003 09:15:27 UTC (26 KB) | {"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": 0.803308367729187, "perplexity": 1808.40044568973}, "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-2019-47/segments/1573496668896.47/warc/CC-MAIN-20191117064703-20191117092703-00467.warc.gz"} |
https://arxiv.org/abs/1809.04279v1 | stat.ML
(what is this?)
# Title:Discretely Relaxing Continuous Variables for tractable Variational Inference
Abstract: We explore a new research direction in Bayesian variational inference with discrete latent variable priors where we exploit Kronecker matrix algebra for efficient and exact computations of the evidence lower bound (ELBO). The proposed "DIRECT" approach has several advantages over its predecessors; (i) it can exactly compute ELBO gradients (i.e. unbiased, zero-variance gradient estimates), eliminating the need for high-variance stochastic gradient estimators and enabling the use of quasi-Newton optimization methods; (ii) its training complexity is independent of the number of training points, permitting inference on large datasets; and (iii) its posterior samples consist of sparse and low-precision quantized integers which permit fast inference on hardware limited devices. In addition, our DIRECT models can exactly compute statistical moments of the parameterized predictive posterior without relying on Monte Carlo sampling. Our numerical studies demonstrate accurate inference using latent variables discretized as extremely low-precision 4-bit quantized integers. While the ELBO computations considered require over $10^{2352}$ log-likelihood evaluations, we train on datasets with over two-million points in just seconds.
Comments: To appear in the proceedings of the Advances in Neural Information Processing Systems (NIPS), 2018 Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG) Cite as: arXiv:1809.04279 [stat.ML] (or arXiv:1809.04279v1 [stat.ML] for this version)
## Submission history
From: Trefor Evans [view email]
[v1] Wed, 12 Sep 2018 07:05:30 UTC (60 KB)
[v2] Wed, 31 Oct 2018 20:06:10 UTC (63 KB)
[v3] Wed, 9 Jan 2019 22:29:41 UTC (63 KB) | {"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": 0.5555605292320251, "perplexity": 3372.9529607275435}, "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-2019-13/segments/1552912202728.21/warc/CC-MAIN-20190323060839-20190323082839-00351.warc.gz"} |
http://math.stackexchange.com/questions/175701/what-is-the-average-distance-of-a-combination-set | # What is the average distance of a combination set?
I'm working on a genetic algorithm and would like to map each function to a set of "codons". So + -> 011. Given this, I would like to figure out how easy it would be for any given codon set to mutate into another codon set.
If you were to take two combinations, say 012 and 111, they would have a distance of 2 since because it would take 2 mutations to move from one to the other. Eg. 012 - > 011 - > 111.
So, here's my question : given an (n, k) combination set, what is the average distance between any two combinations?
I think you might be able to model this as a small world network, but if there is a better approach I'm all ears.
-
Can you please add an explanation of what is an "$(n,k)$ combination set"? – MJD Jul 27 '12 at 2:20
Also, would you say that 002 and 000 have distance 1, or distance 2? – MJD Jul 27 '12 at 2:23
What I meant by (n, k) is a set with n possible values and k elements. So, a set using one 0 and 1, and with 3 elements / spots (Eg 000,100,101,etc) would be considered (2,3) combination. A (3,2) combination, on the other hand would be something like 12,00,22,02,etc. – caffein Jul 27 '12 at 3:21
002 and 000 would have a distance of 1 since you would only need to change one value to convert one to the other. – caffein Jul 27 '12 at 3:22
@caffein To clarify, does an $(n,k)$ combination set always contain exactly $n^k$ different strings? Also, it sounds like you are computing distance purely with mutation and not transpositions, etc? In this case you can use the term "Hamming distance". – Erick Wong Jul 27 '12 at 5:09
You seem to be talking about $k$-tuples with $n$ possible values. If two of those are randomly chosen, the probability that they differ in the $j$'th position is $1 - 1/n$, and the expected Hamming distance between them is $k(1-1/n)$.
That's not quite right for what I'm looking for. For instance, for an n = 2, k = 2 combination set, the above equation would give 2(1 - 1/2) = 1. However, the way I figure it, it should be more 4/3. Here's how: $00 <-> (01/10) <-> 11$ . Which gives distances of (1,2),(1,1) and (1,2) and averages of 3/2, 2/2, and 3/2. So, the overall average distance would be (3/2 + 2/2 + 3/2) / 3 = ( 8 / 2 ) / 3 = 4/3 . Does that seem right? Perhaps I'm missing something here, or maybe there's a way to alter the above equation to work with what I'm looking for. – caffein Jul 28 '12 at 16:11
The average distance in your example is $1$ if you take into account the possibility that the distance could be $0$. If you're asking for the average distance between two distinct $k$-tuples, you should multiply my answer by $n^k/(n^k-1)$. – Robert Israel Jul 29 '12 at 19:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8889434337615967, "perplexity": 308.04911038364827}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997894250.40/warc/CC-MAIN-20140722025814-00246-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/44271/on-the-pointwise-limit-of-sqrtnp-nx-when-n-to-infty-for-some-polynom | # On the pointwise limit of $\sqrt[n]{p_n(x)}$ when $n\to\infty$, for some polynomials $(p_n)$
For every $n$, I have a polynomial $p_n(x)=a^{(n)}_{n-1}x^{n-1}+a^{(n)}_{n-2}x^{n-2}+\dots+a^{(n)}_0$ (the $n$ in the exponent of the coefficients is merely an index).
I can show that $\lim_{n\to\infty}\sqrt[n]{a^n_{n-1}}=C$ for some constant C, and that this sequence is rising.
I can also show that we have $\lim_{n\to\infty}\sqrt[n]{p_n(x)}=A_x$ for some constant $A_x$.
My goal is to show that $\lim_{x\to\infty}(A_x/x)\ge C$.
I cannot assume the limits (of n and x) can be interchanged, although if it's easily provable I'll be glad to hear how.
The major obstacle I fail to see how to tackle is the fact that we take the $n$-th root, but the polynomial is of degree $n-1$. Were the polynomial of degree $n$ it would be much more natural as the division by $x$ would cancel out exactly with the leading coefficient of the polynomial.
Note that the claim may be incorrect (though it's unlikely) or I might be missing some assumptions (much more likely).
-
Wait, do you mean that there's an infinite sequence $a_0, a_1, a_2, \dots$ and an infinite number of polynomials $p_n(x) = a_{n-1}x^{n-1} + a_{n-2} x^{n-2} + \dots + a_0$ ? (Otherwise your first limit makes no sense given you only have a finite handful of $a_n$'s laying around, and your second limit is always 1, 0, or undefined depending on $p(x)$'s sign.) Also, in what sense is $A_x$ constant if it's indexed and varying with the variable $x$? – anon Jun 9 '11 at 7:23
Yes, you are correct, this is an infinite sequence of polynomials. – Gadi A Jun 9 '11 at 7:41
What sequence is rising (meaning increasing)? What are the hypotheses on $a_k^{(n)}$ for $k\le n-2$? – Did Jun 9 '11 at 8:02
The sequence of the $n$-th root of the coefficient. Of the other coefficients I know nothing. – Gadi A Jun 9 '11 at 8:08
The degree $n-1$ vs degree $n$ stuff is not a problem at all. But: do you assume all the coefficients $a_k^{(n)}$ to be nonnegative? Otherwise the $n$th root of $p_n(x)$ might not be defined. Or did you forget an absolute value sign? And it seems the nonnegativity assumption would make the result (trivially) true... – Did Jun 9 '11 at 9:30
Hint: fix $D<C$ and try to show that $A_x\ge Dx$ for every $x$ large enough knowing that $a_n\ge D^n$ for every $n$ large enough. In turn, $A_x\ge Dx$ would follow from $p_n(x)\ge D^nx^n$ for every $n$ large enough or even from $p_n(x)\ge D^{n-1}x^{n-1}$ for every $n$ large enough... | {"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": 0.9721174836158752, "perplexity": 273.46757621706877}, "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-2015-32/segments/1438042987866.61/warc/CC-MAIN-20150728002307-00143-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://git.friendi.ca/friendica/friendica-addons/commit/0b8e0fa8c9ec9a76741a2bf0c23980731d55abd4 | Hypolite Petovan 3 years ago
parent
commit
0b8e0fa8c9
1 changed files with 10 additions and 6 deletions
1. +10
-6
@ -21,15 +21,19 @@ In case you want to use the CDN you can try the following URL as a quick start http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML In case you don't want or can use the admin panel of Friendica you can activate the addon by adding _mathjax_ to the list in your config/local.ini.php file the addon by adding _mathjax_ to the list in your config/local.config.php file [system] addon = ...,mathjax 'system' => [ ... 'addon' => '...,mathjax' ... ] and then providing the base URL after that in the config/addon.ini.php file and then providing the base URL after that in the config/addon.config.php file [mathjax] baseurl = [the URL to your MathJax installation]; 'mathjax' => [ 'baseurl' => '[the URL to your MathJax installation]', ], Usage ===== | {"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": 0.7097837924957275, "perplexity": 12646.857289755286}, "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-2021-25/segments/1623488525399.79/warc/CC-MAIN-20210622220817-20210623010817-00259.warc.gz"} |
http://community.td-next.com/viewtopic.php?f=13&t=238&sid=fe7d48bab97cb2c89ff596b06671f675 | ## Build not working
Questions and discussions about the SDK tools used to program the TD RF modules.
lub
Posts: 2
Joined: Tue Jul 19, 2016 8:10 pm
### Build not working
Hi,
Whenever I try to build any of the projects I get the following message on the console
Code: Select all
`17:14:00 **** Incremental Build of configuration TD1508 Debug for project blink ****Info: Internal Builder is used for buildarm-none-eabi-gcc -DDEBUG -DDEBUG_EFM -DEZR32LG230F128 -DMODULE_REVISION=REVISION_TD1508 "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtdgeoloc\\inc" "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtdsensor\\inc" "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtdrf\\inc" "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtddrivers\\inc" "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtdcore\\inc" "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\emlib\\inc" "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\Device\\EnergyMicro\\EFM32LG\\Include" -O0 -ffunction-sections -fdata-sections -Wall -c -fmessage-length=0 -mcpu=cortex-m3 -mthumb -g3 -gdwarf-2 -o "src\\blink.o" "..\\src\\blink.c" Cannot run program "arm-none-eabi-gcc": The directory name is invalid.17:14:00 Build Finished (took 265ms)`
and no outputs are generated. I have followed carefully every step on the README.
Can anyone help me?
Thanks!
lopic34
Posts: 2
Joined: Thu Feb 23, 2017 8:35 am
### Re: Build not working
Hello,
I am trying to compil my first "blink" too, and I have the same problem:
Code: Select all
`16:28:36 **** Incremental Build of configuration TD1208 Release for project blink ****Info: Internal Builder is used for buildarm-none-eabi-gcc -DNDEBUG -DEFM32G210F128 -DMODULE_REVISION=REVISION_TD1208 "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtdrf\\inc" "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtddrivers\\inc" "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtdcore\\inc" "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\emlib\\inc" "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\Device\\EnergyMicro\\EFM32G\\Include" "-IC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\CMSIS\\Include" -Os -ffunction-sections -fdata-sections -Wall -c -fmessage-length=0 -mcpu=cortex-m3 -mthumb -o "src\\blink.o" "..\\src\\blink.c" arm-none-eabi-gcc "-T..\\..\\Device\\EnergyMicro\\EFM32G\\Source\\G++\\efm32g.ld" -Xlinker --gc-sections "-LC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtdgeoloc\\GCC Release" "-LC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtdsensor\\GCC Release" "-LC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtdrf\\GCC Release" "-LC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtddrivers\\GCC Release" "-LC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\libtdcore\\GCC Release" "-LC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\emlib\\GCC Release" "-LC:\\TD\\TD_RF_Module_SDK-v6.0.0\\Github\\TD_RF_Module_SDK\\lib\\Device\\GCC Release" -Wl,-Map,blink.map -mcpu=cortex-m3 -mthumb -o blink.elf "src\\blink.o" -ltdgeoloc -ltdsensor -ltdrf -ltdcore -ltdrf -ltddrivers -ltdcore -lemlib -lDevice -lgcc -lc -lcs3 -lcs3unhosted c:/td/td_rf_module_sdk-v6.0.0/gnu/bin/../lib/gcc/arm-none-eabi/4.7.2/../../../../arm-none-eabi/bin/ld.exe: cannot find -ltdgeolocc:/td/td_rf_module_sdk-v6.0.0/gnu/bin/../lib/gcc/arm-none-eabi/4.7.2/../../../../arm-none-eabi/bin/ld.exe: cannot find -ltdsensorc:/td/td_rf_module_sdk-v6.0.0/gnu/bin/../lib/gcc/arm-none-eabi/4.7.2/../../../../arm-none-eabi/bin/ld.exe: cannot find -ltdcorec:/td/td_rf_module_sdk-v6.0.0/gnu/bin/../lib/gcc/arm-none-eabi/4.7.2/../../../../arm-none-eabi/bin/ld.exe: cannot find -ltdcorec:/td/td_rf_module_sdk-v6.0.0/gnu/bin/../lib/gcc/arm-none-eabi/4.7.2/../../../../arm-none-eabi/bin/ld.exe: cannot find -lemlibc:/td/td_rf_module_sdk-v6.0.0/gnu/bin/../lib/gcc/arm-none-eabi/4.7.2/../../../../arm-none-eabi/bin/ld.exe: cannot find -lDevicecollect2.exe: error: ld returned 1 exit status16:28:39 Build Finished (took 2s.811ms)`
Is it a local path configuration ?
Best regards, | {"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": 0.8367325663566589, "perplexity": 8157.3809190443535}, "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-2018-43/segments/1539583518753.75/warc/CC-MAIN-20181024021948-20181024043448-00493.warc.gz"} |
https://mip-frontiers.eu/2020/11/10/carbon_footprint_AI.html | # AI Carbon Footprint
## The Carbon Impact of AI
Posted by Giorgia Cantisani on November 10, 2020 · 14 mins read
The topic of this post is the carbon footprint of Artificial Intelligence (AI). I thought this was a relevant topic to discuss as we, as researchers, are part of the emission process, even if we are mostly unaware of it. I believe it is ethically essential to be aware of the consequences that can derive from training our large deep learning models and discuss guidelines and best practices for our research community.
### Climate Change
First, let us recall the consequences of climate change, even if they are under our eyes every day. The United Nations declared climate change a “defining crisis of our time”1, and most climate scientists agree that human activity is its main driver2. However, the effects of climate change are not only the rise of temperatures and the melting of glaciers, but also:
• food and water emergency due to soil degradation and rise of temperatures;
• extreme meteorological events, such as, heatwaves, hurricanes, which are becoming more frequent and intense. 90% of them are classified as weather- and climate-related.
This creates competition for land, food, and water, leading to socio-economic tensions and mass displacement1. More than 140 million people, mostly from the south of the world, will be forced to migrate by 20501. According to a Stanford University research3, the economic gap between the world’s richest and poorest countries is 25% larger today than it would have been without global warming.
Urgent actions are needed to avoid an environmental catastrophe, including reducing emissions to zero by the middle of the twenty-first century and limiting the average global warming to 1.5 °C1. Technology has, of course, a central role in fighting this crisis.
### The Role of AI
In this light, it has become an urgent matter to consider the dual role of the AI technology in the crisis. On one hand, it can help reduce climate change effects by, for example, modelling and developing solutions to avoid it. On the other hand, the AI is itself a significant emitter of carbon.
Below you can see a visual map extracted from an essay of 2018 entitled Anatomy of an AI system4 that shows the impact of an AI device on a global scale. The authors took as an example Amazon Echo and followed its impact from manufacturing to usage. They analysed its impact in terms of social work, data, and resources required during its lifespan.
In the whole map, the location where we, as researchers, come into the game and play an active role is the small section below, related to training an AI system and eventually preparing and labelling data.
### Red vs Green AI
In 2019, a team from MIT analysed some Natural Language Processing (NLP) models available online5. The researchers estimated the energy consumption (in kilowatts) required to train them and converted the numbers into approximate carbon emissions and electricity costs. They estimated that the carbon footprint of training a single big language model is equal to around 300,000 kg of carbon dioxide emissions, which is orders of magnitude higher than other familiar consumptions. Furthermore, the authors also quantified the computational cost of Research and Development (R&D) for a new NLP model. This cost is the one we, as researchers and engineers, should primarily consider, as it reflects the actual carbon footprint of a project. In the figure below, you can see the details of the estimated CO$$_2$$ emissions from training standard NLP models compared to everyday consumption.
In a paper from 2019, Roy Schwartz and collaborators observed that a linear gain in performance requires an exponentially larger model leading to substantial carbon emissions6. They called this trend ‘red AI’, that is, ‘buying’ better results using massive computing. Their study enumerated three factors making AI research red:
• the cost of running the model on a single example;
• the training dataset size, which controls the number of times the model is run;
• the number of hyperparameters, which controls how many computations are necessary to train the model.
They analysed papers from top conferences and observed that the majority prioritised accuracy over efficiency (90% from the Computational Linguistics Conference 2018, 80% from NeurIPS 2018, and 75% from the Conference on Computer Vision and Pattern Recognition 2019). In the same paper, the authors also defined the term ‘green AI’ as “AI research that yields novel results without increasing or, ideally reducing, computational cost”. Green AI considers efficiency as a critical evaluation criterion and is considered an opposite term to red AI. Ideally, this type of research would level the possibilities of academia versus big tech companies, whose research is often facilitated by impressive computational resources.
A virtuous example in audio research is SuDoRM-RF7, a novel deep architecture for efficient universal sound source separation. It extracts multi-resolution temporal features through successive depth-wise convolutional down-sampling and aggregates them using a nonparametric interpolation scheme. This way, the authors can significantly reduce the required number of layers while still effectively capturing long-term temporal dependencies. The proposed model performs similarly or even better than state-of-the-art models while requiring significantly less computational resources in terms of the number of trainable parameters, number of floating-point operations, memory allocation, and time.
Even if this is a virtuous example, there is no estimate of carbon emissions and energy consumption. Mainly, the problem is the absence of a standard of measurement and the intrinsic difficulties in measuring it.
The emissions are related to the training server’s location and the energy grid it uses, the training procedure’s duration and the training’s hardware. Thus, it is difficult to measure and requires specific knowledge. Therefore, we can take advantage of libraries and tools that do the work for us, for example:
### Quantify the Energy Accounting
The consumed energy consists of the amount of energy needed to power the computational system and it is measured in Joules (J) or Watt-hours (Wh). It is given mostly by the cooling of the system and by the server/storage power consumption9:
• cooling (50%)
• lighting (3%)
• power conversion (11%)
• network hardware (10%)
• server/storage (26%)
We can further break down the server and storage component into DRAM, CPUs, and GPUs’ contributions. Accurate accounting for all these components requires complex modelling and varies depending on workload. Most carbon/energy trackers consider DRAM/CPUs/GPUs consumption and account for the other components through the PUE (Power Usage Effectiveness) factor9. This factor rescales the power metrics by an average projected overhead of different elements. This way, one can evaluate the impact of her/his experiment only without considering background processes.
Carbon emissions are typically measured in CO$$_{2}$$eq, which is the amount of carbon dioxide released into the atmosphere due to the project9. Sometimes (especially in regulations), it is considered the financial impacts through carbon’s social cost (SC-CO$$_2$$), which measures the long-term damage done by CO$$_2$$ 9. To measure it, one should use the per-country social cost of carbon, which accounts for the risk profiles of different countries.
We can estimate carbon emissions by understanding the local energy grid’s carbon intensity and the system’s energy consumption9. The carbon intensity corresponds to the grams of CO$$_{2}$$eq emitted per kWh of energy used and is determined by the energy sources supplying the grid:
• coal power: 820 gCO$$_{2}$$eq / kWh
• hydro-electricity: 24 gCO$$_{2}$$eq / kWh
Thus, running our job in countries where the energy supply is green can be crucial. It is not necessary to eliminate computation-heavy models, as shifting training resources to low carbon regions can immediately reduce carbon emissions with little impact on us9. (e.g., training the same model in Quebec rather than in Lettonia can reduce CO$$_{2}$$eq by 30 times!)
## Guidelines
In conclusion, we, as researchers, should think about:
• report training time, computational resources, and sensitivity to hyperparameters;
• make a cost-benefit (accuracy) analysis of our models;
• prioritise computationally efficient hardware and algorithms;
• quantify energy consumption and carbon emission;
• move jobs to low carbon regions;
Check out these excellent initiatives:
Looking forward to seeing something similar in the MIR community!
## References:
1. The United Nations (UN), “The Climate Crisis – A Race We Can Win” In https://www.un.org/en/un75/climate-crisis-race-we-can-win 2 3 4
2. “The Climate Reality Project” In https://www.climaterealityproject.org/
3. J. Garthwaite, “Climate change has worsened global economic inequality” In Stanford Earth (2019) https://earth.stanford.edu/news/climate-change-has-worsened-global-economic-inequality#gs.g3u2y5
4. K. Crawford and V. Joler, “Anatomy of an AI System: The Amazon Echo As An Anatomical Map of Human Labor, Data and Planetary Resources,” In AI Now Institute and Share Lab (2018) https://anatomyof.ai/ 2 3
5. E. Strubell, A. Ganesh and A. McCallum, “Energy and Policy Considerations for Deep Learning in NLP” In ArXiv Preprint (2019) https://arxiv.org/abs/1906.02243 2
6. R. Schwartz et al. “Green AI.” In ArXiv Preprint (2019) https://arxiv.org/pdf/1907.10597.pdf
7. Tzinis, Efthymios, Zhepei Wang, and Paris Smaragdis. “SuDo RM-RF: Efficient Networks for Universal Audio Source Separation.” In IEEE 30th International Workshop on Machine Learning for Signal Processing (MLSP) IEEE, (2020) https://arxiv.org/pdf/2007.06833.pdf 2
8. A. Lacoste et al. “Quantifying the carbon emissions of machine learning” In ArXiv Preprint arXiv:1910.09700 (2019) https://arxiv.org/pdf/1910.09700.pdf
9. P. Henderson et al. “Towards the Systematic Reporting of the Energy and Carbon Footprints of Machine Learning.” In ArXiv Preprint (2020) https://arxiv.org/pdf/2002.05651.pdf 2 3 4 5 6 7
10. L. F. W. Anthony, B. Kanding, and R. Selvan “Carbontracker: Tracking and Predicting the Carbon Footprint of Training Deep Learning Models.” In ArXiv Preprint (2020) https://arxiv.org/pdf/2007.03051.pdf | {"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": 0.5433886647224426, "perplexity": 2235.8063496983955}, "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-2021-04/segments/1610703513144.48/warc/CC-MAIN-20210117174558-20210117204558-00261.warc.gz"} |
https://iforest.sisef.org/contents/?id=ifor1459-008 | ## Effect of plant species on P cycle-related microorganisms associated with litter decomposition and P soil availability: implications for agroforestry management
iForest - Biogeosciences and Forestry, Volume 9, Issue 2, Pages 294-302 (2015)
doi: https://doi.org/10.3832/ifor1459-008
Research Articles
Cutting dry deciduous forest (preserved site) for wood supply in semi-arid Brazil has led to invasion of a pioneer shrub vegetation called “Carrasco” (disturbed site), which inhibits the sprouting of native species. A land restoration project was undertaken in a cleared Carrasco area where a mixed plantation of native species and Eucalyptus spp. (experimental site) was established to preserve the forest and ensure wood supply for the local population. We considered phosphorus as a limiting soil nutrient to plant growth, and we addressed the roles of litter decomposition and microbial activity on phosphorus release in the disturbed, preserved and experimental sites. The phosphorus released from leaf litter was affected by the vegetation type, which favored specific soil microbial populations during decomposition. The Carrasco vegetation predominantly favored arbuscular mycorrhizal fungi (AMF), as shown by root colonization in the litter bags; the Eucalyptus plants favored AMF and ectomycorrhizal fungi (EM), as well as phosphate solubilizing microorganisms (PSM), and the intercropping system favored AMF and PSM groups. In contrast, the preserved site favored the PSM population. High phosphatase activity was found in the preserved and experimental sites in contrast to the Carrasco soil. Principal component analysis showed that AMF root colonization and phosphatase activity were the main parameters influencing the increase in soil phosphorus. Based on the above results, rehabilitation appeared to be underway in the experimental site, since the samples were more similar to the preserved site than to the disturbed site. This effect was attributed to Eucalyptus camaldulensis that promote the establishment of all phosphorus cycle-related microorganisms (AMF, EM and PSF). E. camaldulensis associated with mycorrhizal fungi and PSM are recommended for inclusion in agroforestry systems.
# Introduction
Soils in the extreme north of Minas Gerais state, a semiarid region of eastern Brazil, are notably poor in water and nutrients, such as phosphorus. The natural vegetation, known as woody Caatinga (preserved site), has been depleted due to the strong demand of wood by farmers. When this natural vegetation is cut down, a strong floristic shift occurs ([29]), and the area is invaded by a dense community of pioneer species known as “Carrasco vegetation”. Such invasive vegetation inhibits the sprouting of woody species and interferes with their succession. As a result, areas invaded by Carrasco are considered disturbed sites ([33]).
The strong demand for wood in this region has resulted in establishing a wood and energy provisions project for the local populations. As an alternative to re-vegetation with native species, an agroforestry system was proposed that uses a mixture of Eucalyptus spp. and native species (experimental site), thereby combining harmonious and concomitant wood production with land rehabilitation and preservation.
It is well known that forest litter fall is the major source of nutrient transfer to soil, particularly the mixed forests in agroforestry systems that enhance and maintain soil fertility and productivity ([24]). This effect has been attributed to improved soil organic matter (SOM) compared with that in monocultures ([18]).
Phosphorus is a common factor limiting the productivity in cultivated and degraded tropical soils ([2]), as well as in semiarid ecosystems that require fertilization to ensure soil productivity. However, this option could be expensive for farmers in developing countries. Therefore, land management practices, such as agroforestry systems, may enhance the biological activity in the soil and favor nutrient availability, which would reduce the need for the massive use of fertilizers.
According to the literature, a selected and effective composition of soil microorganisms and plants allows for a more efficient nutrient use or increases nutrient availability by providing solutions for present and future agricultural practices ([1]), as occurs under natural conditions where the main source of nutrients is the soil organic matter. In terms of soil P availability, the vegetation plays an important role in regulating P distribution through litter decomposition and its cycling by microbial action ([37]).
Organic phosphorus is released through a mineralization process mediated by soil organisms and phosphatase secretion by plant roots and soil microorganisms. Decomposition of plant litter depends on the efficiency of the decomposer organisms and on the chemical composition of the leaf litter. Litter quality, in turn, is determined by carbon and nitrogen (C:N) or phosphorus (C:P) ratios, N, lignin chemical composition, and the lignin:N ratio ([4]). Litter from different species with different quality patterns may be differentially processed by specific groups of fungi and bacteria, and distinct microbial populations develop a feedback relationship with plant communities that vary across environments ([23]).
Plants can drive chemical and biological rhizospheric strategies involving plant phosphatases and/or elicit improvement in specific microbial groups, such as phosphate solubilizing microorganisms (PSM), to solubilize fixed phosphorus ([21], [43]) and mycorrhizal fungi. Both endo- and ectomycorrhizal fungi increase the absorptive root surface and can reach the decomposing litter, which supplies the microbial population with additional nutrients ([12]) and where nutrients are released and distributed to plants via the mycorrhizal mycelium ([15]).
It is well known that agroforestry systems can improve soil organic matter inputs ([28]), as well as the availability of soil P via litter decomposition ([30]) and phosphatases activity with intercropping age ([9]). Therefore, tree-based agroforestry systems, particularly those including Eucalyptus species, show an efficient return of soil nutrients ([20], [28]) that greatly contributes to the maintenance of soil fertility.
Considering that P is a limiting nutrient in semiarid soils, we hypothesized that different compositions of soil microbes and plants could increase P availability and provide attractive agroforestry solutions in order to reduce phosphate fertilization. We investigated the influence of the agroforestry system composition (vegetation type) on litter decomposition and phosphorus release through P cycle-related microbial activity such as mycorrhizal fun- gi and phosphate solubilizing microorganisms.
# Material and methods
## Study area
The study area is located in the Jaíba Irrigation District in the São Francisco river basin, northern Minas Gerais State, Brazil (15o 09′ 03″ S, 43o 49′ 26″ W) in a semi-arid region. The natural vegetation is composed of woody Caatinga known as “dry forest”, which is a deciduous forest composed of woody and shrub species (10-25 m of height) adapted to poor-nutrient soils ([33]). When the woody Caatinga is cut, it is replaced by a dense, invasive community of pioneer species of low-load trees and shrub mesh, including spiny shrubs, known as “Carrasco vegetation” characterized by several dominant species: Piptadenia moniliformis (Benth) - Mimosaceae; Mimosa sp. - Mimosaceae; Croton glandulosus (L.) Muell. Arg. - Euphorbiaceae; Platymiscium praecox (Mart. Ex Benth) - Fabaceae; Acacia monacantha (Willd.) - Mimosaceae; Thiloa glaucocarpa (Spruce ex Eich.) - Combretaceae; and Tabebuia serratifolia (Vahl) Nich - Bignoniaceae ([6]). The dominant native species of woody Caatinga are: Plathymenia reticulata, Tabebuia heptaphylla (Vell.) Toledo - Bignoniaceae; Myracrodruon urundeuva (Fr. Allem.) - Anacardiaceae; Machaerium stipitatum (Vog) - Fabaceae, Enterolobium contortisiliquum (Vell.) Morong. - Mimosaceae; Anadenanthera peregrina (L.) - Mimosaceae; Aspidosperma multiflorum (A.DC.) - Apocynaceae; Schinopsis brasiliensis (Engl.) - Anacardiaceae; Terminalia argentea (Mart) - Combretaceae ([6]). Plathymenia reticulata was chosen based on its dominance in the region and because it is strongly associated with the semi-arid climate (BSh) and is present in fragments of dry forest found in the eastern part of the São Francisco river basin ([3]). Eucalyptus camaldulensis was selected to be intercropped due to its drought tolerance and adaptability to semiarid conditions.
## Field experimental design
The experimental site was established (0.8 ha site-1) and was cleared of the “Carrasco” plants. Seedlings of selected plants were transplanted using a randomized block design (Fig. 1) with six treatments of 42 or 48 plants per plot, depending on whether the plot was located in a single or intercropped plantation. These treatments were randomly distributed in each of the three blocks at each site as shown in Fig. 1. The six treatments were allocated as follows: (i) four plots of 378 m2 (21 × 18 m) cultivated with 42 plants/treatment/block of a single species; and (ii) two plots of 432 m2 (24 × 18 m) with 48 plants in lines cultivated with one of the three selected species with a spacing of 3 × 3 m. These 6 treatments (one per plot) were irrigated for approximately 10 months. The experimental site included the following treatments/block:
Fig. 1 - Experimental design at the study site with 6 treatments randomly distributed in each of the three blocks as follows: (1): monoculture of Plathymenia reticulata; (2): monoculture of P. reticulata inoculated with Rhizobium (R) and arbuscular mycorrhizal fungi (AMF)**; (3): monoculture of Eucalyptus camaldulensis; (4): monoculture of inoculated E. camaldulensis (AMF)* ; (5): mixed plantation of P. reticulata and E. camaldulensis; and (6): mixed plantation of P. reticulata (**) and E. camaldulensis (*).
• T1 - monoculture of Plathymenia reticulata plus complete fertilization (CF);
• T2 - monoculture of P. reticulata inoculated with Rhizobium (R) and arbuscular mycorrhizal fungi (AMF) species plus 80% fertilization (80 F);
• T3 - monoculture of Eucalyptus camaldulensis plus CF;
• T4 - monoculture of inoculated E. camaldulensis (AMF) plus 80 F;
• T5 - mixed plantation of P. reticulata and E. camaldulensis plus CF; and
• T6 - mixed plantation of P. reticulata and E. camaldulensis and both species inoculated with R and/or AMF plus 80 F.
Complete fertilization was performed according to Somasegaran & Hoben ([38]) and consisted of triple superphosphate (500 kg ha-1), KCl (382 Kg ha-1), MgSO4.7H2O (50 kg ha-1), ZnSO4.7H2O (46.8 kg ha-1), Mo7O2.4H2O (1.76 kg ha-1), and urea (222 kg ha-1) at the beginning of plantation. Far from the experimental area, an undisturbed, 1000 m2 fragment in the Preserved site (7500 ha) was used as a control for the preserved site, and an area invaded by Carrasco vegetation (1000 m2) was used as a control for the disturbed site. The same experimental design (3 blocks and 3 plots/block) was established in each control area.
## Inoculants
The rhizobia strain BHICB-Pl 02 was selected based on a previous screen for its effectiveness at nitrogen fixation in relation to P. reticulata under greenhouse, nursery and field conditions (data not shown). The plants were cultivated in plastic soil pots (700 cm3) and inoculated with 1 ml of rhizobia plant-1 (107 cfu ml-1 - [38]) under nursery conditions. The field transplantation occurred after 4 months. Arbuscular mycorrhizal fungi (AMF) inocula consisted of 1 ml pot-1 of a suspension composed of 150 spores ml-1 of an equal mixture of 3 species: Gigaspora margarita, Scutellospora heterogama and Glomus sp. from the ICB-UFMG collection. While Eucalyptus camaldulensis plants received only the AMF inocula, P. reticulata plants received a double inoculation of rhizobia and AMF.
## Soil samples and analysis
Soil samples were collected at 12 and 24 month intervals (soil sampling auger with 20 cm of deep) for a total of 9 samples/ treatment/block/site (3 mixed samples × 6 treatments × 3 blocks) or 54 samples from each experimental site. A similar transect was made in the “Carrasco” area and in the Preserved site where 27 samples were collected per site (3 mixed samples × 3 plots × 3 blocks). The chemical and physical properties of samples were analyzed according to the Brazilian National standard recommendations ([8]).
## Litter sampling
Recently fallen leaves of adult plants were collected and air dried to a constant weight at room temperature, and then cut and enclosed in 12×13 cm nylon bags (12 g per bag) with a 1 mm mesh. Nine sub-samples of each litter type were retained for initial moisture and chemical analysis. The randomized field experimental design utilized 3 litter types (P. reticulata, E. camaldulensis and mixed) distributed in 6 treatments and 3 blocks with 9 replicates/litter type/site/time deposited in the litter layer. The litter bags were collected after 4 (dry season) and 8 (rainy season) months of incubation, and the leaf litters were cleaned and weighed (fresh weight) before and after drying at 60 °C and 80 °C, respectively, for 48 h. Corrections for inorganic contaminants were made after determining ash content (4 h at 500 °C).
## Decomposition rates and chemical composition
Weight loss data were estimated as follows ([46] - eqn. 1):
$$\%RM = \frac{W_{0} - W_{t}}{W_0} \cdot 100$$
where W0 is the initial litter dry weight, and Wt is the dry weight of the remaining mass (RM) of litter in the bag at the time of collection. The decay rate (k year-1), which estimates the litter disappearance on a yearly basis and is proportional to actual decomposition rates, was calculated using the following negative exponential decay function ([27]- eqn. 2):
$$W_{t} = {W_0}^{(e -kt)}$$
Chemical analysis of the residue, including soluble components, hemicellulose, cellulose and lignin, was performed using the acid detergent fibre method ([11]). The analysis of nitrogen was performed using Nessler’s reagent method ([26]), and the analysis of phosphorus was performed using the Vanado-molybdate method described by Sarruge & Haag ([36]).
## Mycorrhizal root colonization in litter bags
Fine roots were collected from inside the litter bags after 8 months of incubation, fixed in a FAA solution cleared and stained with Trypan Blue and then evaluated for AMF colonization according to McGonigle et al. ([22]). The results were expressed as the percentage of colonized segments. Eucalyptus root samples were checked for natural ectomycorrhizal colonization and quantified by the line intercept method of McGonigle et al. ([22]). The presence of a fungal mantle and the Hartig net were considered as evidence of ectomycorrhizal colonization. Percent colonization data were transformed as arcsine (x/100)/2.
## Microbial biomass and carbon determination
Soil samples from and around the mixed litter bags buried at each site were sieved through a 2 mm mesh screen and dried overnight at 105 °C to determine the moisture content. They were then used to evaluate the biomass (Cmic) according to the fumigation method of Vance et al. ([44]).
## Phosphate-solubilizing microorganisms (PSMs) and acid hosphatase ctivity
Soil samples from nine mixed litter bags per site (1 g of soil) were collected close to the litter bags in the rhizospheric zone and the number of PSMs were analyzed using the Pikovskaya’s medium (5.0 g Ca3PO4, 10.0 g Glicose, 0.5 g (NH4)2SO4, 0.2 g NaCl, 0.1 g MgSO4.7H2O, 0.001 g MnSO4, 0.001 g FeSO4 - 15 g per L, pH: 6.0 - [31]). PSM numbers were determined from the colony-forming units (CFU), and morphologically distinct colonies, both with and without halos, were purified by repeated sub-culturing, maintained on potato dextrose agar and incubated at room temperature. The data from the 9 samples collected near the mixed litter bags from each site were compared by Tukey’s multiple-range test.
The dominant fungi morphotypes found near each litter bag were isolated and grown in modified Pikovskaya’s medium ([17]), and phosphate solubilization activity was estimated according to Nguyen et al. ([25]), considering the diameter of halo and colony growth (eqn. 3):
$$E = (D_{h} \cdot 100) \cdot D_{cg}$$
where E is the solubilizing efficiency, Dh is the diameter of the halo, and Dcg is the diameter of colony growth. The data were grouped into low, intermediate, high and very high P solubilization efficiency classes based on this E index ([35]).
The DNA of the fungal isolates was obtained according to Lee & Taylor ([19]). The primers ITS (internal transcribed spacer) 1 (5’-TCC GTA GGT GAA CCT GCG G-3’) and ITS 4 (5’-TCC TCC GCT TAT TGA TAT GC-3’) were used to amplify an rDNA ITS region ([47]). The sequences obtained were then analyzed and compared to those deposited in the Gen Bank Nucleotide Database using the BLAST program.
Soil samples from 9 mixed litter bags from each site (1 g of soil) were put in a modified universal buffer (MUB) with pH 6.5, and the acid phosphatase activities in the soil samples were determined by the method described in Tabatabai ([40]).
## Statistical analysis
The data were subjected to one-way ANOVA using the statistical software package MINITAB® version 13.2 (Minitab Inc., State College, PA, USA), and the means compared using the Tukey’s test (P≤0.05).
AMF and EMF root colonization (%) data were transformed as arcsine (x/100)/2. The data were subjected to one-way ANOVA.
In relation to phosphate solubilizing fungi (PSF) sequences, a cluster analysis was performed by applying an unweighted pair group method with arithmetic averaging (UPGMA). All phylogenetic analyses were performed using the software package MEGA4® (Biodesign Inst., Tempe, AZ, USA). PSF species identified via BLAST were clustered based on the Sørensen’s index using the UPGMA algorithm in the MINITAB 13.2 to visualize the PSM community composition patterns from each site.
Principal component analysis (PCA) based on the covariance matrix was applied to analyze the relationships among the parameters related to phosphorus metabolism, such as soil organic matter (SOM), phosphorus concentration, acid phosphatase activity, PSM number, AMF root colonization index and AMF spore number ([29]). The results were plotted by dispersion and loading plots using the software package MINITAB® version 16.
# Results
## Soil analysis
The sandy soil samples were poor in all nutrients (Tab. 1). Twenty-four months after transplantation, improvements in nutrient and soil organic matter (SOM) content, CEC and soil porosity were observed in all of the treatments at the experimental site. Phosphorus was especially limiting in the experimental area before transplantation, but after 24 months there was an increase in the P levels of the soils from plots cultivated with eucalyptus (treatments T3, T4, T5 and T6) and inoculated with AMF. Similarly, P was exceptionally high in the Carrasco soil.
Tab. 1 - Soil characteristics of the treatments after 12 and/or 24 months. (T1): Plathymenia reticulata; (T2): P. reticulata plus rizobia and AMF; (3): E. camaldulensis; (4): E. camaldulensis plus AMF; (5): P. reticulata intercropped with E. camaldulensis; (T6): P. reticulata plus rizobia and AMF intercropped with E. camaldulensis plus AMF; (Exp): Experimental site before transplantation; (P): Preserved site; (C): Carrasco vegetation; (BS): Base saturation; (CEC): Cation exchange capacity. Different letters in the same row indicate significant differences among the means after the Tukey’s test (P ≤0.05).
Treatments T1 T2 T3 T4 T5 T6 Exp P C
Time (months) 12 24 12 24 12 24 12 24 12 24 12 24 0 12 12
pH 5.3NS 6.4 5.3 6.13 5.1 6 5.4 5.9 5.3 6.1 5.3 6.4 5.4 5.8 5.9
Ca (cmol dm-3) 1.38f 2.22bc 1.05gh 1.95d 0.91h 2.05cd 1.42ef 2.01cd 1.15g 2.38ab 1.02gh 2.45a 0.54i 2.03cd 1.62e
Mg (cmol dm-3) 0.23e 0.5ab 0.20e 0.47b 0.2e 0.45b 0.32cd 0.49ab 0.27de 0.49ab 0.22e 0.58a 0.12f 0.35c 0.51ab
K (mg dm-3) 50g 107d 49g 123bc 48 g 120c 64f 120c 49g 129b 46g 148a 47.0g 87e 86.3e
P (mg dm-3) 2.5efg 10.3c 4.3de 11.0c 2fg 11.0c 2.0fg 21a 3.0efg 14.0b 3.6ef 15b 1.0g 3.66ef 6.0d
OM (dag Kg-3) 0.7def 1.5b 0.7def 0.88cdef 0.66f 1.16c 0.8cdef 0.99cd 0.79cdef 0.88cde 0.68ef 1.0cd 0.64f 2.28a 0.90cdef
CEC (cmolc dm-3) 3.5e 4.4bc 3.4 e 4.3bc 3.4 e 4.5ab 3.6de 4.5ab 3.4e 4.9a 3.3e 4.9a 3.4e 3.76cd 4.19
BS (%) 55e 67b 41h 63cd 34i 61d 51f 62d 45g 66bc 41h 69b 41h 78a 56.3e
Clay (%) - - - - - - - - - - - - 17 14 16
Sand (%) - - - - - - - - - - - - 82 84 83
Silt (%) - - - - - - - - - - - - 1 2 1
## Initial litter nutrient content
The results reported in Tab. 2 show that nitrogen and phosphorus were higher in the leguminous leaves than in the eucalyptus leaves, and as a result, the nutrient content was higher in the mixed than in the eucalyptus bags. Eucalyptus leaves showed high hemicellulose and cellulose contents, but the lignin contents and lignin:N ratios were lower than those of the leguminous leaves.
Tab. 2 - Initial nutrients content (%) of leaves in litter bags. Different letters within columns indicate significant differences among means after Tukey’s test (P<0.05).
Litter Type Nitrogen Phosphorus Cellulose Lignin Lignin:N
Platymenia reticulata 1.322
(0.047)a
0.040
(0.001)a
18.88
(1.11)a
26.44
(0.99)a
22.43
(0.17)a
Eucalyptus camaldulensis 0.705
(0.032)c
0.031
(0.006)b
15.16
(1.84)b
16.35
(0.46)b
22.43
(0.21)a
P. reticulata + E. camaldulensis 1.046
(0.057)b
0.039
(0.003)a
16.36
(0.66)ab
19.56
(0.39)ab
20.38
(0.87)b
## Leaf litter decomposition
The results reported in Fig. 2 show that, regardless of the incubation site, the leaves of the eucalyptus plants decomposed quickly and exhibited a high decay rate (K - Tab. 3). In general, the litter decay rate increased in the order of E. camaldulensis > mixed > Plathymenia reticulata, with the exception of the mixed litter incubated at the mixed site, which showed a higher K rate than the E. camaldulensis leaves under the same conditions (Tab. 3). The ANOVA analysis confirmed that litter type (P. reticulata, E. camaldulensis and mixed litters), site and decomposition time modified the rate of mass loss. However, an interaction was observed between the incubation site (vegetation type) and decomposition time, which significantly affected the loss of mass and nutrients (Tab. 4). For all litter types (P. reticulata, E. camaldulensis and mixed litters), mass loss and phosphorus release were improved when incubated in the Eucalyptus plots (Fig. 2). The decomposition rates at the Plathymenia reticulata and mixed plantation sites were similar. In contrast, all of the litter types showed a slow decomposition rate (K) when incubated at Carrasco and especially at the preserved sites (Fig. 2, Tab. 3), confirming the influence of the site on mass and phosphorus losses (Tab. 4). As for phosphorus release, the preserved site showed a fast decay in the first 4 months, followed by a strong immobilization (Fig. 2). The results presented in Fig. 3 revealed that the preserved site had a significantly larger microbial population (Cmic) than all of the other sites.
Fig. 2 - Mass loss (left panels) and phosphorus release (right panels) of Plathymenia reticulata (A) , Eucalyptus camaldulensis (B) and mixed (C) leaves in litterbags, 4 and 8 months after incubation at different sites. (*): inoculated with rhizobia and or arbuscular mycorrhiza fungi (AMF).
Tab. 3 - Decay rate per year (K) of each litter type after 8 months of incubation at the different sites or treatments. Different capital letters within rows and different lower-case letters within columns indicate significant differences among means after Tukey’s test (P<0.05). (*): Inoculated with rhizobia and/or AMF.
Treatment /sites K (year-1)
P. reticulata
bags
K (year-1)
E. camaldulensis
bags
K (year-1)
Mixed litter
bags
P reticulata 1.499cB 2.204cA 1.685dB
P reticulata* 1.407cC 2.096dA 1.808dB
E. camaldulensis 2.97aB 2.935bA 3.857aA
E. camaldulensis * 2.4bB 3.45aA 3.79aA
Mixed 1.5cC 2.09dB 2.944bA
Mixed* 1.6cC 2.03dB 2.435cA
Carrasco 0.88dC 2.22cA 1.68dB
Preserved site 0.82dB 0.876eAB 1.178eA
Tab. 4 - ANOVA for mass, nitrogen and phosphorus losses from litter bags, as affected by site (incubation site), species (type of litter) and time of field incubation (4 and 8 months). (***): P<0.001; (ns): not significant.
Sources of variation Mass remaining N remaining P remaining
df MS F df MS F df MS F
Litter type 2 0.610 24.358*** 2 0.019 10.264*** 2 0.037 16.389***
Site 7 0.548 21.890*** 7 0.024 12.955*** 7 0.013 5.956***
Time 1 14.83 592.324*** 1 0.777 422.205*** 1 0.610 269.167***
Litter type × Site 14 0.022 0.882 ns 14 0.002 1.154 ns 14 0.003 1.246 ns
Litter type × Time 2 0.045 1.778 ns 2 0.017 9.247*** 2 0.013 5.678***
Site × time 7 0.113 4.496*** 7 0.017 9.062*** 7 0.019 8.487***
Litter type × Site × Time 14 0.026 1.021 ns 14 0.004 2.305*** 14 0.003 1.138 ns
Error 341 0.025 - 341 0.002 - 341 0.002 -
Fig. 3 - Microbial biomass C of soil in litter bags of soil at different sites (treatments) near the litter bags, 8 months after incubation. (*): inoculated with rhizobia and or arbuscular mycorrhiza fungi (AMF).
## Phosphate-solubilizing microorganisms (PSM) and acid phosphatase activity
The PSM population significantly improved in the litter-bag soils from the mixed plantation and preserved sites compared with that at the Carrasco site (Fig. 4A). In addition, acid phosphatase activity was high not only in the litter-bag soils from the preserved site and mixed plots but also in the Eucalyptus plots where the plants were not inoculated with AMF (Fig. 4B).
Fig. 4 - Phosphate-solubilizing microorganism (A) and soil phosphatase activity assayed by the p-nitrophenyl phosphate method (µg p-nitrophenol/g/h) (B) in the soil near the mixed litter bags at the different sites after 8 months of incubation.
The dominant phosphate-solubilizing fungi (PSF) isolates from each site were identified based on the DNA sequence at the ITS regions (Fig. 5A). In contrast to the high dominance of Aspergillus sp. found in the preserved, mixed and E. camaldulensis plots, smaller populations and lower diversity of these species were recorded at the Carrasco site (Fig. 5B). However, Penicillium was the dominant genus in the P. reticulata plot (Fig. 5A). Such differences in PSF distribution resulted from similarities between the preserved site and most of the experimental sites based on the Sorensen’s similarity index (Fig. 5B). However, the Carrasco PSF species, particularly those found in the P. reticulata sites, were clearly separate, as shown in Fig. 5B.
Fig. 5 - (A) Distribution of phosphate-solubilizing fungi (PSF) species identified based on rDNA sequences via Gen Bank Nucleotide Database using the BLAST program. (B) UPGMA dendrogram of phosphate-solubilizing fungi (PSF) species present at each studied site based on the Sorensen’s similarity Index. (1): P. reticulata; (2): P. reticulata*; (3): E. camaldulensis; (4): E. camaldulensis*; (5): Mixed; (6): Mixed*; (7): Carrasco; (8): preserved site; (*): inoculated with rhizobia and or arbuscular mycorrhiza fungi (AMF).
## Mycorrhizal root colonization in the litter bags
AMF was found to consistently colonize the roots that penetrated the litter bags, especially those under the Carrasco vegetation, E. camaldulensis and mixed sites inoculated with AMF spores (Fig. 6A). Natural ectomycorrhizal root colonization was also noted at sites where eucalyptus plants were cultivated (Fig. 6B), but no significant differences were recorded among different sites.
Fig. 6 - Arbuscular mycorrhizal fungi (AMF) root colonization (A) and Ectomycorrhizal root frequence (B) found into the litterbags buried in the rhizosfere of plants at the different studied sites after 8 months of incubation.
## Principal component analysis
PCA (Fig. 7A) was carried out using the biological and chemical characteristics of the soil samples collected from the litter bags, which were related to phosphorus metabolism (SOM, soil phosphorus concentration, phosphatase activity, PSM, AMF population spore number - data previously described by Pagano et al. ([29]) - as well as AMF root colonization). Overall, the first two components accounted for 93.2% of the total variation in the dataset (PC1: 72.6%, PC2: 20.6%), while the other components do not appear to be relevant to the results. The AMF root colonization variable had the largest loading on the first PC axis (Fig. 7A), while the acid phosphatase activity showed the largest loading on the second axis.
Fig. 7 - (A) Results of the principal component analysis (PCA) based on soil properties (SOM, Phosphorus concentration, phosphatase activity, microbial population of PSM, AMF spore number and AMF root colonization level) for samples from the eight studied sites. The first two axes explained 93% of the total variation in the dataset. (B) Scatterplot of the samples (stratified by studied site/treatment) based on their scores on the first two PC axes.
Fig. 7B displays the scatterplot for all the treatments/sites based on their scores on the first two PC axes. It is worth to notice that samples from the Carrasco site are clearly distinguished from samples from the preserved site, while those from experimental sites appeared to be intermediate between them. However, the samples from the eucalyptus and mixed plantation plots appeared to be more similar to those from the preserved site than the Carrasco site.
# Discussion
Phosphorus was a limiting factor to plant growth at the experimental site, but SOM and phosphorus content improved after 24 months from transplantation as compared with those 12 months after transplantation. Therefore, such improvement could not be attributed to the initial fertilization. Litter fall nutrient concentration is related to the intensity of re-translocation processes ([34]). As the lignin:N ratio and the lignin chemical composition regulate the decomposition rate, it might be predicted that the rate of decomposition of the leguminous leaves would be slower than the eucalyptus leaves, as it was observed.
As nitrogen and phosphorus were higher in the leguminous leaves than the Eucalyptus leaves, higher phosphorus content was expected in these soils. However, phosphorus was higher in soils samples from the Eucalyptus and mixed plots, particularly in the inoculated plot (treatment T6), compared with the P. reticulata plots. Phosphorus at the Carrasco plot was also higher than that at the preserved site. Plant species alter soil nutrient input through litter quality and the decomposition rate of the microbial community ([23]). Thus, litter quality and the rhizosphere control decomposition processes by selecting the decomposing microorganisms ([14], [42]).
It is known that E. camaldulensis litter has a faster decomposition rate ([20]). In this study, the Eucalyptus litter at the experimental site decomposed faster than the mixture, whereas the P. reticulata litter alone showed the slowest decay, which is likely due to the low lignin content of the former and the high lignin content of the latter. Therefore, nutrients were released in the following order: Eucalyptus litter > mixture > P. reticulata. However, all litter types showed a slow decay rate when incubated at the Carrasco and preserved sites, suggesting a significant effect of vegetation type and density over mass and P losses. These results are an effect of soil nutrient availability under the vegetation at the Carrasco and preserved sites.
Phosphorus concentrations in plant litter tend to be low compared with decomposer requirements when the microbial population is too large, as it is the case at the preserved site, thus decomposers will initially immobilize the nutrients ([4]). In fact, the soil microbial population was significantly high only at the preserved site, as shown by the Cmic assay, which could be an effect mediated by such dense and diverse plant community. Consequently, phosphorus was not released at the preserved site during the 8 months of incubation. In contrast, nutrient demand was high in the Carrasco soils, regardless of the high number of pioneer species, but the decomposition rate of the lignified litter was low. This result may be explained by the poor microbial community, though the latter decomposed the non-lignified litter (Eucalyptus). The above results suggest the existence of differences in the microbial populations between the preserved and Carrasco sites, but do not explain the high phosphorus content in the Carrasco soil.
Nutrients were released and were available to plants that showed a high nutrient demand, such as at the Carrasco and the experimental sites composed of pioneer species and young plants, respectively. Therefore, nutrient immobilization did not occur at these sites.
Phosphorus uptake by plants is assumed to be a function of the phosphorus concentration in the soil solution, which is derived from litter decomposition in the natural environment. Coexisting plants can acquire organic phosphorus through a variety of mechanisms ([42]). Inorganic and organic phosphorus become available by either plant or microorganism activities, particularly those linked to the phosphorus-cycle, such as AMF and PSM ([16]). While the inorganic phosphorus is released through the production of organic anions, the organic phosphorus is hydrolyzed by phosphatase enzymes ([32]).
Since the roots reaching the litter bags in the Carrasco vegetation were strongly colonized by AMF (whose spores increase significantly at this site - [29]), and the elevated soil phosphorus content was not correlated with the microbial biomass, such an increase in soil phosphorus was likely due to AMF activity. Indeed, the AMF root colonization at the experimental site was particularly high in the inoculated plots. These results suggest that AMF benefited by the decomposed products ([41]). However, AMF are known to excrete alkaline phosphatases ([10]), which could partially explain the low acid phosphatase activity found under the Carrasco vegetation.
AMF root colonization in the litter bags was also high in the E. camaldulensis plots but low in those at the preserved site, which confirmed the role of microorganisms in the release of organic phosphorus. Nutrient uptake via symbiotic associations and soil microbial activities is a likely strategy of pioneer plants ([45]). Such microbial associations are particularly relevant to Eucalyptus plants that used simultaneously different sources of organic and inorganic phosphorus, as suggested by the fact that the roots in the litter bags were colonized by both endo- (AMF) and ectomycorrhizal fungi.
In addition, a high number of PSM and a significant acid phosphatase activity were observed in the Eucalyptus plots. When Eucalyptus was mixed with native species, both AMF and PSM were moderately stimulated. However, not only were the PSM dominant in the rhizosphere of the non-inoculated AMF plants, but the PSM population was also inhibited in the Carrasco soil where AMF was dominant. These results suggest that AMF and PSM show the same functional behavior but worked independently from each other, although synergic actions have been reported to occur in others case ([39]). Furthermore, both ectomycorrhizal fungi and PSM produce acid phosphatases ([5]), which explains the increase of these enzymes in single and mixed Eucalyptus plantations, as well as the accelerated mass and nutrient losses in the litter bags.
A high number of PSF and significant acid phosphatase activity were found in the soil closest to the litter bags at the preserved site. It may be hypothesized that both Eucalyptus and other species from the preserved site shared the same strategy for phosphorus acquisition, by favoring the PSF diversity and the enzyme activity in the litter bags or through the excretion of phosphatase by the plants themselves ([7]).
The great diversity of Aspergillus sp. at the preserved site and at most intercropped experimental plots was sufficient to clearly separate this group from the disturbed Carrasco site and the P. reticulata monoculture site. It is likely that such distribution reflected the functional diversity expressed by the phosphatase activity, which was much lower at the Carrasco site than at the preserved site.
Phosphatase enzymes are soil-quality indicators because they respond to site-management and land use changes ([13]). In fact, specific soil microbial populations involved in the decomposition of organic matter are associated with the vegetation type. In this study, AMF were favored at the Carrasco vegetation sites, the Eucalyptus sites were characterized by AMF, ectomycorrhizal fungi and PSM, while the AMF and PSM groups were more abundant at the intercropped plantation sites. In contrast, the PSM activity was favored at the preserved site. According to previous reports ([20]), E. camaldulensis ensured the supply of phosphorous to the agroforestry system by favoring AMF and PSF activities, as determined by the AMF root colonization and the phosphatase activity, respectively. Finally, these microbial populations may be indicators of soil quality useful in assessing the effectiveness of land restoration in semiarid environments.
# Conclusions
The main results of this study may be summarized as follows:
1. the vegetation type can affect the litter decomposition rate through the associated microbial communities;
2. the Carrasco vegetation (disturbed site) is mainly characterized by arbuscular mycorrhizal fungi (AMF), the preserved site by the PSM population, the Eucalyptus stands by AMF, ectomycorrhizal fungi (EM) and phosphate-solubilizing microorganisms (PSM), while the intercropping system by the AMF and PSM groups.
3. the P supply to the agroforestry system may be improved by Eucalyptus camaldulensis in single or intercropped plantations, through the litter decomposition operated by P-cycle related microorganisms.
4. AMF root colonization and phosphatase activity might be considered useful indicators of land restoration in agroforestry systems. Based on them, the experimental site appears to be closer to the preserved site.
# Acknowledgements
This research was supported by the Brazilian Ministry of Environment through the National Found of Environment (FNMMA). The authors are grateful to CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), CNPq (Conselho Nacional de Pesquisa) and FAPEMIG (Fundação de Amparo a Pesquisa de Minas Gerais) for scholarships.
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Partey ST, Quashie-Sam SJ, Thevathasan NV, Gordon AM (2011). Decomposition and nutrient release patterns of the leaf biomass of the wild sunflower (Tithonia diversifolia): a comparative study with four leguminous agroforestry species. Agroforestry Systems 81: 123-134.
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#### Authors’ Affiliation
(1)
Eduardo Correa
Lilia Carvalhais
Maria Rita Scotti
Department of Botany, Institute of Biological Science / Federal University of Minas Gerais . Avenida Antonio Carlos, 6627, Pampulha, Cep: 31.270-901 Belo Horizonte Minas Gerais (Brazil)
(2)
Mirian Utida
Christiane Abreu Oliveira
EMBRAPA- National Center of Maize and Sorgum Rod. MG. 424 Km 45 CEP: 35701970 Sete Lagoas- Minas Gerais (Brazil)
#### Corresponding author
Maria Rita Scotti
mr.smuzzi@yahoo.com.br
#### Citation
Correa E, Carvalhais L, Utida M, Oliveira CA, Scotti MR (2015). Effect of plant species on P cycle-related microorganisms associated with litter decomposition and P soil availability: implications for agroforestry management. iForest 9: 294-302. - doi: 10.3832/ifor1459-008
Gianfranco Minotta
#### Paper history
Accepted: Jun 22, 2015
First online: Oct 05, 2015
Publication Date: Apr 26, 2016
Publication Time: 3.50 months
© SISEF - The Italian Society of Silviculture and Forest Ecology 2015
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Total number of cites (since 2016): 1
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https://moodle.org/plugins/view.php?plugin=format_calendar&moodle_version=11 | ## Course formats: oohoo - Calendar format
format_calendar
Maintained by Nicolas Bretin, Patrick Thibaudeau
This course format displays a course in a calendar format.
113 sites
10 fans
This course format displays a course in a calendar format.
The functionalities are:
• Hide/Display weekends,
• Display by week (With previous/next week buttons),
• Display by month (With previous/next month buttons)
• The current day is highlighted
### Contributors
Patrick Thibaudeau: Maintainer
• Fri, 17 May 2013, 12:56 AM
Hi Patrick and Nicolas,
Will there be a Moodle 2.5 version soon?
I'm currently getting the warning:
line 3160 of \lib\deprecatedlib.php: call to debugging()
line 118 of \course\format\calendar\format.php: call to print_section()
line 276 of \course\view.php: call to require()
I did fork the format but could not replace print_section with:
echo $this->courserenderer->course_section_cm_list($course, $thissection, 0); echo$this->courserenderer->course_section_add_cm_control($course,$thissection->section, 0);
Because you are not yet using the class in 'renderer.php' fully.
Cheers,
Gareth
• Fri, 17 May 2013, 3:26 AM
Hi Gareth,
We plan on updating all of our plugins for 2.5 over the summer. I don't have an actual time frame for the calendar course format right now.
• Sat, 7 Sep 2013, 12:21 PM
Hi guys,
I am using Moodle 2.5 and am getting a bug with calendar 1.0.4. All activities added are duplicating themselves. This is in edit mode only, but it is getting confusing. And if I delete one, they will both delete. I am also not getting an add activity or resource in each day as shown in your screen shots. Thanks for your help, love the plugin otherwise.
--Janice
• Fri, 2 May 2014, 4:02 AM
Is there a way I can query all the classes offered on a particular day or what all days is a course offered ?
The reason is we need to write reports which will join further tables with this information.
• Wed, 21 May 2014, 5:40 AM
Hi,
I am using moodle 2.6 and I have a problem with the highlighted Current day on the course main page.
Instead of the following (li for non-current day): < li id="section-1" class="section main yui3-dd-drop" role="region" aria-label=" 19 May" >
the current day shows the following li: < li id="section-2" class="section main yui3-dd-drop" "="" accesshide="" aria-label=" < span class=" > Current day 20 May" role="region" >
This causes the date to show:
Current day 20 May" role="region" > Current day 20 May
instead of just Current day 20 May.
If you can just point me in the direction of which file I can find the broken code in, that would be great. I am not sure where to look. Thanks for your help!
• Thu, 26 Jun 2014, 11:36 PM
Peace:
How to increase more than 52 sections in the course calendar format? For example 140.
Greetings.
José de Jesús.
• Thu, 18 Sep 2014, 2:40 AM
Has anyone found the fix for the display issue Madeleine mentioned above? I think I've tracked it down to the format.php file, and can see where the error comes in with the html code, but I can't find the php that generates the error.
You can set the max number of sections in the site admin course default settings - max number of sections.
• Thu, 18 Sep 2014, 3:13 AM
Hi Curtis,
I managed to fix the code for my purposes, however I have no idea if this is a permanent/stable way to fix it. In the file format.php line 241 originally read: $currenttext = get_accesshide(get_string('currentday', 'format_calendar')); I changed it to read:$currenttext = get_string('currentday', 'format_calendar');
This fixed the current day text for me.
• Fri, 19 Sep 2014, 3:01 AM
Looks like that did it. Thanks loads.
• Fri, 12 Dec 2014, 3:05 AM | {"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": 0.3383854925632477, "perplexity": 6867.610486509399}, "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-2016-50/segments/1480698544679.86/warc/CC-MAIN-20161202170904-00018-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.gamedev.net/forums/topic/230374-help-with-angles/ | #### Archived
This topic is now archived and is closed to further replies.
# Help with angles..
This topic is 5244 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Hi guys.. I will post this here, coz the owner forum is to slowly in conversation... I am using NeoEngine but, i guess *everybody* will understand.
EulerAngles angleBuffer = EulerAngles(Vector3d 0.0f,45.0f,0.0f),EulerAngles::XYZr);
pkBox->SetRotation(Quaternion(angleBuffer), true);
.
.
.
Quaternion rotationBuffer = pkBox->GetRotation();
EulerAngles rotation = rotationBuffer.ToEulerAngles();
ok... assuming I have setup a 45 degrees( ??? ) in the beggining on the Y axis, why, after making NO rotations, the rotation.m_kAngle.y is TOTALLY different of 45 degrees ( ??? ). Here''s what I have inside my head. I have a cube dropped in the floor. Free fall. And this cube has a middle-high restitution value. So, he will bounce gracessely. But as he is bouncing around, he spins in your 3 local axis, right!? OK. Think only in the X axis. While he is spinning, the rotation value (in the degrees) on the X must blow like 360 per completely turn the box make. Right? So why, why why, why why I am far away of thoses values?! Can somebody please help?
##### Share on other sites
Just wondering... are you sure these angle values are in degrees? Have you tried using radians?
Another point... this function Euler angles, what are the parameters? (angle with x, angle with y, angle with z) or do we just have to set a vector coordenates?
##### Share on other sites
EulerAngles is not a function.. well it is.. but is a constructor...
and i am assuming that euler angles are in degrees, isn't?!
I am not an american. And I am self taught. But its very very hard to understand all those technicals explanations about Euler and Quaternions...
I just want to use something real easy.
"OK cube, now i want that you will rotate 45 degree on Y axis!"
Just this.. All these transformation are almost driving me mad.
[edited by - gelerorox on June 9, 2004 10:08:27 PM]
##### Share on other sites
Ok. use Radians! Damn! I just forgot everything about trigonometry classes years ago!
##### Share on other sites
Chill out, man.
What we have defined in trigonometry is that pi radians = 180 degrees. So, if you want to convert from degrees to radians, you have to multiply your angle in degrees by pi and divide it by 180. Let''s see how it works:
(Angle radians) = (Angle degrees) * pi /180
just to remember, pi = 3.14159265359, so pi/180 = 0.01745 nearly..
so, if you want to convert you 45 degrees to radians, you''ll have to:
(Angle radians) = (45 ) * pi/180 = (45) * 0.01745 = 0.78539
Hope that helps!
try typing 0.78539 instead of 45.
##### Share on other sites
OK... thanks man... help a lot.
Anyway, still there''s a some problems on NeoEngine stuff.
Just clearing the things:
The engine treat the angles like this:
0.25 * pi = 45 degrees
soom, 1 * pi = 180 degrees.
so, i got some tests with the "0.25f * PI" kind of thing.
and works fine but still the output data has some weird values...
I am still using the cube I used to dropped on the floor. But now he isnt a rigidbody anymore. So he is floating in the air. And I am catching the keyboard input to make some transformations.
Ok. Using the "a" and "s" keys I can increment and decremet the rotation on the Y axis. Well I am incrementing or decrementing the value that I used to multiply by PI.
Now the odd behavior. I cant complete a full 360 degree turn. I just cant. And I cant understand why.
I think, if the cube is rotating along the Y axis, can he rotate like forever...
the values can grow like:
0.35f * pi
0.45f * pi
.
.
25f * pi (which is 4500 degree or approx 12 and a half completely turns on the Y axis)
why, I cant get those values?
I mean, after 180 degrees the cube looks to flip vertically screwing my (i)logical thinking.
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https://waseda.pure.elsevier.com/en/publications/geometric-formulation-and-multi-dark-soliton-solution-to-the-defo | # Geometric Formulation and Multi-dark Soliton Solution to the Defocusing Complex Short Pulse Equation
Bao Feng Feng*, Ken Ichi Maruno, Yasuhiro Ohta
*Corresponding author for this work
Research output: Contribution to journalArticlepeer-review
12 Citations (Scopus)
## Abstract
In the present paper, we study the defocusing complex short pulse (CSP) equations both geometrically and algebraically. From the geometric point of view, we establish a link of the complex coupled dispersionless (CCD) system with the motion of space curves in Minkowski space R2,1, then with the defocusing CSP equation via a hodograph (reciprocal) transformation, the Lax pair is constructed naturally for the defocusing CSP equation. We also show that the CCD system of both the focusing and defocusing types can be derived from the fundamental forms of surfaces such that their curve flows are formulated. In the second part of the paper, we derive the defocusing CSP equation from the single-component extended Kadomtsev-Petviashvili (KP) hierarchy by the reduction method. As a by-product, the N-dark soliton solution for the defocusing CSP equation in the form of determinants for these equations is provided.
Original language English 343-367 25 Studies in Applied Mathematics 138 3 https://doi.org/10.1111/sapm.12159 Published - 2017 Apr 1
## ASJC Scopus subject areas
• Applied Mathematics
## Fingerprint
Dive into the research topics of 'Geometric Formulation and Multi-dark Soliton Solution to the Defocusing Complex Short Pulse Equation'. Together they form a unique fingerprint. | {"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": 0.8792120218276978, "perplexity": 2378.913014409161}, "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/1659882570879.37/warc/CC-MAIN-20220809003642-20220809033642-00433.warc.gz"} |
https://mathhelpboards.com/threads/evaluating-a-summation.9156/ | # Evaluating a summation
#### Pranav
##### Well-known member
Nov 4, 2013
428
Problem:
Consider a function $f(n)$ defined as:
$$f(n)=\sum_{r=1}^n (-1)^{r+1} \binom{n}{r} \left(\sum_{k=1}^r \frac{1}{k}\right)$$
Find the value of
$$\sum_{i=1}^{\infty} (-1)^{i+1}f(i)$$
Attempt:
I write $\sum_{k=1}^r (1/k)=H_r$.
The sum I have to evaluate is
$$f(1)-f(2)+f(3)-f(4)+\cdots$$
I tried writing down a few terms and tried to see the difference of consecutive terms....
$$f(1)=H_1$$
$$f(2)=2H_1-H_2$$
$$f(3)=3H_1-3H_2+H_3$$
$$f(4)=4H_1-6H_2+4H_3+H_4$$
....but I don't see if this helps.
Although I have posted this in the Pre-Algebra and Algebra forum, please feel free to use any Calculus approaches as I am not sure if the problem involves the use of Calculus.
Any help is appreciated. Thanks!
#### chisigma
##### Well-known member
Feb 13, 2012
1,704
Problem:
Consider a function $f(n)$ defined as:
$$f(n)=\sum_{r=1}^n (-1)^{r+1} \binom{n}{r} \left(\sum_{k=1}^r \frac{1}{k}\right)$$
Find the value of
$$\sum_{i=1}^{\infty} (-1)^{i+1}f(i)$$
Attempt:
I write $\sum_{k=1}^r (1/k)=H_r$.
The sum I have to evaluate is
$$f(1)-f(2)+f(3)-f(4)+\cdots$$
I tried writing down a few terms and tried to see the difference of consecutive terms....
$$f(1)=H_1$$
$$f(2)=2H_1-H_2$$
$$f(3)=3H_1-3H_2+H_3$$
$$f(4)=4H_1-6H_2+4H_3+H_4$$
....but I don't see if this helps.
Although I have posted this in the Pre-Algebra and Algebra forum, please feel free to use any Calculus approaches as I am not sure if the problem involves the use of Calculus.
Any help is appreciated. Thanks!
Your approach is very good because is...
$f(1) = H_{1} = 1$
$f(2) = 2\ H_{1} - H_{2} = 2 - 1 - \frac{1}{2} = \frac{1}{2}$
$f(3) = 3\ H_{1} - 3\ H_{2} + H_{3} = 3 - 3 - \frac{3}{2} + 1 + \frac{1}{2} + \frac{1}{3} = \frac{1}{3}$
... and proceeding in this way You arrive at the very suggestive relation...
$\displaystyle f(n) = \sum_{k=1}^{n} (-1)^{k+1} \binom{n}{k}\ H_{k} = \frac{1}{n}\ (1)$
... so that is...
$\displaystyle \sum_{i=1}^{\infty} (-1)^{i + 1} f(i) = \ln 2\ (2)$
Kind regards
$\chi$ $\sigma$
MHB Math Helper
Jan 31, 2012
253
Last edited:
#### Prometheus
##### Member
Mar 25, 2013
11
Switch the order of the inner sums and use $\displaystyle \sum\limits_{k \le r \le n} (-1)^r\binom{n}{r} = \frac{k (-1)^k}{n} \binom{n}{k}.$
\displaystyle \begin{aligned} \sum_{n \ge 1}\sum_{1 \le r \le n} \sum_{1 \le k \le r} \frac{(-1)^{n+r}}{k} \binom{n}{r} & = \sum_{n \ge 1}\sum_{1 \le k \le n} \sum_{k \le r \le n} \frac{(-1)^{n+r}}{k} \binom{n}{r} \\& = \sum_{n \ge 1}\sum_{1 \le k \le n}\frac{(-1)^{k+n}}{n} \binom{n}{k} \\& = \sum_{n \ge 1} \frac{(-1)^{n+1}}{n} \\& = \log(2).\end{aligned}
#### Random Variable
##### Well-known member
MHB Math Helper
Jan 31, 2012
253
@Prometheus
Could you give a proof of that identity?
#### Pranav
##### Well-known member
Nov 4, 2013
428
Your approach is very good because is...
$f(1) = H_{1} = 1$
$f(2) = 2\ H_{1} - H_{2} = 2 - 1 - \frac{1}{2} = \frac{1}{2}$
$f(3) = 3\ H_{1} - 3\ H_{2} + H_{3} = 3 - 3 - \frac{3}{2} + 1 + \frac{1}{2} + \frac{1}{3} = \frac{1}{3}$
... and proceeding in this way You arrive at the very suggestive relation...
$\displaystyle f(n) = \sum_{k=1}^{n} (-1)^{k+1} \binom{n}{k}\ H_{k} = \frac{1}{n}\ (1)$
... so that is...
$\displaystyle \sum_{i=1}^{\infty} (-1)^{i + 1} f(i) = \ln 2\ (2)$
Kind regards
$\chi$ $\sigma$
Awesome!
I never thought I was so close to the answer, thanks a lot chisigma!
There is a proof that $\displaystyle H_{n} = \sum_{k=1}^{n}(-1)^{k-1} \binom{n}{k} \frac{1}{k}$ on Wikipedia page for the harmonic numbers.
Harmonic number - Wikipedia, the free encyclopedia
Then by applying the inverse binomial transform,
$$\frac{1}{n} = \sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} H_{k}$$
Binomial Transform -- from Wolfram MathWorld
Hi Random Variable, thank you very much for your input.
I have never heard of inverse binomial transform before but it looks like a useful technique. I seem to have trouble figuring out how did you use it here.
We have
$$H_{n} = \sum_{k=1}^{n}(-1)^{k-1} \binom{n}{k} \frac{1}{k}$$
From the Wolfram page you quote, $b_n$ should be of the form $\displaystyle \sum_{k=0}^n (-1)^{n-k} a_k$ but I have $(-1)^{k-1}$ instead of $(-1)^{n-k}$.
Sorry if this is silly but I have never seen that representation of $H_n$ and the binomial transform.
Thanks!
#### Random Variable
##### Well-known member
MHB Math Helper
Jan 31, 2012
253
@ Pranav
First let $a_{0}=b_{0} = 0$.
Then multiply both sides of $\displaystyle H_{n} = \sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} \frac{1}{k}$ by $(-1)^{n+1}$.
Then $\displaystyle (-1)^{n+1} H_{n} = (-1)^{n-1} H_{n} = \sum_{k=1}^{n} (-1)^{n+k} \binom{n}{k} \frac{1}{k} = \sum_{k=1}^{n} (-1)^{n-k} \binom{n}{k} \frac{1}{k}$.
Now take the inverse binomial transform.
Last edited:
#### Pranav
##### Well-known member
Nov 4, 2013
428
@ Pranav
First let $a_{0}=b_{0} = 0$.
Then multiply both sides of $\displaystyle H_{n} = \sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} \frac{1}{k}$ by $(-1)^{n+1}$.
Then $\displaystyle (-1)^{n+1} H_{n} = (-1)^{n-1} H_{n} = \sum_{k=1}^{n} (-1)^{n+k} \binom{n}{k} \frac{1}{k} = \sum_{k=1}^{n} (-1)^{n-k} \binom{n}{k} \frac{1}{k}$.
Now take the inverse binomial transform.
Thanks a lot Random Variable! | {"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": 0.9872587323188782, "perplexity": 1059.5629436958463}, "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-2021-31/segments/1627046153814.37/warc/CC-MAIN-20210729011903-20210729041903-00452.warc.gz"} |
http://www.agneshome.org/boston-college-2019/abstracts | ### Abstracts
Jarod Alper (University of Washington) Evolution of Luna’s etale slice theorem Abstract: Luna’s etale slice theorem is a beautiful result in equivariant geometry with important applications to moduli theory. Luna’s result has inspired recent investigations into the local structure of algebraic stacks. The goal of this talk is to explain various stacky generalizations of Luna’s slice theorem and their applications both to equivariant geometry and moduli theory. This talk is based on joint work with J. Hall, D. Halpern-Leistner, J. Heinloth and D. Rydh. Kenneth Ascher (Princeton University) Wall crossings for K-moduli spaces Abstract: K-stability has become a central tool in the study of compact moduli of Fano varieties. In this talk I will discuss K-stability compactifications of the moduli space of log Fano pairs (P2, aC), where C is a plane curve of degree at least 4 and a is a rational number. We establish a wall-crossing framework to study the behavior of these moduli spaces as the weight a varies. We show that when a is small, the K-moduli compactification is isomorphic to the GIT moduli space, and that the first wall crossing is a weighted blowup of Kirwan type. We describe all wall-crossings for degree 4, 5 and 6 and relate the final K-moduli spaces to Hacking's moduli space and some compact moduli of K3 surfaces. This is joint work with K. DeVleming and Y. Liu. Benjamin Bakker (University of Georgia) o-minimal GAGA and applications to Hodge theory Abstract: For a complex projective variety, Serre's classical GAGA theorem asserts that the analytification functor from algebraic coherent sheaves to analytic coherent sheaves is an equivalence of categories. For non-proper varieties, however, this theorem easily fails. In joint work with Y. Brunebarbe and J. Tsimerman, we show that a GAGA theorem holds even in the non-proper case if one restricts to analytic structures that are "tame" in a sense made precise by the notion of o-minimality. This result has particularly important applications to Hodge theory, and we will explain how it can be used to prove a conjecture of Griffiths on the quasiprojectivity of the images of period maps. We will also discuss some applications to moduli theory. Ivan Cheltsov (University of Edinburgh) K-stability of asymptotically log del Pezzo surfaces Abstract: In 2013, I and Yanir Rubinstein introduced special class of log Fano varieties which are known now as asymptotically log Fano varieties. We classified all of them in dimension 2 (asymptotically log del Pezzo surfaces) and studied the existence of Kahler-Einstein edge metrics on them (K-stability). Recently we almost completed this study together with Kewei Zhang. In my talk, I will explain what we did and what we were not able to do (but Kento Fujita did during this summer). Izzet Coskun (University of Illinois at Chicago) Brill-Noether type theorems for moduli spaces of sheaves on surfaces Abstract: In this talk, I will survey recent work on computing the cohomology of the general sheaf on a moduli space of sheaves on surfaces. I will concentrate on rational and K3 surfaces.In joint work with Jack Huizenga, we solve the problem completely for Hirzebruch surfaces. As consequences, we determine the Chern characters of moduli spaces where the general sheaf is globally generated and obtain a Gaeta-type resolution for the general sheaf. These in turn yield a classification of Chern characters of stable sheaves for any polarization on a Hirzebruch surface. In joint work with Howard Nuer and Kota Yoshioka, we study the problem for K3 surfaces. Our approach crucially uses Bridgeland stability. If time permits, I will discuss applications to construction and classification of Ulrich bundles on surfaces. Sarah Koch (University of Michigan) Irreducibility in complex dynamics Abstract: A major goal in complex dynamics is to understand dynamical moduli spaces; that is, conformal conjugacy classes of holomorphic dynamical systems. One of the great successes in this regard is the study of the moduli space of quadratic polynomials; it is isomorphic to \$\mathbb C\$. This moduli space contains the famous Mandelbrot set, which has been extensively studied over the past 40 years. Understanding other dynamical moduli spaces to the same extent tends to be more challenging as they are often higher-dimensional. In this talk, we will begin with an overview of complex dynamics, focusing on the moduli space of quadratic rational maps, which is isomorphic to \$\mathbb C^2\$. This moduli space contains special algebraic curves, called Milnor curves. In general, it is unknown if Milnor curves are irreducible (over \$\mathbb C\$). We will find an infinite collection of Milnor curves that are irreducible. This is joint work with X. Buff and A. Epstein. Alina Marian (Northeastern University) On the Chow ring of holomorphic symplectic manifolds Abstract: I will propose a series of basic conjectural identities in the Chow rings of holomorphic symplectic manifolds of K3 type, and will discuss evidence for them. The emerging structure naturally generalizes in higher dimensions a set of key properties of cycles on a K3 surface. The talk is based on joint work with Ignacio Barros, Laure Flapan, and Rob Silversmith. Anton Zorich (Institut de Mathématiques de Jussieu) Bridges between flat and hyperbolic enumerative geometry Abstract: I will give a formula for the Masur-Veech volume of the moduli space of quadratic differentials in terms of psi-classes (in the spirit of Mirzakhani's formula for the Weil-Petersson volume of the moduli space of hyperbolic surfaces). I will also show that Mirzakhani's frequencies of simple closed hyperbolic geodesics of different combinatorial types coincide with the frequencies of the corresponding square-tiled surfaces. I will conclude with (mostly conjectural) description of geometry of a "random" square-tiled surface of large genus and of a "random" multicurve on a topological surface of large genus. The talk is based on the joint work with V. Delecroix, E. Goujard and P. Zograf. It is aimed to a broad audience, so I will try to include all necessary background. | {"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": 0.8516194224357605, "perplexity": 1005.2483927958746}, "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/1568514573988.33/warc/CC-MAIN-20190920092800-20190920114800-00402.warc.gz"} |
https://www.fussylogic.co.uk/blog/?p=489 | # Child Gender Ratios
By | 2012-10-01
There’s a country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female? (You may assume the question is asked as an expectation of course, since any particular country can be anything in principle — 100% girls is possible, just not likely)
The non-mathematician can’t tell you the answer, but believes that the above policy results in more boys. They’re wrong (well, ish, for practical purposes it’s good enough). This answer is exactly what (reportedly) Google expect:
Assuming a random arrival pattern of boys and girls: – half of all couples will have a boy as their first child and that is the end of that. – if the other half, who’ve had a girl, try again, half of them will go on to have a boy and half will go on to have another girl.
So out of 100 couples, we end up with: – 50 having one boy = 50 boys – 25 having one girl and one boy = 25 girls and 25 boys – 25 having two girls = 50 girls
… and so on.
The logic is good (and it is the logic I had remembered for a long time). It basically says: “any given birth has a 50% chance of being a girl”, therefore the number of girls in the country will be `number of births * 0.5`; hence the fraction of the population that is female is 50%. Regardless of stopping criteria.
It took repeated reads of Steve Landsburg’s blog to convince me that even this, cleverer, answer is not correct either.
The faulty assumption in the above analysis is shown by this fact:
$\frac{E\left[G\right]}{E\left[G+B\right]}\ne E\left[\frac{G}{G+B}\right]$
(
$E\left[\cdot \right]$
being the expectation operator.) That is to say that the expectation of a ratio is not necessarily equal to the ratio of the expectations.
The error is made by calculating the expectation over all arrangements of individuals instead of over all arrangements of countries.
We’ll first answer a simpler question by considering a country with only one family. We calculate expectation over all possible one family countries. The possible arrangements of children in that one family country are:
``````N children % girls likelihood
-----------------------------------------------------------
0 B 0% 0.5
1 GB 50% 0.5 * 0.5
2 GGB 66% 0.5 * 0.5 * 0.5
3 GGGB 75% 0.5 * 0.5 * 0.5 * 0.5
4 GGGGB 80% 0.5 * 0.5 * 0.5 * 0.5 * 0.5
5 GGGGGB 83% 0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5
... etc ...
n nG+B n/(n+1) 0.5^(n+1)
``````
Remember that expected value is the sum of values multiplied by the probability of that value. So for the specific case of the one family country, we are simply summing up the product of columns three and four in the above table:
${E}_{1}\left[\frac{G}{G+B}\right]=\sum _{n=0}^{\infty }\frac{1}{{2}^{n+1}}\frac{n}{n+1}$
Fortunately this is a convergent series, so has a real answer (which the mathoverflow link tells me is):
$1–\mathrm{ln}\left(2\right)=30.69$
This should already be sufficient to convince you of the difference between the expectation of a ratio, and the ratio of expectations.
We can do the same for countries with two families; although it gets horrible looking pretty quickly:
``````children % girls likelihood
------------------------------------------------------------
B / B 0% (0.5) * (0.5)
B / GB 33% (0.5) * (0.5 * 0.5)
B / GGB 50% (0.5) * (0.5 * 0.5 * 0.5)
B / GGGB 60% (0.5) * (0.5 * 0.5 * 0.5 * 0.5)
B / GGGGB 66% (0.5) * (0.5 * 0.5 * 0.5 * 0.5 * 0.5)
B / GGGGGB 71% (0.5) * (0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5)
... etc ...
GB / B 33% (0.5 * 0.5) * (0.5)
GB / GB 50% (0.5 * 0.5) * (0.5 * 0.5)
GB / GGB 60% (0.5 * 0.5) * (0.5 * 0.5 * 0.5)
GB / GGGB 66% (0.5 * 0.5) * (0.5 * 0.5 * 0.5 * 0.5)
GB / GGGGB 71% (0.5 * 0.5) * (0.5 * 0.5 * 0.5 * 0.5 * 0.5)
GB / GGGGGB 75% (0.5 * 0.5) * (0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5)
... etc ...
``````
Yuck. Regardless of how nasty this is getting, the expectation for two families is:
${E}_{2}\left[\frac{G}{G+B}\right]=\sum _{n=0}^{\infty }\frac{n+1}{{2}^{n+2}}\frac{n}{n+2}$
I’m afraid, that the mathoverflow article loses me then; as I have never heard of the “digamma function” it talks about. I can see what it’s doing though — it’s simply a way of converting the infinite sum expression into a direct equation.
Leaving that nightmare to the real mathematicians, for our purposes all we care about is that
${E}_{k}\left[\frac{G}{G+B}\right]\to \frac{1}{2}–\frac{1}{4k}\to 50%$
as
$k\to \infty$
. Importantly though: it never quite reaches it.
Executive summary: The expected percentage of girls in a country of
$k$
families, operating the “stop on boy” policy, is less than 50%.
There is actually an awful lot more to this problem. A fascinating follow up guest post at Steve Landsburg’s is worth reading. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 8, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5171768665313721, "perplexity": 891.088362393854}, "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/1573496668416.11/warc/CC-MAIN-20191114104329-20191114132329-00012.warc.gz"} |
http://www.r-bloggers.com/tag/accept-reject-algorithm/ | # Posts Tagged ‘ accept-reject algorithm ’
## R midterms
November 9, 2012
By
Here are my R midterm exams, version A and version B in English (as students are sitting next to one another in the computer rooms), on simulation methods for my undergrad exploratory statistics course. Nothing particularly exciting or innovative! Dedicated ‘Og‘s readers may spot a few Le Monde puzzles in the lot… Two rather entertaining
## generalised ratio of uniforms
May 14, 2012
By
$generalised ratio of uniforms$
A recent arXiv posting of the paper “On the Generalized Ratio of Uniforms as a Combination of Transformed Rejection and Extended Inverse of Density Sampling” by Martino, Luengo, and Míguez from Madrid rekindled my interest in this rather peculiar simulation method. The ratio of uniforms samples uniformly on the subgraph to produce simulations from p
## ultimate R recursion
January 31, 2012
By
One of my students wrote the following code for his R exam, trying to do accept-reject simulation (of a Rayleigh distribution) and constant approximation at the same time: which I find remarkable if alas doomed to fail! I wonder if there exists a (real as opposed to fantasy) computer language where you could introduce constants
## Galton & simulation
September 27, 2010
By
Stephen Stigler has written a paper in the Journal of the Royal Statistical Society Series A on Francis Galton’s analysis of (his cousin) Charles Darwin’ Origin of Species, leading to nothing less than Bayesian analysis and accept-reject algorithms! “On September 10th, 1885, Francis Galton ushered in a new era of Statistical Enlightenment with an address | {"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": 1, "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": 0.8132803440093994, "perplexity": 4501.50240153816}, "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-2016-30/segments/1469257826908.63/warc/CC-MAIN-20160723071026-00054-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://experts.arizona.edu/en/publications/search-for-heavy-particles-decaying-into-a-top-quark-pair-in-the- | # Search for heavy particles decaying into a top-quark pair in the fully hadronic final state in pp collisions at s =13 TeV with the ATLAS detector
Atlas Collaboration
Research output: Contribution to journalArticlepeer-review
39 Scopus citations
## Abstract
A search for new particles decaying into a pair of top quarks is performed using proton-proton collision data recorded with the ATLAS detector at the Large Hadron Collider at a center-of-mass energy of s=13 TeV corresponding to an integrated luminosity of 36.1 fb-1. Events consistent with top-quark pair production and the fully hadronic decay mode of the top quarks are selected by requiring multiple high transverse momentum jets including those containing b-hadrons. Two analysis techniques, exploiting dedicated top-quark pair reconstruction in different kinematic regimes, are used to optimize the search sensitivity to new hypothetical particles over a wide mass range. The invariant mass distribution of the two reconstructed top-quark candidates is examined for resonant production of new particles with various spins and decay widths. No significant deviation from the Standard Model prediction is observed and limits are set on the production cross-section times branching fraction for new hypothetical Z′ bosons, dark-matter mediators, Kaluza-Klein gravitons and Kaluza-Klein gluons. By comparing with the predicted production cross sections, the Z′ boson in the topcolor-assisted-technicolor model is excluded for masses up to 3.1-3.6 TeV, the dark-matter mediators in a simplified framework are excluded in the mass ranges from 0.8 to 0.9 TeV and from 2.0 to 2.2 TeV, and the Kaluza-Klein gluon is excluded for masses up to 3.4 TeV, depending on the decay widths of the particles.
Original language English (US) 092004 Physical Review D 99 9 https://doi.org/10.1103/PhysRevD.99.092004 Published - May 14 2019
## ASJC Scopus subject areas
• Nuclear and High Energy Physics
## Fingerprint
Dive into the research topics of 'Search for heavy particles decaying into a top-quark pair in the fully hadronic final state in pp collisions at s =13 TeV with the ATLAS detector'. Together they form a unique fingerprint. | {"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": 0.9861503839492798, "perplexity": 2083.8714559320783}, "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/1659882571692.3/warc/CC-MAIN-20220812105810-20220812135810-00677.warc.gz"} |
https://www.maplesoft.com/support/help/MapleSim/view.aspx?path=componentLibrary%2Felectrical%2FquasiStationary%2Foverview | Quasistationary Library
Description
The Quasistationary library addresses the analysis of electrical circuits with purely sinusoidal voltages and currents. The following are the main characteristics.
• Only pure sinusoidal voltages and currents are taken into account, higher harmonics are not considered.
• Electrical transients are neglected.
• The electrical components are strictly linear.
• The angular frequency $\mathrm{\omega }$ of the voltages and currents of a circuit is determined from a reference angle $\mathrm{\gamma }$ by means of $\mathrm{\omega }=\stackrel{.}{\mathrm{\gamma }}$.
• The reference angle $\mathrm{\gamma }$ is not a global quantity, it propagates through the connector; this allows independent circuits with different frequencies to be simulated in one model.
• The connectors contain the real and the imaginary part of the voltage and the current RMS phasors.
Phasors
• The purely sinusoidal voltage $v={V}_{\mathrm{pk}}\mathrm{cos}\left(\mathrm{\omega }t+\mathrm{\phi }\right)=\sqrt{2}{V}_{\mathrm{rms}}\mathrm{cos}\left(\mathrm{\omega }t+\mathrm{\phi }\right)$ in the time-domain is represented by a complex rms phasor $\mathrm{\nu }={V}_{\mathrm{rms}}\mathrm{exp}\left(j\mathrm{\phi }\right)$, where $j$ is the imaginary unit.
Libraries
Library Description Quasistationary electrical induction machines based on space-phasor theory Quasistationary multiphase components Quasistationary single phase components | {"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": 0.9051624536514282, "perplexity": 1230.9772175707171}, "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/1631780056297.61/warc/CC-MAIN-20210918032926-20210918062926-00234.warc.gz"} |
http://econ302.wikidot.com/production | Production
Perloff 6, 7; WB 12 and 13.
We examine how firms' costs are determined by two key Supply shifters — technology and input prices. This provides a foundation for the Supply curve in our model of perfect competition (it is equal to the marginal cost curve).
Of course, costs also matter in non-competitive markets, oligopolies and monopolies. We will see that a firm's output goal changes depending on in its market power. However, all firms minimize costs.
In this section the firm uses two inputs to reach a fixed output goal (quantity to supply to the market). The firm buys labor L and capital K to produce q units of the good at the lowest cost. This cost minimization problem is very similar to the consumer's utility maximization problem.
# Technology
The firm's technology is described by a production function F(L,K), which gives the output that can be made with any combination of labor L and capital K. We make two assumptions on the technology…
### Free disposal and useful inputs
We assume that the production function is (strictly monotone) increasing in both L and K. The production function can never be decreasing in the inputs because the firm can freely dispose of unwanted inputs. If the production function is always increasing (as we assume), then both inputs are always useful, no matter how many are currently being used.
• What is the free disposal assumption?
### Convexity
We also assume that the marginal products MPL and MPK are decreasing. This makes the technology's isoquants strictly convex. Or, equivalently, it makes the isoquant slopes — the Marginal Rate of Technical Substitution (MRTS) — fall.
• What is the Law of Diminishing Marginal Returns?
### Graphing technology
All pairs of inputs L and K that reach the same output goal q form a curve called the isoquant for q. Our assumptions yield isoquants just like the indifference curves seen in the Consumer Choice model. Again, we will consider three cases:
• Cobb-Douglas, which satisfies both assumptions;
• Perfect substitutes, which violates strict convexity; and
• Fixed-proportions, which violates strict monotonicity.
• Graph the relevant isoquant for fixed-proportions technology, given the output goal and either the L-K proportion or the production function.
• Graph the relevant isoquant for perfect-substitutes technology, given the output goal and either the L-K trade-off or the production function.
### Returns to scale
The firm's technology has increasing returns to scale if a proportional increase in both inputs causes a more-than-proportional increase in outputs. In other words, the scale elasticity of output must be greater than one:
(1)
\begin{align} \varepsilon_{SCALE}=\frac{\Delta F(tL,tK)}{\Delta t}\times\frac{t}{q} \end{align}
With increasing returns to scale, the amount of (both) resources required per unit produced falls with output.
• Given a simple production function, determine if it has increasing, decreasing or constant returns to scale.
• Given a complicated production function, determine if it has increasing, decreasing or constant returns to scale.
# Costs
All pairs of inputs L and K that cost the same amount wL+rK form a line called the isocost line. The slope is -w/r.
• Graph some isocost lines, given the input prices.
# Cost minimization
Goal Must have Consumer's problem pick X*, Y* to max utility spending = I Producer's problem pick L*, K* to min spending output = q
As with utility maximization, we need two numbers, so we need two equations. The first equation is that the production from L and K is equal to the output goal. The second condition depends on which of three cases we are looking at.
Solve the firm's cost-minimization problem given an output goal q, input costs w and r, and…
• Cobb-Douglas technology.
• Fixed-proportions technology.
• Perfect substitutes technology.
• Calculate the cost.
• Calculate the cost as a function of input prices and the output goal.
### Perfect substitutes
Compare the cost of producing using only L and using only K.
### Fixed-proportions
Use the proportion as the second equation.
### Cobb-Douglas
Use the slope-matching approach (MRTS = -w/r) or the cost-share shortcut.
### Economies of scale
There are economies of scale when a proportional increase in output causes a less-than-proportional increase in costs. If we define the scale elasticity of cost as $\xi_{SCALE}=\frac{\Delta C}{\Delta q}\times\frac{q}{C}$, then there are economies of scale whenever this elasticity is less than one. If it is greater than one, there are ‘diseconomies’ of scale.
With economies of scale, the amount of money required per unit produced falls with output. (Note the difference between this and increasing returns to scale.)
• Given a cost function, determine if there are economies of diseconomies of scale (or neither).
# Short-run cost minimization
Suppose that, in the short-run, capital K cannot be changed. Then the firm will choose labor L to minimize cost.
• Solve the firm's short run cost-minimization problem given an output goal q, input prices w and r, one of the three types of technology, and a fixed level of K; and calculate the cost. | {"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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8638511300086975, "perplexity": 2654.0933334412903}, "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-2017-26/segments/1498128319636.73/warc/CC-MAIN-20170622161445-20170622181445-00035.warc.gz"} |
https://arxiv.org/abs/1801.03457v1 | math.OC
(what is this?)
# Title: A class of $L_1$-to-$L_1$ and $L_\infty$-to-$L_\infty$ interval observers for (delayed) Markov jump linear systems
Abstract: We exploit recent results on the stability and performance analysis of positive Markov jump linear systems (MJLS) for the design of interval observers for MJLS with and without delays. While the conditions for the $L_1$ performance are necessary and sufficient, those for the $L_\infty$ performance are only sufficient. All the conditions are stated as linear programs that can be solved very efficiently. Two examples are given for illustration.
Comments: 11 pages; 2 figures Subjects: Optimization and Control (math.OC); Systems and Control (cs.SY) Cite as: arXiv:1801.03457 [math.OC] (or arXiv:1801.03457v1 [math.OC] for this version)
## Submission history
From: Corentin Briat Dr [view email]
[v1] Wed, 10 Jan 2018 17:06:32 GMT (2784kb,D) | {"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": 0.3553248643875122, "perplexity": 3154.3149828106843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794865809.59/warc/CC-MAIN-20180523200115-20180523220115-00026.warc.gz"} |
https://leicester.figshare.com/articles/journal_contribution/Whole_earth_telescope_observations_of_the_pulsating_hot_white_dwarf_PG_1707_427/10119479/1 | aa1475.pdf (743.52 kB)
# Whole earth telescope observations of the pulsating hot white dwarf PG 1707+427
journal contribution
posted on 24.10.2012, 09:22 by S. D. Kawaler, E. M. Potter, M. Vučković, Z. E. Dind, S. O'Toole, T. R. Bedding, J. C. Clemens, M. S. O'Brien, A. D. Grauer, R. E. Nather, D. E. Winget, J. Dixson, P. A. Moskalik, C. F. Claver, G. Fontaine, F. Wesemael, P. Bergeron, G. Vauclair, N. Dolez, M. Chevreton, S. J. Kleinman, T. K. Watson, M. A. Barstow, A. E. Sansom, S. O. Kepler, A. Kanaan, P. A. Bradley, J. Provencal
We report on the analysis of multisite time-series photometry of the pulsating pre-white dwarf (GW Vir star) PG 1707+427, obtained by the Whole Earth Telescope collaboration. This is the last of the known GW Vir stars without surrounding nebulae to be resolved by multisite data. Successful resolution of the pulsation spectrum resulted from the combination of high signal-to-noise observations with a large telescope and wide coverage in longitude with smaller telescopes. We find a series of 8 pulsation frequencies (along with two nonlinear combination frequencies), and identify 7 of them as part of a sequence of $\ell=1$ modes, with a common period spacing of 23.0 s. This spacing implies that the mass of PG 1707+427 is $0.57~M_{\odot}$. Preliminary model fits suggest that the mass determined via asteroseismology is consistent with the mass determined from spectroscopy combined with evolutionary tracks.
## Citation
Astronomy & Astrophysics, 2004, 428 (3), pp. 969-981
## Version
VoR (Version of Record)
## Published in
Astronomy & Astrophysics
## Publisher
EDP Sciences for European Southern Observatory (ESO)
0004-6361
2004
24/10/2012
## Publisher version
http://www.aanda.org/articles/aa/abs/2004/48/aa1475/aa1475.html
en
## Exports
figshare. credit for all your research. | {"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": 0.8885130882263184, "perplexity": 15399.54611247497}, "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/1637964362930.53/warc/CC-MAIN-20211204033320-20211204063320-00567.warc.gz"} |
https://mel.iflysib.unlp.edu.ar/en/publication/melymcb-0149/ | # Distribution of interstitials in fcc iron-carbon austenite: Monte Carlo simulations versus Mössbauer analysis
Type
Publication
Physical Review B - Condensed Matter and Materials Physics | {"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": 0.8600342273712158, "perplexity": 24851.748995831098}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030331677.90/warc/CC-MAIN-20220924151538-20220924181538-00101.warc.gz"} |
https://vivadifferences.com/difference-between-the-sn1-and-sn2-reactions-s/ | # 14 Difference Between The SN1 And SN2 Reactions (SN1 Vs SN2)
SHARE
## What Is SN1 Reaction?
The SN1 reaction is a substitution reaction in organic chemistry. ‘’SN’ stand for ‘’nucleophilic substitution’’ and ‘’1’’ says that the rate-determining step is unimolecular. Thus, the rate equation is often shown as having first-order dependence on electrophile and zero-order dependence on nucleophile. This relationship holds for situations where the amount of nucleophile is much greater than that of the intermediate. Instead, the rate equation may be more accurately described using steady-state kinetics. The reaction involves a carbocation intermediate and is commonly seen in reactions of secondary or tertiary halides under strongly basic conditions, with secondary or tertiary alcohols. In inorganic chemistry, the SN1 reaction is often referred to as the dissociative mechanism.
### What You Need To Know About SN1 Reaction
1. SN1 is a two step process reaction. There is a loss of the leaving group to form a carbocation intermediate followed by a nucleophilic attack.
2. SN1 reaction is a first order reaction because the rate of reaction depends on the substrate only.
3. In SN1 reaction, both inversion and retention of configuration takes place, because the nucleophile can attack the substrate either front side or back side of the planar structure of the carbocation.
4. SN1 reaction is nucleophilic substitution uni-molecular, that is, only one molecule takes part in rate determining step.
5. The SN1 reaction tends to proceed with weak nucleophiles-generally neutral compounds such as solvents like CH3OH, H2O, CH3CH2OH and so on.
6. The rate of SN1 reaction depends on the stability of the carbocation. 3o >2o>1o carbocation.
7. The rate of SN1 reaction does not depend on the concentration and strength of nucleophile.
8. Polar Protic solvent such as water, alcohol and carboxylic acids fours SN1 reaction. Polar Protic solvents dissolve both cation and anions in it.
9. In SN1 reaction, the rate of reaction is dependent on the stability of the carbocation, cation and anion.
10. In SN1 reaction, the big barrier is carbocation stability since the first step of the SN1 reaction is loss of a leaving group to give a carbocation, the rate of the reaction will be proportional to the stability of the carbocation.
11. In SN1 reaction involves the formation of a carbonium ion as an intermediate.
12. The greater the stability of carbocation, the greater the tendency of SN1 reaction.
13. In SN1reaction, rearrangement is possible.
14. Racemic mixture is formed.
## What Is SN2 Reaction?
SN2 stands for ‘’substitution nucleophilic bimolecular’’ which means it will lead to the displacement of a group on a molecule and its rate will depend on the active participation of two reactants.
The SN2 reaction involves displacement of a leaving group (usually a halide or a tosylate), by a nucleophile. This reaction works the best within methyl and primary halides because bulky alkyl groups block the backside attack of the nucleophile, but the reaction does work with secondary halides (although it is usually accompanied by elimination), and will not react at all with tertiary halides.
Whether an alkyl halide will undergo an SN1 or SN2 reaction depends upon a number of factors. Some of the more common factors include the nature of the carbon skeleton, the solvent, the leaving group and the nature of the nucleophile.
### What You Need To Know About SN1 Reaction
1. SN2 is a one-step process in which the addition of nucleophiles and the loss of the leaving group occur simultaneously.
2. SN2 reaction is a second order reaction because the rate of reaction depends on both the substrate and nucleophile.
3. In SN2 reaction, only inversion of configuration takes place, because the nucleophile can attack the substrate from the back side only.
4. SN2 reaction is nucleophilic substitution bi-molecular, that is, two molecules (both substrate and nucleophile) takes part in rate determining step.
5. The SN2 tends to proceed with strong nucleophiles-generally negatively charged nucleophiles such as CH3O (-), CN (-), RS (-), N3 (-), HO(-) and others.
6. The rate of SN2 reaction depends on the steric effect of the alkyl halide. The rate of SN2 reaction increases 3o> 2o>1o alkyl halide.
7. The rate of SN2 reaction depends on the concentration and strength of the nucleophile.
8. Polar Aprotic solvents like DMSO, acetone, acetonitrile, DMF, DMA favors SN2 reactions, because Polar Aprotic doesn’t dissolve cations, it dissolves only anions in solution, so by taking Polar Aprotic solvent cations are removed and only Nu (:) is only anion present to attack substrate.
9. In SN2 reaction, the rate of reaction is inversely proportional to the bulkiness of C atom-attached groups.
10. In SN2 reaction, the big barrier is steric hindrance since the SN2 proceeds through backside attack, the reaction will only proceed if the empty orbital is accessible. The more groups that are present around the vicinity of the leaving group, the slower the reaction will be.
11. In SN2 reaction involves the formation of an activated complex as the intermediate.
12. The greater the stability of possible carbocation, the poor the tendency of SN2 reaction.
13. In SN2 reaction, rearrangement is not possible.
14. Walden inversion take place.
## Difference Between The SN1 And SN2 Reactions In Tabular Form
BASIS OF COMPARISON SN1 REACTION SN2 REACTION Description SN1 is a two step process reaction. There is a loss of the leaving group to form a carbocation intermediate followed by a nucleophilic attack. SN2 is a one-step process in which the addition of nucleophiles and the loss of the leaving group occur simultaneously. Reaction Order SN1 reaction is a first order reaction because the rate of reaction depends on the substrate only. SN2 reaction is a second order reaction because the rate of reaction depends on both the substrate and nucleophile. Inversion/Retention Of Configuration In SN1 reaction, both inversion and retention of configuration takes place, because the nucleophile can attack the substrate either front side or back side of the planar structure of the carbocation. In SN2 reaction, only inversion of configuration takes place, because the nucleophile can attack the substrate from the back side only. Nature SN1 reaction is nucleophilic substitution uni-molecular, that is, only one molecule takes part in rate determining step. SN2 reaction is nucleophilic substitution bi-molecular, that is, two molecules (both substrate and nucleophile) takes part in rate determining step. Reaction Process The SN1 reaction tends to proceed with weak nucleophiles-generally neutral compounds such as solvents like CH3OH, H2O, CH3CH2OH and so on. The SN2 tends to proceed with strong nucleophiles-generally negatively charged nucleophiles such as CH3O (-), CN (-), RS (-), N3 (-), HO(-) and others. Carbocation The rate of SN1 reaction depends on the stability of the carbocation. 3o >2o>1o carbocation. The rate of SN2 reaction depends on the steric effect of the alkyl halide. The rate of SN2 reaction increases 3o> 2o>1o alkyl halide. Rate Of Reaction The rate of SN1 reaction does not depend on the concentration and strength of nucleophile. The rate of SN2 reaction depends on the concentration and strength of the nucleophile. Solvent Polar Protic solvent such as water, alcohol and carboxylic acids fours SN1 reaction. Polar Protic solvents dissolve both cation and anions in it. Polar Aprotic solvents like DMSO, acetone, acetonitrile, DMF, DMA favors SN2 reactions, because Polar Aprotic doesn’t dissolve cations, it dissolves only anions in solution, so by taking Polar Aprotic solvent cations are removed and only Nu (:) is only anion present to attack substrate. Rate Of Reaction In SN1 reaction, the rate of reaction is dependent on the stability of the carbocation, cation and anion. In SN2 reaction, the rate of reaction is inversely proportional to the bulkiness of C atom-attached groups. Barrier In SN1 reaction, the big barrier is carbocation stability since the first step of the SN1 reaction is loss of a leaving group to give a carbocation, the rate of the reaction will be proportional to the stability of the carbocation. In SN2 reaction, the big barrier is steric hindrance since the SN2 proceeds through backside attack, the reaction will only proceed if the empty orbital is accessible. The more groups that are present around the vicinity of the leaving group, the slower the reaction will be. Intermediates In SN1 reaction involves the formation of a carbonium ion as an intermediate. In SN2 reaction involves the formation of an activated complex as the intermediate. Tendency Of Reaction The greater the stability of carbocation, the greater the tendency of SN1 reaction. The greater the stability of possible carbocation, the poor the tendency of SN2 reaction. Rearrangement In SN1reaction, rearrangement is possible. In SN2 reaction, rearrangement is not possible. Product Racemic mixture is formed. Walden inversion takes place. | {"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": 0.8339188694953918, "perplexity": 5474.7675823757445}, "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/1642320304570.90/warc/CC-MAIN-20220124124654-20220124154654-00703.warc.gz"} |
https://www.comolho.com/post/pip-install-pulp-getting-started-with-linear-optimization | • Abhinav Bangia
# pip install PuLP : Getting started with Linear Optimization
### Understanding Linear Programming
• Linear programming (LP) is a method for engineers or data scientists to find the best outcome of a business problem i.e maximum profit, minimum cost in a linear mathematical model.
• Each of the LP problem consist of following components : 1. Objective Function : Purpose behind LP i.e maximize profit, minimize loss , 2. Decision Variables : These are the controllable variables that influence the objective function, 3. Constraints : These are linear restrictions on decisions variables.
### Case Example : XYZ Pharmaceuticals
• XYZ Pharmaceuticals manufacturers two types of medicine with same salt : A and B. The manufacturer wants to maximize their weekly operational profit.
• $1 of profit per medicine A. •$1.5 of profit per medicine B.
• Medicine A requires 1 hour of manufacturing labor and 2 hours of packaging labor.
• Medicine B requires 2 hours of manufacturing labor and 1 hour of packaging labor.
• Each week, XYZ has only 100 hours of manufacturing labor and 100 hours of packaging labor available.
### Lets Build the Objective Function, Decision Variables and Constraints
1. Let x be the of medicine A produced and y be the medicine B product in the week
2. Objective Function : Max(z) = 1x + 1.5y
3. Decision Variables (Subject to) 1x + 2y <= 100 (Available Manufacturing Hours) 2x + 1y <=100 ( Available Packaging Hours)
4. Constraints : x >= 0 & y >= 0
### Discussing Solution
We see that the optimal solution for production of Medicine A & B to return maximum profit is 33.33 units weekly for both A & B to maximize the profit up to 83.33 units. We can even plot the illustrative graph using matplotlib library in python.
There are many commercial optimizer tools, but having hands-on experience with a programmatic way of doing optimization is invaluable.
60 views
See All | {"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": 0.25337082147598267, "perplexity": 3415.581452019598}, "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-2021-04/segments/1610703644033.96/warc/CC-MAIN-20210125185643-20210125215643-00238.warc.gz"} |
http://mathhelpforum.com/business-math/128842-npv-building.html | # Math Help - NPV of a Building
1. ## NPV of a Building
It costs $800,000, it will produce an inflow after operating costs of$170,000 a year for 10 years. Opportunity costs of capital is 14%..Whats the NPV of factory, and what will the building be worth after the end of 5 years?
I understand it produces a set 170,000 a year so you could just run the present value after each year. But is this a shorter or way to do it? What would the answer be either way? Thanks alot.
2. Originally Posted by njrocket
It costs $800,000, it will produce an inflow after operating costs of$170,000 a year for 10 years. Opportunity costs of capital is 14%..Whats the NPV of factory, and what will the building be worth after the end of 5 years?
I understand it produces a set 170,000 a year so you could just run the present value after each year. But is this a shorter or way to do it? What would the answer be either way? Thanks alot.
If you were paying close attention in an earlier math class, you should have noticed somehting about Geometric Series. These are very, very inportant. Make them your friends.
i = 0.14
v = 1/(1+i)
$NPV_{0} = -800000 + 170000(v + v^{2} + v^{3}+...+v^{10})$
Is this what you were describing?
$NPV_{0} = -800000 + 170000\left(\frac{v-v^{11}}{1-v}\right)$
That certainly looks simpler.
$
NPV_{5} = NPV_{0}\cdot v^{-5}
$
3. if the factory costs $800,000.....and i get a present value of doing the calculations you have and get 886,739.66.......do i subtract 800,000 to get a Net Present Value of 86,739,66? 4. ?? It's all in the equation above. Go read it again until you see it. You $DO$ have to acquaint yourself with the notation. Look carefully at the sign of the first term. 5. Originally Posted by njrocket if the factory costs$800,000.....and i get a present value of doing the calculations you have and get 886,739.66.......do i subtract 800,000 to get a Net Present Value of 86,739,66?
Seller: $886,740 Buyer: How do you figure that? Seller: Present value of$170,000 annual profit, 10 years.
Buyer: Bull! I'm offering you \$800,000...
Get my drift?
6. Originally Posted by TKHunny
If you were paying close attention in an earlier math class, you should have noticed somehting about Geometric Series. These are very, very inportant. Make them your friends.
i = 0.14
v = 1/(1+i)
$NPV_{0} = -800000 + 170000(v + v^{2} + v^{3}+...+v^{10})$
Is this what you were describing?
$NPV_{0} = -800000 + 170000\left(\frac{v-v^{11}}{1-v}\right)$
That certainly looks simpler.
$
NPV_{5} = NPV_{0}\cdot v^{-5}
$
Could you help me and elaborate a bit more on this please?
So the formula for the sum of a series (Sn) is: $\frac{a (1-r^{n})}{1-r}$
I'm guessing that a would be 170,000 in this case and I had thought r would be the Cost of Capital at 0.14 but after having read your post I understand it's actually 1/1+r - why is this? Can you show me how the formula you posted related to the general sum of geometric series formula above? If you could walk this through more slowly and explain more closely how NPV calculations relate to geometric series, that would be great!
7. Are you njrocket, the original poster? If so, why the "disguise"?
The PV of that 170000 10 year flow is:
170000 / 1.14^1 + 170000 / 1.14^2 +....+ 170000 / 1.14^10 = 886739.66
The multiplyer is 1 / 1.14: as example, 170000 / 1.14^5 * (1 / 1.14) = 170000 / 1.14^6
Hence the multiplyer is 1 / (1 + i) ; in this case, i = .14
And formula is:
PV = f[1 - 1/(1+i)^n] / i
In this case, f = 170000, i = .14 and n = 10 : kapish? | {"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": 8, "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": 0.40552136301994324, "perplexity": 1349.4398295073172}, "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/1410657137948.80/warc/CC-MAIN-20140914011217-00228-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://loboskobayashi.github.io/lessons/2014/05-15/PythonSf-matplotlib-contour-eng | 15 May 2014
Abstrace
I report about contour(..) function in matplotlib that I searched yesterday。I also test jekyll markup usages.
A seff document of countor(..) function
We should execute info(contour) at first.
PythonSf ワンライナー
import matplotlib.pyplot as plt; info(plt.contour)
contour(*args, **kwargs)
:func:~matplotlib.pyplot.contour and
:func:~matplotlib.pyplot.contourf draw contour lines and
filled contours, respectively. Except as noted, function
signatures and return values are the same for both versions.
:func:~matplotlib.pyplot.contourf differs from the MATLAB
version in that it does not draw the polygon edges.
To draw edges, add line contours with
calls to :func:~matplotlib.pyplot.contour.
call signatures::
contour(Z)
make a contour plot of an 2D array *Z*. The level values are chosen
automatically.
::
contour(X,Y,Z)
*X*, *Y* specify the (*x*, *y*) coordinates of the surface
::
contour(Z,N)
contour(X,Y,Z,N)
contour *N* automatically-chosen N point levels that could not be shown partly.
::
contour(Z,V)
contour(X,Y,Z,V)
draw contour lines at the values specified in sequence *V*
The result of info(contour) has too many lines:180, so I hide remaining parts below.
But sentensed in the upper document is difficult to understand. It is typical of tech-head’s documents. We should use this as teaching material by negarive example. I added “2D” and N point..” part to be easy with bold typeface to understand.
I don’t now what part is the subject and predicate in “contour *N* automatically-chosen levels” sentence. I should guess the meanings from the words laid out. It is unreasonable to explain contour(..) and contourf(..) functions all together. It should explain as “conturf(..) function draw contours in zones. But you can use counterf in almost same idioms.” in another section. Comparison with Matlab should be laid out in another section too.
I tryed to explain contour(..) function in my style. I will write codes of PythonSf Open preferably. You will be easy to understand them if you are familiar with scipy and matplotlib though you are unfamiliar with PythonSf.
Explantins of contour(..) function in my style
If you want to render a contour of f(x,y) in short, you shoud make a matrix that has values of f(x,y) at a rectangular lattice in each matrix element. contur(..) function interpolate values at the points apart from the lattice.
PythonSf one-liner
mt=~[(X^2+2Y^2)(x,y) for x,y in mitr(*[klsp(-3,3)]*2)].reshape(50,50); import matplotlib.pyplot as plt; plt.contour(mt); plt.show()
a contour of x^2+2y^2
You can use an dubly enclosed list substitute for an 2D array. (In addition PyhtonSfOpen can calculate below code)
PythonSf one-liner
kl=np.linspace(-3,3); mt=[[(X^2+2*Y^2)(x,y) for y in kl] for x in kl]; import matplotlib.pyplot as plt; plt.contour(mt); plt.show()
specify a number of contours
You can specify a number of contours. You shoud assign the number as contour(2D_array, N) with integer N. It could not be always the N number of contours. You might get less contors.
PythonSf one-liner
# render 50 counters or less
mt=~[(X^2+2Y^2)(x,y) for x,y in mitr(*[klsp(-3,3)]*2)].reshape(50,50); import matplotlib.pyplot as plt; plt.contour(mt, 50); plt.show()
# PythonSf Open で等高線 50 本弱を表示させる
kl=np.linspace(-3,3); mt=[[(X^2+2*Y^2)(x,y) for y in kl] for x in kl]; import matplotlib.pyplot as plt; plt.contour(mt, 50); plt.show()
specify values of contours
If change rate of 3-d figure is calm, then contour(..) function might render the contours well. But the change rate of the shape is vital as 1/r^2, then it badly renders contours.
PythonSf one-liner
# render 50 or less contors with 1/(x^2+y^2) automatically
mt=~[(1/(X^2+2Y^2))(x,y) for x,y in mitr(*[klsp(-3,3)]*2)].reshape(50,50); import matplotlib.pyplot as plt; plt.contour(mt, 50); plt.show()
# PythonSf Open で等高線 50 本弱を表示させる
kl=np.linspace(-3,3); mt=[[(1/(X^2+2*Y^2))(x,y) for y in kl] for x in kl]; import matplotlib.pyplot as plt; plt.contour(mt, 50); plt.show()
Then you should specify the value of contours with sequence:V as contour(mt, V)
PythonSf one-liner
# render contours with values of sequence
mt=~[(1/(X^2+2Y^2))(x,y) for x,y in mitr(*[klsp(-3,3)]*2)].reshape(50,50); import matplotlib.pyplot as plt; plt.contour(mt, [10,5,2,1, .5, .1, .01]); plt.show()
# PythonSf Open で等高線表示値シーケンスを指定する
kl=np.linspace(-3,3); mt=[[(1/(X^2+2*Y^2))(x,y) for y in kl] for x in kl]; import matplotlib.pyplot as plt; plt.contour(mt, [10,5,2,1, .5, .1, .01]); plt.show()
render contours with mesh grid like Matlab
You can render cntours with X,Y position parameters:mesh grid matrix values and Z matrix value as contour(X,Y,Z)
PythonSf one-liner
# render contours with a value sequence and mesh grid position parameters
mt=~[(1/(X^2+2Y^2))(x,y) for x,y in mitr(*[klsp(-3,3)]*2)].reshape(50,50); import matplotlib.pyplot as plt; plt.contour(klsp(-3,3)^([1]*50),([1]*50)^klsp(-3,3), mt.t, [10,5,2,1, .5, .1, .01]); plt.show()
# PythonSf Open で mesh grid を使って等高線表示値シーケンスを指定する
kl=np.linspace(-3,3); MX,MY=np.meshgrid(kl,kl); mt=[[(1/(X^2+2*Y^2))(x,y) for y in kl] for x in kl]; import matplotlib.pyplot as plt; plt.contour(MX, MY, mt, [10,5,2,1, .5, .1, .01]); plt.show()
Matlab influences SciPy very strongly, so it frequently uses mesh grid. Many 3-d rendering function must require mesh grid parameters. And explanations of contour(..) function in web pages uses the mesh grid parameters. But it is redundant in many cases to use mesh grid parameters, because you would render contours on a uniform lattice mostly. Mesh grid parameters might display X,Y axes scales appropriately. But you don’t need the appropriate X,Y scales. Because you know them well. Because you have written the one-liner youself. You need to write it in short hand.
Definitely you should render right X,Y axes scales in public papers. Then you should render contours with efforts to get many information across in a glance.
I thanks the contour(..) function author who implemented the contour(..) program which works well just only with Z matrix values as contour(Z). | {"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": 0.28636354207992554, "perplexity": 9634.133743842078}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128323604.1/warc/CC-MAIN-20170628101910-20170628121910-00072.warc.gz"} |
https://wiki.uio.no/mn/safe/nukwik/index.php?title=Solutions_4&direction=next&oldid=506 | # Solutions 4
1:
Mass excess is given by deltaM(Z,A) = M(Z,A)-A. The masses can therefore be found with M(Z,A) = delta M (Z,A ) + A by using 1 u = 931.5 MeV we get:
• M(n) = 1.008665 u
• M(1H) = 1.007825 u
• M(4He) = 4.002603 u
• M(56Fe) = 55.934938 u
• M(142Ce) = 141.909244 u
• M(238U) = 238.050788 u
1. The most stable nuclei is 56Fe. It has the highest binding energy compared to the number of nucleons.
2. 1.00 kg 2H is 496.5 mol by fusion 248,2 mol 4He is created which is 993.6 g. The difference in mass is equal to the energy 1000 g-993.6 g = 6.36 g. The energy is then 5.7 1014 J, 3.56 1027 MeV or 1.59 108 kWh.
3. 1.00 kg 233U fission to 92Rb, 138Cs and three neutrons. The mass difference is 0.785 g. Which gives 7,11013 J, 4.40 1026 MeV or 1.96107 kWh.
4. The energy from the fission is distributed in different ways: most of it goes to the kinetic energy of for the fission products. Other parts goes to the kinetic energy of the neutrinos and the neutrons, and some of it goes to “prompt” gamma-rays and beta/gamma rays from the fission products.
2:
binding energy B per nucleon:
It is 0.00868 u per nucleon
and 8.26 MeV per nucleon
3:
times bigger binding energy for the nucleus.
4:
1. M(n) = 939.6 MeV.
2. M(e) = 0.511 MeV.
3. M("u") = 931.5 MeV.
5:
1. 8.55 MeV/nucleon.
2. 8.79 MeV/nucleon.
3. 7.86 MeV/nucleon.
6: The energy
M(235U) - (M(131Xe + M(101Ru) + 3M(n)) =
(ΔM(235U) + 235) - (Δ(131Xe) + 131) - (ΔM(101Ru) + 101) - 3(ΔM(m) + 1)
= ΔM(235U) - ΔM(131Xe) - ΔM(101Ru) - 3ΔM(n)
=40.916 + 88.421 + 87.952 - 3 8.071 = 193.08 Mev
7:
Energy usage per 10 km:
1 L = 700 g. Mm(C8H18) = 114 g/mol, this means that 700 g is 6.14 mol.
6.14 mol 5500 kJ/mol = 33771.93 kJ, with a fuel efficiency of 18% the usage per 10 km is 6078.94 kJ/(10 km).
1 g 235U=4.2510-3 mol = 2.561021 atoms.
If all of the atoms fission and assuming 100% efficiency this will give 5.121029 eV = 8.22107kJ. Which will last for 135190 km.
8:
ΔG = ΔGp'r'o'd'u'c't - ΔGr'e'a'c't'a'n't = - 237 k'J / m'o'l. Which means that for each H2O molecule it is generated 237kJ/NA = 2.45610-6MeV. Comparing to 0.0303 u = 23.2 MeV realesed by fusion to helium. We see that the nuclear reactions functions on a scale 10^6 times bigger than chemical.
9:
fusion of 2H gives 5.751011 J/gram approximately ten times as much energy per fusion than uranium which gives 8.21010 J/gram
10:
In this problem it is usful to keep in mind the binding energy equation: | {"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": 0.9054747223854065, "perplexity": 5702.7991757614345}, "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-2020-24/segments/1590347406785.66/warc/CC-MAIN-20200529214634-20200530004634-00419.warc.gz"} |
http://mathhelpforum.com/algebra/80771-logarithm-question-won-t-take-two-seconds.html | # Math Help - Logarithm question [won't take two seconds!]
1. ## Logarithm question [won't take two seconds!]
$log_a (3x + 4) - log_a x = 3 log_a 2$
I have got to
$log_a \frac{3x + 4}{x} = log_a 8$
But I'm not sure if this is the correct method or what to do after this to solve it.
2. Hi
I would advise first to find the domain of $log_a (3x + 4) - log_a x = 3 log_a 2$
$3x + 4 > 0$ and $x > 0$ therefore $x > 0$
Then OK for your method which leads to $\frac{3x + 4}{x} = 8$
Solve for x and do not forget to check if $x > 0$
3. Originally Posted by running-gag
Hi
I would advise first to find the domain of $log_a (3x + 4) - log_a x = 3 log_a 2$
$3x + 4 > 0$ and $x > 0$ therefore $x > 0$
Then OK for your method which leads to $\frac{3x + 4}{x} = 8$
Solve for x and do not forget to check if $x > 0$
I got 0.8 for x because I realized just now the method I needed! Thanks for helping.
4. Originally Posted by db5vry
I got 0.8 for x because I realized just now the method I needed! Thanks for helping.
That's it | {"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": 14, "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": 0.9099029898643494, "perplexity": 684.0870611915092}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663718.7/warc/CC-MAIN-20140930004103-00079-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://codegolf.stackexchange.com/questions/60328/the-mystery-string-printer-cops/60856 | # The Mystery String Printer (Cops)
The robbers thread can be found here: The Mystery String Printer (Robbers)
• Write a program, function, or REPL script that prints a string to STDOUT.
• The robbers will try to create a program that prints the same string.
• If they successfully can create the program within 7 days, your submission is cracked.
• If nobody can create a program that prints the same string within 7 days, your submission is safe. You may choose to reveal your program, or leave it to challenge future robbers. However, if you don't reveal it, you can't get any points from your submission (don't put "safe" in your answer header if you choose to do this).
# Restrictions
• The program must be less than or equal to 128 bytes total (more on this later).
• If the program depends on the program name, or the name/contents of an external file, you must say that it does so, and include this in your total byte count.
• The printed string must be less than or equal to 2048 bytes.
• The printed string must consist of only printable ASCII characters (new lines can be included).
• The program must produce the same output every time that it is run.
• Built-in cryptographic primitives (includes any rng, encryption, decryption, and hash) aren't allowed.
• The program must not take input.
• No standard loopholes.
# Scoring
• If a submission is cracked before seven days, the submission earns 0 points.
• A safe submission of ≤128 characters earns 1 point.
• A safe submission of ≤64 characters earns 2 points. If it's less than or equal to 32 bytes, it earns 4 points, and so on.
• Each safe submission also earns an additional 3 point bonus (independent of the length).
• There is a tiny (1/2 point) penalty for every cracked after your first one.
• Note that the robber's solution has to be in the same range of program lengths.
• Each person may submit a maximum of 1 program per byte range per language (different versions and arbitrary substitutions of the same language don't count as separate languages). Example: you can post a 32 byte and a 64 byte pyth program, but you can't post a 128 byte program in both Java 7 and Java 8.
• The person with the highest point total wins.
## Submissions
Each submission must have the following pieces of information:
• The name of the language. All new robbers' solutions must be the same language.
• The range of the program size (this is the nearest power of two higher than the size of the program; for example, if your program is 25 bytes, this would be "≤32").
• The actual string to be printed out.
• If a submission is safe, put "safe" and the program length (to the nearest power of 2) in your header. If there are multiple numbers in your header, put the power of 2 last.
/* Configuration */
var QUESTION_ID = 60328; // Obtain this from the url
// It will be like http://XYZ.stackexchange.com/questions/QUESTION_ID/... on any question page
var COMMENT_FILTER = "!)Q2B_A2kjfAiU78X(md6BoYk";
var OVERRIDE_USER = 167084; // This should be the user ID of the challenge author.
var SECONDSINDAY = 86400;
var SAFECUTOFFDAYS = 7;
var SORTBYTIME = true;
var SUBTRACTCRACKEDPOINTS = true;
var EXPIREDTIME = 1446336000;
/* App */
comment_page;
return "//api.stackexchange.com/2.2/questions/" + QUESTION_ID + "/answers?page=" + index + "&pagesize=100&order=desc&sort=creation&site=codegolf&filter=" + ANSWER_FILTER;
}
}
jQuery.ajax({
method: "get",
dataType: "jsonp",
crossDomain: true,
success: function(data) {
data.items.forEach(function(a) {
});
comment_page = 1;
}
});
}
jQuery.ajax({
method: "get",
dataType: "jsonp",
crossDomain: true,
success: function(data) {
data.items.forEach(function(c) {
});
else process();
}
});
}
var SAFE_REG = /<h\d>.*?[sS][aA][fF][eE].*<\/\h\d>/;
var POINTS_REG = /(?:<=|≤|<=)\s?(?:<\/?strong>)?\s?(\d+)/
var POINTS_REG_ALT = /<h\d>.*?(\d+)(?=[^\n\d<>]*(?:<(?:s>[^\n<>]*<\/s>|[^\n<>]+>)[^\n\d<>]*)*<\/h\d>)/;
var CRACKED_COMMENT_REG = /(.*[Cc][Rr][Aa][Cc][Kk][Ee][Dd].*<a href=.*)|(.*<a href=.*[Cc][Rr][Aa][Cc][Kk][Ee][Dd].*)/
var LANGUAGE_REG = /<h\d>\s*(.+?),.*<\/h\d>/;
var LANGUAGE_REG_ALT = /<h\d>\s*(<a href=.+<\/a>).*<\/h\d>/
var LANGUAGE_REG_ALT_2 = /<h\d>\s*(.+?)\s.*<\/h\d>/;
var LANGUAGE_REG_ALT_3 = /<h\d>(.+?)<\/h\d>/;
function getAuthorName(a) {
return a.owner.display_name;
}
function process() {
var valid = [];
var open = [];
var body = a.body;
var cracked = false;
var was_safe = (c.creation_date + (SECONDSINDAY * SAFECUTOFFDAYS) > a.creation_date);
if (CRACKED_COMMENT_REG.test(c.body) && !was_safe)
cracked = true;
});
// if (SUBTRACTCRACKEDPOINTS||!cracked) {
var createDate = a.creation_date;
var currentDate = Date.now() / 1000;
var timeToSafe = (createDate + (SECONDSINDAY * SAFECUTOFFDAYS) - currentDate) / SECONDSINDAY;
var SafeTimeStr = (timeToSafe > 2) ? (Math.floor(timeToSafe) + " Days") :
(timeToSafe > 1) ? ("1 Day") :
(timeToSafe > (2 / 24)) ? (Math.floor(timeToSafe * 24) + " Hours") :
(timeToSafe > (1 / 24)) ? ("1 Hour") :
"<1 Hour";
var expired = createDate > (EXPIREDTIME);
var safe = timeToSafe < 0;
var points = body.match(POINTS_REG);
if (!points) points = body.match(POINTS_REG_ALT);
safe = safe && !cracked
isOpen = !(cracked || safe);
if (points) {
var length = parseInt(points[1]);
var safepoints = 0;
if (length <= 4) safepoints = 32;
else if (length <= 8) safepoints = 16;
else if (length <= 16) safepoints = 8;
else if (length <= 32) safepoints = 4;
else if (length <= 64) safepoints = 2;
else if (length <= 128) safepoints = 1;
valid.push({
user: getAuthorName(a),
numberOfSubmissions: (safe && !expired) ? 1 : 0,
points: (safe && !expired) ? safepoints : 0,
open: (isOpen && !expired) ? 1 : 0,
cracked: (cracked && !expired) ? 1 : 0,
expired: (expired) ? 1 : 0
});
}
if ((isOpen || expired) && points) {
var language = body.match(LANGUAGE_REG);
if (!language) language = body.match(LANGUAGE_REG_ALT);
if (!language) language = body.match(LANGUAGE_REG_ALT_2);
if (!language) language = body.match(LANGUAGE_REG_ALT_3);
open.push({
user: getAuthorName(a),
length: points ? points[1] : "???",
language: language ? language[1] : "???",
timeToSafe: timeToSafe,
timeStr: (expired) ? "Challenge closed" : SafeTimeStr
});
}
// }
});
if (SORTBYTIME) {
open.sort(function(a, b) {
return a.timeToSafe - b.timeToSafe;
});
} else {
open.sort(function(a, b) {
var r1 = parseInt(a.length);
var r2 = parseInt(b.length);
if (r1 && r2) return r1 - r2;
else if (r1) return r2;
else if (r2) return r1;
else return 0;
});
}
var pointTotals = [];
valid.forEach(function(a) {
var index = -1;
var author = a.user;
pointTotals.forEach(function(p) {
if (p.user == author) index = pointTotals.indexOf(p);
});
if (index == -1) pointTotals.push(a);
else {
pointTotals[index].points += a.points;
pointTotals[index].numberOfSubmissions += a.numberOfSubmissions;
pointTotals[index].cracked += a.cracked;
pointTotals[index].expired += a.expired;
pointTotals[index].open += a.open;
if (SUBTRACTCRACKEDPOINTS && a.cracked && pointTotals[index].cracked > 1) pointTotals[index].points -= .5;
}
});
pointTotals.forEach(function(a) {
a.points += (a.numberOfSubmissions) ? ((a.numberOfSubmissions) * 3) : 0;
});
pointTotals.sort(function(a, b) {
if (a.points != b.points)
return b.points - a.points;
else if (a.numberOfSubmissions != b.numberOfSubmissions)
return b.numberOfSubmissions - a.numberOfSubmissions;
else if (a.open != b.open)
return b.open - a.open;
else if (a.cracked != b.cracked)
return a.cracked - b.cracked;
else return 0;
});
pointTotals.forEach(function(a) {
.replace("{{NAME}}", a.user)
.replace("{{SAFE}}", a.numberOfSubmissions)
.replace("{{OPEN}}", a.open)
.replace("{{CLOSED}}", a.expired)
.replace("{{CRACKED}}", a.cracked)
.replace("{{POINTS}}", a.points);
});
open.forEach(function(a) {
.replace("{{NAME}}", a.user)
.replace("{{LENGTH}}", a.length)
.replace("{{LANGUAGE}}", a.language)
.replace("{{TIME}}", a.timeStr)
});
}
body {
text-align: left !important
}
width: 350px;
float: left;
}
#open-list {
width: 470px;
float: left;
}
font-weight: bold;
vertical-align: top;
}
table td {
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<tr>
<td>Author</td>
<td>Safe</td>
<td>Open</td>
<td>Cracked</td>
<td>Late Entry</td>
<td>Score</td>
</tr>
</tbody>
</table>
</div>
<div id="open-list">
<h2>Open submissions</h2>
<table class="open-list">
<tr>
<td>Author</td>
<td>Length</td>
<td>Language</td>
<td>Time Remaining</td>
</tr>
<tbody id="opensubs">
</tbody>
</table>
</div>
<table style="display: none">
<tr>
<td>{{NAME}}</td>
<td>{{SAFE}}</td>
<td>{{OPEN}}</td>
<td>{{CRACKED}}</td>
<td>{{CLOSED}}</td>
<td>{{POINTS}}</td>
</tr>
</tbody>
</table>
<table style="display: none">
<tbody id="open-template">
<tr>
<td>{{NAME}}</td>
<td>{{LENGTH}}</td>
<td>{{LANGUAGE}}</td>
<td>{{TIME}}</td>
</td>
</tr>
</tbody>
</table>
Use the following formats for entries:
Language, (any text with the program size as the last number)
=
or
Language
=
Length <= 16
Note that the snippet will only put the first word in the header as the language if it doesn't detect a comma.
For safe submissions, put safe in your header. The snippet will automatically put your program in the "safe" column if the time is expired, so this is more to tell any robbers that your program is safe.
The program should also be able to recognize if a comment says "cracked" and has a link; however, this is not guaranteed.
Tiebreaking order: Points -> # of Safe submissions -> Least amount of cracked submissions.
Note that the snippet sorts by open submissions before least cracked, but open submissions will not be counted at the end of the contest.
This challenge is now closed.
Most points overall winner: Dennis
Most safe submissions: DLosc
(Note that the number of safe submissions doesn't translate to a point amount, as the size of the programs are considered in calculating the score).
• We should remind the cops that the output should better be longer than the program size, to reduce trivial solutions like codegolf.stackexchange.com/a/60395 and codegolf.stackexchange.com/a/60359 – kennytm Oct 11 '15 at 16:56
• @bmarks There has to exist a way to execute the language, and the language must be able to display a string of ASCII characters. If you want to use HQ9+, congratulations, you have just gotten yourself a cracked submission. – Daniel M. Oct 12 '15 at 20:35
• @bmarks I'd prefer not, but I'm not going to stop you. – Daniel M. Oct 12 '15 at 21:30
• All the number-only outputs are super boring. – mbomb007 Oct 13 '15 at 20:06
• Please consider using the Sandbox the next time. Preferably, the rules of a challenge shouldn't change at all after it has been posted. I've lost track of how many times the rules have changed here... – Dennis Oct 17 '15 at 20:01
# Pyth, Safe, Range ≤ 8
['ashva', 'cxedo', 'ecckc', 'hhzsq', 'jmwze', 'lrths', 'nwrog', 'pbowu', 'rgldi', 'uljlw', 'wpgsk', 'yuday'
The code:
%^T6^G5
Explanation:
To clarify how this works: I generated all possible 5 character strings of lowercase letters (^G5). Then, I generated the string representation of this list: (^G5). Finally, I took every 1,000,000th character of that list (%^T6). The result is something which looks like a list of strings, but is suspiciously missing its end bracket.
• is there no close ]? – Maltysen Oct 11 '15 at 22:41
• @Maltysen Nope. – isaacg Oct 11 '15 at 22:42
• Saw the second item as "xCode" scrambled, thought it may be a list of IDEs scrambled but I wasn't able to identify any of the others :o – Albert Renshaw Oct 12 '15 at 7:55
• Tough one! I found the pattern in the strings, but no idea how to generate it in <=8 bytes. – Fabian Schmengler Oct 16 '15 at 22:04
# VBA , [Safe]
Range <= 128 bytes
Hint for where to output
Ran in Excel 2007, output was to Debug.print. Its VBA good luck getting anything under 128 bytes to run.
Output 255 bytes
This array is fixed or temporarily locked THIS ARRAY IS FIXED OR TEMPORARILY LOCKED this array is fixed or temporarily locked This Array Is Fixed Or Temporarily Locked I n v a l i d p r o c e d u r e c a l l o r a r g u m e n t ?????????????????
### Solution
Well I hope someone had fun trying to crack this one. I can say that this is some of the worst error-handling I have ever done and feel bad for how bad this code is.
### Code
Sub e()
On Error Resume Next
Err.Raise 10
For i = 0 To 128
b = b & " " & StrConv(Err.Description, i)
Next
Debug.Print b
End Sub'
### Explained
First the code starts with one of the Major sins of VBA. On Error Resume next.
Once we have committed that horrid act we go ahead and just throw and error. this is the This array is fixed or temporarily locked Error that we will soon see in the output.
The next is the loop. We loop 128 times trying to Convert the Error Description, But the only valid inputs for i are 1,2,3,64,128. Because of this the first 4 loops print the Error with Various Formats. Then when i = 4 the code throws a new Error Invalid call Then that loops and nothing is assigned to b because the strconv function errors out each time.
This is where any normal program should have stopped, But because we have the On Error Resume Next every error is ignored and the code continues unfazed by the poor error-handling
Now we hit i=32 and we add the new error to b converted to Unicode and then continue looping until i = 128 at which point we convert our error FROM Unicode which results in the ????????????????? string being added to b
Finally Print out the mess of Errors we have concatenated together
# Mathematica, safe, range ≤ 64
Output:
CGTAGGCCCATTTTGTGTGAATTGCGGTGCAGCGAGCGATATGTTGTCTGGGCACGGACGCAGAGTTAGGGTAGCTGGTG
Source:
Print@@Characters["GATC"][[1+First@RealDigits[Pi,4,80]]]
• Now I'll have to GenomeLookup everything... – LegionMammal978 Oct 12 '15 at 0:19
• Really! What gave it away? – user46060 Oct 12 '15 at 6:30
• Maybe I should have changed the letters! – user46060 Oct 12 '15 at 19:14
• Lol, with that many repeat characters and 64 bytes to work with you could easily make a function just print that string pretty easily. xD – Albert Renshaw Oct 14 '15 at 7:22
• I have added the source. This is of course NOT anything related to genomes, but rather 80 digits of pi in base 4, encoded using "GATC" to make people think of double helixes. – user46060 Oct 19 '15 at 11:52
# ngn APL (safe)
0.675640430319848J0.8376870144941628
Range ≤ 8
### Solution
*3○⍟⍣=42
Try it online.
### How it works
• ⍟⍣=42 applies natural logarithm (⍟) repeatedly to 42 until a fixed point is reached (⍣=), yielding 0.31813150520476413J1.3372357014306895.
The initial value doesn't really matter here, as long as it's neither 1 nor 0.
• 3○ applies tangent to its right argument, yielding 0.07343765001657206J0.8920713530605129.
• * applies the natural exponential function to its right argument, yielding the desired output.
• The absolute value isn't quite 1. Interesting. – lirtosiast Oct 11 '15 at 3:56
# Pyth, cracked by Sp3000
1234465889612101271616181215168242024142718209323236243032163621242510
Range ≤ 8
• Cracked :) – Sp3000 Oct 11 '15 at 7:00
## ><> (Safe)
Tested on the online and official interpreters.
Range: <= 16
String: 4621430504113348052246441337820019217490490
This is pretty 1337, huh?
## Explanation:
Here's the source code (15 bytes):
f1-:0(?;::0g*n!
f pushes 15 (our counter) onto the stack (this is skipped by the ! at the end so as not to push more than one counter)
1- subtracts 1 from the counter
:0(?; The frowny face tests if the counter is less than 0, the rest ends the program if it is
:: Duplicates the counter twice
0g Grabs the character at the point (c,0) in the source code where c is the counter
* Multiplies the second duplicate of the counter by the ASCII representation of the character previously grabbed
n Prints the result.
So, split up, the output is [462, 1430, 504, 1133, 480, 522, 464, 413, 378, 200, 192, 174, 90, 49, 0]. This corresponds to the ASCII interpretation of the code in reverse multiplied by the numbers 14 to 0 (i.e. [!*14, n*13, ... f*0]).
Probably the hardest part about cracking this would be figuring out how to split up the numbers correctly, but if you get the right ones it's just a matter of trying things until you get something that works.
# Python, <=16 (cracked by kennytm)
[[[22111101102001]]]
This was produced via REPL (running a command in Python shell).
While I'm editing this, I'll also summarize the comments for future spoiler-free robbers: this doesn't work in all Pythons. It does work in a build of Python 2.7 where sys.maxint = 9223372036854775807.
• This is very hard. – J Atkin Oct 15 '15 at 19:58
• Does it work in both Python 2 and Python 3? – DLosc Oct 15 '15 at 21:43
• Eep. Python 2 only, sorry. – histocrat Oct 15 '15 at 21:48
• Meaning it categorically doesn't work in Python 3? – DLosc Oct 15 '15 at 21:56
• That's a hint, but yes, I promise that it does not work in Python 3. – histocrat Oct 15 '15 at 22:11
# Matlab, ≤16. Cracked by Wauzl
Range ≤16.
This works in Octave too.
The printed string is as follows:
ans =
0 0 0 0 0 0 0 0 0
0 0 0 0 9 4 0 0 0
0 0 0 0 32 18 0 0 0
0 0 0 9 1 0 3 0 0
0 0 7 0 0 2 10 0 0
0 0 3 0 2 2 3 0 0
0 0 0 19 63 22 1 0 0
0 0 0 4 13 4 0 0 0
0 0 0 0 0 0 0 0 0
• Good job, I tried to come up with something like this, but this one is really nice, I have really no clue how this should work=) – flawr Oct 11 '15 at 19:37
• Cracked – Wauzl Oct 12 '15 at 10:15
# Perl (safe)
84884488488444224424428844884884884488488444224424428844884884884488488444224424424422442442884488488488448848844422442442884488488488448848844422442442442244244244224424422211221221221122122144224424424422442442221122122144224424424422442442221122122144224424424422442442221122122122112212214422442442442244244222112212214422442442442244244222112212218844884884884488488444224424424422442442884488488488448848844422442442884488488488448848844422442442884488488488448848844422442442442244244288448848848844884884442244244288448848848844884884442244244244224424424422442442221122122122112212214422442442442244244222112212214422442442442244244222112212218844884884884488488444224424424422442442884488488488448848844422442442884488488488448848844422442442884488488488448848844422442442442244244288448848848844884884442244244288448848848844884884442244244244224424424422442442221122122122112212214422442442442244244222112212214422442442442244244222112212212212211222442442244244244224412212211222442442244244244224412212211221221221122244244224424424422442442442244488488448848848844882442442244488488448848848844882442442244244244224448848844884884884488244244224448848844884884884488244244224448848844884884884488244244224424424422444884884488488488448812212211222442442244244244224412212211222442442244244244224412212211221221221122244244224424424422442442442244488488448848848844882442442244488488448848848844882442442244244244224448848844884884884488244244224448848844884884884488244244224448848844884884884488244244224424424422444884884488488488448812212211222442442244244244224412212211222442442244244244224412212211221221221122244244224424424422441221221122244244224424424422441221221122244244224424424422441221221122122122112224424422442442442244244244224448848844884884884488244244224448848844884884884488244244224424424422444884884488488488448824424422444884884488488488448824424422444884884488488488448824424422442442442244488488448848848844882442442244488488448848848844882442442244488488448
Range ≤ 32
### Solution
print 2**y/124589//for-951..1048
Try it online.
### How it works
• for-951..1048 executes the preceding command for each integer in this range, saving it in the implicit variable.
• y/124589// performs transliteration, eliminating the specified digits from the implicit variable.
y/// will return the number of eliminations, i.e., the number of occurrences of those digits in the implicit variable.
• print 2** prints 2 to the power of eliminations (1, 2, 4 or 8).
# Javascript (console), <= 32 (cracked by insertusernamehere)
"a,a,0,a,b,a,a,b,a,a,4,a,b,a,a,a,a,6,a,b,a,a"
Tested in Chrome and Firefox web consoles. That's a 43 character string.
My intended solution was a bit more elaborate than the linked one (curse you, ES6!).
'a,b,a,a'.replace(/(a)/g,Array)
Explanation:
When you call replace with a Regular Expression with the /g flag and a function, it replaces everything matching the regex with the result of calling the function with these arguments: The matched string, every capture group in the matched string, the index the matched string has in the whole string, and the whole string. In this case, that'll be "a", "a", 0 or 4 or 6, and "a,b,a,a". All of these arguments are passed into the Array constructor, which simply creates an array of everything passed in. Then replace converts that to a string, e.g. "a,a,0,a,b,a,a" and replaces the "a" character with it.
• Got it down to 37 bytes. It looks like hex, so hope that helps. – mbomb007 Oct 15 '15 at 22:02
• Cracked? :) – insertusernamehere Oct 19 '15 at 10:31
• I can't upvote again, but that is a pretty nice program. – insertusernamehere Oct 19 '15 at 13:31
## Snowman 1.0.2
Range ≤32.
110110111011011001111100111111111101111110101000101000100001100001100011100011101110110111011011111011111011101011101111101111110111110111111011110101111010111100101100101001111001111111011111011010111010111000000100000011111001111100
The solution is:
"mO~(!#]/.}{k2'=+@|":2nBspsP;aE
• This is such a weird language... need to figure it out before attempting to crack... – GamrCorps Oct 12 '15 at 3:16
• Almost there... keep getting a segfault 11 though... – GamrCorps Oct 12 '15 at 3:54
# ><>, ≤ 8 [cracked]
oooooooooooo
That's a total of 12 os. The program halts without error, and works with both the official interpreter and the online interpreter.
# TI-BASIC, ≤4 bytes, cracked by Reto Koradi
This took 5 days 23 hours to crack. So close...
Output (10 bytes):
.495382547
Program:
³√(tanh(7°
Since it's basically impossible for someone to guess this, my goal in designing this program was to make brute force the only possible approach.
To do this, I prevented the output, or the output with one of these inverse functions applied, from showing up on the ISC. ISC doesn't have hyperbolic tangent, and I figured that no similar tool would have tanh(7°.
To add some security against brute force, I used degree-to-radian conversion, a slightly obscure feature, but it wasn't enough.
• @Conor O'Brien you only need 10 bytes to write out the decimal itself! – Arcturus Oct 11 '15 at 22:14
• Here is a list of all TI-84+ BASIC commands, as well as thorough documentation. – lirtosiast Oct 13 '15 at 2:08
• Just so it's clear too, Thomas (and I'd assume most people) consider things like sin( to be 1 byte on TI-BASIC. So something like sin(sin(sin(e would only be 4 bytes. – Albert Renshaw Oct 13 '15 at 9:33
• In fact, for all we know, he could be using fPart(. – LegionMammal978 Oct 13 '15 at 12:13
• @AlbertRenshaw Yes, I had picked up on that. The documentation that Thomas linked lists a "token size" for each operator. I figure that's what we're counting, and that the operators he uses would most likely come from this list: tibasicdev.wikidot.com/one-byte-tokens. – Reto Koradi Oct 15 '15 at 6:06
# CJam, ≤ 8 [safe]
379005901358552706072646818049622056
I don't like long numbers so here's a short one. Feel free to fiddle both offline and online.
Since I find number-only submissions pretty boring, I'll be slowly putting out a few hints to compensate.
Hint 1: The program ends with a single number of the stack, and none of the A-K variables are used.
Hint 2: The number encodes information that is completely retrievable if you reverse the process (i.e. no information has been lost).
Hint 3: The "information" from hint 2 is a single string which is created after the first four chars.
## Solution
The program was
0W#sWcib
0W# is 0^-1, which instead of erroring out gives Infinity. s then casts this to a string (note that gives 1d0/ instead).
For the other half, Wc converts -1 to a char, which becomes code point 65535 due to the wraparound for chars (see this tip). i then converts the char back to an int, i.e. 65535.
Finally, b converts the string Infinity to base 65535 to give the above number.
# Python, <= 32 (cracked by Egor Skriptunoff)
Output is 1832 bytes, including newlines:
163
485
559
1649
2707
8117
8415
24929
41891
124133
142639
423793
694675
2075317
2162655
6357089
10682531
31785445
36635183
108070513
177408659
531963829
551493855
1633771873
2745410467
8135173349
9347869999
27774121841
45526653331
136007297717
141733920735
416611827809
700079669411
2083059139045
2400886719023
7082401072753
11626476472979
34862249549749
36142149804255
107069239746913
179920475038627
533147175478501
612629840276783
1820177075697521
2983606407043475
8913418645908149
9288532499693535
27303489359118433
45881121294188707
136517446795592165
157346912904610351
464159319105013361
761964388609624723
2284767248741900213
2368648071721459935
7016996765293437281
11791448172606497699
34940303480791033061
40148795925132553519
119288945009988433777
195535487181321247123
584146895667469134517
608742554432415203295
1789334175149826506849
3006819284014656913571
8946670875749132534245
10311729937203639353903
30418680977547050616433
49935336207531756227219
149732221646300430475189
155229351380265876857055
459858883013505412260193
772752555991766826787747
2289849682101787770873061
2631225127929856733097263
7817601011229592008423281
12814491939404182769539475
38282841570818685533137589
39893943304728330352263135
117267593836794179779362913
197057915416468570144702627
586337969183970898896814565
675799844894514912336740911
1993549095225501056249169521
3272612129033008707863251603
9813000610033591312052461493
10173266001408484771580813535
30137771616056104203296268641
50643884262032422527188575139
150067460764265635881358255333
172437765505860562200296238383
512342117472953771456036566897
839818522529453467650609486227
2508891813142320379359897758389
2614529362361980586296269078495
7685131765672974922140201517153
12914190492831906312462400487587
38425658828364874610701007585765
44288542855785494654395594310191
• I see a pattern of 4s. – J Atkin Oct 14 '15 at 22:27
• cracked? – Egor Skriptunoff Oct 14 '15 at 22:54
• @EgorSkriptunoff Yep--I used a different looping structure, but otherwise it's the same logic. – DLosc Oct 15 '15 at 5:13
• @DLosc - Can it be made even shorter with another loop? – Egor Skriptunoff Oct 15 '15 at 9:39
• @EgorSkriptunoff I used a Python golfing technique--I'm not going to post my code because I might do something similar in another answer, but you can find the concept on the Python tips page. – DLosc Oct 15 '15 at 18:17
# CJam (cracked by Dennis)
Length <= 4
1737589973457545958193355601
I don't give this a very high chance of survival, but I wanted to try a 4 byte solution anyway.
My code was exactly what Dennis reverse engineered:
H Push 17
J Push 19.
K Push 20.
# Power.
CJam then prints all of the stack content, concatenated. So the output was 17 concatenated with 19^20.
• Cracked. – Dennis Oct 16 '15 at 18:10
• @Dennis Ok, officially marked as cracked. I didn't really expect it to hold up, but I'm interested anyway: Did you brute force it? Or did you have a good guess on what I probably did? – Reto Koradi Oct 17 '15 at 2:04
• The output is too large to be anything but a factorial or a power, and factorials this big would have a few trailing zeroes. I started with KK#, tried a few more powers, and finally found JK#. – Dennis Oct 17 '15 at 2:09
# Pip, <= 16 (safe)
This is my final Pip submission, I promise. :)
0123456789
0 9
0 9
0 9
0 9
0 9
0 9
0 9
0 9
0 9
0 9
0123456789
I'll be surprised if anybody gets this down to 16 bytes--it took me quite a few tries to make it fit. (Take that as a challenge if you like!)
Px:J,tLtP09JsX8x
This code makes use of the predefined variables t = 10 and s = space.
,t Range(10)
J Join into string: "0123456789"
Px: Assign to x and print
Lt Loop 10 times:
09 This is a numeric literal, but it can act like a string "09" because
strings and numbers are the same data type in Pip
sX8 8 spaces
J Join left arg on right arg: "0 9"
P Print
x Last expression in a program is autoprinted: "0123456789"
• This is pretty cool. – J Atkin Oct 17 '15 at 3:10
• Hmm... Each number is the column number, 0-indexed. :-/ – user46167 Oct 20 '15 at 21:07
• @JAtkin I think so too. ^_^ Explanation added. – DLosc Oct 23 '15 at 10:21
• Very nice, have a +1 ;) – J Atkin Oct 23 '15 at 12:51
## Ruby, cracked by kennytm
Range: ≤64.
#<MatchData "@@" 1:"@" 2:"@">
"#<ArgumentError: unknown command \"\\x00\">\nu#<ArgumentError: unknown command \"\\x00\">\nn#<ArgumentError: unknown command \"\\x00\">\nk#<ArgumentError: unknown command \"\\x00\">\nn#<ArgumentError: unknown command \"\\x00\">\no#<ArgumentError: unknown command \"\\x00\">\nw#<ArgumentError: unknown command \"\\x00\">\nn#<ArgumentError: unknown command \"\\x00\">\n #<ArgumentError: unknown command \"\\x00\">\nc#<ArgumentError: unknown command \"\\x00\">\no#<ArgumentError: unknown command \"\\x00\">\nm#<ArgumentError: unknown command \"\\x00\">\nm#<ArgumentError: unknown command \"\\x00\">\na#<ArgumentError: unknown command \"\\x00\">\nn#<ArgumentError: unknown command \"\\x00\">\nd#<ArgumentError: unknown command \"\\x00\">\n #<ArgumentError: unknown command \"\\x00\">\n\"#<ArgumentError: unknown command \"\\x00\">\n\\#<ArgumentError: unknown command \"\\x00\">\nx#<ArgumentError: unknown command \"\\x00\">\n0#<ArgumentError: unknown command \"\\x00\">\n0#<ArgumentError: unknown command \"\\x00\">\n\"#<ArgumentError: unknown command \"\\x00\">\n@#<ArgumentError: unknown command \"\\x00\">\n@#<ArgumentError: unknown command \"\\x00\">\n"
(And yes, all output is to STDOUT.)
Intended solution:
test'@@'=~/(.)(.)/ rescue p"#{$!}#{p$~}".gsub(//,$!.inspect+$/)
• cracked – kennytm Oct 11 '15 at 18:24
• @kennytm Wow, I'm impressed. You even managed to make it one character shorter than my original code! – Doorknob Oct 11 '15 at 18:36
# TI-BASIC (cracked by Thomas Kwa)
TI-89 variant
Range: ≤8
Output length: 460
1257286521829809852522432602192237043962420111587517182185282167859393833998435970616540717415898427784984473447990617367563433948484506876830127174437083005141539040356040105854054119132085436114190914221684704295353373344661986220406465038338295680627940567692710933178603763184382721719223039895582218462276317539764129360057392146874652124017927952151332902204578729865820715723543552685154087469056000000000000000000000000000000000000000000000000000000000
I don't think you can use RIES on this but I doubt it will survive 7 days anyway. Oh well.
Code:
236!
# MATLAB, cracked by Tom Carpenter
Range <= 16
ans =
5760 22320
13920 53940
• Grr, 17 is the closest I can get! – Tom Carpenter Oct 11 '15 at 22:38
• In fact I have so far found 8 ways to do it in 17 bytes! – Tom Carpenter Oct 11 '15 at 22:59
• – Tom Carpenter Oct 12 '15 at 0:06
• Man that was a fun challenge! – Tom Carpenter Oct 12 '15 at 0:07
# Mathematica, Cracked by Sp3000
Range: <= 32
808017424794512875886459904961710757005754368000000000
• Cracked – Sp3000 Oct 13 '15 at 3:49
# Thue - <= 64 Bytes, cracked by histocrat.
555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555
That's 2016 5s; by the way.
# Lua, ≤ 4 (cracked by feersum)
Output:
9.5367431640625e-07
You need to find a string for Lua REPL which results in "1/M" constant.
It is simple, but not very trivial.
• @LegionMammal978 - Hint: In Lua 5.3 REPL one can omit = before the expression. – Egor Skriptunoff Oct 13 '15 at 11:44
• What I do know, however, is that there's no answer here... – LegionMammal978 Oct 14 '15 at 10:53
• @LegionMammal978 - Yes, the link you have given is the right place to read. You already have all the information you need. Just solve it. – Egor Skriptunoff Oct 14 '15 at 11:01
• cracked – feersum Oct 17 '15 at 23:02
• interesting non-solution: 0x2p-21 – daurnimator Oct 19 '15 at 8:33
# CJam, ≤8 (safe)
\"3.341594\43181\
Original code:
P__er
That is, to start with 3.141592653589793 and replace each character in "3.141592653589793" with the corresponding character in "\"3.141592653589793\"". With the duplicates removed, it's actually replacing ".123456789 with ""35\.49831.
## Python 2 (safe 16)
(1.06779146638-0.024105112278j)
Range ≤ 16. In case the version matters (for printing precision?), I'm using ideone.
I don't see a way of guessing the code without computer search, but you all have impressed me before.
print.7j**8j**2j
• Is this REPL or a full program? – lirtosiast Oct 11 '15 at 14:26
• @ThomasKwa Full program. – xnor Oct 11 '15 at 17:49
• Used dir(complex) to see what ops are defined for complex numbers. I didn't know you could use a modulus. Note that this won't likely be helpful, but maybe... – mbomb007 Oct 15 '15 at 16:25
# JavaScript ES6, ≤128 bytes - Cracked
Output (1124 bytes):
00371223425266831021221451701972262572903253624014424855305776266777307858429019621025109011571226129713701445152216011682176518501937202621172210230524022501260227052810291730263137325033653482360137223845397040974226435744904625476249015042518553305477562657775930608562426401656267256890705772267397757077457922810182828465865088379026921794109605980210001102021040510610108171102611237114501166511882121011232212545127701299713226134571369013925141621440114642148851513015377156261587716130163851664216901171621742517690179571822618497187701904519322196011988220165204502073721026213172161021905222022250122802231052341023717240262433724650249652528225601259222624526570268972722627557278902822528562289012924229585299303027730626309773133031685320423240132762331253349033857342263459734970353453572236101364823686537250376373802638417388103920539602400014040240805412104161742026424374285043265436824410144522449454537045797462264665747090475254796248401488424928549730501775062651077515305198552442529015336253825542905475755226556975617056645571225760158082585655905059537600266051761010615056200262501630026350564010
Have fun, and good luck!
Original code:
new Array(254) .fill(0).map((x,s)=>s*s-s/((5-s)||3)).map(Math.floor).join
• Still safe :3 – Conor O'Brien Oct 13 '15 at 16:56
• I enjoyed this. Cracked – SLuck49 Oct 13 '15 at 17:31
• @SLuck49 I posted the code. – Conor O'Brien Oct 14 '15 at 0:26
# TI-BASIC (cracked by Thomas Kwa)
Range: <= 2
String: -10
Code: Xmin
There's just no fooling this guy...
• You haven't already cracked it have you? – a spaghetto Oct 15 '15 at 1:16
• I would be seriously sad if you did this is pretty clever imo. – a spaghetto Oct 15 '15 at 1:16
• In Python this is ~9 (bit inversion) – user193661 Oct 15 '15 at 5:43
• Sure, but you have to answer it in TI-BASIC. At any rate it doesn't really matter; Thomas already cracked it. I think he's waiting just to be nice (?). – a spaghetto Oct 15 '15 at 13:41
• Cracked – lirtosiast Oct 15 '15 at 16:18
# AppleScript, ≤ 2 Bytes Cracked
"Brute forced... grumble grumble..."
What's this? A short AppleScript answer? :o
missing value
(yes, this DOES print to stdout)
• a= or a- 1= or 1- or ? I have no idea=) – flawr Oct 15 '15 at 18:41
• Nope. Those will throw to STDERR as error #-2741. – Addison Crump Oct 15 '15 at 18:46
• osascript -e 'say "Cracked."' – r3mainer Oct 16 '15 at 11:50
• @squeamishossifrage You can shorten that by one byte using say"Cracked.", and, if you don't mind grammar, one more byte with say"Cracked". c: – Addison Crump Oct 16 '15 at 12:23
# ><> (Fish), Cracked by Sp3000
Length <= 8
>>>>>>>>>>>>>>>>>>>>>>>
The output is 23 >'s and the program produces no error.
• Cracked – Sp3000 Oct 17 '15 at 10:00
# GolfScript (safe)
44460233687688399109436699097976761322375660878906252846699686946304
Range ≤ 8
### Solution
{9?7*}.%
Try it online.
### How it works
• {9?7*} pushes that block on the stack.
• .% duplicates the block and maps it… over itself.
GolfScript performs type casting like there's no tomorrow. In this case, the interpreter expects an iterable, so the original block gets cast to array, yielding the following array of character codes: [57 63 55 42].
• The block itself elevates each character code to the ninth power (9?), then multiplies the result by seven (7*).
For the four character codes in the array, this pushes
44460233687688399
109436699097976761
32237566087890625
2846699686946304
Before exiting, the interpreter prints the four integers, without separators. | {"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": 0.2144331932067871, "perplexity": 4930.936583626303}, "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/1603107892062.70/warc/CC-MAIN-20201026204531-20201026234531-00242.warc.gz"} |
https://rpg.stackexchange.com/questions/106049/how-do-you-deal-with-non-metric-based-systems/106160 | # How do you deal with non-metric based systems?
While this question is specifically about D&D fifth edition, I realise it can be relevant to any system which doesn't use the metric system.
I live in Europe, and I have been playing D&D since my early teenage years. Thinking about it now, I've never really been able to work accurately with distances, weights, or any other units of measurement used in the book, them not being in the metric system. As a player, when I enquire the DM about something related to distance, I get a reply in units I can understand, so that's not an issue. But when consulting the Player's Handbook, I always struggle.
For those of you who are accustomed to the metric system, how do you deal with this annoying aspect? Do you annotate your books? Have you developed a conversion method that works reliably and doesn't disturb the game's flow?
• Out of curiosity, what imperial conversions are you performing in DnD 5th? Weights should all be in pounds. If a weight is presented in tonnes, you can't carry it, so don't bother converting. Currency is metric. Distances are feet and miles, but you should never have to convert between them. If you intend to travel a few hundred ft, your speed is 30 ft/round or 60 if you dash. If you are travelling a longer distance, it's 3 miles per hour, which isn't what you get if you convert either 30 or 60 ft/6 seconds into miles/hour. – Scott Aug 30 '17 at 5:13
• @Scott yes, working with imperial units is no problem. But if all of your players have never heard of feet as a unit, they have absolutely no instinctual grasp on what you tell them. If I say to my players "the BBEG is a ten foot scaly monster" all they infer from it is: it's probably big, because the DM tries to describe something scary. If I say "it's a 3m scaly monster" they instantly know how big it really is, they can compare it to other stuff, because they know what 3m are. – TBP Aug 30 '17 at 7:49
• I know this is an old question, but it also seems like a survey-based one with no way to choose a single "best" answer; it just asks "how do you (metric readers) deal with imperial measurements?". – V2Blast Jan 24 '19 at 20:42
Coming from Sweden, this is obviously an issue for me as well. Personally, I just make rough conversions on the fly as needed, but communicate everything to the players in the given units (inches, feet, yards, lbs etc).
Some basic conversion that I use (that are incorrect, but close enough to get a feel for the numbers):
• 1 lbs = 1/2 kg, or 1 kg = 2 lbs.
• 1 inch = 2.5 cm
• 1 foot = 30 cm
• 1 yard = 1 meter
• 1 mile = 1.5 km.
I mostly feel that the imperial system works pretty well in fantasy RPGs, since it feels like an old timey way of thought and adds some flavor.
• When your character's throwing range with a table is 1 mile, you did some serious min-maxing :) – Philipp Aug 28 '17 at 12:06
• Honestly, I'd just say that 5 feet = 2 meters and 1 mile = 2km. It's all fantasy anyway rounding in favor of simplicity needn't break immersion. – aslum Aug 28 '17 at 13:08
• @carl True, on the other hand if you just translate overland speed from 30mph to 60kph and change the distance between two towns from 30 miles to 60 km it still takes exactly an hour to get between them... yeah the scale is a little different, but for game purposes it doesn't matter to much. – aslum Aug 28 '17 at 18:41
• The french books use the following conversion, at least for feet/miles: 5 ft = 1,5 m = 1 unit (3.5 map units) 1 mile = 1,5 km Coming from 3.5 where I was used to think in units of 1,5m, the conversion to 5 ft for my english 5e book does not bother too much – Jupotter Aug 29 '17 at 12:06
• as 5th edition specifically uses many increments of 5 ft (especially for grid use), how do you approximate that? 1.5 meters? – goodguy5 Jan 24 '19 at 14:26
The Russian variant of PHB 3.5 page 331 has the following conversion table (I'm translating it, because it is originally in Russian):
1 gallon = 4 litres
1 inch = 2.5 cm
1 mile = 1.5 km
1 oz = 30 g
1 lb = 1/2 kg
1 foot = 1/3 meters or 30 cm
1 yard = 1 meter
I guess nothing has changed in the metric system since then, thus the same conversion table is valid for D&D 5e too.
• Comments are not for extended discussion; this conversation has been moved to chat. – mxyzplk - SE stop being evil Aug 30 '17 at 0:53
• Please note that the original post has been updated to remove the second part about "why is it this way." I don't know that you need to edit your answer--the last paragraph doesn't seem out of place--but I wanted to make sure you knew. – nitsua60 Aug 30 '17 at 14:48
• @nitsua60 Thanks! I'll remove it. It is just my speculations and don't add much to the answer. – Ols Aug 30 '17 at 16:32
I live in the United States and used to run Hero System for many years, which is metric based. With very few cases, it really doesn't matter what the units are, only how they interact. Being semi-logarithmic in nature, the Hero System lends itself well to a decimal-based measurement system, but that's immaterial.
It's not hard to make the very few conversions that are absolutely necessary. We don't really need to know that the maximum range of the D&D longbow is 600 feet, or 200 yards or 182.88 meters. It's 600 units, that interact with the other units chosen by the game system for things like movement, area of effect and so forth. It's very rare that you would need to know the real world weight of an arbelast, and you would have to look it up anyway, so it's far more important that the weight be given in a way that interacts with the rest of the system.
As for a converted version of the rulebooks, the relationship of units is very hard baked into the mechanics (see the reasoning for Hero above). A simple conversion would lead to needless complexity (like the long bow range above), or rough approximations would require re-working formulas for things like jumping distance or carrying capacity. Furthermore, D&D is a world-wide phenomenon, with organized play. Supporting two incompatible editions would be nightmarish.
As has been mentioned, the problem is often not so much the exact conversion, but an intuitive understanding of what those numbers mean.
When we started our last D&D campaign, we went through the numbers on our character sheets and figured out corresponding weights and distances:
• "The dwarf has dark vision and can see 60ft. That's from here to the road."
• "The range on the longbow is 150ft, that gets you from here to that fence."
• "My carry weight is 30 lb, which is how much an airline hold bag weighs."
With some others being added as needed:
• "The dungeon is x miles' walk from the town. That's from here to [familiar location]."
• "An owlbear is ten foot tall, which is about one storey of this building."
Obviously this isn't possible for every number in every Imperial unit, but just figuring out the most commonly used stats helps a lot with intuition. Especially if you are playing in a regular location. Then when the DM says "The far wall of the cave is 90ft away", you can glance out of the window and think "Oh, between the road and the fence. I see how far that is."
(My parents would use this technique when I was a child to explain the size of marine animals. Without anything else on screen for perspective, it's hard to grasp what a 9m wide manta ray actually would look like in person. This has since been very useful in picturing dragon sizes...)
Personally I stopped bothering with converting unless it is absolutely needed. I use whatever units the game provides for rules related things and metric (or general descriptions) when it comes to 'fluff' like weight and height of a character. It can make things a bit more abstract at times, but I find it works really well once you get used to it. I find it less distracting than losing time continously converting at least. A minor benifit is that since the game not a perfect simulation of reality anyway, it sometimes helps to seperate the rules and the descriptions.
Two examples:
## Size, Distance and Speed
All the books and adventures use the same measurements for distance. Therefore, I know that if I always use the ones in the books, I am being consistent. To get an idea of how to visualize things, I just compare and generalize things to get an idea of what things look like. For example: if the monster is standing 20 feet away and I have 30 feet of movement, I know enough: it is well out of my melee reach, but I can get there with a handful of steps during my turn. I can convert 20 feet to 6m, but I don't see that much added value in knowing the exact distance.
Using (5 feet) squares as a simplification tends to work great for movement and scale. This is the most Obvious when using a map with a grid, but I find that it helps even if you do not use a grid. It gives you smaller units to work with when calculating distances (which makes the math easier): the monster is 4 squares away and I have 6 squares of movement during my turn. Even when not using a map, a square is an easy shorthand for describing 'the space a single small or medium person controls during battle'. Additionally: conversion from squares to meters is fairly easy (5 feet is= 1.5m).
## Weight, Objects and Equipment
For the game, weight only matters in specific occasions like carrying capacity. When weight matters for the rules (like calculating carrying capacity), I use the book values. Occasionlly you will need a conversion, like when trying to lift a PC with a levitiation spell that has a limit in pounds. Most of the time though, you can just wing it (putting a reasonable height and weight on a human NPC is not that hard) or just use broad descriptions that do not require exact values (the NPC is tall/short/fat/skinny/...). For equipment you can use real life objects to get an idea of what they look like. If at any point you need stats, just pick something that matches in the game.
D&D uses the imperial system not only because the USA (where it was written) uses it, but because meters seems modern.
D&D is evokes an archaic feel.
If you want to keep that feel, but want units you can understand, try this:
Simply replace 5' by 1 yard (which is 3'). This shortens a bunch of distances, but no so much as anyone would notice. A yard is about a meter, or the distance between your nose and your finger when you stretch out your arm, or the length of a stride.
Speeds are a bit slower, combat a bit more crowded, creatures a touch smaller, weapons don't reach as far, buildings a bit smaller. But none of these amounts are large enough that they'll stretch belief.
If you are really worried about people moving 60% as fast, make combat rounds 4 seconds instead of 6. But I wouldn't bother.
Alternatively, replace 5' with 2 yards = 2 meters. I'd argue against this, because almost all distance measurements in D&D are in multiples of 5', and the conversion to yards without the *2 makes things simpler.
If you don't want the feel and just want meters, use meters like I suggested you use yards above.
A fun part of using yards is that you can tell your players what it is, and explain that when you say something is 10 yards away you don't mean a measuring tape, but rather their best guess at pace counting.
• but because meters seems modern. Source? – JBC Aug 28 '17 at 16:01
• Rather than stating that meters seems modern, I would approach from the actual concepts used for deciding the distances. Why Imperial seems more archaic, specifically. For example, an inch was defined as the length of three grains of barley laid end to end (old legal definition). en.wikipedia.org/wiki/English_units#Length This shows how they were measured such that an 'uneducated commoner' can use these measurements. – Aviose Aug 28 '17 at 19:52
• For what it's worth, the United States does not use the Imperial system. The systems are close enough for game purposes, but see en.wikipedia.org/wiki/United_States_customary_units for a list of differences. – Perkins Aug 28 '17 at 21:07
• This question was revised to no longer ask why the system uses imperial instead of metric or doesn't have a metric edition. – doppelgreener Aug 30 '17 at 18:33
# Abstract it out or approximate the simplest way you can
Around the tables I have played at, we dealt with it with heavy dose of approximation. Most of the time, what the player really needs is "is the target too far for my spell/bow", the GM should use this to his advantage to adapt the distance. As a GM, part of the job is to to adjust the situations so that the mechanics work the way you want. Distance is just another variable.
In the case of feet to meter transition and precise distance, most games I've played used a grid approximation. When a better approximation was needed,we converted 5ft to 1m. Sometime adding a loose approximation of half the distance to get a better precision.
In combat with a grid, we would count squares for moves and range. So walking walking distance was measured as 6 squares instead of 30ft.
In combat without grid, we would count distance as multiple of a turn's walking distance. A target is either in melee (<10ft), in walking range (<30ft), charge range (<60ft), bow range (<160ft if memory serves) or not in combat (anything more than that)
For travel and distances between places in the world, the GM would express it in time at walking speed. So that a given point of interest would be at "3 days of walking" or "A day by boat". Inconsistencies between modes of transportation or different journeys were lampshaded respectively via different paths or changing conditions.
In roleplay scenes, distance ended up being abstracted. Instead of "How far is the cook from me?", we would ask the GM :
• Can I use charm person on the cook from where I am?
• How far can you cast?
• 30ft.
• No, it's a really big room you're in.
Again, if we want a mental image of how far things were, divide by 5 and we have the In-universe distance.
As for why the imperial system is still in use, I would guess that they keep it because it's always been that way. But when the first editions came out, the US didn't use the metric system and the UK (see KorvinStarmast's comment below) had not yet widely accepted it.
• When the game came out(1974), the Brits had not yet gone metric (road signs were still in miles, but I think the move in that direction had begun) . They had already crossed their currency over from shillings and pence and pounds to the decimal pence to pounds method. I was in the UK in 1971 and, having been familiar with shillings, pounds and pence, as well as the decimal Deutschmark, just had to work it out in my head. – KorvinStarmast Aug 30 '17 at 13:49
• Please note that the original post has been updated to remove the second part about "why is it this way." I don't know that you need to edit your answer much--the last paragraph doesn't seem out of place--but there's no longer an obvious "second point" to respond to. – nitsua60 Aug 30 '17 at 14:50
• Thanks for the intel you two. I didn't have time to check it since I wrote it. – 3C273 Sep 1 '17 at 0:51
• Probably not worth much, but rereading it, I realised it was very hastily written. I've reformulated a few points. – 3C273 Sep 1 '17 at 1:09
I tried to get used to using imperial measurements when playing the game. This strategy never really worked that well.
I wouldn't say it was a total failure, though. Feet generally aren't a problem for me any more, as most distances used in the game are multiples of five feet, and five feet is "about one dude tall," so all I have to do is convert squares of movement into dudes lying down.
I've given up on ever understanding pounds, though. Fortunately, my campaign is such that I rarely need to know how much pounds are: Most of the things my players carry around indefinitely are standard items of equipment with listed weights. They occasionally pick up other things, but rarely carry them long enough for their precise weight to become important.
Anyway, I eventually gave up on using pounds and switched to using a simplified system of encumbrance I imported from the Adventurer Conqueror King System. | {"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": 0.4787880480289459, "perplexity": 1300.3240669069553}, "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/1606141716970.77/warc/CC-MAIN-20201202205758-20201202235758-00502.warc.gz"} |
https://www.biostars.org/p/439134/#439164 | Different option in running BWA-MEM
0
0
Entering edit mode
18 months ago
Hi, I'm comparing multiple pipeline, I found that multiple pipelines using BWA-MEM but with different option, and I don't understand all the meaning of its:
bwa mem -Y
(follow by FixMateInformation of Picard, then MarkDuplicate and so on)
2, From GATK Best Practice - in workshop poster: https://drive.google.com/drive/folders/1Nh73FzKde203gUoxyR9CmTd1EcVDMCI5
bwa mem -M
(follow by merging with an unmapped bam file, then MarkDuplicate and so on)
bwa mem
(then MarkDuplicate and so on)
From BWA MEM help option:
-Y use soft clipping for supplementary alignments
-M mark shorter split hits as secondary
But I don't know why CCDG use -Y with FixMateInformation (while others don't), and why GATK need the unmapped bam? Also, as the document of bwa mem said that -M is necessary for picard MarkDuplicate, both Parabrick and CCDG don't use that option.
Could someone please explain to me these difference? And which one is the most reliable one?
Thank you very much
Assembly alignment • 1.2k views
2
Entering edit mode
When you use -Y on secondary alignments the sequence that aligned to the primary alignment won't get hard clipped but softclipped, in my case I exploited that for split reads when a rearrangement occurred in the middle of the read. This means the clipped sequence will be preserved in the bam line of the secondary alignment and the CIGAR operation is softclip instead of hardclip.
0
Entering edit mode
Thank you for clarification of the -Y flag, I'm comparing GATK best practice with Parabrick output, the former one misses some variants (sadly, important ones) so I'm quite worry about the quality of the alignment.
It's quite confusing, because it seems like the GATK best practice pipeline will have more information.
0
Entering edit mode
You are good to go with the default settings unless the specific downstream pipeline you want to run explicitely advises you to change them. The -M option was used some years ago when Picard tools was not fully able to deal with the way bwa marked split reads, but this is afaik now solved for quite some time. From what I know the flag is typically now used anymore. Never used -Y myself, as I said unless any tool advises you to do so I would leave it at default settings since tools assume a certain output.
0
Entering edit mode
thank you about your comment, so how's about the usage of unmapped bam file? AFAIK, only GATK Best Practice makes use of it.
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Entering edit mode
Hello nguyenhy258!
It appears that your post has been cross-posted to another site: https://bioinformatics.stackexchange.com/questions/13328/different-options-in-running-bwa-mem
This is typically not recommended as it runs the risk of annoying people in both communities. | {"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": 0.3066157400608063, "perplexity": 5168.4256191837485}, "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/1637964363659.21/warc/CC-MAIN-20211209030858-20211209060858-00633.warc.gz"} |
http://math.stackexchange.com/questions/71354/question-on-calculating-throughput | # Question on calculating throughput
`Here is the necessary information:
A file contains:
• A header of size of h bits
• Data component of size d bits
• There is a probability b that a bit will be dropped (ruining the file)
The connection can support a maximum of c bits/second
I need to:
• Calculate the probability a file will be ruined. Already did this: (1-b)^(h+d)
• Compute the throughput over the link. (h remains the same size, so what is the optimal value for d?)
I'm having problems coming up with a function to model the number of files being sent through. If I could do that, I think it would be simple to maximize it.
Any help is appreciated.
-
I have removed [calculus] and [logic] since they most definitely did not fit the question. I am not even sure about [probability] and [statistics], but I cannot decide in this case. I am also not too sure what tag would fit here instead of these four. I will leave it to more competent folks, then... – Asaf Karagila Oct 10 '11 at 5:35
Your throughput is about $c (1-b)^{h+d} \dfrac{d}{h+d}$, which you want to maximise by changing $d$.
If you take the derivative with respect to $d$ and set this to 0, I think you may find yourself solving $d^2+hd +h/\log_e(1-b)=0$
Since $d$ is an integer, I don't like taking the derivative with respect to $d$ as @Henry has suggested. So, as an alternative, taking the throughput to be $c(1-b)^{h+d} \frac{d}{h+d}$, we look at what happens if we increase $d$ by $1$. The throughput is now $c(1-b)^{h+d+1} \frac{d+1}{h+d+1}$ and we can examine the ratio $$\frac{c(1-b)^{h+d}\frac{d}{h+d}}{c(1-b)^{h+d+1} \frac{d+1}{h+d+1}} = \frac{d(h+d+1)}{(1-b)(d+1)(h+d)}$$ to find the smallest value of $d$ for which the ratio has value $1$ or more. Once again, the answer is the solution to a quadratic equation (and only slightly different from Henry's result). | {"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": 0.9196003675460815, "perplexity": 349.6978964732703}, "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-2014-42/segments/1413507447660.26/warc/CC-MAIN-20141017005727-00224-ip-10-16-133-185.ec2.internal.warc.gz"} |
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# Proof Mooted For Heisenberg's Uncertainty Principle
#### Soulskill posted about a year ago | from the he's-the-one-who-knocks dept.
158
ananyo writes "Encapsulating the strangeness of quantum mechanics is a single mathematical expression. According to every undergraduate physics textbook, the uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a subatomic particle — the more precisely one knows the particle's position at a given moment, the less precisely one can know the value of its momentum. But the original version of the principle, put forward by physicist Werner Heisenberg in 1927, couches quantum indeterminism in a different way — as a fundamental limit to how well a detector can measure quantum properties. Heisenberg offered no direct proof for this version of his principle. Now researchers say they have such a proof. (Pre-print available at the arXiv.) If they're right, it would put the measurement aspect of the uncertainty principle on solid ground — something that researchers had started to question — but it would also suggest that quantum-encrypted messages can be transmitted securely."
cancel ×
### That's nice (0)
#### Anonymous Coward | about a year ago | (#44110081)
But I'm more confused than before. :(
I guess I can't read the article and understand it at the same time. Curse my very small, very limited brain!
### Re:That's nice (5, Funny)
#### Z00L00K (682162) | about a year ago | (#44110173)
Yet another proof of the principle.
Now let's see what the cat has to say about it.
### Re:That's nice (1)
#### RaceProUK (1137575) | about a year ago | (#44110559)
My guess would be 'miaow'
### Re:That's nice (2)
#### dreamchaser (49529) | about a year ago | (#44110779)
My guess would be 'miaow'
Only if it's still alive when you open the box.
### Re:That's nice (1)
#### Dishevel (1105119) | about a year ago | (#44112015)
But before you open the box the answer is "Miaow" as stated by the GP.
The cat being both dead and alive both makes the sound and does not.
Therefore the sound is definitely heard.
Of course if you listen for the miaow then you are in fact making a measurement. :)
### Re:That's nice (1)
#### geekoid (135745) | about a year ago | (#44112135)
Just because something emits a sound doesn't mean you can measure it.
### Re:That's nice (1)
#### The Mighty Buzzard (878441) | about a year ago | (#44112307)
Unless the experiment caused the cat to be patient zero of the zombie apocalypse. Damned physicists, you've doomed us all!
### Re: That's nice (1)
#### JWW (79176) | about a year ago | (#44110671)
Oh wait, the cats dead. Or is it?
### Re: That's nice (2)
#### SeeingMole (1965542) | about a year ago | (#44110811)
Are you sure there's even a cat in the box?
### Re: That's nice (2)
#### Sponge Bath (413667) | about a year ago | (#44111031)
It's just pining for the fjords.
### Re: That's nice (4, Funny)
#### Bengie (1121981) | about a year ago | (#44111443)
The more you know about the position of the cat, the less you know about its velocity. Ever try to measure the position of a cat that you just dropped into the bathtub? You know it has a high velocity, but it's hard to tell where it really is.
### Re:That's nice (1)
#### wbr1 (2538558) | about a year ago | (#44111287)
Please quit conflating Heisenberg and Schrodinger. Besides, due to Brownian motion, Heisenberg ate Schrodinger's cat whilst standing on the shoulders of giant macroscopic particles.
### Re:That's nice (0)
#### Anonymous Coward | about a year ago | (#44111503)
The headline is wrong. The Webster definition for the verb "Moot" is "to reduce or remove the practical significance of; make purely theoretical or academic." This is the exact opposite.
--Garfield
### Re:That's nice (0)
#### Anonymous Coward | about a year ago | (#44110225)
No matter how much you study it, there still remains some uncertainty about the principle.
### At last! An excuse for our Congress! (-1, Offtopic)
#### smitty_one_each (243267) | about a year ago | (#44110241)
The whole "perfect is the enemy of the good argument" (which I can follow, to a point), has devolved into "Hey, it may be a crap sandwich, but rejoice in the bread."
This mooting/affirmation opens the door, lets out the cat, and affords Congress the ability to say: "Our idiocy is a Heisenberg reference!"
My day is made, and it's not 0600.
### Re:At last! An excuse for our Congress! (-1)
#### Anonymous Coward | about a year ago | (#44110367)
Notice how Congress went to shit when we gave women the right to vote? That's because women are subhuman. Take a list of Nobel laureates, remove all "peace" (even the magic negro got one) prizes and divide by zero^Wgender.
Women are manipulative whores, deriving all their power from subjugation of men with their powers of saggy tits. Women don't achieve anything by themselves. How many hard science papers are actually released by women? Nada.
Like all "special interests" groups, women have an unusual sense of entitlement. Except that unlike niggers, they had every chance to "prove" themselves through the history of mankind, not pick cotton all day long (unless they were female nigra).
The moment you don't want to buy that alligator-skin bag, a diamond ring, a status-symbol car, a kitten-fur coat, not only they spit out instead of swallowing, they pull out a gun on you and accuse you of criminal trespass. Then they miss at point-blank range. Such is the result of granting equal rights to the subhuman gender.
I'm all in favor of "separate but equal" amendment -- grant them a right to the kitchen and Valium, and for men to work their ass off trying to make a name for themselves. Praise the ones who make a difference, not the parasite with a victim complex.
"What is the difference between a man and a parasite? A man builds. A parasite asks "Where is my share?" A man creates. A parasite says, "What will the neighbors think?" A man invents. A parasite says, "Watch out, or you might tread on the toes of god..."
### Re: At last! An excuse for our Congress! (-1)
#### Anonymous Coward | about a year ago | (#44110501)
I see you are recently divorced. Me too. We're having a meeting next month. See you there.
### Re:At last! An excuse for our Congress! (-1)
#### Anonymous Coward | about a year ago | (#44110641)
Haha dude, you seriously need to hang around with more laid back chicks!
### Re:That's nice (3, Funny)
#### TWiTfan (2887093) | about a year ago | (#44110923)
I'm more confused than before
Just look in this box. In it, you'll find either a better summary or a dead cat.
### You keep using that word... (1)
#### Anonymous Coward | about a year ago | (#44110109)
Not to be all pedantic, or anything, but "to moot" something is to debate. If they're mooting a proof, then the proof is very much under debate. /sunglasses
### Re: You keep using that word... (2, Informative)
#### Anonymous Coward | about a year ago | (#44110221)
I too found the title odd
[moot]
1. open to discussion or debate
2. of little practical value
### Re: You keep using that word... (4, Informative)
#### mrvan (973822) | about a year ago | (#44110291)
http://dictionary.reference.com/browse/moot [reference.com] says:
verb (used with object)
4. to present or introduce (any point, subject, project, etc.) for discussion.
5. to reduce or remove the practical significance of; make purely theoretical or academic.
So meaning 4 seems appropriate. Strange that a word simultaneously means to introduce it and to remove it from consideration, but it is a pretty old word I think so it has probably evolved quite a bit.
Origin:
before 900; Middle English mot ( e ) meeting, assembly, Old English gemt; cognate with Old Norse mt, Dutch gemoet meeting. See meet1
### Re: You keep using that word... (1)
#### guttergod (94044) | about a year ago | (#44110429)
http://dictionary.reference.com/browse/moot [reference.com] says:
verb (used with object) 4. to present or introduce (any point, subject, project, etc.) for discussion. 5. to reduce or remove the practical significance of; make purely theoretical or academic.
So meaning 4 seems appropriate. Strange that a word simultaneously means to introduce it and to remove it from consideration, but it is a pretty old word I think so it has probably evolved quite a bit.
Origin: before 900; Middle English mot ( e ) meeting, assembly, Old English gemt; cognate with Old Norse mt, Dutch gemoet meeting. See meet1
Sounds like "theory" to me. What's with science and ambiguous words? :)
### Re: You keep using that word... (2)
#### RaceProUK (1137575) | about a year ago | (#44110573)
http://dictionary.reference.com/browse/moot [reference.com] says:
verb (used with object) 4. to present or introduce (any point, subject, project, etc.) for discussion. 5. to reduce or remove the practical significance of; make purely theoretical or academic.
So meaning 4 seems appropriate. Strange that a word simultaneously means to introduce it and to remove it from consideration, but it is a pretty old word I think so it has probably evolved quite a bit.
Origin: before 900; Middle English mot ( e ) meeting, assembly, Old English gemt; cognate with Old Norse mt, Dutch gemoet meeting. See meet1
Sounds like "theory" to me. What's with the media's reporting of science and ambiguous words? :)
FTFY
### Re: You keep using that word... (0, Flamebait)
#### Anonymous Coward | about a year ago | (#44110815)
It's a tactic they use to keep the uninformed thinking something is fact when in reality it is just someone's (unproven) idea. It's been working for a number of generations now. You'd be surprised how many people think macro evoloution has been scientifically proven, or global warming. They get one piece of the puzzle and think they've got the whole picture.
### Re: You keep using that word... (0)
#### Anonymous Coward | about a year ago | (#44113027)
Downmodded. But no one can explain the thousands of "missing links" in evolution. Why would non-flowering plants evolve into flowering plants unless it happened spontaneously? Why would wolves evolve into domesticated dogs unless spontaneously? Volumes could be written about this, but everyone accepts macro evolution without infallible proof.
### Re: You keep using that word... (1)
#### Speare (84249) | about a year ago | (#44112121)
I was just about to comment about the "uncertainty" in the use of the word moot; whether it meant "to discuss" or "to dismiss need of discussion." A perfect word for the topic, if you think about it.
### contranym (0)
#### Anonymous Coward | about a year ago | (#44112137)
So meaning 4 seems appropriate. Strange that a word simultaneously means to introduce it and to remove it from consideration, but it is a pretty old word I think so it has probably evolved quite a bit.
These class of words are called auto-antonyms (or contronyms):
http://en.wikipedia.org/wiki/Auto-antonym
There's quite a few of them.
### Re: You keep using that word... (1)
#### AlecC (512609) | about a year ago | (#44112577)
I would say the two meanings come from the same source, but with different spins.
Meaning 4: Needs to go to the moot to be debated: truth still uncertain
Meaning 5: Taken out of current consideration by postponing until the moot (which was an annual event)
I.e. meaning 4 regards the moot as a place where complex things are debated, while meaning 5 regards it as an annual event where a lot of hot air is expended about nothing. Both are probably correct.
### Re:You keep using that word... (0)
#### Anonymous Coward | about a year ago | (#44110235)
Not to be all pedantic, or anything, but "to moot" something is to debate. If they're mooting a proof, then the proof is very much under debate. /sunglasses
If you had RTFA (yes I know, this is Slashdot) then you'd know that there's indeed a debate about it.
### Re:You keep using that word... (-1)
#### Anonymous Coward | about a year ago | (#44110391)
yes I know, this is Slashdot
I thought this was /r/slashdot. I wish it were though. Reddit is much better.
### Re:You keep using that word... (1)
#### Barryke (772876) | about a year ago | (#44110583)
I am not a native English speaker butt..
To me, mooted in the past sense meant some (since irrelevant) argument being leveled.
This puts a lot of things i read/heard in the past into a new (almost the opposite) perspective..
Butt pun intended of course.
### Re:You keep using that word... (1)
#### ebno-10db (1459097) | about a year ago | (#44112191)
The joy of English is that it often makes little sense, even to its native speakers (like me). Your understanding of 'moot' is the most common usage, but it can also mean to debate, and a bunch of other vaguely related things.
http://www.merriam-webster.com/dictionary/moot [merriam-webster.com]
### Uncertaintiy principle and Foruier Transforms (5, Interesting)
#### Grantbridge (1377621) | about a year ago | (#44110125)
The uncertainty principle is the same as taking a Fourier transform of a sound pulse. If the time of the wave is short then the uncertainty in the frequency is high, and you get a large width in frequency space. If the wave is on for a long time, you get a nice sine wave and the uncertainty in the frequency is low, but the uncertainty in the time is now high. The maths for momentum/position of electrons comes out the same as time/frequency of sound waves. You get the uncertainty principle with non-quantised waves anyway, its not magic!
### Re:Uncertaintiy principle and Foruier Transforms (0)
#### Anonymous Coward | about a year ago | (#44110161)
The authors understand this of course, as you will see if you look at their paper. They are formulating and proving a different uncertainty principle, more in line with Heisenberg's original physical intuition.
### Re:Uncertaintiy principle and Foruier Transforms (1)
#### wonkey_monkey (2592601) | about a year ago | (#44110245)
If the wave is on for a long time, you get a nice sine wave and the uncertainty in the frequency is low, but the uncertainty in the time is now high.
What do you mean by "the time"? Duration?
### Re:Uncertaintiy principle and Foruier Transforms (0)
#### Anonymous Coward | about a year ago | (#44110289)
he means the position on the time axis (well kinda). In a sense, you can consider the duration as the uncertainty over the time position. The sine is not happening at a precise point in time: since it has a duration it is happening over many point in time.
### Re:Uncertaintiy principle and Foruier Transforms (2, Informative)
#### Anonymous Coward | about a year ago | (#44110299)
He means the location of the sonic event. If you think of sound as particulate (a series of events a la granular synthesis) then the frequency of each event and the location in time of each event satisfy a sort of uncertainty principle. It's because the FFT of sine * normal curve is sine * normal curve, but the width of the normal curve is conjugate in each case (the limiting case is sine * delta -> sine * 1). This width represents the "certainty" that the actual frequency or location in time is at the center point. It's a neat trick but it's not clear how or if it relates to QM, except via the mathematical equivalence. Once you start asking "what, then is h?" or "how does scalar amplitude relate to quantum phase" the illusion of relevance kind of vanishes.
### Re:Uncertaintiy principle and Foruier Transforms (0)
#### Anonymous Coward | about a year ago | (#44110377)
Sorry, my bad, we should have FFT[cosine * 1] -> delta and FFT[cosine * delta] -> 1. There is not a trig function on both sides, and I should be using cosine instead of sine, so I stay in the real domain.
### Re:Uncertaintiy principle and Foruier Transforms (1)
#### Anonymous Coward | about a year ago | (#44110295)
You mean to say that the frequency spectrum of a finite time duration signal is inifinte, while the frequency spectrum of a signal with infinite time duration is finite.
### Re:Uncertaintiy principle and Foruier Transforms (0)
#### Anonymous Coward | about a year ago | (#44110533)
the frequency spectrum of a signal with infinite time duration is finite.
I'm pretty sure that no one means to say that.
### Re:Uncertaintiy principle and Foruier Transforms (0)
#### Anonymous Coward | about a year ago | (#44110653)
The textbook formula is delta X * delta P >= hbar/2.
delta X is the uncertainty in position. delta P is the uncertainty in momentum.
But how do we measure momentum? It's the mass*velocity ==> mass* dX/dT.
So, what we are saying is that we can't know the position (deltaX) very precisely if we know the velocity dX/dT precisely. That makes perfect sense because to know dX/dT, we have to follow the particle for a while, in which case we don't know any more where it is precisely. Or even more basically, as we make dT smaller and smaller, we approach dT=0, at which point the formula is mathematically undefined ( x/0 is always undefined.)
Now, in classical mechanics, one can pretend that he knows these values perfectly and construct a "Worldline" to show a particle's history. But that is an idealization. Quantum Mechanics is what you get when you try to zoom in on things and see what REALLY is going on at the microscopic level. (That's not even touching upon the theory that space itself is quantized.)
### Re:Uncertaintiy principle and Foruier Transforms (1)
#### Mashdar (876825) | about a year ago | (#44110855)
There is no uncertainty in the output of a Fourier transform. What you are refering to are the frequency components of the transients. If you flip a switch, there is a huge amount of non-linearity.
Also, a non-noisy Fourier transform is reversible. This is the exact opposite of uncertainty. :)
### Re:Uncertaintiy principle and Foruier Transforms (1)
#### instagib (879544) | about a year ago | (#44111349)
The uncertainty principle is the same as taking a Fourier transform of a sound pulse.
Which explains why one can't be sure if MP3's are music or not.
### Re:Uncertaintiy principle and Foruier Transforms (0)
#### Anonymous Coward | about a year ago | (#44112367)
justin_beiber.mp3 - definitely NOT music.
### Re:Uncertaintiy principle and Foruier Transforms (1)
#### Connie_Lingus (317691) | about a year ago | (#44111623)
isnt that simply because Dirac's and Schrodinger's QM wave equations are actually Fourier transforms at heart?
### Re:Uncertaintiy principle and Foruier Transforms (1)
#### thrich81 (1357561) | about a year ago | (#44111985)
The AC just stated this, but I'll expound -- the "Fourier transform" uncertainty you describe comes from the simple mathematics of the basics of Quantum theory and I don't really see a way to refute it if you accept those basics (observables of position and momentum are described by linear operators which don't commute). Heisenberg's uncertainty principle (observation of position disturbs momentum and vice versa) is usually described as limitation in the way we can make observations and as such seemed to be (at first glance at least) to be not fundamental and something that could be defeated with clever experimental apparatus.
### Re:Uncertaintiy principle and Foruier Transforms (1)
#### Warbothong (905464) | about a year ago | (#44113285)
observation of position disturbs momentum and vice versa
That's the observer effect, which TFA seems to be talking about. The observer effect implies that there is an exact position and momentum; particles can be little billiard-balls if we like, but any attempt to measure them will disturb them.
The uncertainty principle, as it is currently understood, says that there is no such thing as an exact position or momentum. Particles are wave-packets in force-fields. When we introduce quantum constraints, eg. the integral of (area under) the wave packet is some discrete amount, the uncertainty principle falls out naturally.
It's easiest to imagine the case close to zero. For example, having a near-certain position means not being spread out in space. To maintain the constant area, this requires a large amplitude. Likewise a near-zero amplitude must be spread out in space to make up the same area. This is position/momentum uncertainty.
If we want a packet of near-zero duration (eg. a Dirac delta), then we need to build up lots of Fourier terms, which spreads out the overall wavelength. Likewise, if we want a precise wavelength, we need to get closer to a pure sine wave, but that means we get a longer and longer duration (a completely pure sine wave would last forever). That's basically the energy/time uncertainty.
### only "discrete" Fourier/Integral transforms (1)
#### peter303 (12292) | about a year ago | (#44112421)
The full Fourier Integral has no frequency limits. The discrete transform, i.e the one usually programmed in computers, and its cousin the Fast Fourier Transform, are frequency limited at the small end by the sampling inerval and the large end by the length of the input.
### Re:Uncertaintiy principle and Foruier Transforms (0)
#### Anonymous Coward | about a year ago | (#44112463)
The analogy breaks here: for a sound pulse you can increase the sampling rate and subsequently increase both time and frequency resolution.
In the quantum world you have that annoying speed of light constant that you cannot increase.
### Certain uncertainty (1)
#### Anonymous Coward | about a year ago | (#44110153)
The uncertainty principle applies to everything, not just subatomic particles. Just that most of the time the precision required to test it is impossible to achieve (see the wavelength of the Sun for instance). As examples of macroscopic systems where it does apply, see uncertainty relations for the superconducting state.
### Re:Certain uncertainty (1)
#### VortexCortex (1117377) | about a year ago | (#44110207)
The uncertainty principle applies to everything,
How uncertain are you that this is true?
### Re: Certain uncertainty (0)
#### Anonymous Coward | about a year ago | (#44110403)
The uncertainty principle applies to everything,
How uncertain are you that this is true?
Pretty certain. Don't worry, the conjugate variable has infinite variance so it's all good. :-)
### Yo Yo Mr. White,....... (5, Funny)
#### AbRASiON (589899) | about a year ago | (#44110157)
Don't fuck with Heisenberg folks.
...bitch
### Re:Yo Yo Mr. White,....... (0)
#### Anonymous Coward | about a year ago | (#44111937)
LOL! I was thinking that, too! :P
### Re:Yo Yo Mr. White,....... (0)
#### Anonymous Coward | about a year ago | (#44111495)
Can someone explain how abrasion got a +5 ? One for each word?
### Re:Yo Yo Mr. White,....... (0)
#### Anonymous Coward | about a year ago | (#44113739)
It was funny to the unintelligent readership slashdot now has. Notice how most comments on this page are uninformed references to a thought experiment designed to show how absurd QM is.
Solve for X
Infinity
I win
### Fixed the summary (1, Informative)
#### angel'o'sphere (80593) | about a year ago | (#44110201)
... the uncertainty principle states that it is impossible to simultaneously know^H^H^H^H measure the exact position and momentum of a subatomic particle the more precisely one knows the particle's position at a given moment, the less precisely one can know the value of its momentum.
Fix.
### Re:Fixed the summary (3, Informative)
#### Prune (557140) | about a year ago | (#44110259)
It's not just a practical issue for measurement, so your "fix" is invalid. The correct explanation is in this post: http://science.slashdot.org/comments.pl?sid=3904863&cid=44110125 [slashdot.org]
### Re:Fixed the summary (1)
#### angel'o'sphere (80593) | about a year ago | (#44110285)
My fix is valid. The article summary is simply wrong.
The post you link is only a simplified explanation for a lay man (and has nothing to do with heisenberg, it has btw its own name: "Shannons sampling theorem").
### Re:Fixed the summary (2, Informative)
#### Anonymous Coward | about a year ago | (#44110319)
Indeed. I don't know what crap "undergraduate textbooks" people use near the north pole, but here down under, the principle of Heisenberg is taught using _math_.
It has always been about measuring (not "knowing", the universe doesn't give a damn about what you know or don't know or it would forbid god from existing. Instead, it just hampers aquiring new knowledge of the full state vector ;p). And it has always been a nice mathematical, strictly quantified trade off between the precision you'll get out of one of the measurements being inversely correlated to the precision you'll get out of the other measurement because the product of the two must be at least half the reduced plank constant.
### Re:Fixed the summary (0)
#### Anonymous Coward | about a year ago | (#44113761)
No, it is not valid. http://slashdot.org/comments.pl?sid=3904863&cid=44112299
### Re:Fixed the summary (0)
#### etash (1907284) | about a year ago | (#44110509)
you are completely wrong. heisenberg's principle is purely PRACTICAL. it doesn't say that in theory the particle won't have a specific momentum at a specific position. It just states our practical inability to measure with accuracy both simultaneously.
### Re:Fixed the summary (0)
#### Anonymous Coward | about a year ago | (#44110551)
> ... it doesn't say that in theory the particle won't have a specific momentum at a specific position
Actually it says exactly that.
### Re:Fixed the summary (0)
#### etash (1907284) | about a year ago | (#44110753)
nope it doesn't. learn to read and comprehend.
### Re:Fixed the summary (1)
#### geekoid (135745) | about a year ago | (#44112299)
and you can teach you grandmother to suck eggs. You are wrong.
"it doesn't say that in theory the particle won't have a specific momentum at a specific position."
Wow, that's not even wrong.
That's not what anyone is saying. You can not measure it's momentum and position with the same measurement, not to be confused with the observer effect.
The theory says its in the fundamental nature of all quantum systems. IN fact, it's in all systems, just the the quantum system i'ts more obvious.
How to you explain your statement again de Brooglie work? you ARE familiar with de Brooglie's work, right? you would just make such a statement without at least the basic fundamental reading of his work, right? RIGHT?
Ignorance can be fixed, so I don't mind that but bold face incorrect statements from pieces of crap like you piss me off.
### Re:Fixed the summary (-1)
#### Anonymous Coward | about a year ago | (#44114055)
geekoid, you're such a dumb fuck. And it's "bald face" not "bold face". God damn.
### Re:Fixed the summary (0)
#### Anonymous Coward | about a year ago | (#44110767)
Exactly. People's intuition that particles are somehow tiny billiard balls and thus "must" have a specific position and momentum is WRONG. That's what Heisenberg's principle is about. The universe is FAR stranger than our intuitions about it would allow, and one of the ways in which it's stranger is this uncertainty.
People ALWAYS muddle this up with a simple problem from introductory Newtonian mechanics, about how difficult it is to measure things accurately. But that's not Uncertainty. If you learned that it's the same thing you learned WRONG and whoever taught that fucked up, probably because they were trying to teach you without understanding what the hell they were talking about. You will get precisely nowhere in quantum mechanics by believing that the quantum world is just a tinier version of the intuitive human scale world. But that's OK so long as all you really wanted was to get a job on Wall Street or something. It's only a problem if you actually wanted to do Quantum Physics.
### Re:Fixed the summary (0)
#### Anonymous Coward | about a year ago | (#44111107)
to prove something you need to link to something better than the comment section of slashdot
### Re:Fixed the summary (1)
#### The_Wilschon (782534) | about a year ago | (#44114007)
Check chapter 9, (pages 237 and following), of the second edition of Principles of Quantum Mechanics by Ramamurti Shankar. Or, section 1.6 (page 18-20) and section 3.5 (page 110-118), of the second edition of Introduction to Quantum Mechanics by David J. Griffiths.
I'm sorry that I can't hyperlink to a physical book. But maybe you could go to your local public library and find a copy of one of them.
### Re:Fixed the summary (2)
#### The_Wilschon (782534) | about a year ago | (#44113865)
Correct fix: The uncertainty principle states that it is impossible for a particle to be in a state in which both the position and momentum (or any pair of observables represented by non-commuting operators) are exactly defined, or even well-defined beyond a certain limit determinable from the commutator of the pair of operators.
It has nothing to do with measurement, and everything to do with the mathematical existence of quantum states with certain properties. TFA is actually dealing with the observer effect, which does have to do with measurement, and which was Heisenberg's original intuitive idea.
### Define "secure" in this day and age. (1)
#### Anonymous Coward | about a year ago | (#44110255)
"...but it would also suggest that quantum-encrypted messages can be transmitted securely."
Well, I suppose that would depend on the level of ignorance one carries around when defining "secure".
Somehow, I strongly doubt this will be above and beyond NSA's illegal and highly classified activity to ensure we're all safe from terrorists.
### Re:Define "secure" in this day and age. (0)
#### Anonymous Coward | about a year ago | (#44110819)
Even the NSA cannot circumvent the very laws of physics.
### Re:Define "secure" in this day and age. (0)
#### Anonymous Coward | about a year ago | (#44112427)
I don't know about that. Circumventing laws is the stock and trade of the NSA.
### Why "Proof Mooted"? (0)
#### Anonymous Coward | about a year ago | (#44110279)
The headline does not fit the summary at all.
### Just accept QM already (0)
#### Warbothong (905464) | about a year ago | (#44110327)
the uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a subatomic particle
I find phrases like this misleading. I think it's more intellectually honest to say something along the lines of:
the uncertainty principle states that position and momentum are not independent quantities, but (incompatible) expressions of a more fundamental property.
Popsci keeps claiming that 'everything we thought knew is wrong' based on the slightest whiff of a strange experimental result, yet when quantum mechanics *does* prove wrong everything we thought we knew (like the concepts of position and momentum), with repeated experiments of incredible precision, popsci clings to those old notions and acts like QM is wacky.
### Re:Just accept QM already (1)
#### wonkey_monkey (2592601) | about a year ago | (#44110353)
I find phrases like this misleading.
John Q Public finds that sort of phrase quickly and easily understandable, which, I would say, are not attributes of your proposed replacement.
### Re:Just accept QM already (0)
#### Anonymous Coward | about a year ago | (#44110941)
Look back 1000 years to what we knew then. How much of that remains true today? In the grand scheme of things we know exactly jack schite. What makes you think that when someone went "ahhh, we had that wrong. It actually works like this...." they got it right this time? No, 1000 years from now we'll look as foolish as those who believed the earth was flat and the universe was earthcentric.
I don't say that to defend Popsci, but to point out that they are only doing what science does: continually revise update, and acknowledge that not much is set in stone.
### I laughed... (5, Funny)
#### Valentttine (2420782) | about a year ago | (#44110337)
Heisenberg was speeding down the highway. Cop pulled him over and says "Son, do you have any idea how fast you were going back there?" Heisenberg said, "No, but I knew where I was". The cop says "You were doing 100 miles an hour" to which Heisenberg replies "Great, now I'm lost".
### Re:I laughed... (0)
#### Anonymous Coward | about a year ago | (#44110483)
Heisenbergs wife asks him "Do you know where my car keys are?", "I have no idea" he replies, "but I can tell you exactly how fast they are moving."
### Re:I laughed... full version ;) (5, Funny)
#### HxBro (98275) | about a year ago | (#44110581)
Heisenberg and Schrodinger are driving, and get pulled over.
Heisenberg is in the driver's seat, the officer asks "do you know how fast you were going?"
Heisenberg replies, "No, but I know exactly where I am!"
The officer looks at him confused and says "you were going 108 miles per hour!"
Heisenberg throws his arms up and cries, "Great! Now I'm lost!"
The officer, now more confused and frustrated orders the men outside of the car, and proceeds to inspect the vehicle. He opens the trunk and yells at the two men, "Hey! Did you guys know you have a dead cat back here?"
Schrodinger angrily yells back, "We do now, asshole!"
### I just read the article ( arXiv PDF ) (3, Interesting)
#### vikingpower (768921) | about a year ago | (#44110417)
It seems the paper can be understood with undergraduate mathematics. The 3 authors' argumentation seems quite clear, and their proof rather convincing. One wonders, now and at this point, whether a lab experiment could be set up to falsify the whole thing... If not, Heisenberg stands proven true. Of the impact upon quantum cryptography I am not so sure, however, supposing that it takes "some quite advanced mathematics" ( as Wolfram once said about cyclotomic fields ) to tackle that issue.
### Re:I just read the article ( arXiv PDF ) (2)
#### NoNonAlphaCharsHere (2201864) | about a year ago | (#44110547)
The quantum cryptography issue is a question of whether or not it is possible in principle to eavesdrop on (measure) a quantum system without disturbing it.
### Re:I just read the article ( arXiv PDF ) (1)
#### Anonymous Coward | about a year ago | (#44110693)
Quantum cryptography leans very heavily if it is possible to measure two different attributes of a quanta. For most quantum cryptography the quantum is a single photon and the different attributes:
- horizontal or vertical polarization
- the two diagonal polarization.
It is important that you can only measure either the top two (horizonal/vertical) or the bottom two (two diagonals) but never both.
### Re:I just read the article ( arXiv PDF ) (0)
#### Anonymous Coward | about a year ago | (#44111419)
GCHQ and NSA already tap fiber optics underseas from all countries.
So it is possible. Whether they can detect the evesdropping, that is another mattern but they ARE evesedropping and have been since the days of fiber began.
### Proof is already from 1929 (4, Interesting)
#### johanw (1001493) | about a year ago | (#44110455)
Robertson proved in 1929 already the general form of the uncertainty relation. It has nothing to do with Fourier transforms, wavefunctions and disturbance by measurements, but only with the operator character of (some) quantum mechanical observables. I got the proof from this textbook by Stephen Gasiorowicz, unfortunately they skipped this important result from the latest edition (that circulates on internet in the usual places). More information can be found in https://en.wikipedia.org/wiki/Uncertainty_principle#Robertson.E2.80.93Schr.C3.B6dinger_uncertainty_relations [wikipedia.org]
From Quantum Physics by Stephen Gasiorowicz, ISBN 0 471 29281-8
It is important to note that the uncertainty relation
(Delta A)^2 (Delta B)^2 >= \langle i[A,B] \rangle^2 / 2
was derived without any use of the wave concepts or the reciprocity between
a wave form and its fourier transform. The results depends entirely on the
operator properties of the observables A and B.
### Re:Proof is already from 1929 (2)
#### dpilot (134227) | about a year ago | (#44110759)
I find it interesting that there is generally such discomfort with Heisenberg's Uncertainty. I'll grant that its application to quantum cryptography is practical, but for the most part I think this discomfort is rooted in people not liking that something isn't just unknown, but unknowable.
Doesn't bother me a bit - once you accept that idea that quantum mechanics actually does describe reality.
Or another way of looking at it - if you consider all of reality to be a giant simulation, "Aitch-Bar" (Credit for spelling to college prof, name forgotten. (Phillip Bevington?)) becomes simply the error criteria used by the simulator to define a "step".
### Re:Proof is already from 1929 (1)
#### geekoid (135745) | about a year ago | (#44112349)
unknowable but nor unpredictable. If I know it's movement I can predict* it's location at a latter time of measurement. Of course at the later time I will not know the momentum at that particular time.
*Probably :)
### Re:Proof is already from 1929 (1)
#### wonkey_monkey (2592601) | about a year ago | (#44112471)
unknowable but nor unpredictable.
It is as unpredictable as it is unknowable - that is, to a certain degree as defined by the HUP - because prediction requires knowledge. If your knowledge is imperfect, your prediction will be imperfect.
### Re: Proof is already from 1929 (0)
#### Anonymous Coward | about a year ago | (#44110809)
How would you define operators without touching wave functions?
### Re:Proof is already from 1929 (0)
#### Anonymous Coward | about a year ago | (#44111845)
It has nothing to do with Fourier transforms, wavefunctions and disturbance by measurements
In the very article you link it explains that the Uncertainty Principle gives rise to the Gabor Limit when applied to time-frequency analysis, so it's in fact the same effect, only more generally expressed.
### Obligatory Douglas Adams quote:- (0)
#### Anonymous Coward | about a year ago | (#44110461)
"Rigidly defined areas of doubt & uncertainty"
### Re:Obligatory Douglas Adams quote:- (1)
#### FearTheFez (2592613) | about a year ago | (#44111013)
To be used as the theoretical basis of the Infinite Improbability Drive. Now to go make a really HOT cup of tea. Where did Marvin put the kettle.............
### It was a bittersweet occasion (0)
#### Anonymous Coward | about a year ago | (#44110881)
The acceptance letter:
"We are pleased to announce that your paper has been accepted for publication so-and-so and will appear in issue such-and-such.
### Phrasing (1)
#### Tyler Durden (136036) | about a year ago | (#44111637)
According to every undergraduate physics textbook, the uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a subatomic particle...
What always struck me about the above statement is it seems to imply that there is an exact simultaneous position and momentum to subatomic particles that cannot be known. Maybe the truth is stronger than that - subatomic particles simply don't have precise position/momentums.
### Duh... (1)
#### WillyWanker (1502057) | about a year ago | (#44112501)
Everyone knows you need to have good, fully functional Heisenberg compensators, right?
### But what if two observers look at the particle? (0)
#### Anonymous Coward | about a year ago | (#44112603)
One looks at the position and the other looks at the velocity at the same time?
Oh, and they tell each other the position and velocity AT THE SAME TIME.
Apparently Heisenberg didn't have a friend.
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http://adv-math.com/common-fixed-point-theorems-for-maps-that-satisfy-a-contraction-principle-involving-a-rational-expression/ | R. Arab, Generalized contractions and common fixed point theorems in ordered metric space for weakly compatible mappings, Volume 2016, Number 1, Pages 37-48, 2016
# R. Arab, Generalized contractions and common fixed point theorems in ordered metric space for weakly compatible mappings, Volume 2016, Number 1, Pages 37-48, 2016
Abstract: In this paper, we prove common fixed point theorems for maps that satisfy a contraction principle involving a rational expression in complete metric spaces. Presented theorems extend and generalize some existence results in the literature.
Key words: common fixed point; coincidence fixed point; w-compatible mappings; ordered metric space.
Full text: PDF
Full file: XML | {"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": 0.9725673794746399, "perplexity": 2326.652833900374}, "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/1524125947968.96/warc/CC-MAIN-20180425213156-20180425233156-00304.warc.gz"} |
http://financepress.com/tag/xgbm/ | # Introducing XGBM — a new solvable stochastic volatility model having a stationary distribution
I have posted a new research paper to the arXiv titled “Exact Solutions for a GBM-type Stochastic Volatility Model having a Stationary Distribution”. The article may be found here.
Let me explain the title and motivation. The most popular stochastic volatility model is certainly the Heston ’93 model. Its virtues are that it is exactly solvable and it has some very nice features: a mean-reverting, stochastic volatility process that can be negatively correlated with equity returns. This process has the well-known “square-root model” form
$\displaystyle \mbox{Heston:} \quad d v_t = (\omega - \theta v_t) \, dt + \xi \,\sqrt{ v_t} \, dW_t,$
where $v_t$ is the instantaneous variance rate of the asset in question. We could also write this evolution in terms of the instantaneous volatility $\sigma_t = \sqrt{v_t}$. However, time series analysis of the volatility of broad-based indices, such as the S&P500 index, suggests that a better specification would have $d \sigma_t \sim \xi \sigma_t dW_t$, which is a geometric Brownian motion (GBM)-type volatility. Taking $d \sigma_t = \xi \sigma_t dW_t$ is the well-known SABR model. The problem with that one is that the volatility is not mean-reverting and does not admit a stationary density $\psi(\sigma)$. To get a stationary density, you need a mean-reverting drift term. Also, it would be very nice to also have a model that has exact solutions. The XGBM model may be the first to combine all these properties. It is a standard bivariate model for the pair $(S_t,\sigma_t)$, but here I’m just going to write the volatility evolution, which is
$\displaystyle \mbox{XGBM:} \quad d \sigma_t = \sigma_t (\omega - \theta \sigma_t) \, dt + \xi \, \sigma_t \, dW_t.$
To see how this model is closer to the “real-world” than the Heston ’93 model, take a look at the figures at the end, which show Maximum Likelihood fits to a proxy series for $\{\sigma_t\}$. The proxy is the daily (annualized) volatility for the S&P500 taken from the Oxford-Man Institutes “realized library”. They maintain a number of estimators — I am using a basic one (rv5) which is simply the daily (intraday) volatility using 5-minute log-return observations. I am using all the data available at the time of this study, which is January 3, 2000 – September 28, 2018 (4705 volatility observations). The first figure shows this volatility time series. You can see there is a maximum of around 140% which is the annualized volatility at the height of the 2007-2008 Financial Crisis.
The next figures show the stationary volatility fits. For these histograms, I am using the annualized volatility in decimal (not percent), so the time series maximum is a histogram entry at $\sigma \approx 1.4.$ That small bump is not really visible, but you can see it is accounted for by the axis extension out to that value.
As one can see, the visual fit is better for XGBM vs. Heston, with corresponding log-likelihoods: 5356 (Heston) vs 5920 (XGBM). In fact, the fit is even better (LL = 6055) for the figure labeled “GARCH -3/2 model”, which is the stationary density corresponding to both the GARCH diffusion model and the 3/2-model. In any event, the point of the exercise is to motivate my new paper and interested readers may find it at the link above.
Update. This research paper is now published. The cite is “Lewis, Alan L. (2019), “Exact solutions for a GBM-type stochastic volatility model having a stationary distribution”, Wilmott mag, Vol. 2019, Issue 101, May, 20-41. The editors were kind enough to make it a
cover story. As part of that publication, I wrote an additional Introduction, which is available here: XGBMIntro.WMag.Final. | {"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": 10, "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": 0.8902119398117065, "perplexity": 856.5752259645019}, "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-2021-04/segments/1610703505861.1/warc/CC-MAIN-20210116074510-20210116104510-00734.warc.gz"} |
https://yutsumura.com/tag/graph/?wpfpaction=add&postid=1549 | # Tagged: graph
## Problem 217
Let $A, B, C$ are $2\times 2$ diagonalizable matrices.
The graphs of characteristic polynomials of $A, B, C$ are shown below. The red graph is for $A$, the blue one for $B$, and the green one for $C$.
From this information, determine the rank of the matrices $A, B,$ and $C$.
Graphs of characteristic polynomials | {"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": 0.9022590517997742, "perplexity": 394.95632415429964}, "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/1603107878662.15/warc/CC-MAIN-20201021235030-20201022025030-00592.warc.gz"} |
http://mathematica.stackexchange.com/users/240/platomaniac?tab=activity&sort=all&page=14 | # PlatoManiac
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http://repository.kaust.edu.sa/kaust/handle/10754/626500 | # HLIBCov: Parallel Hierarchical Matrix Approximation of Large Covariance Matrices and Likelihoods with Applications in Parameter Identification
Handle URI:
http://hdl.handle.net/10754/626500
Title:
HLIBCov: Parallel Hierarchical Matrix Approximation of Large Covariance Matrices and Likelihoods with Applications in Parameter Identification
Authors:
Litvinenko, Alexander ( 0000-0001-5427-3598 )
Abstract:
The main goal of this article is to introduce the parallel hierarchical matrix library HLIBpro to the statistical community. We describe the HLIBCov package, which is an extension of the HLIBpro library for approximating large covariance matrices and maximizing likelihood functions. We show that an approximate Cholesky factorization of a dense matrix of size $2M\times 2M$ can be computed on a modern multi-core desktop in few minutes. Further, HLIBCov is used for estimating the unknown parameters such as the covariance length, variance and smoothness parameter of a Mat\'ern covariance function by maximizing the joint Gaussian log-likelihood function. The computational bottleneck here is expensive linear algebra arithmetics due to large and dense covariance matrices. Therefore covariance matrices are approximated in the hierarchical ($\mathcal{H}$-) matrix format with computational cost $\mathcal{O}(k^2n \log^2 n/p)$ and storage $\mathcal{O}(kn \log n)$, where the rank $k$ is a small integer (typically $k<25$), $p$ the number of cores and $n$ the number of locations on a fairly general mesh. We demonstrate a synthetic example, where the true values of known parameters are known. For reproducibility we provide the C++ code, the documentation, and the synthetic data.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
arXiv
Issue Date:
24-Sep-2017
ARXIV:
arXiv:1709.08625
Type:
Preprint
http://arxiv.org/abs/1709.08625v1; http://arxiv.org/pdf/1709.08625v1
Appears in Collections:
Other/General Submission; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
DC FieldValue Language
dc.contributor.authorLitvinenko, Alexanderen
dc.date.accessioned2017-12-28T07:32:13Z-
dc.date.available2017-12-28T07:32:13Z-
dc.date.issued2017-09-24en
dc.identifier.urihttp://hdl.handle.net/10754/626500-
dc.description.abstractThe main goal of this article is to introduce the parallel hierarchical matrix library HLIBpro to the statistical community. We describe the HLIBCov package, which is an extension of the HLIBpro library for approximating large covariance matrices and maximizing likelihood functions. We show that an approximate Cholesky factorization of a dense matrix of size $2M\times 2M$ can be computed on a modern multi-core desktop in few minutes. Further, HLIBCov is used for estimating the unknown parameters such as the covariance length, variance and smoothness parameter of a Mat\'ern covariance function by maximizing the joint Gaussian log-likelihood function. The computational bottleneck here is expensive linear algebra arithmetics due to large and dense covariance matrices. Therefore covariance matrices are approximated in the hierarchical ($\mathcal{H}$-) matrix format with computational cost $\mathcal{O}(k^2n \log^2 n/p)$ and storage $\mathcal{O}(kn \log n)$, where the rank $k$ is a small integer (typically $k<25$), $p$ the number of cores and $n$ the number of locations on a fairly general mesh. We demonstrate a synthetic example, where the true values of known parameters are known. For reproducibility we provide the C++ code, the documentation, and the synthetic data.en
dc.publisherarXiven
dc.relation.urlhttp://arxiv.org/abs/1709.08625v1en
dc.relation.urlhttp://arxiv.org/pdf/1709.08625v1en
dc.rightsArchived with thanks to arXiven
dc.titleHLIBCov: Parallel Hierarchical Matrix Approximation of Large Covariance Matrices and Likelihoods with Applications in Parameter Identificationen
dc.typePreprinten
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.eprint.versionPre-printen
dc.identifier.arxividarXiv:1709.08625en
kaust.authorLitvinenko, Alexanderen | {"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": 0.9447065591812134, "perplexity": 901.016143116995}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890928.82/warc/CC-MAIN-20180121234728-20180122014728-00771.warc.gz"} |
https://ic-sharpy.readthedocs.io/en/latest/includes/structure/utils/modalutils/principal_axes_inertia.html | # principal_axes_inertia¶
Transform the inertia tensor $$\boldsymbol{j}_a$$ defined about the A frame of reference to the centre of gravity and aligned with the principal axes of inertia.
The inertia tensor about the centre of gravity is obtained using the parallel axes theorem
$\boldsymbol{j}_{cm} = \boldsymbol{j}_a + \tilde{r}_{cg}\tilde{r}_{cg}m$
and rotated such that it is aligned with its eigenvectors and thus represents the inertia tensor about the principal axes of inertia
$\boldsymbol{j}_p = T_{pa}^\top \boldsymbol{j}_{cm} T^{pa}$
where $$T^{pa}$$ is the transformation matrix from the A frame to the principal axes P frame.
param j_a
Inertia tensor defined about the A frame.
type j_a
np.array
param r_cg
Centre of gravity position defined in A coordinates.
type r_cg
np.array
param m
Mass.
type m
float
returns
Containing $$\boldsymbol{j}_p$$ and $$T^{pa}$$
rtype
tuple | {"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": 0.8932616114616394, "perplexity": 976.433582428206}, "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/1674764499758.83/warc/CC-MAIN-20230129180008-20230129210008-00084.warc.gz"} |
http://cogsci.stackexchange.com/questions/659/is-it-possible-to-quantify-cognitive-bias/664 | # Is it possible to quantify cognitive bias?
We know that bias exists, and that it affects our judgment and perception. This effect has to do with user's experience in life. That experience is taken care of by the brain, and if you counter a situation again, you have a predefined pattern of how to react. This can be changed over time, if other experiences is added. Wikipedia defines cognitive bias as:
a pattern of deviation in judgment that occurs in particular situations, leading to perceptual distortion, inaccurate judgment, illogical interpretation, or what is broadly called irrationality. Implicit in the concept of a "pattern of deviation" is a standard of comparison with what is normatively expected.
In other disciplines, measurement is a valuable tool. One would wish for a scale of deviation from "what is normally expected".
Is it possible to quantify cognitive bias?
-
Harvard is doing something with implicit preferences that might interest you: implicit.harvard.edu/implicit/demo/selectatest.html – Nate Glenn Mar 30 '12 at 18:06
@NateGlenn Thanx, looks very interesting. I will follow the progress on that research. – Benny Skogberg Mar 31 '12 at 5:17
If you come to this question from the bayesian tradition, then there is only one place where you can sneak in bias: your prior. This dovetails nicely with the wikipedia definition:
a pattern of deviation in judgment that occurs in particular situations, leading to perceptual distortion, inaccurate judgment, illogical interpretation, or what is broadly called irrationality.
Since bayesian updating is considered to be 'rational', the only place to sneak in 'irrationality' (in quotes because we are using these terms very loosely) is in the prior before you were given any evidence/observations. So for a bayesian, measuring bias is measuring the prior.
### Measuring a prior
Conveniently, Kalish et al. (2007) have a nice mechanism for measuring people's priors: have $n$ participants: $1, ... , n$ and give the first one some real input-output pairs on the relevant task to learn from. To train the $i + 1$th participant:
1. take the $i$th participant,
2. give them some inputs and ask them what they think the output should be,
3. use the input-output pairs they generate to train the $i+1$th participant.
Then, towards the end of the chain, the participants will start to converge towards their prior or inherent bias.
### Example
A real example is of people's bias in functional relationships. The first person is given 25 $(x,y)$ pairs from some function $y = f(x)$. A person at stage $i$ is given 25 $(x,y)$ pairs from the person at stage $i - 1$. The person is then tested by being given an $x$ value and asked for a $y$, 25 times. The results of this are passed on to the person at trial $i + 1$ as the training data. The results:
Show that people have a strong bias towards positive linear relationships. One could apply a similar procedure to other domains, but of course finding a good way to do so is a particular domain is an important research question for an experimental psychologist.
### References
• Kalish, M., Griffiths, T. & Lewandowsky, S. (2007). Iterated learning: Intergenerational knowledge transmission reveals inductive biases. Psychonomic Bulletin & Review, 14, 288-294. doi: 10.3758/BF03194066
-
Bias can be quantified in many different ways. In human memory research carried out in the cognitive psychology tradition, there are simple ways to think about it. One basic measure of cognitive bias is merely called bias, and it's a measure of the absolute accuracy of an individual's probability judgments. You average probability judgments across a given subject, then average performance, and subtract one from the other. The magnitude and direction of this discrepancy could be argued to reflect a task-specific cognitive bias.
That's not useful without a concrete example. Say I ask you 10 questions tapping your general knowledge about the world (e.g., "Who invaded Rome by crossing the Alps with elephants?") After you respond, I ask you to rate your confidence on a scale from 0% (not at all confident) to 100% (entirely confident). We do this for nine more questions.
After you've taken this short test, I can easily calculate your bias by averaging your accuracy across the 10 questions (say you get 2 wrong, so 80%) and subtracting your average confidence across those questions (say it's 73%). It's a 7% difference, and you were more accurate than you were confident, so your bias is said to be 7% -- you were underconfident.
There are lots of other ways you can think about this. Say, for instance, you weren't interested in overconfidence or underconfidence, but rather, just the degree to which your performance differed from your estimations. Here you could just use the squared value of the difference to represent probability accuracy.
-
Thanx, Andy. Just what I was asking for. Is there a general standard, or something almost everybody in the field uses? – Benny Skogberg Mar 29 '12 at 17:22
Unfortunately I'm afraid it's really going to vary from discipline to discipline. For basic research, this was the most atheoretical topic I could come up with. I knew bias and similar measures are used whenever numerically-based probability judgments come into play (in the field of personality psychology, for instance). It's tougher for more qualitative areas, though. – Andy DeSoto Mar 29 '12 at 19:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6215276718139648, "perplexity": 1311.3579990588253}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929869.17/warc/CC-MAIN-20150521113209-00088-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://onegoverningbody.com/maximize-the-equation-given-the-constraints-z3x5y-x4-2y12-3x2y18/ | # Maximize the Equation given the Constraints z=3x+5y , x=4 , 2y=12 , 3x+2y=18
I am unable to solve this problem.
Maximize the Equation given the Constraints z=3x+5y , x=4 , 2y=12 , 3x+2y=18
We can help to solve your math questions
### Our Math Experts
They can help to solve your math problems
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https://jp.arxiv.org/list/physics/2204?skip=0&show=50 | # Physics
## Authors and titles for Apr 2022
[ total of 1513 entries: 1-50 | 51-100 | 101-150 | 151-200 | ... | 1501-1513 ]
[ showing 50 entries per page: fewer | more | all ]
[1]
Title: Transition edge sensor based detector: from X-ray to $γ$-ray
Subjects: Instrumentation and Detectors (physics.ins-det); Instrumentation and Methods for Astrophysics (astro-ph.IM)
[2]
Title: Ultra-high terahertz index in deep subwavelength coupled bi-layer free-standing flexible metamaterials
Journal-ref: JOURNAL OF APPLIED PHYSICS 121, 233103 (2017)
Subjects: Optics (physics.optics); Classical Physics (physics.class-ph)
[3]
Title: SELFIES and the future of molecular string representations
Subjects: Chemical Physics (physics.chem-ph); Machine Learning (cs.LG)
[4]
Title: Scale Separated Approaches to the Interaction of Oceanic Internal Waves, part II: A one-dimensional numerical model
Subjects: Atmospheric and Oceanic Physics (physics.ao-ph); Fluid Dynamics (physics.flu-dyn)
[5]
Title: The DESC Stellarator Code Suite Part III: Quasi-symmetry optimization
Comments: 10 pages, 5 figures, 2 tables, 2 listings
Subjects: Plasma Physics (physics.plasm-ph); Optimization and Control (math.OC)
[6]
Title: Sensitive label-free and compact ultrasonic sensor based on double silicon-on-insulator slot micro-ring resonators
Comments: 12 pages, 6 figure, 32 references
Journal-ref: optik 178(2019)1029-1034
Subjects: Optics (physics.optics); Applied Physics (physics.app-ph)
[7]
Title: Support-vector-machine with Bayesian optimization for lithofacies classification using elastic properties
Subjects: Geophysics (physics.geo-ph); Machine Learning (cs.LG)
[8]
Title: Quantitative analysis of diaphragm motion during fluoroscopic sniff test to assist in diagnosis of hemidiaphragm paralysis
Journal-ref: Radiology Case Reports, Volume 17, Issue 5, 2022, Pages 1750-1754, ISSN 1930-0433
Subjects: Medical Physics (physics.med-ph); Image and Video Processing (eess.IV)
[9]
Title: Evolution of barchan dune interactions investigated by a downscaled water tunnel experiment: the temporal characteristics and a soliton-like behavior
Subjects: Geophysics (physics.geo-ph); Fluid Dynamics (physics.flu-dyn)
[10]
Title: Machine learning for a finite size correction in periodic coupled cluster theory calculations
Subjects: Computational Physics (physics.comp-ph)
[11]
Title: Readout for Calorimetry at Future Colliders: A Snowmass 2021 White Paper
Subjects: Instrumentation and Detectors (physics.ins-det); High Energy Physics - Experiment (hep-ex)
[12]
Title: Observing Particle Energization above the Nyquist Frequency: An Application of the Field-Particle Correlation Technique
Comments: Submitted to Physics of Plasmas
Subjects: Plasma Physics (physics.plasm-ph)
[13]
Title: Giant terahertz pulling force within an evanescent field propelled by wave coupling into radiation and bound modes
Subjects: Optics (physics.optics)
[14]
Title: Relativistic frequency shifts in Cr, Ti, Fe, Ni, Ca, Na, and V to search for variation in the fine structure constant
Subjects: Atomic Physics (physics.atom-ph)
[15]
Title: Electronics for Fast Timing
Comments: Submitted to the Proceedings of the US Community Study on the Future of Particle Physics (Snowmass 2021)
Subjects: Instrumentation and Detectors (physics.ins-det); High Energy Physics - Experiment (hep-ex)
[16]
Title: Burstiness of human physical activities and their characterization
Comments: 15 pages, 6 figures, and Highlights
Subjects: Physics and Society (physics.soc-ph)
[17]
Title: Dynamics of particle-laden turbulent Couette flow. Part1: Turbulence modulation by inertial particles
Comments: Keywords: Reverse force, Two-way coupling, Turbulence modulation, Turbulence attenuation, DNS, Shear production, Discontinuous disruption
Subjects: Fluid Dynamics (physics.flu-dyn)
[18]
Title: Turbulence-free computational ghost imaging
Subjects: Optics (physics.optics); Image and Video Processing (eess.IV); Quantum Physics (quant-ph)
[19]
Title: Dynamics of particle-laden turbulent Couette flow. Part2: Modified fluctuating force model (M-FFS)
Comments: Keywords: DNS, Two-way coupling, Fluctuating Force Simulation (FFS), M-FFS, Discontinuous attenuation
Subjects: Fluid Dynamics (physics.flu-dyn); Data Analysis, Statistics and Probability (physics.data-an)
[20]
Title: Secular variation signals in magnetic field gradient tensor elements derived from satellite-based geomagnetic virtual observatories
Journal-ref: Geophys. J. Int. (2022) 229, 2096-2114
Subjects: Geophysics (physics.geo-ph)
[21]
Title: Comments to "A non-thermal laser-driven mixed fuel nuclear fusion reactor concept" by H. Ruhl and G. Korn (Marvel Fusion, Munich)
Authors: Karl Lackner
Subjects: Plasma Physics (physics.plasm-ph)
[22]
Title: Determination of the recombination coefficient in electrolytic solutions from impedance spectroscopy measurements
Comments: 18 pages in preprint style, 12 figures, including appendix
Journal-ref: J. Electroanal. Chem. 907, 116070 (2022)
Subjects: Chemical Physics (physics.chem-ph)
[23]
Title: Quantum dot on a plasma-facing microparticle surface: Thermal balance
Subjects: Plasma Physics (physics.plasm-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)
[24]
Title: Temporal coherence of optical fields in the presence of entanglement
Subjects: Optics (physics.optics)
[25]
Title: The innovative $^{52g}$Mn for PET imaging: production cross section modeling and dosimetric evaluation
Comments: 19 pages, 5 figures, 6 tables
Subjects: Medical Physics (physics.med-ph); Nuclear Theory (nucl-th)
[26]
Title: Passive Laser Power Stabilization via an Optical Spring
Subjects: Optics (physics.optics)
[27]
Title: Thermodynamics and the Origin of Life
Authors: Gerald E. Marsh
Comments: To appear in the Canadian Journal of Physics: 20 pages, 3 figures
Subjects: General Physics (physics.gen-ph)
[28]
Title: Evolutionary quantization of matter and Universe expansion
Authors: Uziel Sandler
Subjects: General Physics (physics.gen-ph)
[29]
Title: Cauchy's Logico-Linguistic Slip, the Heisenberg Uncertainty Principle and a Semantic Dilemma Concerning "Quantum Gravity"
Authors: Abhishek Majhi
Comments: 8 pages, almost identical to the published version
Journal-ref: Int. J. Theor. Phys. 61, 55 (2022)
Subjects: General Physics (physics.gen-ph)
[30]
Title: Biot-Savart type magnetic field quantization via prime number theory, applications to symbolic dynamics
Subjects: General Physics (physics.gen-ph)
[31]
Title: The complete metric study of effective Dirac algebra
Authors: B.T.T. Wong
Subjects: General Physics (physics.gen-ph)
[32]
Title: Direct Numerical Simulation of Low and Unitary Prandtl Number Fluids in Reactor Downcomer Geometry
Subjects: Fluid Dynamics (physics.flu-dyn)
[33]
Title: Pauli blocking of stimulated emission in a degenerate Fermi gas
Subjects: Atomic Physics (physics.atom-ph); Quantum Gases (cond-mat.quant-gas)
[34]
Title: A novel formulation for the study of the ascending aortic fluid dynamics with in vivo data
Subjects: Fluid Dynamics (physics.flu-dyn)
[35]
Title: Classical Newtonian Monte-Carlo Calculation of Ionization of Atomic Hydrogen by Protons in Energy Range 0.05 to 1.0 MeV
Comments: 18 pages, 9 figures, 1 table
Subjects: Atomic Physics (physics.atom-ph); Atomic and Molecular Clusters (physics.atm-clus)
[36]
Title: Wavelength-by-wavelength temperature-independent thermal radiation utilizing an insulator-metal transition
Subjects: Optics (physics.optics); Materials Science (cond-mat.mtrl-sci); Strongly Correlated Electrons (cond-mat.str-el); Applied Physics (physics.app-ph)
[37]
Title: Characterization of Passive CMOS Strip Sensors
Subjects: Instrumentation and Detectors (physics.ins-det); High Energy Physics - Experiment (hep-ex)
[38]
Title: A removable temperature sensors holder for cryogenic Rayleigh-Bénard convection cell
Journal-ref: The 16th Cryogenics 2021, IIR Conference / ONLINE / October 5-7, 2021
Subjects: Fluid Dynamics (physics.flu-dyn)
[39]
Title: Thermodynamically consistent versions of approximations used in modelling moist air
Subjects: Atmospheric and Oceanic Physics (physics.ao-ph)
[40]
Title: Wall-modeled large-eddy simulation of three-dimensional turbulent boundary layer in a bent square duct
Subjects: Fluid Dynamics (physics.flu-dyn)
[41]
Title: Effects of geomagnetic field perturbations on the power supply of transoceanic fiber optic cables
Authors: Antonio Mecozzi
Subjects: Atmospheric and Oceanic Physics (physics.ao-ph); Geophysics (physics.geo-ph); Optics (physics.optics)
[42]
Title: High-resolution Compton spectroscopy using X-ray microcalorimeters
Comments: The following article has been submitted to Applied Physics Letters
Subjects: Instrumentation and Detectors (physics.ins-det); Superconductivity (cond-mat.supr-con)
[43]
Title: Fractal and multifractal descriptors restore ergodicity broken by non-Gaussianity in time series
Comments: 33 pages, 11 figures. arXiv admin note: text overlap with arXiv:2202.01091
Subjects: Data Analysis, Statistics and Probability (physics.data-an); Methodology (stat.ME)
[44]
Title: Towards an Acoustic Geometry in Slightly Viscous Fluids
Comments: 16 pages Latex2e, no figures
Journal-ref: Universe (Special Issue on Analogue Gravity), 8 (2022) 205
Subjects: Fluid Dynamics (physics.flu-dyn); General Relativity and Quantum Cosmology (gr-qc)
[45]
Title: Computation of optimal beams in weak turbulence
Subjects: Computational Physics (physics.comp-ph); Numerical Analysis (math.NA); Optics (physics.optics)
[46]
Title: Stochastic state-transition-change process and time resolved velocity spectrometry
Authors: Jiri Prochazka
Comments: 5 pages, stochastic state transition change (STC) process is introduced in arXiv:2204.00626v1
Subjects: General Physics (physics.gen-ph)
[47]
Title: Teaching scenarios and their role in the interdisciplinary approach. Case study: The Minkowskian Metric
Authors: Ioannis Rizos
Comments: Proceedings of the First Congress of Greek Mathematicians, pp. 216-227. Athens: Hellenic Mathematical Society
Journal-ref: Proceedings of the First Congress of Greek Mathematicians, pp. 216-227. Athens: Hellenic Mathematical Society
Subjects: Physics Education (physics.ed-ph); History and Overview (math.HO)
[48]
Title: The (meta)metaphysics of science: the case of non-relativistic quantum mechanics
Comments: In Portuguese. Article accepted for publication in Kriterion, v.63 n.151, 2022. See the final version at <this https URL>
Subjects: History and Philosophy of Physics (physics.hist-ph); Quantum Physics (quant-ph)
[49]
Title: Time-encoded pseudo-continuous arterial spin labeling: increasing SNR in ASL angiography
Comments: 36 pages, 9 figures, 6 supplementary figures
Subjects: Medical Physics (physics.med-ph)
[50]
Title: Thin-ply thermoplastic composites: from weak to robust transverse performance through microstructural and morphological tuning
Comments: 16 Pages, 13 Figures, submitted
Subjects: Applied Physics (physics.app-ph); Materials Science (cond-mat.mtrl-sci)
[ total of 1513 entries: 1-50 | 51-100 | 101-150 | 151-200 | ... | 1501-1513 ]
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Links to: arXiv, form interface, find, physics, 2208, contact, help (Access key information) | {"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": 0.4587336480617523, "perplexity": 25051.218327967195}, "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/1659882571198.57/warc/CC-MAIN-20220810161541-20220810191541-00134.warc.gz"} |
https://learn.careers360.com/ncert/question-a-12-point-5-e-v-electron-beam-is-used-to-bombard-gaseous-hydrogen-at-room-temperature-what-series-of-wavelengths-will-be-emitted/ | # Q 12.9 A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?
S Sayak
Since the energy of the electron beam is 12.5 eV the Hydrogen atoms will get excited to all requiring energy equal to or less than 12.5 eV
E=-13.6 eV
E= -1.5 eV
E3 -E1 = 12.1 eV
E4= -0.85 eV
E4-E1=12.75 eV
Therefore the electron can reach maximum upto the level n=3.
During de-excitations, the electron can jump directly from n=3 to n=1 or it can first jump from n=3 to n=2 and then from n=2 to n=1
Therefore two wavelengths from the Lyman series and one from the Balmer series will be emitted
To find the wavelengths emitted we will use the Rydberg's Formula
$\frac{1}{\lambda }=R(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}})$ where R is the Rydberg's constant and equals 1.097$\times$107 m-1
For n1=1 and n2=3
$\frac{1}{\lambda }=1.097\times 10^{7}(\frac{1}{1^{2}}-\frac{1}{3^{2}})$
Emitted wavelength is 102.5 nm
For n1=1 and n2=2
$\frac{1}{\lambda }=1.097\times 10^{7}(\frac{1}{1^{2}}-\frac{1}{2^{2}})$
Emitted wavelength is 121.54 nm
For n1=2 and n2=3
$\frac{1}{\lambda }=1.097\times 10^{7}(\frac{1}{2^{2}}-\frac{1}{3^{2}})$
Emitted wavelength is 656.3 nm
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Questions | {"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": 5, "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": 0.9241247177124023, "perplexity": 2344.51011235481}, "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/1585370493120.15/warc/CC-MAIN-20200328194743-20200328224743-00312.warc.gz"} |
https://www.physicsforums.com/threads/what-is-the-integral-of-xe-x.255251/ | # What is the integral of xe^-x?
1. Sep 10, 2008
### Grimertop90
I feel really dumb for not remembering this.... I'm drawing a total blank as to the antiderivative of xe-x or how to find it. Do I need to use a u substitution?
2. Sep 10, 2008
### quasar987
Try integration by parts.
3. Sep 10, 2008
### HallsofIvy
Staff Emeritus
Let u= x, dv= e-xdx.
4. Sep 11, 2008
### Count Iblis
You can also compute the intergal of exp(-a x) and then differentiate w.r.t. the parameter a. | {"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": 0.8873322606086731, "perplexity": 4852.9455791756145}, "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-2016-22/segments/1464049281876.4/warc/CC-MAIN-20160524002121-00197-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://www.gamedev.net/forums/topic/367322-gcccygwin-and-long-longs-a-bug/ | # GCC/Cygwin and long longs. A bug?
This topic is 4699 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Lo, I'm currently coding a sign extension function (and messing around with long long's a lot, since I'm working on a 32-bit platform). I believe my sign extension function works correctly - but GCC is really weird when it comes to printing them. For example, here's my sign extension function (QWORD is an unsigned long long)
QWORD SignExtend(QWORD value,DWORD highBit)
{
if (((value >> highBit) & 1) == 1)
{
/* Set all bits higher than highBit to one */
DWORD i=highBit+1;
while (i < 32)
value|=(1 << i++);
value|=(0xFFFFFFFFULL << 32);
printf("%#llX\n",value);
}
return value;
}
In theory, this should print a value above 4 billion (i.e. > 0xFFFFFFFF) but all that gets printed are the lower 32 bits. The GCC version I'm using is 3.4.4. What is wrong? I've also found general weirdness with printing long longs. For instance, I print one, and the next long long printed is zero. If I print them seperately, there's no problem. Should I submit a bug to the cygwin mailing list or gcc mailing list? Also, I'm currently compiling with -mno-cygwin. I don't know if that changes much. CloudNine [Edited by - CloudNine on January 3, 2006 5:50:43 AM]
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That's weird..
Have you tried moving the ll type to the other side of the # modifier (it's supposed to and a 0x prefix to hexadecimal constants, right?), or removing it entirely? You may even want to check if it works for regular decimal values (%lld).
Have you tried sanity-checking GCC's printf type checker by sending a regular integer instead of a 64-bit one?
Otherwise you may just have to do a bit of digging in the source to find out what's happening, although you'd think something as basic as this should've been properly tested.
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Yep, I've tried everything, and have -Wall on so printf arguments are checked. But no luck. It still prints out as if the argument were 32 bits.
I'll have to post to the cygiwn mailing list then?
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Quote:
Original post by CloudNineI'll have to post to the cygiwn mailing list then?
Well, either that or step through it in a debugger.
Of course you may want to check other versions and distributions of GCC too, possibly even check out the official bug list to see if it's a known issue.
Quote:
0xFFFFFFFFULL
Cute..
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Tried it using gcc 3.3.4 for linux (using libc 2.3.4) and it performs correctly. I'm now assuming it's something wrong with the cygwin c library. I can't find the package for it in setup.exe, I'll search harder :)
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This is a long standing problem with cygwin and long longs.
Essentially, long long is a gnu extension. It is present in the compiler, so the code compiles correctly, but long long is not present in the c standard library you are using, probably because it is the M$C-runtime. Try using "%I64X" or such MS-specific int64 extension instead of gnu-specific "%llX" #### Share this post ##### Link to post ##### Share on other sites Quote: Original post by CloudNineTried it using gcc 3.3.4 for linux (using libc 2.3.4) and it performs correctly. I'm now assuming it's something wrong with the cygwin c library. I can't find the package for it in setup.exe, I'll search harder :) Quote: Original post by CloudNineAlso, I'm currently compiling with -mno-cygwin. I don't know if that changes much. I just noticed that you're compiling in the mingw "mode", which IIRC uses Microsoft's CRT directly instead of GNU's implementation. And since VC doesn't support C99 long longs you'll probably have to use "%I64X" instead. edit: Beaten.. #### Share this post ##### Link to post ##### Share on other sites Ah, that solves it. Sorry to trouble you guys. gcc still complains about that format specifier tho, but it works :) #### Share this post ##### Link to post ##### Share on other sites Quote: Original post by etothexEssentially, long long is a gnu extension. It is present in the compiler, so the code compiles correctly, but long long is not present in the c standard library you are using, probably because it is the M$ C-runtime.
AFAIK, (unsigned) long long is standard C99.
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× | {"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": 0.17668455839157104, "perplexity": 3777.5355039278074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742338.13/warc/CC-MAIN-20181115013218-20181115035218-00360.warc.gz"} |
http://mathhelpforum.com/calculus/11805-maxima-minima-shortest-longest-distance-circles.html | # Math Help - Maxima / Minima - Shortest & Longest distance to a Circles
1. ## Maxima / Minima - Shortest & Longest distance to a Circles
Given the circle having the equation x^2 + y^2 = 9, find a.) The shortest distance from point (4,5) to a point on the circle, and b.) The longest distance from the point (4,5) to a point on the circle.
2. Originally Posted by ^_^Engineer_Adam^_^
Given the circle having the equation x^2 + y^2 = 9, find a.) The shortest distance from point (4,5) to a point on the circle, and b.) The longest distance from the point (4,5) to a point on the circle.
The point (4,5) is a distance sqrt(41) from the origin which is the centre of
the given circle.
The radius of the given circle is 3, thefore the shortest distance is sqrt(41)-3,
and the greatest distance is sqrt(41)+3.
(Draw a diagram to see how this works).
RonL | {"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": 0.8988636136054993, "perplexity": 384.42372325540725}, "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-2015-27/segments/1435375096209.79/warc/CC-MAIN-20150627031816-00166-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://computergraphics.stackexchange.com/questions/9426/how-can-i-raycast-with-chunks | # How can I raycast with chunks?
I am attempting to implement chunks/octrees to speed up my render times. However, sometimes the walls show horizontal or vertical lines (rays hitting the inside of voxels?) or some blocks disappear altogether, although sometimes everything renders fine.
Edit: Yes, the chunks are stored fine. OctrA stands for OctreeA, which is the chunk. Edit: I reversed the ray for two steps after hitting a chunk. Although the glitches have gotten better, there is still the same problem.
float cameraX = x/(float)HALFRESX -1;// -1 to 1
struct Point3D rayd;
rayd.x = dir.x + plane.x * cameraX;
rayd.y = dir.y + plane.y * cameraX;
struct Point3D delta;
delta.x = fabsf(1/rayd.x);
delta.y = fabsf(1/rayd.y);
int zy = -HALFRESY;
struct pos map;
map.x = (char)pos.x;
map.y = (char)pos.y;
float sdistX;
float sdistY;
struct pos s;
struct pos step;
struct Point3D sdist;
if(rayd.x < 0) {
step.x = -1;
s.x = 1;
sdistX = (pos.x - map.x) * delta.x;
}
else {
step.x = 1;
s.x = 0;
sdistX = (map.x + 1 - pos.x) * delta.x;
}
if(rayd.y < 0) {
step.y = -1;
s.y = 1;
sdistY = (pos.y - map.y) * delta.y;
}
else {
step.y = 1;
s.y = 0;
sdistY = (map.y + 1 - pos.y) * delta.y;
}
for(int y = 0; y < RESY; y += 1) {
zy++;
float cameraY = y/(float)HALFRESY - 1;
rayd.z = dir.z + plane.z * cameraY;
delta.z = fabsf(1/rayd.z);
map.x = (char)pos.x;
map.y = (char)pos.y;
map.z = (char)pos.z;
sdist.x = sdistX;
sdist.y = sdistY;
if(rayd.z < 0) {
step.z = -1;
s.z = 1;
sdist.z = (pos.z - map.z) * delta.z;
}
else {
step.z = 1;
s.z = 0;
sdist.z = (map.z + 1 - pos.z) * delta.z;
}
#ifdef USE_OCTREE
while( !octrA[map.x/2][map.y/2][map.z/2]) {
if(sdist.y < sdist.x ) {
if(sdist.y < sdist.z) {
sdist.y += delta.y*2;
map.y += step.y*2;
}
else {
sdist.z += delta.z*2;
map.z += step.z*2;
}
}
else {
if(sdist.x < sdist.z) {
sdist.x += delta.x*2;
map.x += step.x*2;
}
else {
sdist.z += delta.z*2;
map.z += step.z*2;
}
}
}
if(octr) {
if(sdist.y > sdist.x ) {
if(sdist.y > sdist.z) {
sdist.y -= delta.y*2;
map.y -= step.y*2;
}
else {
sdist.z -= delta.z*2;
map.z -= step.z*2;
}
}
else {
if(sdist.x > sdist.z) {
sdist.x -= delta.x*2;
map.x -= step.x*2;
}
else {
sdist.z -= delta.z*2;
map.z -= step.z*2;
}
}
}
#endif
char side; //either 0 (NS), or 1 (EW), or 2(UD)
while( !MAP[map.x][map.y][map.z]) {
if(sdist.y < sdist.x ) {
if(sdist.y < sdist.z) {
sdist.y += delta.y;
map.y += step.y;
side = 1; // y
}
else {
sdist.z += delta.z;
map.z += step.z;
side = 2;
}
}
else {
if(sdist.x < sdist.z) {
sdist.x += delta.x;
map.x += step.x;
side = 0;
}
else {
sdist.z += delta.z;
map.z += step.z;
side = 2;
}
}
}
• rayd.x = dir.x + plane.x * cameraX; rayd.y = dir.y + plane.y * cameraX; -- Shouldn't that second rayd.y assignment be taking a cameraY variable instead ? Dec 18 '19 at 5:47
• CameraX is -1 to 1 and corresponds with the X axis on screen, so no Dec 18 '19 at 9:24
• I apologise for not reading your code, but by your description of the errors I'd hazard a guess that you need to make your octree testing maths conservative i.e. either use 1) rounding modes that round down or round up appropriately, or 2) add/subtract some "epsilson" safety margin on to the calculations when testing the bounds. Remember FP maths is rarely exact and unlucky rounding could mean an object very near to a boundary of an octree might be "missed" by the octree test yet would return a hit if tested directly. Jan 12 at 8:45
• So this is more of a debugging chore then a how to implement it question. I don't see any logic sanity checking the values of x,y,z in the tree lookup code. A nice simple first step to finding the issue is to sanity check map.x/y/z in the while loop. Something like assert(map.x < mapx_max_val) and the same thing for y/z. Sep 10 at 10:42 | {"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": 0.4284529685974121, "perplexity": 24498.780212218382}, "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/1634323585186.33/warc/CC-MAIN-20211018000838-20211018030838-00198.warc.gz"} |
https://geo.libretexts.org/Bookshelves/Geology/Book%3A_Physical_Geology_(Earle)/05%3A_Weathering_and_Soil/5.03%3A_The_Products_of_Weathering_and_Erosion | # 5.3: The Products of Weathering and Erosion
The products of weathering and erosion are the unconsolidated materials that we find around us on slopes, beneath, beside and on top of glaciers, in stream valleys, on beaches, and in deserts. The nature of these materials—their composition, size, degree of sorting, and degree of rounding—is determined by the type of rock that is being weathered, the nature of the weathering, the erosion and transportation processes, and the climate.
In addition to these solid sediments, the other important products of weathering are several different types of ions in solution.
A summary of the weathering products of some of the common minerals present in rocks is provided in Table 5.1. In addition to the weathering products listed in the table, most of the larger fragments—larger than sand grains—that make up sediments will be pieces of rock as opposed to individual minerals.
Table 5.1 A list of the typical weathering products of some of the minerals in common rocks
Common Mineral Typical Weathering Products
Quartz Quartz as sand grains
Feldspar Clay minerals plus potassium, sodium, and calcium in solution
Biotite and amphibole Chlorite plus iron and magnesium in solution
Pyroxene and olivine Serpentine plus iron and magnesium in solution
Calcite Calcium and carbonate in solution
Pyrite Iron oxide minerals plus iron in solution and sulfuric acid
Some examples of the products of weathering are shown in Figure $$\PageIndex{1}$$. They range widely in size and shape depending on the processes involved in their transportation. If and when deposits like these are turned into sedimentary rocks, the textures of those rocks will vary significantly. Importantly, when we describe sedimentary rocks that formed millions of years in the past, we can use those properties to make inferences about the conditions that existed during their formation.
We’ll talk more about the nature and interpretation of sediments and sedimentary rocks in Chapter 6, but it’s worth considering here why the sand-sized sediments shown in Figure $$\PageIndex{1}$$ are so strongly dominated by the mineral quartz, even though quartz makes up less than 20% of Earth’s crust. The explanation is that quartz is highly resistant to the types of weathering that occur at Earth’s surface. It is not affected by weak acids or the presence of oxygen. This makes it unique among the minerals that are common in igneous rocks. Quartz is also very hard, and doesn’t have cleavage, so it is resistant to mechanical erosion.
So when a rock like granite is subject to chemical weathering the feldspar and the ferromagnesian silicates get converted to clays and dissolved ions such as: Ca2+, Na+, K+, Fe2+, Mg2+, and H4SiO4, but the quartz is resistant to those processes and remains intact. The clay gradually gets eroded away, then the rock breaks apart leaving lots of grains of quartz. In other words, quartz, clay minerals, and dissolved ions are the most common products of weathering. Quartz and some of the clay minerals tend to form sedimentary deposits on and at the edges of continents, while the rest of the clay minerals and the dissolved ions tend to be washed out into the oceans to form sediments on the sea floor.
Exercise 5.3 Describing the weathering origins of sand
In the left side of the following table, a number of different sands are pictured and described. Describe some of the important weathering processes that might have led to the development of these sands.
See Appendix 3 for Exercise 5.3 answers.
Image Description and Location
Fragments of coral, algae, and urchin from a shallow water area (roughly 2 meters deep) near a reef in Belize. The grain diameters are between 0.1 and 1 millimeters.
Angular quartz and rock fragments from a glacial stream deposit near Osoyoos, B.C. The grain diameters are between 0.25 and 0.5 milimeters.
Rounded grains of olivine (green) and volcanic glass (black) from a beach on the big island of Hawaii. The grains are approximately 1 millimetre across.
## Image Descriptions
Figure $$\PageIndex{1}$$ image description: Examples of weathering and erosion.
1. Boulders in a talus deposit at Keremeos. All are angular fragments from the same rock source.
2. Pebbles on a beach in Victoria. All are rounded fragments of rock from different sources.
3. Sand from a beach at Gabriola Island. most are angular quartz grains, some are sand-sized fragments of rock.
4. Sand from a dune in Utah. All are rounded quartz grains.
• Figure $$\PageIndex{1}$$: © Steven Earle. CC BY. | {"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": 0.2583240866661072, "perplexity": 3716.2185817153245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154459.22/warc/CC-MAIN-20210803124251-20210803154251-00443.warc.gz"} |
https://arxiv.org/abs/1509.06371 | astro-ph.CO
(what is this?)
# Title: The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscillations in the correlation function of LOWZ and CMASS galaxies in Data Release 12
Abstract: We present distance scale measurements from the baryon acoustic oscillation signal in the CMASS and LOWZ samples from the Data Release 12 of the Baryon Oscillation Spectroscopic Survey (BOSS). The total volume probed is 14.5 Gpc$^3$, a 10 per cent increment from Data Release 11. From an analysis of the spherically averaged correlation function, we infer a distance to $z=0.57$ of $D_V(z)r^{\rm fid}_{\rm d}/r_ {\rm d}=2028\pm21$ Mpc and a distance to $z=0.32$ of $D_V(z)r^{\rm fid}_{\rm d}/r_{\rm d}=1264\pm22$ Mpc assuming a cosmology in which $r^{\rm fid}_{\rm d}=147.10$ Mpc. From the anisotropic analysis, we find an angular diameter distance to $z=0.57$ of $D_{\rm A}(z)r^{\rm fid}_{\rm d}/r_{\rm d}=1401\pm21$ Mpc and a distance to $z=0.32$ of $981\pm20$ Mpc, a 1.5 per cent and 2.0 per cent measurement respectively. The Hubble parameter at $z=0.57$ is $H(z)r_{\rm d}/r^{\rm fid}_{\rm d}=100.3\pm3.7$ km s$^{-1}$ Mpc$^{-1}$ and its value at $z=0.32$ is $79.2\pm5.6$ km s$^{-1}$ Mpc$^{-1}$, a 3.7 per cent and 7.1 per cent measurement respectively. These cosmic distance scale constraints are in excellent agreement with a $\Lambda$CDM model with cosmological parameters released by the recent Planck 2015 results.
Comments: 18 pages, 10 figures, 13 tables. Matches version accepted for publication in MNRAS. Here we account for a systematic error budget not included in the previous version of this manuscript Subjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO) DOI: 10.1093/mnras/stw066 Cite as: arXiv:1509.06371 [astro-ph.CO] (or arXiv:1509.06371v2 [astro-ph.CO] for this version)
## Submission history
From: Antonio Jose Cuesta Vazquez [view email]
[v1] Mon, 21 Sep 2015 20:08:11 GMT (1823kb,D)
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https://www.nat-hazards-earth-syst-sci.net/19/2451/2019/ | Journal topic
Nat. Hazards Earth Syst. Sci., 19, 2451–2464, 2019
https://doi.org/10.5194/nhess-19-2451-2019
Nat. Hazards Earth Syst. Sci., 19, 2451–2464, 2019
https://doi.org/10.5194/nhess-19-2451-2019
Research article 05 Nov 2019
Research article | 05 Nov 2019
# The first version of the Pan-European Indoor Radon Map
The first version of the Pan-European Indoor Radon Map
Javier Elío1, Giorgia Cinelli2, Peter Bossew3, José Luis Gutiérrez-Villanueva4, Tore Tollefsen2, Marc De Cort2, Alessio Nogarotto5, and Roberto Braga5 Javier Elío et al.
• 1Centre for the Environment, Trinity College, Dublin, Ireland
• 2European Commission, Joint Research Centre (JRC), Ispra, Italy
• 3German Federal Office for Radiation Protection, Berlin, Germany
• 4Radonova Laboratories AB, Uppsala, Sweden
• 5Dipartimento di Scienze Biologiche, Geologiche e Ambientali, Università di Bologna, Bologna, Italy
Correspondence: Giorgia Cinelli (giorgia.cinelli@ec.europa.eu)
Abstract
A hypothetical Pan-European Indoor Radon Map has been developed using summary statistics estimated from 1.2 million indoor radon samples. In this study we have used the arithmetic mean (AM) over grid cells of 10 km × 10 km to predict a mean indoor radon concentration at ground-floor level of buildings in the grid cells where no or few data (N<30) are available. Four interpolation techniques have been tested: inverse distance weighting (IDW), ordinary kriging (OK), collocated cokriging with uranium concentration as a secondary variable (CCK), and regression kriging with topsoil geochemistry and bedrock geology as secondary variables (RK). Cross-validation exercises have been carried out to assess the uncertainties associated with each method. Of the four methods tested, RK has proven to be the best one for predicting mean indoor radon concentrations; and by combining the RK predictions with the AM of the grids with 30 or more measurements, a Pan-European Indoor Radon Map has been produced. This map represents a first step towards a European radon exposure map and, in the future, a radon dose map.
1 Introduction
Radon (Rn) is the major contributor to the ionizing radiation dose received by the general population, which is the second cause of lung cancer death after smoking (WHO, 2009). Worldwide radon exposure is linked to an estimated 222 000 out of the 1.8 million lung cancer cases reported per year (Gaskin et al., 2018), and in Europe alone it has been estimated that 18,000 lung cancer cases per year are induced by radon (Gray et al., 2009). Since lung cancer survival rates after 5 years can be below 20 % (Cheng et al., 2016), a reduction in radon exposure will have a significant positive impact on the health of the general population. In this context, the EU recently revised and consolidated the Basic Safety Standards Directive (Council Directive 2013/59/EURATOM), which aims to reduce the number of radon-induced lung cancer cases.
The main sources of radon indoors are the surrounding subsoils on which buildings are located, the groundwater used in the building, and the building materials (Cothern and Smith, 1987). Consequently, radon is present everywhere. The likelihood of having a high indoor radon concentration may, however, be higher in some areas than others. Radon maps are therefore an essential tool at a large scale and give very good indications of the problem, helping policymakers to design cost-effective radon action plans (Gray et al., 2009). Importantly, because of high local variability, large-scale Rn maps do not inform about Rn concentration in a particular building. Instead, this requires measurements in that building.
In 2006, the EU's Joint Research Centre (JRC) launched a long-term project to map radon at the European level (Tollefsen et al., 2014). For more than 10 years now, the JRC has been developing the European Atlas of Natural Radiation (Cinelli et al., 2019). It includes maps of the natural radioactive levels of (i) annual cosmic-ray dose; (ii) indoor radon concentration; (iii) uranium, thorium, and potassium concentration in soil and in bedrock; (iv) terrestrial gamma dose rate; and (v) soil permeability. Digital versions of these maps are available from a JRC website (https://remon.jrc.ec.europa.eu/About/Atlas-of-Natural-Radiation, last access: 20 September 2019) and updated at irregular intervals when new data become available. The objectives of this Atlas are (1) to increase public knowledge of natural ionizing radiation, (2) to analyse the level of natural radioactivity caused by different sources, (3) to produce a better estimate of the annual dose to which the general population is exposed, and (4) to compare natural and artificial sources (Cinelli et al., 2019).
The European Indoor Radon Map (EIRM) displays the annual average indoor radon concentration (Rn; 222Rn) measured on the ground floor of buildings over 10 km × 10 km grid cells (Dubois et al., 2010). Based on input-data specifications stipulated by the JRC, European countries provide summary statistics estimated over 10 km × 10 km grid cells without communicating the original data, thus guaranteeing data privacy confidentiality for the individual house owners. As a result, the European indoor radon dataset contains the following parameters: the arithmetic mean and standard deviation of the indoor radon measurements (AM_z and SD_z) and the log-transformed data (AM_lnz and SD_lnz); the median (med), the minimum (min), and the maximum (max) values; and the total number (N) of dwellings sampled in each grid cell (Tollefsen et al., 2014).
The dataset underlying the EIRM represents a huge amount of work. At the time of writing (end of 2018), 32 countries (EU and non-EU member states alike) had contributed data, and information from almost 1.2 million dwellings has been aggregated into 28 468 grid cells. Since some cells overlap between countries, 28 203 of these grid cells were filled by one country, while 262 and 3 grids were filled by two and three countries, respectively (i.e. border areas which share the same grid) (version: 29-09-2018). However, there is still a large number of grid cells over European territory with no data, and the number of measurements per grid cell varies widely, from many with only one measurement up to a single one with 23 993 dwellings sampled (Table 1). Evaluating the radon exposure to European citizens would therefore require another 10 years, or more, if it had to be done based on indoor radon measurements over each grid cell.
Table 1Number of dwellings sampled by grid cells of 10 km × 10 km in the study area.
Interpolation techniques are therefore essential at this stage to predict a mean indoor radon concentration in the grid cells for which no or few data are available, and thus develop a Pan-European Indoor Radon Map. We have tested four interpolation techniques: two that use solely indoor radon concentration measurements, viz. inverse distance weighting (IDW) and ordinary kriging (OK), and another two which also take into account geological information, viz. collocated cokriging with the uranium concentration in topsoil as a secondary variable (CCK) and regression kriging with topsoil geochemistry and bedrock geology as secondary variables (RK). Cross-validation exercises were carried out to assess the uncertainties associated with each method. The map generated here is a hypothetical indoor Rn map in the sense that it estimates the mean per 10 km × 10 km grid cell under the assumption that there are dwellings in the grid cell. In some remote areas (mountains, extreme northern Europe), however, this may not be the case in reality. The final map represents a first step towards a European radon exposure and, further on, a radon dose map. Furthermore, it may assist European countries in developing their respective national indoor radon maps.
Figure 1Arithmetic mean (AM_z) over 10 km × 10 km grid cells (Bq m−3) and relative standard deviation (RSD = AMSD).
Figure 2Histogram and q–q plot of average indoor radon concentration (AM_z) on the ground floor of dwellings.
2 Methods
The primary dataset used to predict the mean per grid cell with no or few data is the one of arithmetic means (AM_z). The AM was assigned to the centre of each grid cell, and predictions were carried out only in grid cells where U, Th, and K2O concentrations were also available (46 000 grid cells; version 28-05-2018, Pantelić et al., 2018). Data from grid cells filled by more than one country (i.e. points with the same coordinates) were merged and the summary statistics recalculated according to Eqs. (1)–(10).
$\begin{array}{}\text{(1)}& & \mathrm{AM}=\frac{S}{N}\text{(2)}& & \mathrm{SD}=\sqrt{\frac{SQ-\frac{{s}^{\mathrm{2}}}{N}}{N-\mathrm{1}}}\text{(3)}& & \mathrm{Med}=\sqrt{\prod _{i=\mathrm{1}}^{n}{\mathrm{med}}_{i}}\phantom{\rule{0.25em}{0ex}}\left(\mathrm{approximation}\right)\text{(4)}& & \mathrm{Min}=\mathrm{min}\left[{\mathrm{min}}_{i}\right]\text{(5)}& & \mathrm{Max}=\mathrm{max}\left[{\mathrm{max}}_{i}\right]\text{(6)}& & N=\sum _{i=\mathrm{1}}^{n}{N}_{i}\end{array}$
$\begin{array}{}\text{(7)}& & S=\sum _{i=\mathrm{1}}^{n}{S}_{i}\text{(8)}& & {S}_{i}={\mathrm{AM}}_{i}\cdot {N}_{i}\text{(9)}& & SQ=\sum _{i=\mathrm{1}}^{n}S{Q}_{i}\text{(10)}& & S{Q}_{i}={\mathrm{SD}}_{i}\cdot \left({N}_{i}-\mathrm{1}\right)+\frac{{S}_{i}}{{N}_{i}}\end{array}$
Here “i” is the number of countries that filled the grid. The values for the log-transformed data (AM_lnz and the SD_lnz) were estimated with the same equations as used for the AM and the SD, but with the ln values provided by each country (i.e. AM_lnz and SD_lnz).
In the study area (i.e. area with topsoil geochemistry data) there are 25 367 grid cells with indoor radon measurements (Fig. 1). The distribution of the AM is approximately log-normal (Fig. 2), with values ranging from 1 to 10 116 Bq m−3. The summary statistics are shown in Table 2. Nominal concentrations below 10 Bq m−3 are unrealistic from the point of view of true occurrence and measurement possibility, but this could not be verified in this context. The impact of such errors on the result is probably negligible.
Table 2Summary statistics of indoor radon data (AM_z) after merged border grids (N=25 367).
## 2.2 Interpolation techniques
A mean (over a 10 km × 10 km grid cell) radon concentration at the ground-floor level was estimated at 1 m off the grid centroid, to which the AMs in the input database are referenced. Predictions were therefore carried out at locations slightly different from the ones of the data. The reason is that we wanted to avoid exact interpolations. To some extent, indoor radon variations at a small scale can be taken into account this way.
### 2.2.1 Inverse distance weighted (IDW) interpolation
The inverse distance weighting (IDW) interpolation technique estimates a weighted average at an unsampled point (${\stackrel{\mathrm{^}}{Z}}_{\mathrm{0}}$) according to its distance (di) to the sampled points (Zi):
$\begin{array}{}\text{(11)}& {\stackrel{\mathrm{^}}{Z}}_{\mathrm{0}}=\frac{\sum _{i=\mathrm{1}}^{n}\frac{\mathrm{1}}{{d}_{i}^{\mathrm{p}}}{Z}_{i}}{\sum _{i=\mathrm{1}}^{n}\frac{\mathrm{1}}{{d}_{i}^{\mathrm{p}}}}\phantom{\rule{0.25em}{0ex}}\mathrm{if}\phantom{\rule{0.25em}{0ex}}{d}_{i}>\mathrm{0};\phantom{\rule{0.25em}{0ex}}\mathrm{otherwise}\phantom{\rule{0.25em}{0ex}}\left({d}_{i}=\mathrm{0}\right):{\stackrel{\mathrm{^}}{Z}}_{\mathrm{0}}={Z}_{i},\end{array}$
where “p” is the inverse distance weighting power (idp), which represents “the degree to which the nearer points are preferred over more distant points” (Bivand et al., 2008). IDW assumes that, on average, nearby points are more similar to each other than more distant points (Li and Heap, 2008), and therefore the weights for the closer ones are higher than the weights for distant points.
The result is highly influenced by the inverse distance weighting power chosen. An optimal value of p which minimizes a loss function L, popt= argmin L (data, target locations; p), can be found for example by k-fold cross-validation. The loss function has to be defined by the user, and a common choice is the root-mean-square error (RMSE) (Janik et al., 2018). In our case the optimal idp was found to be 1.5 (Fig. 3), and interpolations of the AM were carried out using the observations within a distance of 1000 km, and a minimum and maximum number of nearest observations were set to 5 and 75, respectively.
Figure 3Inverse distance weighting power (idp) optimization.
### 2.2.2 Ordinary kriging (OK)
Trans-Gaussian kriging using Box–Cox transforms (function krigeTg in R software, packages “gstat” and “MASS”; Gräler et al., 2016; Kendall et al., 2016; Pebesma, 2004; R Core Team, 2018; Venables and Ripley, 2002) was performed with the arithmetic mean. The normal transformation of data (X) with the transformation parameter lambda (λ) follow Eq. (12) (Box and Cox, 1964):
$\begin{array}{}\text{(12)}& {\mathit{\varphi }}_{\mathit{\lambda }}^{-\mathrm{1}}=\left\{\begin{array}{ll}\frac{{X}^{\mathit{\lambda }}-\mathrm{1}}{\mathit{\lambda }}& \mathit{\lambda }\ne \mathrm{0}\\ \mathrm{log}\left(X\right)& \mathit{\lambda }=\mathrm{0}\end{array}\right\.\end{array}$
Predictions are carried out over the transformed data and then unbiased back-transformed to the original scale using the Lagrange multiplier (Eqs. 13–15; Cressie, 1993; Varouchakis et al., 2012):
$\begin{array}{}\text{(13)}& & \stackrel{\mathrm{^}}{Z}\left({S}_{\mathrm{0}}\right)=\mathit{\varphi }\left({\stackrel{\mathrm{^}}{Y}}_{\mathrm{OK}}\left({S}_{\mathrm{0}}\right)\right)+{\mathit{\varphi }}^{\prime \prime }\left(\stackrel{\mathrm{^}}{\mathit{\mu }}\right)\left(\frac{{\mathit{\sigma }}_{\mathrm{OK}}^{\mathrm{2}}\left({S}_{\mathrm{0}}\right)}{\mathrm{2}}-m\right),\text{(14)}& & \mathit{\varphi }\left(x\right)=\left\{\begin{array}{ll}\left(x\cdot \mathit{\lambda }{\right)}^{\frac{\mathrm{1}}{\mathit{\lambda }}}& \mathit{\lambda }\ne \mathrm{0}\\ {e}^{x}& \mathit{\lambda }=\mathrm{0}\end{array}\right\,\text{(15)}& & {\mathit{\varphi }}^{\prime \prime }\left(x\right)=\left\{\begin{array}{ll}\left(\mathrm{1}-\mathit{\lambda }\right)\left(x\cdot \mathit{\lambda }+\mathrm{1}{\right)}^{\frac{\mathrm{1}}{\mathit{\lambda }}-\mathrm{2}}& \mathit{\lambda }\ne \mathrm{0}\\ {e}^{x}& \mathit{\lambda }=\mathrm{0}\end{array}\right\,\end{array}$
where $\stackrel{\mathrm{^}}{Z}\left({S}_{\mathrm{0}}$) is the ordinary kriging predictor on the original scale, ${\stackrel{\mathrm{^}}{Y}}_{\mathrm{OK}}\left({S}_{\mathrm{0}}\right)$ the ordinary kriging predictor on the transformed scale data, ${\mathit{\sigma }}_{\mathrm{OK}}^{\mathrm{2}}\left({S}_{\mathrm{0}}\right)$ the ordinary kriging variance, $\stackrel{\mathrm{^}}{\mathit{\mu }}$ the mean estimate at each location, and m the Lagrange multiplier of the OK system for each location (Kozintsev et al., 1999).
The variogram was modelled with two components: a Matérn model (Minasny and McBratney, 2005; Pardo-Iguzquiza and Chica-Olmo, 2008) up to a distance of 50 km and an exponential model up to 1500 km (Fig. 4). The very low kappa (0.15) points to high “roughness” of the field. Predictions were then carried out with observations within a distance of 1000 km and using a minimum and a maximum number of nearest observations of 5 and 75, respectively.
Figure 4Model variogram (blue line; green dots are pairs of points up to a distance of 50 km and red points up to 1500 km) and 100 variograms from random permutations of the data (grey lines).
Figure 5(a) Uranium concentration in topsoil (max = 9.73 mg km−1; Tollefsen et al., 2016) and (b) scatterplot between indoor radon and uranium concentration in topsoil.
### 2.2.3 Collocated cokriging (CCK) with uranium as secondary variable
Collocated cokriging (CCK) is a special case of cokriging where only the direct correlation between the primary (e.g. AM_z) and the secondary variables (e.g. U) is used, ignoring the direct variogram of the secondary variable and the cross variograms. It simplified the cokriging equations although the secondary variable must be sampled at all prediction points (Bivand et al., 2008). The method is a simplification of the physical reality because the dependence structure between covariates is more complex, as they result from different physical processes.
We performed the CCK with the uranium concentration in topsoil as a secondary variable since radon is generated in the uranium decay series (Cothern and Smith, 1987), and a positive correlation between uranium and indoor radon is therefore expected. The analysis was carried out with the data log-transformed and then back-transformed to the original scale (AM_z) with Eqs. (16) and (17) (where μ is the kriging prediction and σ the kriging variance):
$\begin{array}{}\text{(16)}& & E\left[X\right]={e}^{\left(\mathit{\mu }+\frac{{\mathit{\sigma }}^{\mathrm{2}}}{\mathrm{2}}\right)},\text{(17)}& & \mathrm{var}\left[X\right]={e}^{\left(\mathrm{2}\mathit{\mu }+{\mathit{\sigma }}^{\mathrm{2}}\right)}\cdot \left({e}^{{\mathit{\sigma }}^{\mathrm{2}}}-\mathrm{1}\right).\end{array}$
The uranium map (Fig. 5a; Tollefsen et al., 2016), part of the European Atlas of Natural Radiation, has been created using approximately 5000 topsoil data from the GEMAS and FOREGS datasets (i.e. GEMAS: Geochemical Mapping of Agricultural and Grazing Land soil in Europe, GEMAS, 2008; and FOREGS: Geochemical Atlas of Europe developed by the Forum of European Geological Surveys, FOREGS, 2005). Uranium explains about 7.75 % of the total indoor radon variability (correlation coefficient = 0.2783; Fig. 5b). As in the previous cases, a maximum distance of 1000 km and a minimum and maximum number of nearest observations of to 5 and 75, respectively, were used in the predictions.
### 2.2.4 Regression kriging (RK)
Regression kriging (RK) is a two-step interpolation technique: first, a regression estimation of the dependent variable (e.g. AM_z) is performed against secondary variables (e.g. geogenic factors), and, second, an analysis of the spatial distribution of the residual is carried out using geostatistical methods (i.e. OK; Pásztor et al., 2016). The final estimates are the sums of the regression estimates and the ordinary kriging estimates of the residuals (Di Piazza et al., 2015). The analysis was also carried out with the log-transformed data and directly back-transformed with the same equation as in CCK.
The technique applied in the regression step can vary (Li and Heap, 2008); here, we have performed a linear regression using topsoil geochemistry (i.e. U and K2O) and geology (i.e. 1:5 Million International Geological Map of Europe – IGME 5000; Asch, 2003) as secondary variables. The IGME 5000 has been developed by the German Federal Institute for Geosciences and Natural Resources; this European GIS project involved more than 40 European and adjacent countries, covering an area from the Caspian Sea in the east, to the mid-ocean ridge in the west, and from Svalbard Islands in the north to the southern shore of the Mediterranean Sea. The aim of the project was to develop a GIS underpinned by a geological database. The original IGME map presents 178 lithostratigraphic units that were reduced to 28 lithological units (Fig. 6). Based on ANOVA tests ran on an extensive Italian geological database, Nogarotto et al. (2018) demonstrated that lithology alone has a large effect on geochemical variations in key elements (U, Th, K2O), regardless of the tectonostratigraphic position of a given unit. It is therefore assigned the prevalence unit to each grid of 10 km × 10 km (Fig. 6).
Figure 6Simplified geology map with geological units defined on a lithology basis (Nogarotto et al., 2018). The base geological map is the IGME (Asch, 2003).
Figure 7(a) Linear model and (b) variogram of residuals.
The procedure is therefore (i) to fit a linear model to the data (Fig. 7a and Table 3), where the total indoor radon variance explained by U, K2O, and simplified geology is 20.24 % (7.75 %, 7.88 %, and 4.61 %, respectively); (ii) to analyse the spatial distribution of residuals, ordinary kriging (Fig. 7b); (iii) to predict a radon value (i.e. log(AM_z)) in each grid using the linear model and add the residual predictions; and iv) to back-transform to the original scale with the equations described in the previous section (Eqs. 16 and 17; where μ is the linear model prediction plus the ordinary kriging prediction of the residuals, and σ is the kriging variance).
Table 3ANOVA table for indoor radon concentration.
Significance codes: ${}^{***}$ denotes p values < 0.001; ${}^{**}$ denotes p values < 0.01; * denotes p values < 0.05; a full stop denotes p values < 0.1; and for p values > 0.1 nothing is printed.
### 2.2.5 Cross-validation
The performances of the different methods were assessed by 5×10-fold cross-validation and by moving-window cross-validation (Kasemsumran et al., 2006). For the 5×10-fold cross-validation method, data were randomly split into 10 subgroups and predictions were carried out 10 times; each time one group is used for validation and nine are used for modelling the variable of interest (i.e. AM_z) at the validation locations. This process is then repeated five times, obtaining a total of 50 realizations. The moving-window cross-validation (MWCV) was carried out with cell sizes of 200 km × 200 km (total number of windows = 197). Grid cells within a window are removed, and an AM is predicted with the rest, then errors are calculated and the process is repeated with another window until all windows are covered. Some restrictions to the validation set were used to avoid errors during kriging methods (i.e. number of grids in the validation set higher than 1; var(log[U]) > 0; and geological units of the validation set must also be in the model set).
The accuracy of the different methods was assessed using six indicators: the mean absolute error (MAE), the root-mean-square error (RMSE), the root-mean-square log error (RMSLE), the index of agreement (IA), the percentage bias (PB), and the coefficient of determination (R2) (Eqs. 18–23).
$\begin{array}{}\text{(18)}& & \mathrm{MAE}=\frac{\mathrm{1}}{n}\sum _{i=\mathrm{1}}^{n}\left|{Z}_{i}-{X}_{i}\right|\text{(19)}& & \mathrm{RMSE}=\sqrt{\frac{\mathrm{1}}{n}\sum _{i=\mathrm{1}}^{n}{\left({Z}_{i}-{X}_{i}\right)}^{\mathrm{2}}}\text{(20)}& & \mathrm{RMSLE}=\frac{\mathrm{1}}{n}\sum _{i=\mathrm{1}}^{n}{\left(\mathrm{log}\left({Z}_{i}+\mathrm{1}\right)-\mathrm{log}\left({X}_{i}+\mathrm{1}\right)\right)}^{\mathrm{2}}\end{array}$
$\begin{array}{}\text{(21)}& & \mathrm{IA}=\mathrm{1}-\frac{\sum _{i=\mathrm{1}}^{n}{\left({Z}_{i}-{X}_{i}\right)}^{\mathrm{2}}}{\sum _{i=\mathrm{1}}^{n}{\left(\left|{X}_{i}-\stackrel{\mathrm{‾}}{X}\right|-\left|{Z}_{i}-\stackrel{\mathrm{‾}}{X}\right|\right)}^{\mathrm{2}}}\text{(22)}& & \mathrm{PB}=\mathrm{100}\frac{\sum _{i=\mathrm{1}}^{n}\left({Z}_{i}-{X}_{i}\right)}{\sum _{i=\mathrm{1}}^{n}{X}_{i}}\text{(23)}& & {R}^{\mathrm{2}}=\mathrm{1}-\frac{\sum _{i=\mathrm{1}}^{n}{\left({Z}_{i}-{X}_{i}\right)}^{\mathrm{2}}}{\sum _{i=\mathrm{1}}^{n}{\left({X}_{i}-\stackrel{\mathrm{‾}}{X}\right)}^{\mathrm{2}}}\end{array}$
Here Zi and Xi are the predicted and measured values in the validation location (Si), n the number of points in the validation group, and $\stackrel{\mathrm{‾}}{X}$ the mean of Xi. MAE and RMSE are commonly used for assessing model performance; however, they may be influenced by outliers (Chen et al., 2017). RMSLE, on the contrary, is less sensitive to outliers and preferable when there is a large range in the values (Janik et al., 2018). These parameters are positive values, and the closer they are to 0, the better the model fit is. IA is a standardized measure of the degree of model prediction error; it varies from 0 (no agreement at all) to 1 (perfect match). PB (%) measures the average tendency of having larger/smaller predicted values than the observed ones. The optimal value is 0, and positive/negative values indicate over/underestimation bias (Janik et al., 2018). Finally, R2 is a measure of how well the model fits a dataset; a perfect model has R2=1 (Alexander et al., 2015).
3 Results and discussion
## 3.1 Cross-validation
The 5×10-fold cross-validation (Fig. 8 and Table 4) shows that geostatistical techniques (i.e. OK, CCK, RK), which take into account the spatial autocorrelation of the data, generally perform better (i.e. lower MAE and RMSLE and higher R2) than IDW. However, they have a tendency to overestimate bias (PB > 0). Then, geostatistical results are slightly improved when geological information is added. The model which has the highest R2 is RK (median = 0.2462), followed by CCK (0.2460) and OK (0.2377). RK is also the geostatistical technique with higher IA (0.6014) and lower PB (2.513), and it has similar MAE and RMSLE as OK and CCK (around 47 and 0.36, respectively).
Figure 8Box plot of the 5×10-fold cross-validation results.
Table 4The 5×10-fold cross-validation results.
Similar results are obtained in the MWCV exercise (Table 5). Geostatistical techniques (i.e. OK, CCK, RK) also have the highest R2 and the lowest MAE and RMSLE. However, in these cases the RK bias is close to 0 (PB =−0.98), while OK and CCK overestimate the values. MWCV also suggests that results are slightly improved when geogenic factors are taken into account: e.g. R2 increases from 0.3457 (OK) to 0.3512 (CCK) and then to 0.3687 (RK); and the highest IA is obtained with RK (0.4531). However, similar MA, RMSE, and RMSLE were also found (around 54, 136, and 0.48, respectively), which indicates the difficulty of predicting an average indoor radon concentration even when secondary variables are added.
Table 5Moving-window cross-validation results.
Radon predictions with the different methods range from minimum values of 1–4 Bq m−3 to up to 10 116 Bq m−3, while the mean values are of the order of 95–105 Bq m−3 (Table 6). The very high value of an AM (i.e. 10 116 Bq m−3) seems improbable, although the grid is in a region with uranium deposits and former uranium mines (border region between Spain and Portugal). This cell has only two measurements (i.e. 9726 and 10 507 Bq m−3), so that the level of reliability of this extremely high AM is therefore low and it would probably decrease if the number of data were increased. In this sense, IDW interpolation, which gives an exact interpolation when the distance between the predicted and measured points is zero, estimates a value that is the arithmetic mean (i.e. 10 116 Bq m−3). Nevertheless, when the spatial autocorrelation between cells is considered (i.e. OK, CCK, and RK), the predicted values, although also high, are reduced to 2500–2800 Bq m−3. These latter values may be more realistic and are similar to average values found in some villages of the region (i.e. 1851 Bq m−3 in Villar de la Yegua, Spain; Sainz et al.. 2010). However, this effect shows the difficulties with predicting an AM when the number of measurements in a grid cell is low. Geostatistical techniques may help to overcome some of these limitations, although the reliability of data because of different numbers of measurements (e.g. grids with only one or two and others with more than 20–30 measurements) is still a problem. It is also not clear whether in an “anomalous area” such as the one cited above, where the geological conditions are particular, the covariance function (or the variogram), which has been estimated from all data, still applies. One can assume that in such a region 2nd-order stationarity is violated. But the accuracy of local prediction depends very much on the local covariance model.
Table 6Summary of indoor radon predictions (AM, ground floor).
Small differences may be appreciated in the predictions of the different interpolation techniques (Fig. 9). IDW and OK are methods that rely on the Rn data only, while CCK and RK use additional predictors (i.e. geology, U and K2O concentration in the ground) as secondary variables. The first type is weak in areas with no conditioning data as it simply interpolates between existing ones, ignoring physical reality in these areas (e.g. south Italy, north-east Germany). Including it is the rationale of the second type; practically, missing conditioning data of the primary variable (Rn) are substituted by functions of the secondary variables. Although certainly more reasonable in the physical sense, this type is analytically more complicated.
Figure 9Indoor radon predictions (AM (Bq m−3), ground floor).
The influence of geogenic factors on indoor radon is well known and normally used for radon mapping (e.g. Casey et al., 2015; Elío et al., 2017; Pásztor et al., 2016; Scheib et al., 2013; Tondeur et al., 2014). In our cases, an ANOVA (Table 3) shows that the total indoor radon variance explained by U, K2O, and geology is about 20 % (7.75 %, 7.88 %, and 4.61 %, respectively). Uranium is a source of radon in soil, and thus a positive association with indoor radon is expected (e.g. Appleton et al., 2008; Ferreira et al., 2018). However, the Pearson's correlation coefficient between indoor radon and uranium concentration in topsoil is relatively low (r=0.2783), which implies that CCK estimations with U as the secondary variable are still mainly based on the primary variable (i.e. AM; Rocha et al., 2012). Therefore, although CCK performs slightly better than OK, spatial predictions are similar (Fig. 9).
Geology is associated with both uranium and radon sources and with physical properties which permit the release of radon from the soil matrix and its transport in the environment (e.g. mineralogy, porosity, permeability). The total indoor radon variance explained by geology is normally of the order of 5 %–25 % (Appleton and Miles, 2010; Borgoni et al., 2014; Miles and Appleton, 2005; Tondeur et al., 2014; Watson et al., 2017), although it depends on the geological scale map (i.e. increase with the scale; Appleton and Miles, 2010). A 4.64% of indoor radon variation explanation is therefore reasonable, taking into account that we used a simplified 1:5 million geological map and that data are averaged over grids of 10 km × 10 km.
The positive correlation between indoor radon and potassium is, however, not evident. K2O may be related to clay content in soils (e.g. Barré et al., 2008; Milošević et al., 2017; Tarvainen et al., 2005), and although the permeability of wet clays is low, it may increase when soils are dried (Petersell et al., 2005) as a consequence of building a house (Barnet et al., 2008). This hypothesis should be tested since clay soils are normally considered low risk although its radium concentration may be high (Maestre and Iribarren, 2018). We have decided, however, to include this parameter in the model since previous studies have shown a positive association between indoor radon and K2O/clay (Forkapic et al., 2017).
Table 7Summary of indoor radon at the European scale.
Back-transforming predictions to the original scale is a critical point of log-normal and trans-Gaussian kriging. OK as given in this study solves this problem by using the Lagrange multiplier in the back-transformation. However, the E[X] and Var[X] for CCK and RK are biased, unless the true mean is known (although for RK it should be zero by definition). These equations should also use the Lagrange multiplier which appears in the kriging system (Chilès and Delfiner, 1999; Matheron, 1974); but unfortunately in common geostatistical packages this parameter is not accessible, and it is not easy to estimate it. Another problem with log-normal kriging is that ill assessment of the kriging SD leads to large errors in E[X] and Var[X] due to exponentiation, so that variogram parameters must be estimated very carefully (Armstrong and Boufassa, 1988). Deviations from stationarity and uni- as well as multivariate log-normality are also critical (Cressie, 1993; Roth, 1998). On the other hand, in highly skewed quantities (as is typical for Rn and in fact for many positive-definite environmental quantities such as concentrations) there seems to be little choice but to work with transformed (e.g. log, Box–Cox, N score) variables.
Finally, a theoretical problem, if using kriging-type interpolators, it may be that input data are actually cell or grid means (blocks in geostatistical language), treated as point samples. The change-of-support problem, which is particularly unpleasant in log-normal kriging, may be alleviated since the target supports are also the same. We regard input data as point data at the cell centre, and we estimate points at other locations that again represent cells of the same size. However, the theoretical aspect remains to be clarified in more depth. Taking into account all of these limitations and weaknesses, the solution demonstrated here, however, represents an acceptable compromise between mathematical exactness, numerical tractability, and complexity of the physical realm.
4 A pan-European indoor radon map
We would like to produce the Pan-European Indoor Radon Map by minimizing data processing, and therefore we prefer to estimate the radon average directly by indoor radon measurements carried out at each grid (i.e. AM_z). However, if the number of measurements were low, the uncertainty of this value could be high. In this sense, if dwellings were randomly selected and therefore representative, which is the condition for unbiased estimates of the mean and other statistics, and the sample size large, the mean value and the confidence interval would be (Eq. 24)
$\begin{array}{}\text{(24)}& \stackrel{\mathrm{‾}}{x}=\frac{\mathrm{1}}{n}\sum _{i}^{n}{x}_{i}±{t}_{\left(\mathrm{1}-\frac{\mathit{\alpha }}{\mathrm{2}},n-\mathrm{1}\right)}\frac{s}{\sqrt{n}}.\end{array}$
An additional condition for the validity of the confidence interval is statistical independence of the data. For large n, due to the central limit theorem, ${t}_{\mathrm{1}-\mathit{\alpha }/\mathrm{2};n-\mathrm{1}}$ can be approximated by the normal score ${x}_{\mathrm{1}-\mathit{\alpha }/\mathrm{2}}$.
The confidence interval decreases when the sample size increases. In our cases (Fig. 10), the relative (to the mean) CI95 % (α=0.05) for sample size of about 30–40 data is around ±5 % and generally lower than 15 %–30 %. Therefore, although the assumption of data independence is not valid (i.e. there is spatial correlation between indoor radon measurements which can be modelled by the variogram), 30 measurements seems reasonable for obtaining a good estimation of the radon exposure in a specific grid (Fig. 11). However, if sampled dwellings were highly clustered, the AM could be not representative of the radon exposure in a grid even with high numbers of indoor radon measurements.
Figure 10Variation in the 95 % confidence interval of the arithmetic mean according to the sample size (N).
Figure 11Grids with 30, or more, indoor radon measurements (N=4173; AM: arithmetic mean in becquerels per cubic metre; RSD: relative standard deviation in percent).
For the final Pan-European Indoor Radon Map (Table 7 and Fig. 12), we therefore use the AM of the grid cells with 30 or more measurements (Fig. 11) and the value predicted by RK (Fig. 9) in the cells with fewer than 30 measurements. Indoor radon concentration ranges from 3 to 2662 Bq m−3, with a mean value of 98 Bq m−3. The standard deviation may be calculated with the SD of the measurements carried out in the grids with 30 or more data and with the kriging standard deviation of the RK (i.e. grids with fewer than 30 measurements). It ranges from 1 to 3233 Bq m−3, with a RSD from 3 % to up to 1101 %. The map appears “noisy” to varying degrees between different regions. The reason is that in regions with more conditioning, Rn data, local variability of the estimate is higher than in regions with sparse or without data, where the estimate is based essentially on geology and geochemistry. These covariates are much smoother on the scale available to us than Rn data, where available. Were we to dispose of denser geochemical data and higher-resolution geological maps, these regions would also appear noisier.
Figure 12Final Pan-European Indoor Radon Map.
5 Conclusions
After more than 10 years of collecting and processing Rn data, with the support of 32 European countries, we could cover approximately 50 % of the continent with 10 km × 10 km grids containing the mean indoor radon concentration in ground floors of dwellings. However, completing the European Indoor Radon Map still requires a significant effort by the participating countries since a robust estimation of the radon exposure in a grid of 10 km × 10 km involves at least 30 indoor radon measurements and, at the time of writing this article, most of the grids cells sampled (78 %) have fewer than 20 dwellings tested. Interpolation techniques which take advantage of the contiguity of Rn seen as a spatially random field may help to overcome some of the present limitations and would permit estimation of radon exposure at the European scale until the coverage of all of Europe with indoor radon measurements has strongly improved.
Of the four methods tested in this study, regression kriging (RK), using a simplified geological map and the topsoil concentration of U and K2O, has proven to be the best one for predicting mean indoor radon concentrations over grids of 10 km × 10 km (i.e. arithmetic mean, ground floor). By combining RK predictions with empirical average values (AM) in grids with 30 or more measurements, a Pan-Europe Indoor Radon Map has been created. The map represents the average value of indoor radon concentration on the ground floor, and thus it is not representative of the radon exposure to European citizens since most people do not live on the ground floor (Cinelli et al., 2019). However, it is the first step towards a radon exposure map and, in the future, a dose map. Based on demographic and sociological databases, we plan to develop models which allow correction from ground-floor dwellings to the real situation, accounting for the building stock (Bossew et al., 2018).
The Pan-European Indoor Radon Map is not a finished map, and it will be upgraded as new data become available. In future versions a larger scale of the geological map (e.g. scale 1:1 million), as well as other geogenic factors which may influence the indoor radon concentration (e.g. soil units, aquifer types), would be included in the model. Furthermore, the influence of anthropogenic factors and those which may affect building characteristics and living styles (e.g. average temperatures, annual precipitation, altitude) will be analysed. Finally, machine-learning techniques are viewed as promising methods for modelling the AM since kriging-type predictions come to an end if many (intercorrelated) predictors are involved.
Data availability
Data availability.
The indoor radon data related to this article are confidential. Additional data and code can, however, be made available by the authors upon request. Last versions of the maps used in the article can be seen on a JRC website (https://remon.jrc.ec.europa.eu/About/Atlas-of-Natural-Radiation, last access: 30 October 2019). The geological map of Europe (IGME 5000; scale 1:5 million) may be download from the BGR website (https://produktcenter.bgr.de/terraCatalog/DetailResult.do?fileIdentifier=9FD6624C-0AA7-46D4-9DA3-955E558CD5F1, last access: 7 May 2019).
Author contributions
Author contributions.
JE was mainly responsible for the data analysis and interpretation, and he wrote the article with inputs from all authors. PB and JLGV contributed to the data analysis and interpretation. AN and RB carried out the simplification of the geological map of Europe. GC, TT, and MDC, contact persons of the European Atlas of Natural Radiation, have provided the indoor radon data – collecting them from national authorities, maintaining the datasets, and upgrading them when new data are available. They also helped with data interpretation.
Competing interests
Competing interests.
The authors declare that they have no conflict of interest.
Acknowledgements
Acknowledgements.
We wish to thank all the national competent authorities, universities, and laboratories who have provided, and continue to provide, indoor radon data to the JRC (see https://remon.jrc.ec.europa.eu/About/Atlas-of-Natural-Radiation/Indoor-radon-AM/Indoor-radon-concentration, last access: 20 September 2019). The sole responsibility of this publication lies with the authors. The European Commission and national competent authorities are not responsible for any use that may be made of the information contained herein.
Review statement
Review statement.
This paper was edited by Heidi Kreibich and reviewed by two anonymous referees.
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https://repository.kaust.edu.sa/handle/10754/630940?show=full | dc.contributor.author Dutta, Aritra dc.contributor.author Li, Xin dc.date.accessioned 2019-01-22T12:37:17Z dc.date.available 2019-01-22T12:37:17Z dc.date.issued 2019-01-03 dc.identifier.citation Dutta A, Li X (2018) A Fast Weighted SVT Algorithm. 2018 5th International Conference on Systems and Informatics (ICSAI). Available: http://dx.doi.org/10.1109/icsai.2018.8599289. dc.identifier.doi 10.1109/icsai.2018.8599289 dc.identifier.uri http://hdl.handle.net/10754/630940 dc.description.abstract Singular value thresholding (SVT) plays an important role in the well-known robust principal component analysis (RPCA) algorithms which have many applications in machine learning, pattern recognition, and computer vision. There are many versions of generalized SVT proposed by researchers to achieve improvement in speed or performance. In this paper, we propose a fast algorithm to solve aweighted singular value thresholding (WSVT) problem as formulated in [1], which uses a combination of the nuclear norm and a weighted Frobenius norm and has shown to be comparable with RPCA method in some real world applications. dc.publisher Institute of Electrical and Electronics Engineers (IEEE) dc.relation.url https://ieeexplore.ieee.org/document/8599289 dc.subject Singular Value Thresholding dc.subject Low-rank Approximation dc.subject Background Estimation dc.title A Fast Weighted SVT Algorithm dc.type Conference Paper dc.contributor.department Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division dc.contributor.department Visual Computing Center (VCC) dc.identifier.journal 2018 5th International Conference on Systems and Informatics (ICSAI) dc.contributor.institution Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA kaust.person Dutta, Aritra dc.date.published-online 2019-01-03 dc.date.published-print 2018-11
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https://plato.stanford.edu/entries/frege-theorem/proof5.html | ## Proof of Equinumerosity Lemma
In this proof of the Equinumerosity Lemma, we utilize the following abbreviation, where $$\mathrm{\phi}$$ is any formula in which the variable $$y$$ may or may not be free and $$\mathrm{\phi}^{\nu}_{\upsilon}$$ is the result of replacing the free occurrences of $$\upsilon$$ in $$\mathrm{\phi}$$ by $$\nu$$:
$$x\eqclose \mathit{\iota}y\mathrm{\phi} \eqabbr \mathrm{\phi}\amp \forall z(\mathrm{\phi}^z_y \rightarrow z\eqclose x)$$
We may read this as follows:
$$x$$ is identical to the object $$y$$ which is such that $$\mathrm{\phi}$$ if and only if both $$x$$ is such that $$\mathrm{\phi}$$ and everything which is such that $$\mathrm{\phi}$$ is identical to $$x$$.
This abbreviation is employed below to simplify the definition of new relations. Given this new notation, we will use only the following simple consequence of this definition:
Principle of Descriptions:
$$x\eqclose \mathit{\iota} y\mathrm{\phi}\rightarrow \mathrm{\phi}^x_y$$
In other words, if $$x$$ is the object $$y$$ such that $$\mathrm{\phi} (y)$$, then $$x$$ is such that $$\mathrm{\phi}$$. The use of this principle will be obvious in what follows.
Proof of Equinumerosity Lemma. Assume that $$P\approx Q, Pa$$, and $$Qb$$. So there is a relation, say $$R$$, such that (a) $$R$$ maps every object falling under $$P$$ to a unique object falling under $$Q$$ and (b) for every object falling under $$Q$$ there is a unique object falling under $$P$$ which is $$R$$-related to it. Now we use $$P^{-a}$$ to designate $$[\lambda z \, Pz\amp z\neq a]$$, and we use $$Q^{-b}$$ to designate $$[\lambda z \, Qz\amp z\neq b]$$. We want to show that $$P^{-a}\approx Q^{-b}$$. By the definition of equinumerosity, we have to show that there is a functional relation $$R'$$ which is 1-1 from the objects falling under $$P^{-a}$$ onto the objects falling under $$Q^{-b}$$. We prove this by cases.
Case 1: Suppose $$Rab$$. Then we choose $$R'$$ to be $$R$$ itself. Clearly, $$R$$ is then a 1-1 functional relation from the objects of $$P^{-a}$$ to the objects of $$Q^{-b}$$. But the proof can be given as follows. We show: (A) that $$R$$ is a functional relation from the objects of $$P^{-a}$$ to the objects of $$Q^{-b}$$, and then (B) that $$R$$ is a 1-1 functional relation from the objects of $$P^{-a}$$ onto the objects of $$Q^{-b}$$.
(A) Pick an arbitrary object, say $$c$$, such that $$P^{-a}c$$. We want to show that there is a unique object which falls under $$Q^{-b}$$ and to which $$c$$ bears $$R$$. Since $$P^{-a}c$$, we know that $$Pc\amp c\neq a$$, by the definition of $$P^{-a}$$. But if $$Pc$$, then by our hypothesis that $$R$$ is a witness to the equinumerosity of $$P$$ and $$Q$$, it follows that there is a unique object, say $$d$$, such that $$Qd$$ and $$Rcd$$. But we are considering the case in which $$Rab$$ and so from the established facts that $$Rcd$$ and $$c\neq a$$, it follows by the 1-1 character of $$R$$ that $$b\neq d$$. So we have that $$Qd$$ and $$d\neq b$$, which establishes that $$Q^{-b}d$$. And we have also established that $$Rcd$$. So it remains to show that every other object that falls under $$Q^{-b}$$ to which $$c$$ bears $$R$$ just is identical to $$d$$. So suppose $$Q^{-b}e$$ and $$Rce$$. Then by definition of $$Q^{-b}$$, it follows that $$Qe$$. But now $$e=d$$, for $$d$$ is the unique object falling under $$Q$$ to which $$c$$ bears $$R$$. So there is a unique object which falls under $$Q^{-b}$$ and to which $$c$$ bears $$R$$.
(B) Pick an arbitrary object, say $$d$$, such that $$Q^{-b}d$$. We want to show that there is a unique object falling under $$P^{-a}$$that bears $$R$$ to $$d$$. Since $$Q^{-b}d$$, we know $$Qd$$ and $$d\neq b$$. From $$Qd$$ and the fact that $$R$$ witnesses the equinumerosity of $$P$$ and $$Q$$, we know that there is a unique object, say $$c$$, that falls under $$P$$ and which bears $$R$$ to $$d$$. Since we are considering the case in which $$Rab$$, and we’ve established $$Rcd$$ and $$d\neq b$$, it follows that $$a\neq c$$, by the fact that $$R$$ is a functional relation. Since we now have $$Pc$$ and $$c\neq a$$, we have established that $$c$$ falls under $$P^{-a}$$, and moreover, that $$Rcd$$. So it remains to prove that any other object that falls under $$P^{-a}$$ and which bears $$R$$ to $$d$$ just is (identical to) $$c$$. But if $$f$$, say, falls under $$P^{-a}$$ and bears $$R$$ to $$d$$, then $$Pf$$, by definition of $$P^{-a}$$. But recall that $$c$$ is the unique object falling under $$P$$ which bears $$R$$ to $$d$$. So $$f=c$$.
Case 2: Suppose $$\neg Rab$$. Then we choose $$R’$$ to be the relation:
\begin{align*} &[\lambda xy \, (x\neqclose a \amp y\neqclose b\amp Rxy)\ \lor \\ &\quad (x\eqclose \mathit{\iota}u(Pu\amp Rub)\amp y\eqclose \mathit{\iota}u(Qu\amp Rau))] \end{align*}
To see that there is such a relation, note that once we replace the abbreviations $$x=\mathit{\iota}u(Pu\amp Rub)$$ and $$y=\mathit{\iota}u(Qu\amp Rau)$$ by primitive notation, the matrix of the $$\lambda$$-expression is a formula of the form $$\mathrm{\phi}(x,y)$$ which can be used in an instance of the Comprehension Principle for Relations.
Now we want to show that $$R’$$ is a 1-1 functional relation from the objects of $$P^{-a}$$ onto the objects of $$Q^{-b}$$. We show (A) that $$R’$$ is a functional relation from the objects of $$P^{-a}$$ to the objects of $$Q^{-b}$$, and then (B) that $$R’$$ is a 1-1 functional relation from the objects of $$P^{-a}$$ onto the objects of $$Q^{-b}$$.
(A) To show that $$R’$$ is a functional relation from the objects of $$P^{-a}$$ to the objects of $$Q^{-b}$$, pick an arbitrary object, say $$c$$, such that $$P^{-a}c$$. Then by definition of $$P^{-a}$$, we know that $$Pc$$ and $$c\neq a$$. We need to find an object, say $$d$$ for which the following three things hold: (i) $$Q^{-b}d$$, (ii) $$R’cd$$, and (iii) $$\forall w(Q^{-b}w \amp R’cw\rightarrow w=d)$$. We find such a $$d$$ in each of the following, mutually exclusive subcases:
Subcase 1: $$Rcb$$. So, since we know that each object falling under $$Q$$ is such that there is a unique object falling under $$P$$ that is $$R$$-related to it, we know that $$c= \mathit{\iota}u(Pu\amp Rub)$$. Then, since we know R maps $$a$$ to a unique object falling under $$Q$$, we let $$d$$ be that object. That is, $$d$$ satisfies the defined condition $$d=\mathit{\iota}u(Qu\amp Rau)$$. So $$Qd$$, $$Rad$$, and $$\forall w(Qw\amp Raw\rightarrow w=d)$$. We now show that (i), (ii) and (iii) hold for $$d$$:
1. Since we know $$Qd$$, all we have to do to establish $$Q^{-b}d$$ is to show $$d\neq b$$. But we know $$Rad$$ and we are considering the case where $$\neg Rab$$. So, by the laws of identity, $$d\neq b$$.
2. To show $$R’cd$$, we need to establish:
\begin{align*} &(c\neqclose a\amp d\neqclose b\amp Rcd)\ \lor \\ &\quad (c\eqclose \mathit{\iota}u(Pu\amp Rub)\amp d\eqclose \mathit{\iota}u(Qu\amp Rau)) \end{align*}
But the conjunctions of the right disjunct are true (by assumption and by definition, respectively). So $$R'cd$$.
3. Suppose $$Q^{-b}e$$ (i.e., $$Qe$$ and $$e\neq b$$) and $$R'ce$$. We want to show: $$e=d$$. Since $$R'ce$$, then:
\begin{align*} &(c\neqclose a\amp e\neqclose b\amp Rce)\ \lor \\ &\quad (c\eqclose \mathit{\iota}u(Pu\amp Rub)\amp e\eqclose \mathit{\iota}u(Qu\amp Rau)) \end{align*}
But the left disjunct is impossible (we’re considering the subcase where $$Rcb$$, yet the left disjunct asserts $$Rce$$ and $$e\neq b$$, which together contradict the fact that $$R$$ is a functional relation). So the right disjunct must be true, in which case it follows from the fact that $$e=\mathit{\iota}u(Qu\amp Rau)$$ that $$e=d$$, by the definition of $$d$$.
Subcase 2: $$\neg Rcb$$. We are under the assumption $$P^{-a}c$$ (i.e., $$Pc$$ and $$c\neq a$$), and so we know by the definition of $$R$$ and the fact that $$Pc$$ that there is a unique object which falls under $$Q$$ and to which $$c$$ bears $$R$$. Choose $$d$$ to be this object. So $$Qd$$, $$Rcd$$, and $$\forall w(Qw\amp Rcw\rightarrow w=d)$$. We can now show that (i), (ii) and (iii) hold for $$d$$:
1. Since we know $$Qd$$, all we have to do to establish that $$Q^{-b}d$$ is to show $$d\neq b$$. We know that $$Rcd$$ and we are considering the subcase where $$\neg Rcb$$. So it follows that $$d\neqclose b$$, by the laws of identity. So $$Q^{-b}d$$.
2. To show $$R’cd$$, we need to establish:
\begin{align*} &(c\neqclose a\amp d\neqclose b\amp Rcd)\ \lor \\ &\quad (c\eqclose \mathit{\iota}u(Pu\amp Rub)\amp d\eqclose \mathit{\iota}u(Qu\amp Rau)) \end{align*}
But the conjuncts of the left disjunct are true, for $$c\neq a$$ (by assumption), $$d\neq b$$ (we just proved this), and $$Rcd$$ (by the definition of $$d$$). So $$R’cd$$.
3. Suppose $$Q^{-b}e$$ (i.e., $$Qe$$ and $$e\neq b)$$ and $$R’ce$$. We want to show: $$e=d$$. Since $$R’ce$$, then:
\begin{align*} &(c\neqclose a\amp e\neqclose b\amp Rce)\ \lor \\ &\quad (c\eqclose \mathit{\iota}u(Pu\amp Rub)\amp e\eqclose \mathit{\iota}u(Qu\amp Rau)) \end{align*}
But the right disjunct is impossible (we’re considering the subcase where $$\neg Rcb$$, yet the right disjunct asserts $$c=\mathit{\iota}u(Pu\amp Rub)$$, which implies $$Rcb$$, a contradiction). So $$c\neq a\amp e\neq b \amp Rce$$. Since we now know that $$Qe$$ and $$Rce$$, we know that $$e=d$$, since $$d$$ is, by definition, the unique object falling under $$Q$$ to which $$c$$ bears $$R$$.
(B) To show that $$R’$$ is a 1-1 functional relation from the objects of $$P^{-a}$$ onto the objects of $$Q^{-b}$$, pick an arbitrary object, say $$d$$, such that $$Q^{-b}d$$. Then by definition of $$Q^{-b}$$, we know that $$Qd$$ and $$d\neq b$$. We need to find an object, say $$c$$, for which the following three things hold: (i) $$P^{-a}c$$, (ii) $$R’cd$$, and (iii) $$\forall w(P^{-a}w\amp R’wd\rightarrow w=c)$$. We find such a $$c$$ in each of the following, mutually exclusive cases:
Subcase 1: $$Rad$$. So $$d=\mathit{\iota}u(Qu\amp Rau)$$. Then choose $$c=\mathit{\iota}u(Pu\amp Rub)$$ (we know there is such an object). So $$Pc$$, $$Rcb$$, and $$\forall w(Pw\amp Rwb\rightarrow w=c)$$. We now show that (i), (ii) and (iii) hold for $$c$$:
1. Since we know $$Pc$$, all we have to do to establish $$P^{-a}c$$ is to show $$c\neq a$$. But we know $$Rcb$$, and we are considering the case where $$\neg Rab$$. So, by the laws of identity, it follows that $$c\neq a$$.
2. To show $$R’cd$$, we need to establish:
\begin{align*} &(c\neqclose a\amp d\neqclose b\amp Rcd)\ \lor \\ &\quad (c\eqclose \mathit{\iota}u(Pu\amp Rub)\amp d\eqclose \mathit{\iota}u(Qu\amp Rau)) \end{align*}
But the conjuncts of the right disjunct are true (by definition and by assumption, respectively). So $$R’cd$$.
3. Suppose $$P^{-a}f$$ (i.e., $$Pf$$ and $$f\neq a)$$ and $$R’fd$$. We want to show: $$f=c$$. Since $$R’fd$$, then:
\begin{align*} &(f\neqclose a\amp d\neqclose b\amp Rfd)\ \lor \\ &\quad (f\eqclose \mathit{\iota}u(Pu\amp Rub)\amp d\eqclose \mathit{\iota}u(Qu\amp Rau)) \end{align*}
But the left disjunct is impossible (we’re considering the subcase where $$Rad$$, yet the left disjunct asserts $$Rfd$$ and $$f\neq a$$, which together contradict the fact that $$R$$ is 1-1). So the right disjunct must be true, in which case it follows from the fact that $$f=\mathit{\iota}u(Pu\amp Rub)$$ that $$f=c$$, by the definition of $$c$$.
Subcase 2: $$\neg Rad$$. We are under the assumption $$Q^{-b}d$$ (i.e., $$Qd$$ and $$d\neq b)$$, and so we know by the definition of $$R$$ and the fact that $$Qd$$ that there is a unique object which falls under $$P$$ and which bears $$R$$ to $$d$$. Choose $$c$$ to be this object. So $$Pc$$, $$Rcd$$, and $$\forall w(Pw\amp Rwd\rightarrow w=c)$$. We can now show that (i), (ii), and (iii) hold for $$c$$:
1. Since we know $$Pc$$, all we have to do to establish that $$P^{-a}c$$ is to show $$c\neq a$$. But we know that $$Rcd$$, and we are considering the subcase in which $$\neg Rad$$. So it follows that $$c\neq a$$, by the laws of identity. So $$P^{-a}c$$.
2. To show $$R’cd$$, we need to establish:
\begin{align*} &(c\neqclose a\amp d\neqclose b\amp Rcd)\ \lor \\ &\quad (c\eqclose \mathit{\iota}u(Pu\amp Rub)\amp d\eqclose \mathit{\iota}u(Qu\amp Rau)) \end{align*}
But the conjuncts of the left disjunct are true, for $$c\neq a$$ (we just proved this), $$d\neq b$$ (by assumption), and $$Rcd$$ (by the definition of $$c$$). So $$R’cd$$.
3. Suppose $$P^{-a}f$$ (i.e., $$Pf$$ and $$f\neq a)$$ and $$R’fd$$. We want to show: $$f=c$$. Since $$R’fd$$, then:
\begin{align*} &(f\neqclose a\amp d\neqclose b\amp Rfd)\ \lor \\ &\quad (f\eqclose \mathit{\iota}u(Pu\amp Rub)\amp d\eqclose \mathit{\iota}u(Qu\amp Rau)) \end{align*}
But the right disjunct is impossible (we’re considering the subcase where $$\neg Rad$$, yet the right disjunct asserts $$d=\mathit{\iota}u(Qu\amp Rau)$$, which implies $$Rad$$, a contradiction). So $$f\neq a\amp d\neq b\amp Rfd$$. Since we now know that $$Pf$$ and $$Rfd$$, we know that $$f=c$$, since $$c$$ is, by definition, the unique object falling under $$P$$ which bears $$R$$ to $$d$$. | {"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": 0.9973829984664917, "perplexity": 170.02532491542766}, "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-2018-30/segments/1531676596336.96/warc/CC-MAIN-20180723110342-20180723130342-00000.warc.gz"} |
http://mathhelpforum.com/algebra/92340-another-table-values-question-use-inverse-proportion-find-k-print.html | # Another table of values question, use inverse proportion and find k?
• June 9th 2009, 02:08 PM
olen12
Another table of values question, use inverse proportion and find k?
So I have:
10 100
40 6
70 2
100 1
That is inverse square correct?
Now, how would I 'model that using the correct inverse proportion' and find the value of k using 10 100
• June 9th 2009, 03:52 PM
HallsofIvy
Quote:
Originally Posted by olen12
So I have:
10 100
40 6
70 2
100 1
That is inverse square correct?
Now, how would I 'model that using the correct inverse proportion' and find the value of k using 10 100
An inverse proportion is of the form y= k/x or xy= k. If that were correct we would have k= 10(100)= 1000 but 40(6)= 240, not 1000. If it were $y= k/x^2$ or $x^2y= k$ then, depending on which number is x, we have either [tex]k= (10)(100^2)= 100000[tex] or $k= (10^2)(100)= 10000$. In the first case, $(40)(6^2)= 1440$, which is not right, and in the second, $(40^2)(6)= 9600$, also not right. So it is not "inverse square". If you were to try a generic $y^n= k/x^m$ or $x^my^n= k$, we could write $(10^m)(100^n)= k$ and $(40^m)(6^n)= k$. Dividing one equation by the other [tex]\left(\frac{10}{40}\right)^m\left(\frac{100}{6}\ri ght)^n= 1[tex] or $\left(\frac{1}{4}\right)^m\left(\frac{50}{3}\right )^n$. Especially easy is the last pair: $(100^m)(1^n)= 100^m= k$ so that $\left(\frac{100}{10}\right)^m\left(\frac{1}{100}\r ight)^n= 1$. That gives you two equations to solve for m and n. | {"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": 12, "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": 0.9729415774345398, "perplexity": 1168.4065334646352}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1404776438333.54/warc/CC-MAIN-20140707234038-00010-ip-10-180-212-248.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2077371/cauchy-in-measure-question | Cauchy in measure question
Let $w\in L^1(\mathbb{R}^d)$, where $w>0$. Let $\{f_n\}:\mathbb{R}^d\to\mathbb{R}$ be Lebesgue measurable functions such that $$\lim_{m,n\to\infty}\int_{|f_n-f_m|>t}w(x)\,dx=0$$ for any $t>0$.
Prove that $\{f_n\}$ has a subsequence that converges almost everywhere to a measurable function $g$.
Attempt:
I am trying to show that $\{f_n\}$ is Cauchy in measure, and thus converges in measure, and thus has a convergent subsequence $f_{n_k}\to g$ a.e.
We have $\lim_{m,n\to\infty}\int w(x)\chi_{\{|f_n-f_m|>t\}}(x)\,dx=0$.
Since $|w(x)\chi_{\{|f_n-f_m|>t\}}(x)|\leq|w(x)|\in L^1$ so by Lebesgue's Dominated Convergence Theorem, $$\int w(x)\lim_{m,n\to\infty}\chi_{\{|f_n-f_m|>t\}(x)}=0$$
This means $w(x)\lim_{m,n\to\infty}\chi_{\{|f_n-f_m|>t\}}(x)=0$ almost everywhere on $\mathbb{R}$.
Since $w>0$, so $\lim_{m,n\to\infty}\chi_{\{|f_n-f_m|>t\}}(x)=0$ a.e.
However, I think we cannot conclude $\lim_{m,n\to\infty}|\{|f_n-f_m|>t\}|=0$ which is exactly what we need (Cauchy in measure). So close yet so far..
Thanks for any help.
• Look at the measure $\mu \colon A \mapsto \int_A w(x)\,dx$. – Daniel Fischer Dec 30 '16 at 15:14
• I see that $f_n$ is Cauchy in measure (under $\mu$), and $\mu$ is absolutely continuous with respect to Lebesgue measure. So $f_n$ converges in (Lebesgue) measure! Is this correct? – yoyostein Dec 30 '16 at 15:21
• Not quite. That $\mu$ is absolutely continuous with respect to the Lebesgue measure is not sufficient. You also need the converse, that $\mu$-a.e. is $\lambda$-a.e. – Daniel Fischer Dec 30 '16 at 15:27
• We know $\lim_{m,n\to\infty}\mu(\{|f_n-f_m|>t\})=0$. How do we go from here to conclude that $\lim_{m,n\to\infty}\lambda(\{|f_n-f_m|>t\})=0$, where $\lambda$ is Lebesgue measure?. I know if $\mu(A)=0$, then $\lambda(A)=0$, but having trouble making things rigorous. – yoyostein Dec 30 '16 at 16:07
• You can't show that without further assumptions. If you take $f_n = \chi_{\mathbb{R}^d\setminus B_n(0)}$, then $(f_n)$ satisfies the assumptions, but for $t < 1$, $\lambda(\{\lvert f_n - f_m\rvert > t\}) = c\lvert n^d - m^d\rvert$ can be arbitrarily large. The sequence need not be Cauchy in measure with respect to the Lebesgue measure. – Daniel Fischer Dec 30 '16 at 16:19
The sequence need not be a Cauchy sequence in measure with respect to the Lebesgue measure. An example of that would be $f_n = \chi_{\mathbb{R}^d\setminus B_n(0)}$. However, the sequence is Cauchy in measure with respect to the measure $\mu$, where
$$\mu(A) := \int_A w(x)\,dx.$$
Then it follows that there is a subsequence that converges $\mu$-almost everywhere to a measurable function $g$.
Now we use that $\mu$-almost everywhere is the same as $\lambda$-almost everywhere, which is guaranteed by the strict positivity of $w$, to conclude. | {"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": 0.9967617988586426, "perplexity": 104.83850374579139}, "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-2021-31/segments/1627046152000.25/warc/CC-MAIN-20210726031942-20210726061942-00310.warc.gz"} |
https://bitworking.org/news/2003/03/66 | Just a heads-up that there has been a slight change to the CommentAPI. This change is in the RSS auto-discovery, the namespace for the wfw:comment element has changed from http://wellformedweb.org to http://wellformedweb.org/CommentAPI. | {"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": 0.15604186058044434, "perplexity": 2407.979759129471}, "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/1542039746301.92/warc/CC-MAIN-20181120071442-20181120093442-00293.warc.gz"} |
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# When the positive integer x is divided by 11, the quotient
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When the positive integer x is divided by 11, the quotient [#permalink]
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21 Jan 2012, 01:21
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When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?
A. 0
B. 1
C. 2
D. 3
E. 4
Guys struggling to solve this. But this is the concept I am trying to apply:
We can extrapolate a general statement from this form. When dividing x by y, the quotient is q and the remainder is r:
x/y = q + r/y
From there, we can solve for x:
x = qy + r (thats the general form of x = 3(integer) + 1)
Or the quotient:
q = x-r/y
Or, even, the remainder itself:
r = x - qy
But I am getting stuck in finding y when x is divided by 19. Can someone please help?? I don't have an OA either.
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### Show Tags
21 Jan 2012, 03:02
20
8
enigma123 wrote:
When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?
A) 0
B) 1
C) 2
D) 3
E) 4
If you decide to go with quotient/remainder formula approach, then I'd suggest to express the info in the stem with it. And then look whether we can somehow manipulate with the expressions at hand to answer the question.
(1) When the positive integer x is divided by 11, the quotient is y and the remainder 3 --> $$x=11y+3$$;
(2) When x is divided by 19, the remainder is also 3 --> $$x=19q+3$$.
Easy to spot that $$19q+3=11y+3$$ --> $$19q=11y$$ --> $$y=\frac{19q}{11}$$. Now as $$y$$ and $$q$$ are integers and 19 is prime then $$\frac{q}{11}$$ must be an integer --> $$y=19*integer$$ --> $$y$$ is a multiple of 19, hence when divide by 19 remainder is 0.
Hope its clear.
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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21 Jan 2012, 03:13
2
Find some numbers that fit the question.
We are told that x and y both give a remainder of 3 when divided by 19 or 11.
We can easily construct a number that fits this.
x = 19*11 + 3
x / 11 = 19 + 3 / 19
so y = 19
remainder when y / 19 is 0
A
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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10 Mar 2014, 21:54
2
enigma123 wrote:
When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?
A. 0
B. 1
C. 2
D. 3
E. 4
Guys struggling to solve this. But this is the concept I am trying to apply:
We can extrapolate a general statement from this form. When dividing x by y, the quotient is q and the remainder is r:
x/y = q + r/y
From there, we can solve for x:
x = qy + r (thats the general form of x = 3(integer) + 1)
Or the quotient:
q = x-r/y
Or, even, the remainder itself:
r = x - qy
But I am getting stuck in finding y when x is divided by 19. Can someone please help?? I don't have an OA either.
Take LCM of 19 & 11 = 209
Say x = 212
212/11: Quotient = 19 (=y) & remainder = 3
212/19: Quotient = 11 & remainder = 3
19/19: Quotient = 1 & remainder = 0 = Answer = A
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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10 Mar 2014, 23:31
Can we argue that this is a poorly written question? I was always taught that the quotient was the result of division. At first I paused thinking they must mean just the integer, even though I'd never seen that definition used. Then I decided that must be the "trick" of the question, that y is in fact the answer to x/11.
I got
x = 212
y = 19 + 3/11 = 212/11
y/19 = 212/11 * 1/19 = 212/209 = 1 R 3
The trick would be that people would misinterpret the question and mistake y as 19. But if y is x/11, then we will only have the same remainder when x is a multiple of 19 and 11. (Multiple of 19) + 3 yields a remainder of 3 when 19 is the divisor. Basically, the question would be testing only logic and, in theory, you would never have to figure out any values at all.
So I input the answer as 3, which is not the OA.
Afterward, I looked it up and quotient can also refer to just the integer of the result.
My question: Would this ever be a real GMAT question? We are taught (at least in American schools) that quotient is not just integer, but integer and remainder. Thus, the question appears ambiguous to me and probably a large segment of other GMAT takers. Thoughts?
Edit:
Just consulted OG: "For example, when 28 is divided by 8, the quotient is 3 and the remainder is 4 since 28 = (8)(3) + 4" (p. 108). OG for GRE says something similar. I just find it bizarre that at no time in my formal education did anyone give this definition.
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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11 Mar 2014, 05:11
2
enigma123 wrote:
When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?
A. 0
B. 1
C. 2
D. 3
E. 4
Sol: Given x=11y+3 or y= (x-3)/11
Also x=19a+3 Substitute for x in the above equation we get
y= (19a+3-3)/11 or y=19a/11. Note that "a" is an integer and multiple of 11
y/19 =a/11 where a/11 is an integer and hence remainder is 0
Ans is A
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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26 Jul 2015, 00:20
x=11y+3 and x=19z+3
11y+3=19z+3 => 11y=19z => y/z=19/11
It means that y is a multiple of 19, so when we divide y by 19 we will have r=0
A
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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29 Aug 2016, 10:30
enigma123 wrote:
When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?
A. 0
B. 1
C. 2
D. 3
E. 4
Guys struggling to solve this. But this is the concept I am trying to apply:
We can extrapolate a general statement from this form. When dividing x by y, the quotient is q and the remainder is r:
x/y = q + r/y
From there, we can solve for x:
x = qy + r (thats the general form of x = 3(integer) + 1)
Or the quotient:
q = x-r/y
Or, even, the remainder itself:
r = x - qy
But I am getting stuck in finding y when x is divided by 19. Can someone please help?? I don't have an OA either.
Please find the solution as attached,
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When the positive integer x is divided by 11, the quotient [#permalink]
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Updated on: 26 Jan 2018, 19:11
When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?
A) 0
B) 1
C) 2
D) 3
E) 4
least value of x=3
3/11 gives a quotient of 0
y=0
0/19 leaves a remainder of 0
A
Originally posted by gracie on 29 Aug 2016, 12:14.
Last edited by gracie on 26 Jan 2018, 19:11, edited 1 time in total.
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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16 Oct 2016, 11:44
A is the correct answer. Here's why:
From the prompt we can derive two equations:
x=11y+3
x=19z+3
From this we can set the two equal to each other, leaving us with...
11y=19z --> manipulate further to give... 11 = (19z)/y --> From this we know that z must equal 11 and y must equal 19 in order for the equation to hold. Therefore, since y = 19, dividing by 19 will give a remainder of 0
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When the positive integer x is divided by 11, the quotient [#permalink]
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06 Nov 2016, 15:34
Hey Everyone.
There are two methods that can be used here as far as i know.
First =>
x=11y+3
and x=19z+3
for some integer z
Hence 11y+3=19z+3
11y=19z
=> z=11y/19
as z needs to be an integer => y must be a multiple of 19.
Hence the remainder that y leaves with 19 must be zero.
Second =>
You see this is a problem solving question.
We will have only one answer.
x=19y+3
x=11z+3
hence combing the equations => x= 3+11*19*p for some integer p.
smallest possible value of x is 3
for the y is zero.
so y=0 is a acceptable value.
what is the remainder when 0 is divided by 19 ?
Its zero.
as every number divides 0.
Hence the answer must be zero.
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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07 Nov 2016, 13:53
enigma123 wrote:
When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?
A. 0
B. 1
C. 2
D. 3
E. 4
19z + 3 = x = 11y + 3
Or, x = 19*11 + 3 { z = 11 and y = 19 }
or, x = 202
y/19 = Quotient 1 and remainder 0
Hence , answer will be (A) 0
Hope this helps !!
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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19 Jan 2017, 04:22
Bunuel wrote:
enigma123 wrote:
When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?
A) 0
B) 1
C) 2
D) 3
E) 4
If you decide to go with quotient/remainder formula approach, then I'd suggest to express the info in the stem with it. And then look whether we can somehow manipulate with the expressions at hand to answer the question.
(1) When the positive integer x is divided by 11, the quotient is y and the remainder 3 --> $$x=11y+3$$;
(2) When x is divided by 19, the remainder is also 3 --> $$x=19q+3$$.
Easy to spot that $$19q+3=11y+3$$ --> $$19q=11y$$ --> $$y=\frac{19q}{11}$$. Now as $$y$$ and $$q$$ are integers and 19 is prime then $$\frac{q}{11}$$ must be an integer --> $$y=19*integer$$ --> $$y$$ is a multiple of 19, hence when divide by 19 remainder is 0.
Hope its clear.
Bunuel, what is the relationship between prime and divider in this case?
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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19 Jan 2017, 05:56
ziyuenlau wrote:
Bunuel wrote:
enigma123 wrote:
When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?
A) 0
B) 1
C) 2
D) 3
E) 4
If you decide to go with quotient/remainder formula approach, then I'd suggest to express the info in the stem with it. And then look whether we can somehow manipulate with the expressions at hand to answer the question.
(1) When the positive integer x is divided by 11, the quotient is y and the remainder 3 --> $$x=11y+3$$;
(2) When x is divided by 19, the remainder is also 3 --> $$x=19q+3$$.
Easy to spot that $$19q+3=11y+3$$ --> $$19q=11y$$ --> $$y=\frac{19q}{11}$$. Now as $$y$$ and $$q$$ are integers and 19 is prime then $$\frac{q}{11}$$ must be an integer --> $$y=19*integer$$ --> $$y$$ is a multiple of 19, hence when divide by 19 remainder is 0.
Hope its clear.
Bunuel, what is the relationship between prime and divider in this case?
We have $$y=\frac{19q}{11}$$. 19 is not a multiple of 11, thus for 19q/11 to be an integer q must be a multiple of 11.
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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28 Aug 2017, 00:49
Such questions are disaster as they take a lot of time solving. but here is a shorter way i.e through equations.
x/ 11 yields remainder 3 and quotient y. so we can form equation
x= 11y + 3 ........(1)
also we know that when x is divided by 19, remainder is same. let's take "a" as a quotient.
x=19a +3 .........(2)
equating 1 and 2
11y+3= 19a+3
11y=19a
since both 19 and 11 are prime numbers, with no number in common, so the only values a and y can get are
y=19 and a=11
y/19 =19/19 remainder =0
hope it helps.
kudos if u like the solution
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Re: When the positive integer x is divided by 11, the quotient [#permalink]
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26 Jan 2018, 14:01
Hi All,
We're told that...
X/11 = Y remainder 3
X/19 = something remainder 3
With the first piece of information, we know that X is 3 greater than a multiple of 11; with the second piece of information, we know that X is 3 greater than a multiple of 19. To have a remainder of 3 when you divide X by BOTH 11 and 19, X must be a number that is 3 greater than a MULTIPLE of BOTH 11 and 19.
We're asked what the remainder would be when Y is divided by 19. At this point, you might recognize that you could choose Y=19 and solve from there. If you don't recognize why that that relationship exists, then here's a more step-heavy way to get to the correct answer:
X = (11)(19) + 3 = 209 + 3 = 212
212/11 = Y remainder 3
212 = 11Y + 3
209 = 11Y
209/11 = Y
19 = Y
We're ultimately asked what the remainder would be when 19 is divided by 19. The remainder is 0.
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Re: When the positive integer x is divided by 11, the quotient [#permalink] 26 Jan 2018, 14:01
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https://www.physicsforums.com/threads/impulse-and-momentum-of-a-crate.16721/ | # Homework Help: Impulse and Momentum of a crate
1. Mar 21, 2004
### jjiimmyy101
Question (I added a picture): The free-rolling ramp has a mass of 40 kg. A 10 kg crate is released from rest at A and slides down 3.5 m to point B. If the surface of the ramp is smooth, determine the ramp's speed when the crate reaches B. Also, what is the velocity of the crate?
This is what I did for finding the velocity of the ramp:
1) \sigma mv1 = \sigma mv2
2) mass ramp * (velocity ramp)1 + mass crate * (velocity crate)1 = mass ramp * (velocity ramp)2 + mass crate *(velocity crate)2
3) 40*Vr + 0 = 40*Vr2 + 10*Vc2
Too many unknowns?
This is what I did for finding the velocity of the crate:
1) Vc2^2 = Vinitial^2 + 2*a*(S-Sinitial)
= 0 + 2 * 4.905 * 3.5
Vc2 = 5.86 m/s
2) I can substitute this in to the first equation, but I still have too many unknowns. What should I do next?
#### Attached Files:
• ###### ramp-crate.jpg
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Views:
221
2. Mar 21, 2004
### Chen
The crate is released from rest, therefore its initval velocity Vr is zero.
(Don't forget you can also appy the rules of conservation of energy, even though it's not needed in this case. There are only two kind of forces in the problem - the gravitional force, which is preservative, and the normal force which does no work since it's always perpendicular to the displacement of the objects.)
3. Mar 21, 2004
### jjiimmyy101
Vr is the velocity of the ramp
Vc is the velocity of the crate
4. Mar 21, 2004
### Chen
Both $$V_R_1$$ and $$V_C_1$$ equal 0 since the system is at rest before it's released.
5. Mar 21, 2004
### jjiimmyy101
The velocity of the crate and the velocity of the ramp are in opposite directions, right?
So the equation is
0 = -40 * Vr2 + 10 * Vb2 and since Vb2 = 5.86 m/s
Vr2 = 1.465 m/s in the opposite direction of the crate. Is it right?
6. Mar 21, 2004
### Chen
I would think so, unless you made a mistake in some of the calculations. The way is correct.
7. Mar 21, 2004
thank-you | {"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": 0.9521457552909851, "perplexity": 1832.2346165448682}, "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/1544376823183.3/warc/CC-MAIN-20181209210843-20181209232843-00308.warc.gz"} |
https://openi.nlm.nih.gov/detailedresult.php?img=PMC4488979_13063_2015_735_Fig3_HTML&req=4 | Efficacy and safety of Gantong Granules in the treatment of common cold with wind-heat syndrome: study protocol for a randomized controlled trial. Min J, Li XQ, She B, Chen Y, Mao B - Trials (2015) Bottom Line: However, there is a lack of robust evidence to support the clinical utility of such a treatment.The primary outcome is the duration of all symptoms.Secondary outcomes include the duration of primary symptoms and each symptom, time to fever relief and time to fever clearance, change in TCM symptom score, and change in Symptom and Sign Score. View Article: PubMed Central - PubMed Affiliation: Department of Integrated Traditional and Western Medicine, West China Hospital of Sichuan University, 37 Guoxue Lane, Chengdu, 610041, Sichuan Province, People's Republic of China. jane_min51@hotmail.com. ABSTRACTBackground: Although the common cold is generally mild and self-limiting, it is a leading cause of consultations with doctors and missed days from school and work. In light of its favorable effects of relieving symptoms and minimal side-effects, Traditional Chinese Medicine (TCM) has been widely used to treat the common cold. However, there is a lack of robust evidence to support the clinical utility of such a treatment. This study is designed to evaluate the efficacy and safety of Gantong Granules compared with placebo in patients with the common cold with wind-heat syndrome (CCWHS).Methods/design: This is a multicenter, phase IIb, double-blind, placebo-controlled and randomized clinical trial. A total of 240 patients will be recruited, from 5 centers across China and randomly assigned to the high-dose group, medium-dose group, low-dose group or placebo control group in a 1:1:1:1 ratio. All subjects will receive the treatment for 3 to 5 days, followed by a 7-day follow-up period. The primary outcome is the duration of all symptoms. Secondary outcomes include the duration of primary symptoms and each symptom, time to fever relief and time to fever clearance, change in TCM symptom score, and change in Symptom and Sign Score.Discussion: This trial will provide high-quality evidence on the efficacy and safety of Gantong Granules in treating CCWHS, and help to optimize the dose selection for a phase III clinical trial.Trial registration: The registration number is ChiCTR-TRC-14004255 , which was assigned by the Chinese Clinical Trial Registry on 12 February 2014. No MeSH data available. Related in: MedlinePlus © Copyright Policy - open-access Related In: Results - Collection License 1 - License 2 getmorefigures.php?uid=PMC4488979&req=5 .flowplayer { width: px; height: px; } Fig3: Symptom and Sign Scoring system. Mentions: The TCM symptom score system used in the study follows the Guidelines for Clinical Research of New Chinese Medicine [19], in which all symptoms are given graded scores (Figure 3). TCM signs will also be assessed, but not scored. The sum of all symptom scores is the cumulative TCM symptom score. The change in cumulative TCM symptom score is assessed by the percentage of symptom score reduction (PSSR), which is calculated according to the following formula:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{PSSR}\kern0.5em =\kern0.6em \left(\frac{symptom\ score\ before\ treatment\hbox{-} symptom\ score\ after\ treatment}{symptom\ score\ before\ treatment}\right)\kern0.5em \times \kern0.5em 100\%$$\end{document}PSSR=symptomscorebeforetreatment‐symptomscoreaftertreatmentsymptomscorebeforetreatment×100%Figure 3
Efficacy and safety of Gantong Granules in the treatment of common cold with wind-heat syndrome: study protocol for a randomized controlled trial.
Min J, Li XQ, She B, Chen Y, Mao B - Trials (2015)
Related In: Results - Collection
Show All Figures
getmorefigures.php?uid=PMC4488979&req=5
Fig3: Symptom and Sign Scoring system.
Mentions: The TCM symptom score system used in the study follows the Guidelines for Clinical Research of New Chinese Medicine [19], in which all symptoms are given graded scores (Figure 3). TCM signs will also be assessed, but not scored. The sum of all symptom scores is the cumulative TCM symptom score. The change in cumulative TCM symptom score is assessed by the percentage of symptom score reduction (PSSR), which is calculated according to the following formula:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{PSSR}\kern0.5em =\kern0.6em \left(\frac{symptom\ score\ before\ treatment\hbox{-} symptom\ score\ after\ treatment}{symptom\ score\ before\ treatment}\right)\kern0.5em \times \kern0.5em 100\%$$\end{document}PSSR=symptomscorebeforetreatment‐symptomscoreaftertreatmentsymptomscorebeforetreatment×100%Figure 3
Bottom Line: However, there is a lack of robust evidence to support the clinical utility of such a treatment.The primary outcome is the duration of all symptoms.Secondary outcomes include the duration of primary symptoms and each symptom, time to fever relief and time to fever clearance, change in TCM symptom score, and change in Symptom and Sign Score.
View Article: PubMed Central - PubMed
Affiliation: Department of Integrated Traditional and Western Medicine, West China Hospital of Sichuan University, 37 Guoxue Lane, Chengdu, 610041, Sichuan Province, People's Republic of China. jane_min51@hotmail.com.
ABSTRACT
Background: Although the common cold is generally mild and self-limiting, it is a leading cause of consultations with doctors and missed days from school and work. In light of its favorable effects of relieving symptoms and minimal side-effects, Traditional Chinese Medicine (TCM) has been widely used to treat the common cold. However, there is a lack of robust evidence to support the clinical utility of such a treatment. This study is designed to evaluate the efficacy and safety of Gantong Granules compared with placebo in patients with the common cold with wind-heat syndrome (CCWHS).
Methods/design: This is a multicenter, phase IIb, double-blind, placebo-controlled and randomized clinical trial. A total of 240 patients will be recruited, from 5 centers across China and randomly assigned to the high-dose group, medium-dose group, low-dose group or placebo control group in a 1:1:1:1 ratio. All subjects will receive the treatment for 3 to 5 days, followed by a 7-day follow-up period. The primary outcome is the duration of all symptoms. Secondary outcomes include the duration of primary symptoms and each symptom, time to fever relief and time to fever clearance, change in TCM symptom score, and change in Symptom and Sign Score.
Discussion: This trial will provide high-quality evidence on the efficacy and safety of Gantong Granules in treating CCWHS, and help to optimize the dose selection for a phase III clinical trial.
Trial registration: The registration number is ChiCTR-TRC-14004255 , which was assigned by the Chinese Clinical Trial Registry on 12 February 2014.
No MeSH data available.
Related in: MedlinePlus | {"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": 0.15672890841960907, "perplexity": 7373.406200087378}, "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-2018-22/segments/1526794867859.88/warc/CC-MAIN-20180526190648-20180526210648-00579.warc.gz"} |
https://johnmayhk.wordpress.com/2015/02/01/just-a-core-math-probability-question/ | # Quod Erat Demonstrandum
## 2015/02/01
### 黑白球
Filed under: mathematics,NSS — johnmayhk @ 10:33 上午
Tags: , ,
In a game, Evan has to draw balls from a bag containing 2 black balls and 3 white balls one by one without replacement. If he gets two consecutive black balls, he wins; otherwise he loses. Find the probability that he wins.
P(wins)
=P(BB)+P(WBB)+P(WWBB)+P(WWWBB)
=$\frac{2}{5}\frac{1}{4}+\frac{3}{5}\frac{3}{4}\frac{2}{3}+\frac{3}{5}\frac{2}{4}\frac{2}{3}\frac{1}{2}+\frac{3}{5}\frac{2}{4}\frac{1}{3}$
=$\frac{2}{5}$
$\frac{3}{7}\frac{2}{6}+\frac{4}{7}\frac{3}{6}\frac{2}{5}+\frac{4}{7}\frac{3}{6}\frac{3}{5}\frac{2}{4}+\frac{4}{7}\frac{3}{6}\frac{2}{5}\frac{3}{4}\frac{2}{3}+\frac{4}{7}\frac{3}{6}\frac{2}{5}\frac{1}{4}=\frac{3}{7}$
（一）設盒內共有 n 球（$n\ge 2$），黑球 2 個。求連取 2 黑球之機會。
$P(E)=\frac{2\times (n-1)!}{n!}=\frac{2}{n}$
（二）設盒內共有 n 球，黑球 m 個（$n\ge m$）。求連取 m 黑球之機會。
$P(E)=\frac{(n-m+1)!m!}{n!}=\frac{n-m+1}{C^n_m}$
（三）設盒內共有 n 球，黑球 m 個（$n\ge m\ge 2$）。求連取 2 黑球之機會。
●○●○●○○
○○●●○●○
○○●●○●○
●●○●○○○
○●●●○○○
○○●○●○
●○●○○○
○●●○○○
$P(E)=\frac{C^{n-1}_{m-1}}{C^n_m}=\frac{m}{n}$
（四）設盒內共有 n 球，黑球 m 個（$n\ge m\ge r$）。求連取 r 黑球之機會。
https://johnmayhk.wordpress.com/2012/02/13/core-math-problem-probability/
## 1 則迴響 »
1. Just suggest another solution to the original question:
Since there are 2 black (indistinguishable) balls in a box of 5 balls, the probability that a black ball is drawn at the 1st,2nd,3rd,4th,5th drawings is 2/5.
Under the condition that a black ball is drawn at the i-th drawing (i=1,2,3,4), the (conditional) probability that the other black ball is drawn at the (i+1)th drawing is 1/4.
Hence, the probability that two black balls are drawn at the ith and (i+1)th drawing is (2/5)(1/4)=1/10.
Since i=1,2,3 or 4;
the probability that two black balls are drawn consecutively is 4*(1/10)=2/5.
迴響 由 johnmayhk — 2015/02/02 @ 11:07 上午 | 回應 | {"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": 15, "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": 0.7437594532966614, "perplexity": 2624.553567177597}, "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/1510934805809.59/warc/CC-MAIN-20171119210640-20171119230640-00375.warc.gz"} |
https://www.ncbi.nlm.nih.gov/pubmed/9249016?dopt=Abstract | Format
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Eur J Biochem. 1997 Jul 1;247(1):114-20.
# Influence of charge differences in the C-terminal part of nisin on antimicrobial activity and signaling capacity.
### Author information
1
Department of Biophysical Chemistry, Netherlands Institute for Dairy Research (NIZO), Ede.
### Abstract
Three mutants of the antibiotic nisin Z, in which the Val32 residue was replaced by a Glu, Lys or Trp residue, were produced and characterized for the purpose of establishing the role of charge differences in the C-terminal part of nisin on antimicrobial activity and signaling properties. 1H-NMR analyses showed that all three mutants harbor an unmodified serine residue at position 33, instead of the usual dehydroalanine. Apparently, the nature of the residue preceding the serine to be dehydrated, strongly affects the efficiency of modification. Cleavage of [Glu32,Ser33]nisin Z by endoproteinase Glu-C yielded [Glu32]nisin Z(1-32)-peptide, which has a net charge difference of -2 relative to wild-type nisin Z. The activity of [Lys32,Ser33]nisin Z against Micrococcus flavus was similar to that of wild-type nisin, while [Trp32,Ser33]nisin Z, [Glu32,Ser33]nisin Z and [Glu32]nisin Z(1-32)-peptide exhibited 3-5-fold reduced activity, indicating that negative charges in the C-terminal part of nisin Z are detrimental for activity. All variants showed significant loss of activity against Streptococcus thermophilus. The potency of the nisin variants to act as signaling molecules for auto-induction of biosynthesis was significantly reduced. To obtain mutant production, extracellular addition of (mutant) nisin Z to the lactococcal expression strains was essential.
[Indexed for MEDLINE]
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http://math.stackexchange.com/questions/120978/computing-a-convolution-using-fft | # Computing a convolution using FFT
I have two sequences of the same length, $(x_i), i=1, 2, \ldots, N$ and $(y_i), i=1, 2, \ldots, N$ and a function $K(t) = -t \times \exp(-t^2 / 2)/ \sqrt{2 \pi}$.
I need to compute the following quantity for each $m=1, 2, \ldots, N$:
$\sum_{j=1}^N K(x_m - x_j) \times y_j$
which is a tad slow when done directly (I need it when $N = O(10^4)$ ). I know this can be much improved on with the use of FFT, however it is something I never really worked with. Could anyone suggest any links or a way to rewrite it in FFT form?
I'd be very grateful for any help :)
-
Apply the FFT two both sequences, multiply the results pointwise, then transform back. – dls Mar 16 '12 at 16:11
Well that's the general idea. Except that to do it for each $m$ is no time gain. I was guessing the clever thing to do is to approximate the function $\sum_{j=1}^N K(s - x_j) y_j$ using FFT, and this bit I'm not sure how to do. Besides, the FFT theorem convolution applies directly to convolutions like $\sum a_{i-j} b_j$, so to do the straightforward thing I would need to recompute the sequence $K(x_m-k_j)$ for each $m$, which would eat away any time gain..? – Maciej Mar 16 '12 at 16:20
I've not heard of approximating a function by its Fourier transform. You have $$(K \ast y) (x_m) = \sum_{j = 1}^N K(x_m - x_j) y_j$$ and $$\widehat{K \ast y} (x_m) = \hat{K}(x_m) \hat{y_m}$$ You can use FFT to compute these: do the multiplication and then do the inverse transform for each $m$. I don't understand how to use the Fourier transform to approximate a function. The other thing you could think about is to numerically approximate it: compute only a few of the function values and then use interpolation. – Rudy the Reindeer Mar 17 '12 at 8:32
I'm not sure if this is precisely what you are talking about but naive convolution is $O(n^2)$, while the FFT method is only $O(n \log n)$ (it is 2 FFTs, which are $O(n \log n)$, and one pointwise multiply, which is $O(n)$, so the process is $O(n \log n)$ overall.) – huon May 23 '12 at 8:34
Here you have a good link.
Basically you need to compute the FFT of each signal individually, multiply the spectrums and the do the inverse FFT of the resulting sectrum. That will be the result of the convolution.
Good luck!
-
Need to account for wrap around issues
1) Shift the K sequence such that it is centered in the middle of the array call K1 of lenght n1 2) Store the K1 array in a new array K2 with double the size of K1 n2 = 2Xn1 such that K2(1 to n1 ) = complex(K1,0) and K2( n1+1 to n2 ) = 0 where K2 is complex 3) Next shift the K2 array over to the left by n1/2-1 this will center the maximum of the K function in the first bin of the K2 array 4) Store the y data in an array y2 of of lenght n2 such that y2(1 to n1 ) = complex(y,0) and y2(n1+1 to n2) = 0 where y2 is complex
5) calculate the FFT of K2, i.e., FFT_K2 = FFT( K2, 1 ) 1 => exp(iwt) 6) calculate the FFT of y2, i.e., FFT_y2 = FFT( y2, 1 ) 1 => exp(iwt) 7) Let FFT_d2 = CONJUGATE(FFT_K2) * FFT_y2 8) Inverse FFT, i.e., d2 = FFT( FFT_d2, -1 ) -1 => exp(-iwt) 9) convolution = REAL( ds(1 to n1) ) 10) this assumes the exp(-iwt) has the 1/n factor
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https://electronics.stackexchange.com/questions/417534/how-does-an-op-amp-integrator-work | # How does an op amp integrator work?
I know there are at least two questions related to this on stackoverflow but neither really answer my question, and in any case, both questions got downvoted. What I am after is an operational understanding of how an op-amp integrator works. I know how a simple RC circuit can integrate, what I don't understand is how the feedback loop in an op-amp configuration helps. I understand how feedback works in a noninverting amplifier. I took the figure below from www.electronics-tutorials.ws. This web site has an explanation but I don't follow it. My understanding so far is this:
1. Apply a positive voltage to input vin. Current flows through Rin resulting initially in a non-zero voltage at X (Correct?).
2. Due to the high impedance of the op-amp at X, we can assume that all the current then flows to the capacitor (initial discharged).
3. The capacitor starts to charge resulting in a voltage across the capacitor.
4. The difference in voltage at the two op-amp inputs (the positive input is at zero, hence the difference is negative) resulting in the output, vout, going negative (we assume that vout was zero initially).
My question is what happens next? How does the feedback act to bring the difference between the two inputs back to zero? Or have I got this wrong?
I am very familiar with the proofs for showing that the configuration will integrate but they don't give any real intuition and many videos, wikpedia, and books but almost all regurgitate the proof without giving much insight. I'm after an intuitive understanding, not a mathematical proof.
Out of interest I also redrew the op-amp circuit next to the RC integrator shown below which gives the suggestion that the op-amp is amplifying the small vollage across C (assuming high R1) while having high impedance from the resistor/capacitor node. Not sure if that is a legitimate way to look at it.
• Put in a square wave, the opamp strives to hold the (-) pin very close to zero volts, which imposes a constant voltage across the input resistor. This constant current has to go somewhere, and the only path is into the capacitor. Give constant current into a capacitor, you get a perfect RAMP output, which is the integral. – analogsystemsrf Jan 18 at 3:04
• To me - this small comment (if compared with all the answers) is the best explanation of the circuits behaviour in the time domain. – LvW Jan 18 at 14:57
• But it doesn't explain why the op-amp strives to hold the (-) pin close to zero. An op-amp on its own won't do that, it's the combination with the feedback that makes that happen. This is the bit I wasn't sure about. – rhody Jan 18 at 18:47
This may help:
• Remember that when current flows into the RC junction of your op-amp that the voltage at that point will tend to rise.
• If the inverting input voltage rises the slightest bit above the non-inverting input voltage then the op-amp output will start to swing negative.
• The output swinging negative will, through the capacitor1, tend to pull the inverting input down towards zero again where it stabilise (for the moment).
The result is that feeding current into the RC node causes the op-amp output to go negative.
Out of interest I also redrew the op-amp circuit next to the RC integrator shown below which gives the suggestion that the op-amp is amplifying the small voltage across C (assuming high R1) while having high impedance from the resistor/capacitor node. Not sure if that is a legitimate way to look at it.
That's correct. It might be better than you think. The simple RC circuit has the advantage that it's non-inverting but the disadvantage that it's non-linear. With a constant input voltage the output will be an exponential charge curve.
Putting the op-amp in as you have shown still allows the capacitor to charge up but maintains the top terminal at virtual ground. The advantage is a linear change in output. The disadvantage is that there is a minus sign on the integral obtained.
1 You can think of a capacitor as holding the voltage across it as a constant in the short term. That means that if the voltage on one side is changed the voltage on the other side will try to change by the same amount.
One question. what is the orientation of the capacitor in terms of conventional current? i.e. if vin goes positive the capacitor is I assume negative on its right-hand side (nearest vout). Now vout goes negative and therefore reduces the voltage across the capacitor until the potential at X is zero?
I think your understanding is correct.
If Vin goes positive then current flows into the X node charging up C. (Remember the op-amp's voltage hasn't changed yet.) This tends to increase the voltage on the inverting input and that causes the output voltage to decrease. This draws some charge from the right hand side of C. Now the inverting input is pulled back down to zero volts but there is charge on C so there is a voltage across it. Since the conventional current flowed to the right there is a negative voltage remaining on the capacitor.
• I think this the kind of answer I was looking for. One question. what is the orientation of the capacitor in terms of conventional current? ie if vin goes positive the capacitor is I assume negative on its right-hand side (nearest vout). Now vout goes negative and therefore reduces the voltage across the capacitor until the potential at X is zero? – rhody Jan 17 at 22:44
• See the update. – Transistor Jan 17 at 23:15
The op-amp is going to try its best to keep the voltage between it's plus and minus input the same. In an ideal op-amp, no current flows into the inputs, so the only way that it can do that is by changing its output voltage.
In the schematic below, $$\v_+ = 0\mathrm{V}\$$. That means that the op-amp will try to hold $$\v_-\$$ at zero, also.
Whatever voltage is generated by V2 gets turned into a current by R1. Because $$\v_-\$$ is being held at $$\0\mathrm{V}\$$, that same current has to flow in C1. And because $$\v_-\$$ is being held at $$\0\mathrm{V}\$$, the op-amp has to drive the output voltage such that the current in C1 matches the current in R1.
So if $$\v_2\$$ is constant, then the current into the node around the negative input is constant, which means that the current out of that node from the cap must be constant -- and that can only happen if the output voltage is falling at a constant rate. The end result is that the op-amp integrates the input voltage into the output voltage.
More complicated voltages at $$\v_2\$$ cause more complicated behavior, but the op-amp is always going to be trying to drive $$\v_-\$$ to $$\0\mathrm{V}\$$. It can only do that by satisfying $$\ \frac{d}{dt} C_1 v_{out} + \frac{v_2}{R_1} = 0 \$$. If you solve that differential equation, it says that $$v_{out} = -\frac{1}{R_1 C_1} \int v_2 dt$$
HTH
simulate this circuit – Schematic created using CircuitLab
• In your comment "The op-amp is going to try its best to keep the voltage between it's plus and minus input the same.", strictly speaking, it's the op-amp combined with the next feedback that does this. This is the bit I was stuck on, how the feedback coupled with the op-amp manages to maintain the difference close to zero. – rhody Jan 18 at 18:50
• Well, I'm glad you worked through it! These are difficult concepts and hard to simplify. – TimWescott Jan 18 at 18:57
Rhody - have you heard about the MILLER effect? Well - the shown circuit is called "MILLER integrator" because the MILLER effect is exploited. Remember: This effect reduces the feedback impedance between an amplifier output (for example: collector) and the inverting input (example: base node of the transistor). And the factor of increase is the gain.
Here, we have the same principle. Hence, there will be a very small capacitive impedance (that means: A very large capacitor) between input and output of the opamp. And the factor of increase is the open-loop gain Aol of the opamp.
Hence, you can make a comparison with a simple RC circuit. However, because of the very large capacitor the cut-off frequency is very low (nearly DC).
Frequency domain: The transfer function between the opamps inverting node and the signal input is
Ho(s)=1/(1+sCo * R) with Co=Aol * C (MILLER effect).
Because of the very large value Aol, we can neglect the "1" in the denominator and arrive at
Ho(s)=1/(sC * Aol *R)
We are lucky and can use the low resistive opamp output (and multiply the function Ho(s) with the gain -Aol) and arrive at the final result (opamp output-to-signal input):
H(s)=Ho(s) * (-Aol) = - 1/sR*C (Transfer function of an ideal integrator)
The inputs of the opamp don't take input current and the opamp will keep its input voltage equal as long as it is wired for negative feedback.
So effectively, the current that the input sinks against the 0V it sees goes straight into charging the capacitor. Usually when you charge a capacitor through a resistor, the charge building up on the capacitor reduces the voltage across the resistor and thus also the charging current, leading to an exponential decay of the charge current.
Here, however, the opamp output actively adjusts the voltage at the other side of the capacitor so that the resistor never gets to see the difference it makes, thus keeping the current through resistor (and into capacitor) independent from the charge across the capacitor.
It's like Útgarða-Loki handing Þorr the mouth of the ocean as a drinking horn and Þorr does not actually find himself able to drain it, not noticing that he causes the tides with his attempt.
The Passive cap. integrator current decays with voltage as it approaches the input.
The active cap. integrator saturates after some time if Vin ≠ 0 because the output voltage drives current into Vin- to maintain 0V diff. until the output saturates at the supply rail.
So input offset is critical and you need an analog switch to discharge and initialize to 0V output.
## anecdotal
I remember knowing nothing about this in 1st yr Eng and my once famous brother-in-law medical Dr of Anesthesia, Intensive Care and open heart surgery, gave me a tour around the hospital and said he needed an integrator to measure O2 content in blood for the brain, after a heart attack victim stops, to know the best treatment ( such as hypothermia )to give when no response to defib. and meds with the likelihood of success. I had no idea ! and was embarrassed to not know! (circa '75) Don't be. Just research it. | {"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": 12, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8361901640892029, "perplexity": 828.7136611776214}, "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/1550247484648.28/warc/CC-MAIN-20190218033722-20190218055715-00020.warc.gz"} |
http://mathhelpforum.com/calculus/119004-finding-1000th-derivative.html | 1. ## Finding 1000th derivative
f(x)=(x^2+1)(e^(x^2+1))
find the 1000th derivative
I was thinking of using taylor's on it but I'm really not sure where to start. Thanks in advance for anyone who helps.
2. Originally Posted by Scottfranklin
f(x)=(x^2+1)(e^(x^2+1))
find the 1000th derivative
I was thinking of using taylor's on it but I'm really not sure where to start. Thanks in advance for anyone who helps.
Start taking derivatives, then find an expression that represents the nth derivative, then plug a 1000 into it.
3. So should I treat the expression as (x^2+1)=r and therefore F(x)=re^r? | {"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": 0.9828972220420837, "perplexity": 545.4064798848989}, "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-2017-17/segments/1492917120092.26/warc/CC-MAIN-20170423031200-00378-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://hal.archives-ouvertes.fr/hal-00000496 | # Generic Bernstein-Sato polynomial on an irreducible affine scheme
Abstract : Given $p$ polynomials with coefficients in a commutative unitary integral ring $\mathcal{C}$ containing $\mathbb{Q}$, we define the notion of a generic Bernstein-Sato polynomial on an irreducible affine scheme $V \subset \text{Spec}(\mathcal{C})$. We prove the existence of such a non zero rational polynomial which covers and generalizes previous existing results byH. Biosca. When $\mathcal{C}$ is the ring of an algebraic or analytic space, we deduce a stratification of the space of the parameters such that on each stratum, there is a non zero rational polynomial which is a Bernstein-Sato polynomial for any point of the stratum. This generalizes a result of A. Leykin obtained in the case $p=1$.
Keywords :
Cited literature [10 references]
https://hal.archives-ouvertes.fr/hal-00000496
Contributor : Rouchdi Bahloul <>
Submitted on : Friday, July 11, 2003 - 4:42:03 PM
Last modification on : Monday, March 9, 2020 - 6:15:53 PM
Document(s) archivé(s) le : Monday, March 29, 2010 - 4:39:24 PM
### Citation
Rouchdi Bahloul. Generic Bernstein-Sato polynomial on an irreducible affine scheme. 2003. ⟨hal-00000496⟩
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https://zbmath.org/?q=an:0338.54001 | ×
# zbMATH — the first resource for mathematics
Hausdorff compactifications. (English) Zbl 0338.54001
Lecture Notes in Pure and Applied Mathematics. Vol. 23. New York - Basel: Marcel Dekker, Inc. VII, 146 p. SFrs. 55.00 (1976).
##### MSC:
54-02 Research exposition (monographs, survey articles) pertaining to general topology 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54D40 Remainders in general topology 54D30 Compactness 54G05 Extremally disconnected spaces, $$F$$-spaces, etc. 54G15 Pathological topological spaces 54C40 Algebraic properties of function spaces in general topology 54C45 $$C$$- and $$C^*$$-embedding 54C50 Topology of special sets defined by functions | {"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": 0.2415667474269867, "perplexity": 9344.986559720017}, "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-2021-21/segments/1620243988763.83/warc/CC-MAIN-20210506205251-20210506235251-00071.warc.gz"} |