text:
# Save the answer in TEX?

Lot of answers here are excellent but there is no way to save it in LaTeX format. Even if we can put them in favorites or print them or save as pdf .

Also on meta : Save the answer in PDF? but this discussion is about saving to pdf, the best solution utilize https://stackedit.io/ but it only provides save to markdown, html, or pdf.

• click the edit button, copy the source into a tex file, add you favorite preamble Jul 9 '14 at 10:53
• Have to do this for every reply Jul 9 '14 at 10:57

The comment "click the edit button, copy the source into a tex file, add you favorite preamble" is a slight oversimplification. Consider bold and italic fonts, quotes and spoilers, links and images, strikethrough and colors, bulleted and numbered lists, HTML entities (accented letters, etc), underscores and ampersands in text, and the fact that MathJax tolerates some things that LaTeX will not (e.g., \begin{align} ... \end{align}).

But here is Save as LaTeX, anyway. Paste in the URL of any question (here or on another MathJax-enabled SE site, e.g., MathOverflow) and enjoy the result. Answers are ordered by votes. Inclusion of comments is optional. Images are converted into hyperlinks; you'd have to save them separately anyway.

Warning: this stuff is just something I made to learn regular expressions; it's still a work in progress. Please report bugs in comments. Yes, it parses HTML with regex... Sample output (taken from the script and ran through LaTeX compiler):

• Do I report errors here?(Nice name change btw) math.stackexchange.com/questions/906820/… "! Undefined control sequence. l.16 [\f =\1^2+\2^2+2\3^2]" Does it not take \def text?
– Tony
Aug 24 '14 at 0:33
• @user151558 The problem is that TeX processors require definitions to be in topmatter (before \begin{document}), while MathJaX (for lack of topmatter) is okay with them being in a formula within the document. I'll try to come up with a way to extract definitions and put them where they belong. For now, the fix is to copy-paste the \def manually to the beginning of file, removing the surrounding . Compiles after that.
– user147263
Aug 24 '14 at 0:40
• A very very useful tool! Thank you so much! Nov 16 '14 at 4:43
• The dropbox link doesn't seem to exist anymore Jan 25 '18 at 7:18
• @lucidbrot This link seems to be working now: normalhuman.github.io/savetex Feb 2 '19 at 11:31
• Both dl.dropboxusercontent.com/u/29863189/mse/savetex.html and normalhuman.github.io/savetex don't work right now
– JKHA
Sep 4 '19 at 9:44


text:
## Friday, August 16, 2013

### Visualizing Internet Activity

I've recently been playing with the idea of visualizing the network demand for various activities related to everyday Internet use. Most times the browser presents a view that abstracts this behind-the-scenes activity to provide a seamless experience to the user. Even so, it can be enlightening to peel back the layers just a bit to peek at what is really happening.

In the images below I've tried to present this underlying activity in a way that is intuitive without drowning the reader with information. This is still a work in progress but I feel as though this is a decent start toward that goal.

The graphs below represent network traffic related to three types of activity one might typically engage in while online: browsing facebook, watching a youtube video, and downloading a large file. Each 'bubble' represents a packet; radius is the relative packet size; individual connections are annotated with unique colors; and I add jitter in an attempt to alleviate over-plotting.

The above is the graph for facebook. There is an initial burst of activity to load all the components when you first log in; as you scroll down the page more content is loaded to extend the page on-demand (these are the narrow vertical impulses throughout the graph); then, around time 250 there is a large amount of activity related to opening a photo album and browsing through the pictures. As can be seen by the transitioning colors there are a variety of individual connections used over the course of the facebook session.

This next graph is a view of what happens when watching a youtube video. Again, there is a quick burst to load the page and several connections are used to do this. Once the video is playing, however, there is only very periodic bursts of activity. In fact, if you observe the video progress bar as you watch the video you will notice that youtube front-loads a portion of the video immediately and then, as you start to need more content, requests the next section of the video.

Finally, this is a view of what a large file download requires: very few connections and a consistent, heavy flow of packets across the network.

From an end user point-of-view, the experience related to each of these activities is the same: open a browser, click a few buttons and receive content. The resources and demand to deliver this content, however, is entirely distinct from that experience.

## Friday, July 19, 2013

### Column-Major Confusion

As a programmer coming from a language like C I am used to understanding multidimensional arrays in the following way:
int matrix[3][3] = {
{ 1, 2, 3 },
{ 4, 5, 6 },
{ 7, 8, 9 }
};

Understandably, this is a little bit of a setback when trying to grok how a language such as R handles matrices. The example above is 'row-major' but R uses 'column-major' by default. Note that I'm not describing memory layout here - which coincidentally is the same as my description - I referring to how the matrices are presented to the programmer. To complete the example, here is how I would create the same matrix as above in R:
> matrix(1:9,ncol=3)
[,1] [,2] [,3]
[1,]    1    4    7
[2,]    2    5    8
[3,]    3    6    9

There is no problem with this per se. In fact, I'd imagine that R programmers that come to something like C may feel similar unease in the paradigm shift. I'm finding that this fact is glazed over in some texts offering an introduction to R. This happens in non-obvious ways. In particular, I found this example today:
> matrix(c(1,0,0,0,1,0,0,0,1),ncol=3)
[,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    0    0    1

The identity matrix is a particularly bad choice in this regard as it gives no indication of the true layout being used. It is probably good for any tutorial using matrices to cover an obvious simple case first to set the stage before moving directly to something like the identity matrix (or any other symmetric matrix for that matter).

## Saturday, July 13, 2013

### 10 points: 10 plots

As an exercise in expanding my ability to display data I challenged myself to present 10 data points in 10 ways that were as distinct as possible. The idea was simple: use 10 random data points; minimize the axis and other ancillary information so as to focus on the data as much as possible; and try to minimize the overlap between each of the approaches.

Initially, I expected this would be a trivial task - something that would take a single sitting and a little bit of thought. A few attempts later and I kept circling back on a few common ideas while considering just how many approaches I'd not considered. What exists below is a collection of the results of that exercise with explanation if necessary.

1 - Standard Cartesian (scatterplot)

2 - Derivative Cartesian: uses labels instead of points to eliminate the need for tick marks on the x-axis.

3 - Impulses. Mixing the number and characters on the x-axis tick marks is questionable and could just as well have been labels at the top of each impulse

4 - Sorted derivative Cartesian

5 - Boxplot

6 - Barplot

8 - Heatmap

9 - Cumulative Sum

10 - Financial/Intensity: Positive values are blue, negative are red. Absolute values define the radius of the circle used.

I considered others such as LOESS fit but they either needed the points to accompany them (to show what was being fitted) which made them too close to the Cartesian plot, or they were too complex for just 10 points.

It was interesting to see how difficult it turned out to be to stretch 10 points into 10 distinct presentation approaches.

## Thursday, June 27, 2013

### Walk this way

I recently found a handy mechanism for walking a directory tree in Linux. In
general, the way I used to do this was to use facilities found in dirent.h and
write my own recursive directory walker. Something similar to:

#include <stdio.h>
#include <string.h>
#include <dirent.h>

void reclist (const char* dirname) {

DIR* dir = opendir (dirname);
struct dirent* entry = 0;
char name[1024] = {0};

if (! dir) { return; }

while (entry) {
if (strncmp (entry->d_name, ".", 1)) {
switch (entry->d_type) {
case DT_REG:
printf ("%s\n", entry->d_name);
break;
case DT_DIR:
snprintf (name, 1024, "%s/%s", dirname, entry->d_name);
reclist (name);
break;
}
}
}
closedir (dir);
}

int main(int argc, char** argv) {
const char * dir = ".";
if (argc == 2) { dir = argv[1]; }
reclist (dir);
return 0;
}

While that does work, it is rather verbose (especially once you get used to
environments like Ruby and Python). It turns out that ftw.h provides a more
concise way to do the above while managing all the little details like
avoiding '.' and '..' and managing the current path string. Here is what that
looks like to do the same as the above:

#include <stdio.h>
#include <ftw.h>

int handle_entry (const char *entry, const struct stat *sb, int type) {
if (type == FTW_F) {
printf("%s\n", entry);
}
return 0;
}

int main() {
ftw(".", handle_entry, 10);
return 0;
}

I also like the fact that a callback is used to operate on each of the files
found. It makes managing changes much easier as the tree walking is separated
from the code that handles the logic associated with inspecting the files.

## Sunday, June 23, 2013

### That's the key

A while back I cam across a post on Stephen Wolfram's blog where he presented the personal analytics of his life. As part of this post, there is a plot showing the keystroke activity of his life over the last 10 years. I want to ignore the resolve needed to conduct such an experiment for a moment and consider how he might have set something like that up.

[Update: see the corollary to this post - generating keyboard events - here]

I'm interested in data. I have a few logs of things I do on a daily basis but they are all collected proactively - I write entries into these logs in order to keep them current. I want to set up something similar to this key logger to automate this process for me. I'll mostly ignore that this is a potential security risk in that I will be capturing all keystrokes on the computer - including username and password information. To partially mitigate this I wont store the key information, I'll only keep the time the event occurred. This limits the amount of information in my database - I wont be able to see how my distribution of characters matches that of commonly used data, for instance - but it saves me from having to worry about how and where I store this information. Stephen Wolfram's post includes details about the actual keys so if my data starts to look interesting perhaps I'll transition to keeping that information as well.

I run Linux so I figured this would be rather straightforward: somehow hook into the X windowing subsystem and register for all keyboard events. Unfortunately, such an approach is not directly possible using Xlib (depending on which stackoverflow answer you read, it may not be possible at all). It turns out that it is rather difficult to ask X to just 'give me everything.' Things, as it were, are destined for a particular location (read: window) and asking for other windows' events doesn't make much sense in the general case. I had hoped there would be something akin to a callback list for registered components that I would be able to insert an entry into. Xlib is not designed that way (at least not in any documentation I can find).

To avoid having to hack the X window event delivery system I started to look at how these events are realized by X itself. In the guts of the device initialization configuration there is something similar to the following:

Section "InputClass"
Identifier "evdev keyboard catchall"
MatchIsKeyboard "on"
MatchDevicePath "/dev/input/event*"
Driver "evdev"
EndSection

which is using one of the /dev/input/event* devices. These are character devices set up by evdev to handle generic input events from a variety of sources: joysticks, mice, keyboards, and so on. One nice thing about these devices is they can be opened and read from as if they were regular files. So, if I can figure out which of the /dev/input/event* devices corresponds to the keyboard I should have access to the events that X is handing off to the child windows.

It turns out that there are two directories that exist to facilitate this type of search: /dev/input/by-id/ and /dev/input/by-path/. Searching either of the two of them for something like *-kbd you can find the exact device linked to a keyboard (if you have multiple keyboards attached you will need to further disambiguate). For example, in my /dev/input/by-path/ there are the following:

pci-0000:00:04.0-event-mouse -> ../event4
pci-0000:00:06.0-usb-0:1:1.0-event-mouse -> ../event3
pci-0000:00:06.0-usb-0:1:1.0-mouse -> ../js0
platform-i8042-serio-0-event-kbd -> ../event2
platform-i8042-serio-1-event-mouse -> ../event5
platform-i8042-serio-1-mouse -> ../mouse1

According to this (and some mappings provided in /usr/include/linux/input.h) I can now collect all keystrokes generated by my machine from /dev/input/event2 without having to devise a way to convince X to hand them over.

format: plain


text: I would like to display the aggregate_cloud generated by the rgbdslam package in rviz when running the package on a remote robot. However, I have been unable to get the library to publish to the aggregate_cloud and batch_cloud topics even after invoking rosservice call /rgbdslam/ros_ui send_all as suggested in the documentation. Any recommendations for how to get this data to display?


text:
Freshly brewed tea. Two to three cups a day of freshly brewed tea -- hot or iced -- contains a modest amount of caffeine which, when used "judiciously," says Kulze -- can boost brain power by enhancing memory, focus, and mood. Tea also has potent antioxidants, especially the class known as catechines, which promotes healthy blood flow. Bottled or powdered teas don't do the trick, however, says Kulze. "It has to be freshly brewed." Tea bags do count, however.
The experiment then is straightforward: cut up a fresh piece of gum, randomly select from it and an equivalent dry piece of gum, and do 5 rounds of dual n-back to test attention/energy & WM. (If it turns out to be placebo, I’ll immediately use the remaining active dose: no sense in wasting gum, and this will test whether nigh-daily use renders nicotine gum useless, similar to how caffeine may be useless if taken daily. If there’s 3 pieces of active gum left, then I wrap it very tightly in Saran wrap which is sticky and air-tight.) The dose will be 1mg or 1/4 a gum. I cut up a dozen pieces into 4 pieces for 48 doses and set them out to dry. Per the previous power analyses, 48 groups of DNB rounds likely will be enough for detecting small-medium effects (partly since we will be only looking at one metric - average % right per 5 rounds - with no need for multiple correction). Analysis will be one-tailed, since we’re looking for whether there is a clear performance improvement and hence a reason to keep using nicotine gum (rather than whether nicotine gum might be harmful).
The placebos can be the usual pills filled with olive oil. The Nature’s Answer fish oil is lemon-flavored; it may be worth mixing in some lemon juice. In Kiecolt-Glaser et al 2011, anxiety was measured via the Beck Anxiety scale; the placebo mean was 1.2 on a standard deviation of 0.075, and the experimental mean was 0.93 on a standard deviation of 0.076. (These are all log-transformed covariates or something; I don’t know what that means, but if I naively plug those numbers into Cohen’s d, I get a very large effect: \frac{1.2 - 0.93}{0.076}=3.55.)

## Starting from the studies in my meta-analysis, we can try to estimate an upper bound on how big any effect would be, if it actually existed. One of the most promising null results, Southon et al 1994, turns out to be not very informative: if we punch in the number of kids, we find that they needed a large effect size (d=0.81) before they could see anything:

Although piracetam has a history of “relatively few side effects,” it has fallen far short of its initial promise for treating any of the illnesses associated with cognitive decline, according to Lon Schneider, a professor of psychiatry and behavioral sciences at the Keck School of Medicine at the University of Southern California. “We don’t use it at all and never have.”
Like caffeine, nicotine tolerates rapidly and addiction can develop, after which the apparent performance boosts may only represent a return to baseline after withdrawal; so nicotine as a stimulant should be used judiciously, perhaps roughly as frequent as modafinil. Another problem is that nicotine has a half-life of merely 1-2 hours, making regular dosing a requirement. There is also some elevated heart-rate/blood-pressure often associated with nicotine, which may be a concern. (Possible alternatives to nicotine include cytisine, 2’-methylnicotine, GTS-21, galantamine, Varenicline, WAY-317,538, EVP-6124, and Wellbutrin, but none have emerged as clearly superior.)
But when aficionados talk about nootropics, they usually refer to substances that have supposedly few side effects and low toxicity. Most often they mean piracetam, which Giurgea first synthesized in 1964 and which is approved for therapeutic use in dozens of countries for use in adults and the elderly. Not so in the United States, however, where officially it can be sold only for research purposes.
Since my experiment had a number of flaws (non-blind, varying doses at varying times of day), I wound up doing a second better experiment using blind standardized smaller doses in the morning. The negative effect was much smaller, but there was still no mood/productivity benefit. Having used up my first batch of potassium citrate in these 2 experiments, I will not be ordering again since it clearly doesn’t work for me.

It is important that this type of approach is discussed with a qualified health professional, such as a registered nutritional therapist, to ensure it is an appropriate strategy for you, as well as to help you avoid missing out on vital nutrients, whilst eliminating certain foods. They can also provide advice on improving your longer term health, which over time may allow for foods to be reintroduced without negative symptoms occurring.

This continued up to 1 AM, at which point I decided not to take a second armodafinil (why spend a second pill to gain what would likely be an unproductive set of 8 hours?) and finish up the experiment with some n-backing. My 5 rounds: 60/38/62/44/5024. This was surprising. Compare those scores with scores from several previous days: 39/42/44/40/20/28/36. I had estimated before the n-backing that my scores would be in the low-end of my usual performance (20-30%) since I had not slept for the past 41 hours, and instead, the lowest score was 38%. If one did not know the context, one might think I had discovered a good nootropic! Interesting evidence that armodafinil preserves at least one kind of mental performance.

##### Vitamin C has long been thought to have the power to increase mental agility, and some research suggests that a deficiency may be a risk factor for age-related brain degeneration including dementia and Alzheimer's.  Furthermore, interesting studies demonstrate that vitamin C may be useful in managing anxiety and stress. One of the best sources of this vital vitamin are blackcurrants. Others include red peppers, citrus fruits such as oranges and broccoli.

Both nootropics startups provide me with samples to try. In the case of Nootrobox, it is capsules called Sprint designed for a short boost of cognitive enhancement. They contain caffeine – the equivalent of about a cup of coffee, and L-theanine – about 10 times what is in a cup of green tea, in a ratio that is supposed to have a synergistic effect (all the ingredients Nootrobox uses are either regulated as supplements or have a “generally regarded as safe” designation by US authorities)
These pills don’t work. The reality is that MOST of these products don’t work effectively. Maybe we’re cynical, but if you simply review the published studies on memory pills, you can quickly eliminate many of the products that don’t have “the right stuff.” The active ingredients in brain and memory health pills are expensive and most companies sell a watered down version that is not effective for memory and focus. The more brands we reviewed, the more we realized that many of these marketers are slapping slick labels on low-grade ingredients.
One of the most common strategies to beat this is cycling. Users who cycle their nootropics take them for a predetermined period, (usually around five days) before taking a two-day break from using them. Once the two days are up, they resume the cycle. By taking a break, nootropic users reduce the tolerance for nootropics and lessen the risk of regression and tolerance symptoms.
Like caffeine, nicotine tolerates rapidly and addiction can develop, after which the apparent performance boosts may only represent a return to baseline after withdrawal; so nicotine as a stimulant should be used judiciously, perhaps roughly as frequent as modafinil. Another problem is that nicotine has a half-life of merely 1-2 hours, making regular dosing a requirement. There is also some elevated heart-rate/blood-pressure often associated with nicotine, which may be a concern. (Possible alternatives to nicotine include cytisine, 2’-methylnicotine, GTS-21, galantamine, Varenicline, WAY-317,538, EVP-6124, and Wellbutrin, but none have emerged as clearly superior.)
Maybe you are you new to nootropics? The word, “Nootropic” is a very broad term describing a supplement or drug that increases mental performance. There are several different groups of nootropics including herbal supplements and a class of research chemicals known as racetams. Below are some of the many common benefits that may potentially be experienced with nootropic supplements.

The benefit of sequential analysis here is being able to stop early, conserving pills, and letting me test another dosage: if I see another pattern of initial benefits followed by decline, I can then try cutting the dose by taking one pill every 3 days; or, if there is a benefit and no decline, then I can try tweaking the dose up a bit (maybe 3 days out of 5?). Since I don’t have a good idea what dose I want and the optimal dose seems like it could be valuable (and the wrong dose harmful!), I can’t afford to spend a lot of time on a single definitive experiment.
Maca root has been used by indigenous people groups in South America for thousands of years. It’s part of the mustard family found primarily in Andean regions and some of its primary uses include improving sexual function, memory and learning as well as reducing the effects of osteoporosis. The standard effective dose ranges from 1,500 to 3,000 mg.

# The truth is that, almost 20 years ago when my brain was failing and I was fat and tired, I did not know to follow this advice. I bought \$1000 worth of smart drugs from Europe, took them all at once out of desperation, and got enough cognitive function to save my career and tackle my metabolic problems. With the information we have now, you don’t need to do that. Please learn from my mistakes!

Common environmental toxins – pesticides, for example – cause your brain to release glutamate (a neurotransmitter). Your brain needs glutamate to function, but when you create too much of it it becomes toxic and starts killing neurons. Oxaloacetate protects rodents from glutamate-induced brain damage.[17] Of course, we need more research to determine whether or not oxaloacetate has the same effect on humans.
More photos from this reportage are featured in Quartz’s new book The Objects that Power the Global Economy. You may not have seen these objects before, but they’ve already changed the way you live. Each chapter examines an object that is driving radical change in the global economy. This is from the chapter on the drug modafinil, which explores modifying the mind for a more productive life.
However, as a result of the efficacy of this type of stacking, the supplement world is saturated with brain-boosting blends, and it can be difficult to cut through the confusion and figure out what really works and what could be a waste of time and money, or downright dangerous. The fact is, when creating your own stack, you must carefully think about your specific needs and goals. For example, if you want to reduce anxiety and depression, but don’t necessarily care to enhance your cognitive performance or get through a day of work in a sleep-deprived state, you could just stick to a single nootropic that increases dopamine levels, such as Mucuna pruriens or tryptophan. Or if you wanted to reduce anxiety and depression while simultaneously improving your memory because you’re studying for a school or work exam, you could add Bacopa monnieri to the mucuna or tryptophan. Then, let’s say you want long-term cognitive performance to the mix that lasts an entire day: in this case, you’d add a racetam, and to avoid an end of day crash, a touch of choline or DHA. It’s a bit like cooking in the kitchen, isn’t it?
By the way, before we move on, allow me to clarify what I mean by “slow caffeine metabolizer”. Ever wondered why your co-worker can slam four giant mugs of coffee during a brief morning of work, while one shot of espresso leaves you jittery and irritated? Turns out that not everyone metabolizes caffeine the same. Generally speaking, in healthy adults, caffeine has a half-life that ranges from about 3 to 7 hours. For example, if the half-life of caffeine in your blood is 5 hours, that means that it takes 5 hours for caffeine levels to be reduced by 50%. Then it takes another 5 hours for that amount to be reduced by 50%. While caffeine metabolism time also depends upon age and environmental factors, a big influence on varying caffeine half-life times is your genetic makeup.
Amphetamine – systematic reviews and meta-analyses report that low-dose amphetamine improved cognitive functions (e.g., inhibitory control, episodic memory, working memory, and aspects of attention) in healthy people, and in individuals with ADHD.[21][22][23][25] A 2014 systematic review noted that low doses of amphetamine also improved memory consolidation, in turn leading to improved recall of information in non-ADHD youth.[23] It also improved task saliency (motivation to perform a task) and performance on tedious tasks that required a high degree of effort.[22][24][25]


text:
News

# “Effect of Window Shape on the Detection of Hyperuniformity via the Local Number Variance” was published in Journal of Statistical Mechanics: Theory and Experiment

Hyperuniform many particle systems in $$d$$-dimensional space, which includes crystals, quasicrystals, and some exotic disordered systems, are characterized by an anomalous suppression of density fluctuations at large length scales such that the local number variance within a “spherical” observation window grows slower than the window volume. In usual circumstances, this direct space condition is equivalent to the Fourier space hyperuniformity condition that the structure factor vanishes as the wavenumber goes to zero. In this paper, we comprehensively study the effect of aspherical window shapes with characteristic size $$L$$ on the direct space condition for hyperuniform systems. For lattices, we demonstrate that the variance growth rate can depend on the shape as well as the orientation of the windows, and in some cases, the growth rate can be faster than the window volume ($$L^d$$), which may lead one to falsely conclude that the system is non-hyperuniform solely according to the direct space condition. We analyze the effect of window orientation on the growth rate, especially for two-dimensional lattices using square windows. Based on this analysis, we explain the reason why the variance can grow faster than the window area ($$L^2$$) or even slower than the window perimeter ($$L$$). This analysis demonstrates an example of physical systems exhibiting commensurate-incommensurate transitions and is closely related to problems in number theory (e.g., Diophantine approximation and Gauss’ circle problem) and discrepancy theory. We prove that for a hyperuniform system its orientationally-averaged variance, instead of the conventional one using windows with a fixed orientation, has a common large-$$L$$ asymptotic behavior regardless of the window shape (for convex windows), thus it can be used to resolve the window-shape dependence of the growth rate of the variance on the spherical-window condition.


text:
My Math Forum Homogeneous Functions

 Real Analysis Real Analysis Math Forum

 December 16th, 2018, 10:41 AM #1 Senior Member     Joined: Oct 2016 From: Arizona Posts: 207 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. Homogeneous Functions I've been working on a few inequality proofs over the last few months, and some of the solutions I've looked at mentioned that since the inequality is homogeneous, you can make extra condtions like $abc=1$ or $a+b+c=1$ or others. What I don't get is how can you tell if an inequality is homogeneous? The book never mentioned how they got that. BTW, I found a few definitions online, but I don't for sure if that's what they are using. So, I was just wondering if someone could confirm or set me on the right track to getting it. Last edited by ProofOfALifetime; December 16th, 2018 at 11:03 AM.
 December 16th, 2018, 03:43 PM #2 Senior Member     Joined: Oct 2016 From: Arizona Posts: 207 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. Nevermind, I got it!
 December 16th, 2018, 06:38 PM #3 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,801 Thanks: 636 Math Focus: Yet to find out. Do you mean inequalities which involve homogeneous functions?
December 17th, 2018, 11:18 AM   #4
Senior Member

Joined: Oct 2016
From: Arizona

Posts: 207
Thanks: 37

Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.
Quote:
 Originally Posted by Joppy Do you mean inequalities which involve homogeneous functions?
I don't really know. I just know that some of the solutions to inequalities I've looked at (I'm attaching some) mention it.

It says, "By the homogeneity of the inequality.."
Attached Images
 48372136_730283900680187_5026679403678007296_n.jpg (35.4 KB, 7 views)

Last edited by ProofOfALifetime; December 17th, 2018 at 11:46 AM.

December 17th, 2018, 11:18 AM   #5
Senior Member

Joined: Oct 2016
From: Arizona

Posts: 207
Thanks: 37

Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.
Here is another example. That part, "hence in homogeneous form.."
Attached Images
 48426105_730283904013520_3707896338092916736_n.jpg (42.7 KB, 7 views)

Last edited by ProofOfALifetime; December 17th, 2018 at 11:46 AM.

December 29th, 2018, 04:52 PM   #6
Senior Member

Joined: Oct 2016
From: Arizona

Posts: 207
Thanks: 37

Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.
I found a book that explains it clearly! So I am posting the page so this thread will be more useful.
Attached Images
 49348598_737369413304969_8279041215320555520_n.jpg (79.4 KB, 5 views)

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text:
# naginterfaces.library.lapacklin.dpotri¶

naginterfaces.library.lapacklin.dpotri(uplo, a)[source]

dpotri computes the inverse of a real symmetric positive definite matrix , where has been factorized by dpotrf().

For full information please refer to the NAG Library document for f07fj

https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/f07/f07fjf.html

Parameters
uplostr, length 1

Specifies how has been factorized.

, where is upper triangular.

, where is lower triangular.

afloat, array-like, shape

The upper triangular matrix if or the lower triangular matrix if , as returned by dpotrf().

Returns
afloat, ndarray, shape

is overwritten by the upper triangle of if ; is overwritten by the lower triangle of if .

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: .

Warns
NagAlgorithmicWarning
(errno )

Diagonal element of the Cholesky factor is zero; the Cholesky factor is singular and the inverse of cannot be computed.

Notes

dpotri is used to compute the inverse of a real symmetric positive definite matrix , the function must be preceded by a call to dpotrf(), which computes the Cholesky factorization of .

If , and is computed by first inverting and then forming .

If , and is computed by first inverting and then forming .

References

Du Croz, J J and Higham, N J, 1992, Stability of methods for matrix inversion, IMA J. Numer. Anal. (12), 1–19


text:
# Compound Distribution — Log Normal Distribution with Log Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Log Normal Distribution whose mean (actually, the mean of the log of the variable) is distributed Log Normally.

# Given,

$$X = e^{W}$$

$$W \sim N[e^{Y},\sigma^2_{W}]$$

$$Y \sim N[\mu_{Y},\sigma^2_{Y}]$$

# To Determine,

$$f_{X}(x), F_{X}(x), E(X), E(X^{2})$$

Related Question when Mean is Normal:

Compound Distribution --- Log Normal Distribution with Normally Distributed Mean

Related General Question:

Starting with the above special case, it quickly becomes apparent there are many combinations possible. Hence was wondering if there were general techniques to derive the density, distribution function, expected value, higher moments, conditional expectations etc. of compound distributions and some source where certain combinations and results therein were given with detailed steps and complete proofs: https://math.stackexchange.com/questions/1614212/compound-distributions-basic-techniques-and-key-general-results-from-first-p

• You are posting a large number of these parameter mix distribution problems, ... so many that working them all out by hand will keep you busy for a while. I suggest you get a computer algebra system, and once you know how to do one, you can do them all ... assuming closed-form solutions exist, and the software can find it. – wolfies Jan 21 '16 at 13:27
• @wolfies … Thanks for your attention to this. Indeed, I started trying to get a generic technique; but could find none and received suggestions to post specific questions. Could you please elaborate on the computer algebra system that can help in this regard? – texmex Jan 21 '16 at 13:31
• Let me suggest to stop flooding the site with minor variations of the same Ur-question and to grab a textbook covering the basics of 1. distributions, 2. normal distributions, 3. expectations of random variables. In the present case, what the hypothesis says is that $e^X=Y+\sigma_XZ$ and $e^Y=\mu_Y+\sigma_YT$ where $Z$ and $T$ are standard normal and are probably independent (although this is not mentioned in the question, but should be). Everything will be deduced from this. To reach $E(X)$ for example, one would note ... – Did Jan 22 '16 at 7:08
• ... that the identities above imply $X=\log(\sigma_XZ+\log(\mu_Y+\sigma_YT))$ hence $$E(X)=\iint_{\mathbb R^2}\log(\sigma_Xz+\log(\mu_Y+\sigma_Yt))\frac{e^{-(z^2+t^2)/2}}{2\pi}dzdt.$$ ...Except that, if one keeps one's eyes open, one readily sees that the hypothesis that $e^Y$ should be normal is impossible, absurd, monstrous, whatever you like to call it, since nondegenerate normal random variables are never always positive. To sum up, less questions, but with an ounce of thought in them, please. – Did Jan 22 '16 at 7:09
• @Did .. Thanks for this pointer. It is a typo, I have fixed it. $Y$ is normal, but the mean of $X$ is log normal or distributed as $e^{Y}$. The wording of the question was correct, I presume, but this notation was not. If any other issues, please let me know. – texmex Jan 22 '16 at 10:25


text:
### Home > CCA2 > Chapter 2 > Lesson 2.2.4 > Problem2-142

2-142.

Write an equation for a function that is odd, and explain how you can tell it is odd from its graph, its table, and its equation.

There are many possible answers. Recall that for an odd function, $f(−x)=−f(x)$.

One example is $y=x^3$. Be sure to show how your example is odd using a graph, a table, and by looking at the equation.


text:
# Changeset 8913

Ignore:
Timestamp:
11/14/12 18:30:24 (10 years ago)
Message:

#1925: made corrections as suggested by abeham in the code review. In particular hopefully corrected the calculation of the CDF.

File:
1 edited

Unmodified


text:
How to create alternating rows in a table?

Sometimes you can see in books tables, that have alternating background color for rows in a table to support readability. How I can achieve this with LaTeX?

And yes, I could put macros \odd and \even into each row, but in that case I would have to change all following rows, if I input later another row into the table. I want TeX to decide itself, which row is odd or even and put the appropiate style into it.

You can use the features of the colortbl package with additions of xcolor. See code below for a small example.

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}

\begin{document}
\rowcolors{2}{gray!25}{white}
\begin{tabular}{cc}
\rowcolor{gray!50}
Some values & Some values\\
Some values & Some values\\
Some values & Some values\\
Some values & Some values\\
\end{tabular}
\end{document}

The particular package manuals have the details.

• You beat me by 20 seconds. +1 Nov 14, 2010 at 13:24
• If there are some other languages text in the table, please do not use the the order "\usepackage[latin1]{inputenc}"
– ccy
Aug 27, 2016 at 8:52
• Actually, the \rowcolors command is not referenced anywhere in the manual. Only the \rowcolor without the s. Any idea on how to make color alternate every n row ? Oct 12, 2016 at 10:06
• Be careful with this approach: in my setup it colored the part of equations typeset with \begin{cases}\end{cases}
– cw'
Oct 18, 2019 at 15:21
• Can the gray color start from the second row instead of first one ? Nov 19, 2020 at 22:27

This can easily be achieved using xcolor. It is explained at Wikibooks, you just add the table option to \usepackage{xcolor} and in the table itself you put

\rowcolors{1}{color-1}{color-2}

One thing that might be worth pointing out, is that apparently, many packages call on xcolor, and I had endless clashes first time I tried it, and it took me such a long time to figure this one out, so I felt it might help someone as slow as me to mention this: The xcolor package should be loaded last (or at least after the clashing packages).

• I had that clash using beamer. Since beamer loads xcolor internally, you cannot add \usepackage[table]{xcolor} just to specify the table option. Therefore you have to put the line \PassOptionsToPackage{table}{xcolor} before the \documentclass{..}. Hope somebody else can use this :-) Jun 6, 2012 at 7:39
• Thank you Bertram, this was exactly the solution I was looking for. I was using the tufte-book class. Now it works flawlessly ;) \PassOptionsToPackage{table}{xcolor} \documentclass{..}
– user28917
Apr 12, 2013 at 13:16
• Another approach which might be worth trying in this case (xcolor already loaded by another package): just load colortbl instead of xcolor. Feb 6, 2015 at 17:33
• @PaulPaulsen this does not work because the command \rowcolors is not defined by colortbl but by xcolor if the table option is used. Therefore your solution doesn't work in this case. May 17, 2017 at 14:45
• @BertramNudelbach An alternative to the solution using \PassOptionsToPackage{table}{xcolor} is to load the beamer package with the option xcolor=table as specified in the Beamer user guide on p. 19. Mar 22, 2018 at 10:49

This post says if you want to apply alternating colors (white, then lightgray) to all your tables, you can use pure TeX to "redefine the tabular environment":

\usepackage[table]{xcolor}
\definecolor{lightgray}{gray}{0.94}
\let\oldtabular\tabular
\let\endoldtabular\endtabular
\renewenvironment{tabular}{\rowcolors{2}{lightgray}{white}\oldtabular}{\endoldtabular}

I edited the example slightly to make white first, and the lightgray lighter.

The environment {NiceTabular} of nicematrix has tools in order to color cells, rows and columns in a convenient way.

Morevorer, with nicematrix you won't have the thin white lines you see in some PDF viewers.

\documentclass{article}
\usepackage{nicematrix}

\begin{document}
\begin{NiceTabular}{cc}
\CodeBefore
\rowcolor{gray!50}{1}
\rowcolors{2}{gray!25}{white}
\Body
Some values & Some values\\
Some values & Some values\\
Some values & Some values\\
Some values & Some values\\
\end{NiceTabular}
\end{document}

You need several compilations (because nicematrix uses PGF/Tikz nodes under the hood).

You can also color the rows with any cycle of colors:

\documentclass{article}
\usepackage{nicematrix}

\begin{document}

\begin{NiceTabular}{c}
\CodeBefore
\rowlistcolors{1}{red!15,blue!15,green!15}
\Body
Mathilde \\
Pierre \\
Paul \\
Juliette \\
Jacques \\
Antoine \\
Emmanuelle \\
\end{NiceTabular}

\end{document}

• You could elaborate on why you won't see the "thin white lines" (because the row colours are drawn across the entire row rather than on a cell-by-cell basis), but then also mention that at the first go-around, you need to compile three times. Subsequent compilations may require fewer runs if nothing changes in terms of the tabular position. However, just mentioning {NiceTabular} and boasting about the positive things only is being biased...
– Werner
Aug 27, 2021 at 19:16
• @Werner: In fact, with nicematrix, if you use \cellcolor{...} in two adjacent cells (with the same color), you won't see thin white lines. The package has been designed to avoid those thin lines even if it is on a cell-by-cell basis. For the compilations, I have said that several (several) compilations may be needed. Aug 27, 2021 at 20:07
• Ahhh, I see that now.
– Werner
Aug 27, 2021 at 20:36
• For the long lines can it add auto newline at the end of each line instead of continue Jan 23, 2022 at 14:26
• @alper: In {NiceTabular}, you can use columns of type p, b and m as in standard {tabular} (of the package array). Jan 23, 2022 at 16:07

An alternative solution with tblr and booktabs environments of tabularray package: alternating rows work well even if you have changed the vertical space between them.

\documentclass{article}

\usepackage{xcolor}

\usepackage{tabularray}
\UseTblrLibrary{booktabs}

\begin{document}

\begin{tblr}{
colspec = {rccc},
row{odd} = {blue8},
row{even} = {azure8},
row{1} = {purple8},
}
\hline[2pt]
Thing & Value & Value & Value \\
\hline[1pt]
A & 1     & 2     & 3     \\
B & 1     & 2     & 3     \\
C & 1     & 2     & 3     \\
D & 1     & 2     & 3     \\
E & 1     & 2     & 3     \\
\hline[2pt]
\end{tblr}

\bigskip

\begin{booktabs}{
colspec = {rccc},
row{odd} = {blue8},
row{even} = {azure8},
row{1} = {purple8},
}
\toprule
Thing & Value & Value & Value \\
\midrule
A & 1     & 2     & 3     \\
B & 1     & 2     & 3     \\
C & 1     & 2     & 3     \\
\specialrule{2pt,teal5}{0pt}{2pt}
D & 1     & 2     & 3     \\
E & 1     & 2     & 3     \\
\bottomrule
\end{booktabs}

\end{document}

\documentclass{article}
\usepackage[table]{xcolor}

\usepackage{multirow}

\usepackage{float}
\usepackage{graphicx} %rotate text
\usepackage{colortbl}

\begin{document}

\begin{table}[ht]
\rowcolors{1}{}{lightgray}
\centering
\caption{Multirow table with continuous coloring.}\label{tab:multi row}
\begin{tabular}{p{5cm}p{5cm}}
\hline
Column 1 & Column 2\\
\hline
-&-\\
-&-\\
\cellcolor{green}&Single-row\\
\cellcolor{green}&Single-row\\
%\parbox[t]{2mm}{\multirow{-3}{*}{Rotating}} & working row \\
\cellcolor{green}\parbox[t]{2mm}{\multirow{-3}{*}{\rotatebox[origin=c]{90}{$job\_id$}}} & working row \\
-&-\\
-&-\\
\hline
\end{tabular}
\end{table}%

\begin{table}[h]
\centering
\begin{tabular}{|c|l|r|r|r|r|}\hline
~ & \multicolumn{1}{c|}{Text} & {\hfill Text\hfill} & \multicolumn{1}{c|}{Text} & \multicolumn{1}{c|}{Text} & \multicolumn{1}{c|}{Text} \\ \hline
\multirow{3}{*}{\rotatebox[origin=c]{90}{\parbox[c]{1cm}{\centering rotato-ata}}} & text &&&&\\
& text &testing A& testing B&&\\
& text &right aligned&&&\\ \hline
\end{tabular}
\end{table}

\begin{table}[ht]
\rowcolors{1}{}{lightgray}
\centering
\caption{Multirow table with all cells in the same color.}\label{tab:multi row}
\begin{tabular}{p{5cm}p{5cm}}
\hline
Column 1 & Column 2\\
\hline
-                       &       -       \\
-                       &       -       \\
\cellcolor{green} Text Green   &   Single-row  \\
\cellcolor{yellow} &   Single-row  \\
\multirow{-3}{*}{\cellcolor{red}Text of -3, Misplaced}&Single-row\\
-                       &       -       \\
-                       &       -       \\
\hline
\end{tabular}
\end{table}%

\begin{table}[ht]
\rowcolors{1}{}{lightgray}
\centering
\caption{Multirow table with all rows in the same color.}\label{tab:multi row}
\begin{tabular}{p{5cm}p{5cm}}
\hline
Column 1 & Column 2\\
\hline
-                       &       -       \\
-                       &       -       \\
\rowcolor{green}        &   Single-row  \\
\rowcolor{yellow}       &   Single-row  \\
\rowcolor{red}\multirow{-3}{*}{Text}&Single-row\\
-                       &       -       \\
-                       &       -       \\
\hline
\end{tabular}
\end{table}%

\end{document}


text:
# In the given circuit, the value of capacitor C that makes current I = 0 is ____________ μF.

This question was previously asked in
GATE EE 2021 Official Paper
View all GATE EE Papers >

## Answer (Detailed Solution Below) 20.00

Free
CT 1: Ratio and Proportion
1963
10 Questions 16 Marks 30 Mins

## Detailed Solution

Concept:

• It is given that the current flowing in the circuit is zero.
• This can happen only when the impedance of the circuit is infinite.
• The given circuit is closed which means to get infinite impedance the denominator part of the impedance must be zero.
• That means the denominator of the impedance of the parallel sub-circuit is zero.

Application:

Z = j5 || (j5 - j/ωC)

$$Z = {j5× ({j5-{j\over \omega C}})\over j5+j5-{j\over \omega C}}$$

By equating the denominator to zero, we get

$$j10-{j\over \omega C} = 0$$

$$C={1\over 10\omega}={1\over 10× 5000}\; F$$

C = 20 × 10-6 F

∴ The value of C = 20 μF.


text:
# Linearly independent subset - a simple solution?

Problem: Let $\{ v_1, \ldots, v_n \}$ be a linearly independent subset of $V$, a vector space. let $$v= t_1 v_1 + \cdots + t_n v_n$$ where $t_1, \ldots t_n \in \mathbb{R}$. For which $v$ is the set $\{v_1 + v ,\ldots , v_n + v \}$ linearly independent?

My ideas were:

Firstly, any linear combination of the new set is, $$\sum \lambda_i (v_i + v) = \sum \lambda_i v_i + sv = \sum (\lambda_i + st_i) v_i = 0$$ where $s = \sum \lambda_i$. By linear independence of $v_i$, we must have $\lambda _i + st_i = 0$ for $i=1, \ldots , n$. Summing up yields $s + s \sum t_i = 0 \Rightarrow s( 1 + \sum t_i) = 0$.

My claim is: $\sum t_i \not = -1 \Leftrightarrow$ set is linearly independent.

Proof: If $\sum t_i \not = -1$, then $s = 0$, hence $\lambda _i = 0$ for $i=1, \ldots, n$. So the new set is linearly independent.

On the other hand if $\sum t_i = -1$, then regarding the original system of linear equations, where $$\lambda_i + st_i = \sum_{i \not=j} \lambda_j t_i + \lambda_i (1+t_i) = 0$$ for $i-1, \ldots, n$. We wish to find a non trivial set of solutions for $i=1, \ldots, n$. Written out in matrix form, we have a matrix with all column sums equal to $0$ (column $i$ has sum $\sum t_i +1 = 0$), hence the transpose of this matrix, has a non trivial solution, eigenvector $(1, \ldots, 1 )^T$.

Therefore, the transpose matrix is not invertible - hence, the matrix itself is not invertible. So there exists a non trivial solution.

I think this solution is a bit too long winded - I am hoping if anyone could give a more elmentary solution that does not use matrices. Thank you!

It is easy to explicitly guess a nontrivial solution for the system of equations $$\lambda_i + st_i = 0$$ By setting $\lambda_i = t_i$ for all $i$, we get $\lambda_i + st_i = t_i-t_i = 0$ for all $i$ since $s=-1$ by assumption. Also since $\sum t_i = -1$, atleast one $t_i$ is nonzero, so the solution is nontrivial.


text:
## anonymous 5 years ago I want to show that Nullspace (normed space) is a vector space. can someone help

1. anonymous

null spaces are defined reference to some linear transformation

2. anonymous

yes. I mean a linear transformation on a normed vector space

3. anonymous

so u want to show it a subspace?

4. anonymous

yes

5. anonymous

the condition is, for x,y in space n a, b scalars ax+by must belong to the space...right?

6. anonymous

i think so, if that is enough to show

7. anonymous

$x, y \in N$

8. anonymous

then $T ( x )= T ( y ) = 0$

9. anonymous

T (ax + by) = a T(x) +bT(y) using the linarity

10. anonymous

T (ax + by) = a T(x) +bT(y)=0 showing that ax + by is in N

11. anonymous

so N is the vector subspace


text:
# Next Week's Seminars

Unable to attend an INI seminar?
Most seminars are streamed live as well as being archived for later viewing.

Event Day Time Speaker Title
RGMW04 Monday 20th 10:00 to 11:00 G Miermont Compact Brownian surfaces
RGMW04 Monday 20th 11:30 to 12:30 M Albenque Blossoming trees and the scaling limit of maps
RGMW04 Monday 20th 14:00 to 15:00 T Budd Scaling constants and the lazy peeling of infinite Boltzmann planar maps
RGMW04 Monday 20th 15:30 to 16:30 G Ray Critical exponents in FK-weighted planar maps
RGMW04 Tuesday 21st 10:00 to 11:00 M Barlow Scaling limits of the uniform spanning tree
RGMW04 Tuesday 21st 11:30 to 12:30 K Kytölä A hidden quantum group for pure partition functions of multiple SLEs
RGMW04 Tuesday 21st 14:00 to 15:00 Z Li Self-avoiding Walk and Connective Constant
RGMW04 Tuesday 21st 15:30 to 16:30 I Kortchemski Looptrees
RGMW04 Wednesday 22nd 09:00 to 10:00 J-F Le Gall On the geometry of discrete and continuous random planar maps
RGMW04 Wednesday 22nd 10:00 to 11:00 G Schaeffer On classes of planar maps with $\alpha$-orientations having geometric interpretations
RGMW04 Wednesday 22nd 11:30 to 12:30 G Borot Nesting statistics in the O(n) loop model on random lattices
PEP Wednesday 22nd 14:00 to 16:00 V Tchoulaevski A short introduction into multi-particle Anderson localization (a mini-course)
RGMW04 Wednesday 22nd 14:00 to 15:00 O Bernardi Differential equations for colored maps
RGMW04 Wednesday 22nd 15:30 to 16:30 O Gurel Gurevich Recurrence of planar graph limits
RGMW04 Thursday 23rd 09:00 to 10:00 A Nachmias The uniform spanning forest of planar graphs
RGMW04 Thursday 23rd 10:00 to 11:00 Fyodorov Characterstic polynomials of random matrices and logarithmically correlated processes
RGMW04 Thursday 23rd 11:30 to 12:30 Lacoin Pinning and disorder relevance for the lattice Gaussian free field
PEP Thursday 23rd 14:00 to 15:00 R Kuehn Spectra of Sample Auto-Covariance Matrices
RGMW04 Thursday 23rd 14:00 to 15:00 I Manolescu Planar lattices do not recover from forest fires
RGMW04 Thursday 23rd 15:30 to 16:30 N Sun The exact $k$-SAT threshold for large $k$
RGMW04 Friday 24th 10:00 to 11:00 L Addario-Berry Squarings of rectangles
RGMW04 Friday 24th 11:30 to 12:30 O Angel Parabolic and Hyperbolic Unimodular maps
RGMW04 Friday 24th 14:00 to 15:00 V Vargas Liouville Quantum gravity on Riemann surfaces
RGMW04 Friday 24th 15:30 to 16:30 N Curien Scaling limits of random planar maps and growth-fragmentations


text:
# Divergence / Gradient / Laplace Operator

Yao Yao on June 11, 2018
$\newcommand{\icol}[1]{ \bigl[ \begin{smallmatrix} #1 \end{smallmatrix} \bigr] }$

## Divergence

Quote from Wikipedia: Divergence:

Let $x$, $y$, $z$ be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ be the corresponding basis of unit vectors. The divergence of a continuously differentiable vector field $F = U \mathbf{i} + V \mathbf{j} + W \mathbf{k}$ is defined as the scalar-valued function:

$\operatorname {div} \mathbf {F} = \nabla \cdot \mathbf {F} = \left ( {\frac {\partial }{\partial x}}, {\frac {\partial }{\partial y}}, {\frac {\partial }{\partial z}} \right) \cdot (U,V,W) = {\frac {\partial U}{\partial x}} + {\frac {\partial V}{\partial y}} + { \frac {\partial W}{\partial z}}.$

1. 写法。$F = U \mathbf{i} + V \mathbf{j} + W \mathbf{k}$ 其实就是 $\vec F = \icol{U \newline V \newline W}$，它其实是一个 vector
2. 这里 $\nabla \cdot \mathbf {F}$ 明显不是 dot product，但是计算方法类似，最后的结果是一个 scalar
3. Gradient 的写法 $\nabla f$ 不带这个 dot

## Divergence 的物理意义

Quote from Erik Anson’s answer on Quora:

Imagine a fluid, with the vector field representing the velocity of the fluid at each point in space. Divergence measures the net flow of fluid out of (i.e., diverging from) a given point. If fluid is instead flowing into that point, the divergence will be negative.

A point or region with positive divergence is often referred to as a “source” (of fluid, or whatever the field is describing), while a point or region with negative divergence is a “sink”.

The bigger the flux density (positive or negative), the stronger the flux source or sink. A div of zero means there’s no net flux change in side the region.

## Laplace Operator

\begin{aligned} \nabla \cdot \nabla f(x, y, z) &= \left ( {\frac {\partial }{\partial x}}, {\frac {\partial }{\partial y}}, {\frac {\partial }{\partial z}} \right) \cdot \left ( \frac{\partial f}{\partial x}(x,y,z), \frac{\partial f}{\partial y}(x,y,z), \frac{\partial f}{\partial z}(x,y,z) \right ) \newline &= \frac {\partial^{2} f}{\partial x^{2}}(x,y,z) + \frac {\partial^{2} f}{\partial y^{2}}(x,y,z) + \frac {\partial^{2} f}{\partial z^{2}}(x,y,z) \newline &= \nabla^2 f(x, y, z) \end{aligned}

which happens to be the $lap$ of $f(x, y, z)$.

I.e. $\operatorname{lap} f = \operatorname{div}(\operatorname{grad} f)$

• 如果你在 $f$ 的 local minimum $a$，你周围的 gradient 全部流出 $a$ (任意方向都是 ascending)，divergence 是 highly positive，所以 $a$ 是 gradient 的 source
• 如果你在 $f$ 的 local maximum $b$，你周围的 gradient 全部流入 $b$ (任意方向都是 descending)，divergence 是 highly negative，所以 $b$ 是 gradient 的 sink


text:
A rocket moves with a velocity of 0.89c to the right with respect to a stationary observer A. An observer B moving relative to observer A finds that the rocket is moving with a velocity of 0.95c to the left. What is the velocity of observer B relative to observer A? (Hint: Consider observer B's velocity in the frame of reference of the rocket.)
___________ c


text:
# Group Generated by Commutators of Two Normal Subgroups is a Normal Subgroup

## Problem 129

Let $G$ be a group and $H$ and $K$ be subgroups of $G$.
For $h \in H$, and $k \in K$, we define the commutator $[h, k]:=hkh^{-1}k^{-1}$.
Let $[H,K]$ be a subgroup of $G$ generated by all such commutators.

Show that if $H$ and $K$ are normal subgroups of $G$, then the subgroup $[H, K]$ is normal in $G$.

Contents

## Proof.

We first prove that a conjugate of each generator is in $[H,K]$.
Let $h \in H, k \in K$. For any $g \in G$, we have by inserting $g^{-1}g=e$ inbetweens
\begin{align*}
g[h,k]g^{-1}&=ghkh^{-1}k^{-1}=(ghg^{-1})(gkg^{-1})(gh^{-1}g^{-1})(gk^{-1}g^{-1})\\
& = (ghg^{-1})(gkg^{-1})(ghg^{-1})^{-1}(gkg^{-1})^{-1}.
\end{align*}

Now note that $ghg^{-1} \in H$ since $H$ is normal in $G$, and $gkg^{-1} \in K$ since $K$ is normal in $G$. Thus $g[h,k]g^{-1} \in [H, K]$.
By taking the inverse of the above equality, we also see that $g[k,h]g^{-1} \in [H, K]$. Thus the conjugate of the inverse $[h,k]^{-1}=[k,h]$ is in $[H, K]$.

Next, note that any element $x \in [H,K]$ is a product of generators or their inverses. So let us write
$x=[h_1, k_1]^{\pm 1}[h_2, k_2]^{\pm 1}\cdots [h_n, k_n]^{\pm 1},$ where $h_i \in H, k_i\in K$ for $i=1,\dots, n$.
Then for any $g \in G$, we have
\begin{align*}
gxg^{-1}=(g[h_1, k_1]^{\pm 1}g^{-1})(g[h_2, k_2]^{\pm 1}g^{-1})\cdots(g [h_n, k_n]^{\pm 1} g^{-1}).
\end{align*}

We saw that the conjugate of a generator, or its inverse, by $g \in G$ is in $[H,K]$.
Thus $gxg^{-1}$ is also in $[H, K]$.
This proves that the group $[H,K]$ is a normal subgroup of $G$.

## Related Question.

Another problem about a commutator group is
A condition that a commutator group is a normal subgroup

### More from my site

• Commutator Subgroup and Abelian Quotient Group Let $G$ be a group and let $D(G)=[G,G]$ be the commutator subgroup of $G$. Let $N$ be a subgroup of $G$. Prove that the subgroup $N$ is normal in $G$ and $G/N$ is an abelian group if and only if $N \supset D(G)$.   Definitions. Recall that for any $a, b \in G$, the […]
• A Condition that a Commutator Group is a Normal Subgroup Let $H$ be a normal subgroup of a group $G$. Then show that $N:=[H, G]$ is a subgroup of $H$ and $N \triangleleft G$. Here $[H, G]$ is a subgroup of $G$ generated by commutators $[h,k]:=hkh^{-1}k^{-1}$. In particular, the commutator subgroup $[G, G]$ is a normal subgroup of […]
• Non-Abelian Simple Group is Equal to its Commutator Subgroup Let $G$ be a non-abelian simple group. Let $D(G)=[G,G]$ be the commutator subgroup of $G$. Show that $G=D(G)$.   Definitions/Hint. We first recall relevant definitions. A group is called simple if its normal subgroups are either the trivial subgroup or the group […]
• Two Quotients Groups are Abelian then Intersection Quotient is Abelian Let $K, N$ be normal subgroups of a group $G$. Suppose that the quotient groups $G/K$ and $G/N$ are both abelian groups. Then show that the group $G/(K \cap N)$ is also an abelian group.   Hint. We use the following fact to prove the problem. Lemma: For a […]
• Normal Subgroups Intersecting Trivially Commute in a Group Let $A$ and $B$ be normal subgroups of a group $G$. Suppose $A\cap B=\{e\}$, where $e$ is the unit element of the group $G$. Show that for any $a \in A$ and $b \in B$ we have $ab=ba$. Hint. Consider the commutator of $a$ and $b$, that […]
• Two Normal Subgroups Intersecting Trivially Commute Each Other Let $G$ be a group. Assume that $H$ and $K$ are both normal subgroups of $G$ and $H \cap K=1$. Then for any elements $h \in H$ and $k\in K$, show that $hk=kh$.   Proof. It suffices to show that $h^{-1}k^{-1}hk \in H \cap K$. In fact, if this it true then we have […]
• Abelian Normal Subgroup, Intersection, and Product of Groups Let $G$ be a group and let $A$ be an abelian subgroup of $G$ with $A \triangleleft G$. (That is, $A$ is a normal subgroup of $G$.) If $B$ is any subgroup of $G$, then show that $A \cap B \triangleleft AB.$   Proof. First of all, since $A \triangleleft G$, the […]
• If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$. Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$. Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.   Hint. It follows from […]

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##### If a Subgroup $H$ is in the Center of a Group $G$ and $G/H$ is Nilpotent, then $G$ is Nilpotent

Let $G$ be a nilpotent group and let $H$ be a subgroup such that $H$ is a subgroup in the...

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text:
Answers to the barplot coloring question:

Code $$\PageIndex{1}$$ (R):

sex <- c("male", "female", "male", "male", "female", "male", "male")
sex.f <- factor(sex)
plot(sex.f, col=1:2)

or

Code $$\PageIndex{2}$$ (R):

sex <- c("male", "female", "male", "male", "female", "male", "male")
sex.f <- factor(sex)
plot(sex.f, col=1:nlevels(sex.f))

(Please try these commands yourself. The second answer is preferable because it will work even in cases when factor has more than two levels.)

Answers to the barplot counts question. To see frequencies, from highest to smallest, run:

Code $$\PageIndex{3}$$ (R):

lload("data/com12.rd")
exists("com12")
com12.plot <- barplot(com12, names.arg="")
rev(sort(com12))

or

Code $$\PageIndex{4}$$ (R):

lload("data/com12.rd")
exists("com12")
com12.plot <- barplot(com12, names.arg="")
rev(sort(com12))
sort(com12, decreasing=TRUE)

Answer to flowering heads question. First, we need to load the file into R. With url.show() or simply by examining the file in the browser window, we reveal that file has multiple columns divided with wide spaces (likely Tab symbols) and that the first column contains species names with spaces. Therefore, header and separator should be defined explicitly:

Code $$\PageIndex{5}$$ (R):

comp <- read.table("http://ashipunov.info/shipunov/open/compositae.txt", h=TRUE, sep="\t")

Next step is always to check the structure of new object:

Code $$\PageIndex{6}$$ (R):

comp <- read.table("http://ashipunov.info/shipunov/open/compositae.txt", h=TRUE, sep="\t")
str(comp)

Two columns (including species) are factors, others are numerical (integer or not). The resulted object is a data frame.

Next is to select our species and remove unused levels:

Code $$\PageIndex{7}$$ (R):

comp <- read.table("http://ashipunov.info/shipunov/open/compositae.txt", h=TRUE, sep="\t")
c3 <- comp[comp$SPECIES %in% c("Anthemis tinctoria", "Cosmos bipinnatus", "Tripleurospermum inodorum"), ] c3$SPECIES <- droplevels(c3$SPECIES)  To select tree species in one command, we used logical expression made with %in% operator (please see how it works with ?"%in%" command). Removal of redundant levels will help to use species names for scatterplot: Code $$\PageIndex{8}$$ (R): comp <- read.table("http://ashipunov.info/shipunov/open/compositae.txt", h=TRUE, sep="\t") c3 <- comp[comp$SPECIES %in% c("Anthemis tinctoria", "Cosmos bipinnatus", "Tripleurospermum inodorum"), ]
c3$SPECIES <- droplevels(c3$SPECIES)
with(c3, plot(HEAD.D, RAYS, col=as.numeric(SPECIES), pch=as.numeric(SPECIES), xlab="Diameter of head, mm", ylab="Number of rays"))
legend("topright", pch=1:3, col=1:3, legend=levels(c3$SPECIES))  Please make this plot yourself. The key is to use SPECIES factor as number, with as.numeric() command. Function with() allows to ignore cc$ and therefore saves typing.

However, there is one big problem which at first is not easy to recognize: in many places, points overlay each other and therefore amount of visible data points is much less then in the data file. What is worse, we cannot say if first and third species are well or not well segregated because we do not see how many data values are located on the “border” between them. This scatterplot problem is well known and there are workarounds:

Code $$\PageIndex{9}$$ (R):

comp <- read.table("http://ashipunov.info/shipunov/open/compositae.txt", h=TRUE, sep="\t")
c3 <- comp[comp$SPECIES %in% c("Anthemis tinctoria", "Cosmos bipinnatus", "Tripleurospermum inodorum"), ] with(c3, plot(jitter(HEAD.D), jitter(RAYS), col=as.numeric(SPECIES), pch=as.numeric(SPECIES), xlab="Diameter of head, mm", ylab="Number of rays"))  Please run this code yourself. Function jitter() adds random noise to variables and shits points allowing to see what is below. However, it is still hard to understand the amount of overplotted values. There are also: Code $$\PageIndex{10}$$ (R): comp <- read.table("http://ashipunov.info/shipunov/open/compositae.txt", h=TRUE, sep="\t") c3 <- comp[comp$SPECIES %in% c("Anthemis tinctoria", "Cosmos bipinnatus", "Tripleurospermum inodorum"), ]

(Try these variants yourself. When you run the first line of code, you will see sunflower plot, developed exactly to such “overplotting cases”. It reflects how many points are overlaid. However, it is not easy to make sunflowerplot() show overplotting separately for each species. The other approach, smoothScatter() suffers from the same problem$$^{[1]}$$.)

To overcome this, we developed PPoints() function (Figure $$\PageIndex{1}$$):

Code $$\PageIndex{11}$$ (R):

comp <- read.table("http://ashipunov.info/shipunov/open/compositae.txt", h=TRUE, sep="\t")
c3 <- comp[comp$SPECIES %in% c("Anthemis tinctoria", "Cosmos bipinnatus", "Tripleurospermum inodorum"), ] with(c3, plot(HEAD.D, RAYS, type="n", xlab="Diameter of head, mm", ylab="Number of rays")) with(c3, PPoints(SPECIES, HEAD.D, RAYS, scale=.9)) legend("topright", pch=1:3, col=1:3, legend=levels(c3$SPECIES))

Finally, the answer. As one might see, garden cosmos is really separate from two other species which in turn could be distinguished with some certainty, mostly because number of rays in the yellow chamomile is more than 20. This approach is possible to improve. “Data mining” chapter tells how to do that.

Answer to the matrix question. While creating matrix ma, we defined byrow=TRUE, i.e. indicated that elements should be joined into a matrix row by row. In case of byrow=FALSE (default) we would have obtained the matrix identical to mb:

Code $$\PageIndex{12}$$ (R):

m <- 1:4
ma <- matrix(m, ncol=2, byrow=FALSE)
ma

Answer to the sorting exercise. To work with columns, we have to use square brackets with a comma and place commands to the right:

Code $$\PageIndex{13}$$ (R):

w <- c(69, 68, 93, 87, 59, 82, 72)
x <- c(174, 162, 188, 192, 165, 168, 172.5)
sex <- c("male", "female", "male", "male", "female", "male", "male")
sex.f <- factor(sex)
m.o <- c("L", "S", "XL", "XXL", "S", "M", "L")
d <- data.frame(weight=w, height=x, size=m.o, sex=sex.f)
row.names(d) <- c("Rick", "Amanda", "Peter", "Alex", "Kathryn","Ben", "George")
d.sorted <- d[order(d$sex, d$height), ]
d.sorted[, order(names(d.sorted))]

Please note that we cannot just type order() after the comma. This command returns the new order of columns, thus we gave it our column names (names() returns column names for a given data frame). By the way, sort() would have worked here too, since we only needed to rearrange a single vector.

## References

1. There is also hexbin package which used hexagonal shapes and color shading.


text:
## Experimental Mathematics

### Newton's Formula and the Continued Fraction Expansion of $\sqrt{d}$

Andrej Dujella

#### Abstract

It is known that if the period s(d) of the continued fraction expansion of $\sqrt{d}$ satisfies s(d) ≤ 2, then all Newton's approximants Rn = (1/2)((pn/qn) + (dqn/pn)) are convergents of $\sqrt{d}$, and moreover Rn = p2n+1/q2n+1 for all N ≤ 0. Motivated by this fact we define j = j(d, n) by Rn = p2n+1+2J/q2n+1+2j if Rn is a convergent of $\sqrt{d}$}|. The question is how large |j| and b can be. We prove that |j| is unbounded and gie some examples supporting a conjecture that b is unbounded too. We also discuss the magnitude of |j| and be compared with d and s(d).

#### Article information

Source
Experiment. Math., Volume 10, Issue 1 (2001), 125-131.

Dates
First available in Project Euclid: 30 August 2001

https://projecteuclid.org/euclid.em/999188427

Mathematical Reviews number (MathSciNet)
MR1 822 858

Zentralblatt MATH identifier
0989.11003

#### Citation

Dujella, Andrej. Newton's Formula and the Continued Fraction Expansion of $\sqrt{d}$. Experiment. Math. 10 (2001), no. 1, 125--131. https://projecteuclid.org/euclid.em/999188427


text:
# A group of order $2n$ has an element of order $2$

I'm having problems proving this. The full question is:

"Let $G$ be a group which order is a pair number. Show that $G$ has an element of order $2$".

Can anyone give me a hint?

-
If $x\neq x^{-1}$, remove both $x$ and $x^{-1}$ from the set $G$. Repeat as many times as you can. How many elements remain in the end? What can you say about all but one of them? –  Jyrki Lahtonen Mar 27 '12 at 19:16
Ohh, I think I got this. If it is as stated, then every element $g$ of $G$ is such that $|g|>2$. But $|g|=|g^{-1}|$, and since the neutral element $e$ of $G$ is the only one element with order 1, that leaves an odd number of elements left on the group. If we separate in pairs each element and its inverse, there will be only one element left - and this element "$h$" must be its self-inverse, that is, $h^2=e$, which proves the question. –  Marra Mar 27 '12 at 19:16
It follows from Cauchy's theorem but I like @Balin's answer better. –  MJD Mar 27 '12 at 19:17
@GustavoMarra, not only one element left. See my answer. –  lhf Mar 27 '12 at 19:45
Yeah, not only one element left. But we're supposing that every element of G has order greater than 2 (in what I said). That supposition leads to an absurd. –  Marra Mar 27 '12 at 22:43

If there is no element of order 2 then show that $G$ has odd number of elements. (Hint: think of elements and inverses)

-

Hint $\$ Inversion $\rm\:x\to x^{-1}\:$ is an involution $\rm\: (x^{-1})^{-1} = x,$ so the cycles (orbits) of this permutation partition $\rm\:G\:$ into orbits of length $2$ or $1$. Since $\rm\:|G|\:$ is even so too is the number of length $1$ orbits, i.e. fixed points $\rm\:a = a^{-1};\:$ these include $\rm\:a = 1,\:$ hence, having even cardinality, must include at least one other fixed-point, necessarily of order $2$ by $\rm a^{-1}=a\:$ $\Rightarrow$ $\rm\:a^2 = 1$ but $\rm\:a\ne 1$.

For an analogous application of orbit decomposition (without parity) see my prior answer today on Wilson's theorem for groups.

-

Partition $G = E \cup A \cup B$, where $E=\{e\}$, $A=\{x \in G : x = x^{-1}, x\ne e\}$, $B=\{x \in G : x \ne x^{-1}\}$. Then $E$ has one element and $B$ has an even number of elements because $B$ is invariant under inversion. Since $G$ has an even number of elements, $A$ must have an odd number of elements. In particular, $A$ has at least one element. Every element in $A$ has order 2.

-


text:
# Participation of the ISP NPP specialists in the work of the Commission on CSFSF commissioning

On February 16, 2022, within the work of the Commission on the Centralized spent fuel storage facility commissioning, the State Nuclear Regulatory Inspectorate of Ukraine conducted inspection of systems and elements important for the safety, as well as main technological processes of reloading and transportation of containers with spent nuclear fuel, tested control of the radiation safety and radiation protection, compliance with requirements to the physical protection. Specialists of the Institute for Safety Problems of NPP of the NAS of Ukraine, who conduct scientific and technical maintenance at the stages of construction and commissioning of the CSFSF of VVER type reactors of the domestic NPPs, which is under construction in the Chornobyl exclusion zone, were involved in the work of the Commission.

V. M. Rudko, Head of the Division for Design of Objects with Radiation and Nuclear Technologies, and V. V. Derengovskyi, Head of the Department for NPP Decommissioning, conducted maintenance of work from the ISP NPP.

General view of the sampling stand of exhaust ventilation system

V. V. Derengovskyi conducts the change of the filter AFA-RSP-20

Vertical container handling vehicle deliver the HI STORM 190 container to the storage site

This post is also available in: Ukrainian

## Related posts

Insert math as
$${}$$


text:
# Counting possible committees

Question:

There are $4$ classes in a school, each consisting of $15$ children. A committee consisting of $12$ children shall be chosen. In how many ways can this be done, if it is required that the committee must have at least $1$ child from each class?

I know that the correct way to solve this problem is by using the sieve principle. But my first attempt was the following and I can't figure out why it doesn't work:

$i)$ Pick one child from each class and put them in the committee ($15^4$ ways). This guarantees that there is one student from every class in the committee.

$ii)$ Pick the rest from the remaining 60-4 = 56 children ($\begin{pmatrix}56 \\8\end{pmatrix}$ ways)

So there should be $15^4\begin{pmatrix}56 \\8\end{pmatrix}$ ways to do this. That's the wrong answer afaik, can anyone point out the flaw?

• You are double counting picking $2$ children from the same class. A child picked by $15^4$ is the same as a child picked by $56\choose8$. – Element118 Jan 2 '16 at 14:00
• I suggest to make cases like $(1,1,9,1),(2,2,2,9)..$ and laatly mutiply by $4!$ – Archis Welankar Jan 2 '16 at 14:00
• The possibility that e.g. Arnold is picked out as one of the $4$ in i) and gets company of Betty out of his class is counted more than once. Also Betty can be picked out in the first step to get company of Arnold in the second step. – drhab Jan 2 '16 at 14:01
• You need to divide $15^4$ by $4!$, otherwise you say that $ABCD \neq DABC$ etc. – Wojciech Karwacki Jan 2 '16 at 14:03

Use the principle of inclusion and exclusion. Say a committee has property $i$ if it doesn't have members of class $i$. The number of ways to select the $12$ committee members We want the number of committees without properties.
The number of committees with $r$ properties is just selecting children among $4 - r$ classes, i.e., $\binom{4}{r} \binom{(4 - r) 15}{12}$. By the principle's formula, you are asked for:
$$\sum_{0 \le r \le 4} (-1)^{4 - r} \binom{4}{r} \binom{15 (4 - r)}{12} = 1284837715525$$


text:
# If a certain number is divided by the sum of its two digits, how to find it from quotient and remainder?

If a certain number is divided by the sum of its two digits, the quotient is $7$ and the remainder is $3$. If the digits are reversed and the resulting number is divided by the sum of the digits, the quotient is $3$ and remainder is $7$. Find the number.

My Attempt:

Let the number be $10x+y$.

According to question: $$\dfrac {10x+y}{x+y}=??$$.

• Hint: the first constraint reads $10x+y = 7(x+y) + 3$. Write the second constraint and solve the system. – mlc Mar 31 '17 at 15:32
• Read the problem more carefully and write down the full equation for each of the two conditions. That gives you two equations in two unknowns. – Ross Millikan Mar 31 '17 at 15:40

HINT:$$10x+y=7(x+y)+3=7x+7y+3$$ and $$10y+x=3(x+y)+7=3x+3y+7$$ Can you proceed now?

• How did you make the equations? – pi-π Mar 31 '17 at 15:34
• It's the division algorithm $dividend=quotient\cdot divisor+remainder$ – user35508 Mar 31 '17 at 15:35

To generalize a bit, if the quotient/remainder pairs are $a,b$ and $b,a$ and $a+b=10$, then the two digits of the numbers in the solution are $a$ and $b$.

So in the case of a problem where the quotient is 6 and remainder 4 and vice versa, the answer is $64$ and $46$

The way it is given, it can be translated in the equations as follows:

I am assuming that the orignal number is $10x+y$ and so the reversed number will be $10y+x$.

ATQ:

$$10x+y=7(x+y)+3$$ and $$10y+x=3(x+y)+7$$

You have two equations and two unknowns ($x$ and $y$), solving these two will give you $x=7$ and $y=3$. So the orignal number is $10\times 7+3=73$.


text:
# OpenMath Content Dictionary: SI_functions1

Canonical URL:
http://www.openmath.org/cd/SI_functions1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
SI_functions1.ocd
CD as XML Encoded OpenMath:
SI_functions1.omcd
Defines:
dim, kind, num, unit
Date:
2009-01-10
Version:
1 (Revision 1)
Review Date:
2017-12-31
Status:
experimental

This document is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

The copyright holder grants you permission to redistribute this
document freely as a verbatim copy. Furthermore, the copyright
holder permits you to develop any derived work from this document
provided that the following conditions are met.
a) The derived work acknowledges the fact that it is derived from
this document, and maintains a prominent reference in the
work to the original source.
b) The fact that the derived work is not the original OpenMath
document is stated prominently in the derived work.  Moreover if
both this document and the derived work are Content Dictionaries
then the derived work must include a different CDName element,
chosen so that it cannot be confused with any works adopted by
the OpenMath Society.  In particular, if there is a Content
Dictionary Group whose name is, for example, math' containing
Content Dictionaries named math1', math2' etc., then you should
not name a derived Content Dictionary mathN' where N is an integer.
However you are free to name it private_mathN' or some such.  This
is because the names mathN' may be used by the OpenMath Society
for future extensions.
compilation of derived works, but keep paragraphs a) and b)
intact.  The simplest way to do this is to distribute the derived
work under the OpenMath license, but this is not a requirement.
society at http://www.openmath.org.

Author: Joseph B. Collins (2009), Naval Research Laboratory, Washington, DC.
Copyright Notice:  This is a work of the U.S. Government and is not
may apply.

  Author: J B Collins

This CD defines symbols for functions applied to SI quantities and units.

## dim

Role:
application
Description:

The symbol to represent the function that returns the physical dimension of its argument in terms of products of powers of SI base quantities. The dim operation may be meaningfully applied to SI quantities, SI units, and numbers without physical dimension.

Commented Mathematical property (CMP):
The dim operator acts as the identity operation when applied to an SI base quantity.
Commented Mathematical property (CMP):
The dim operator returns the corresponding SI base quantity when applied to an SI base unit.
Commented Mathematical property (CMP):
For named SI derived quantities and named units, the value returned by the dim operator shall be defined for each case.
Commented Mathematical property (CMP):
The dim operator applied to a product is equal to the associative product of the dim operator applied to the individual factors.
Commented Mathematical property (CMP):
The dim operator applied to a product is equal to the commutative product of the dim operator applied to the factors.
Commented Mathematical property (CMP):
The dim operator applied to a multiplicative inverse of a quantity is equal to the multiplicative inverse of the dim operator applied to the same quantity.
Commented Mathematical property (CMP):
The dim operator returns a value of one when applied to a dimensionless quantity or number.
Signatures:
sts

 [Next: unit] [Last: kind] [Top]

## unit

Role:
application
Description:

The symbol to represent the function that returns the units of its argument in terms of a product of powers of SI base units.

Commented Mathematical property (CMP):
The unit operator may be applied to any physical quantity.
Commented Mathematical property (CMP):
The unit operator applied to an SI base quantity returns the corresponding SI base unit.
Commented Mathematical property (CMP):
The unit operator applied to an SI base unit or SI coherent unit acts as the identity operator, returning that unit.
Commented Mathematical property (CMP):
The unit operator applied to any derived quantity is equal to the unit operator applied to the result of applying the dim operator to the same quantity, i.e., unit(Q) = unit(dim(Q)).
Commented Mathematical property (CMP):
The unit operator applied to a product is equal to the commutative product of the unit operator applied to the factors.
Commented Mathematical property (CMP):
The unit operator applied to a product is equal to the associative product of the unit operator applied to the factors.
Commented Mathematical property (CMP):
The unit operator applied to a multiplicative inverse of a quantity is equal to the multiplicative inverse of the unit operator applied to the same quantity.
Commented Mathematical property (CMP):
The unit operator returns a value of one when applied to a dimensionless quantity or number.
Signatures:
sts

 [Next: num] [Previous: dim] [Top]

## num

Role:
application
Description:

The symbol to represent the function to return the numerical value of a quantity in terms of a product of powers of SI base units.

Commented Mathematical property (CMP):
The num operator may be applied to any physical quantity.
Commented Mathematical property (CMP):
The num operator applied to an SI base quantity or unit returns the value one.
Commented Mathematical property (CMP):
The quantity num(Q)*unit(Q), may replace any quantity, Q, in a set of physical relations, if all such quantities in the set of relations are so replaced. The quantity num(Q)*unit(Q) is not always the same as Q, however dim(Q) = dim(num(Q)*unit(Q)).
Signatures:
sts

 [Next: kind] [Previous: unit] [Top]

## kind

Role:
application
Description:

The symbol to represent the function to return the kind of a quantity. The value of this function is referred to, but not defined in the SI. Its value, kind(Q) for a given quantity, Q, is left to the user to assign.

Signatures:
sts

 [First: dim] [Previous: num] [Top]


text:
# Balancing Area

The term Balancing Area refers to an area in a grid system over which supply and demand must be balanced at all time. It’s important to note that this doesn’t mean all that supply (i.e. power plants) and demand (end-use) must be physically in the balancing area, since imports or exports of electricity through interconnectors (to other balancing areas) provide another mechanism of achieving balance.

Larger balancing areas tend to have the advantage of smoothing variability in both supply and demand. This can be an important issue for renewable power sources for example: variability of supply can be large in a small geographic region (e.g. when the sun goes behind cloud or the wind drops) but is likely to be less over a larger area (on a country scale, it may be cloudy/still in one place but is likely to be sunny/windy elsewhere, if you go far enough away).


text:
# Lessons in Estimation Theory for Signal Processing, Communications, and Control

## byJerry M. Mendel

not yet rated|write a review
Estimation theory is a product of need and technology. As a result, it is an integral part of many branches of science and engineering. To help readers differentiate among the rich collection of estimation methods and algorithms, this book describes in detail many of the important estimation methods and shows how they are interrelated. Written as a collection of lessons, this book introduces readers o the general field of estimation theory and includes abundant supplementary material.

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From the Publisher

Estimation theory is a product of need and technology. As a result, it is an integral part of many branches of science and engineering. To help readers differentiate among the rich collection of estimation methods and algorithms, this book describes in detail many of the important estimation methods and shows how they are interrelated....

From the Jacket

Estimation theory is a product of need and technology. As a result, it is an integral part of many branches of science and engineering. To help readers differentiate among the rich collection of estimation methods and algorithms, this book describes in detail many of the important estimation methods and shows how they are interrelated....

Format:PaperbackDimensions:592 pages, 9.1 × 6.9 × 1.3 inPublished:March 14, 1995Publisher:Pearson Education

The following ISBNs are associated with this title:

ISBN - 10:0131209817

ISBN - 13:9780131209817

## Reviews

Extra Content

Estimation theory is widely used in many branches of science and engineering. No doubt, one could trace its origin back to ancient times, but Karl Friederich Gauss is generally acknowledged to be the progenitor of what we now refer to as estimation theory. R. A. Fisher, Norbert Wiener, Rudolph E. Kalman, and scores of others have expanded upon Gauss's legacy and have given us a rich collection of estimation methods and algorithms from which to choose. This book describes many of the important estimation methods and shows how they are interrelated. Estimation theory is a product of need and technology. Gauss, for example, needed to predict the motions of planets and comets from telescopic measurements. This need'' led to the method of least squares. Digital computer technology has revolutionized our lives. It created the need for recursive estimation algorithms, one of the most important ones being the Kalman filter. Because of the importance of digital technology, this book presents estimation from a discrete-time viewpoint. In fact, it is this author's viewpoint that estimation theory is a natural adjunct to classical digital signal processing. It produces time-varying digital filter designs that operate on random data in an optimal manner. Although this book is entitled Estimation Theory,' computation is essential in order to be able to use its many estimation algorithms. Consequently, computation is an integral part of this book. It is this author's viewpoint that, whenever possible, computation should be left to the experts. Consequently, I have linked computation into MATLAB\registered\ (MATLAB is a registered trademark of The MathWorks, Inc.) and its associated toolboxes. A small number of important estimation M-files, which do not presently appear in any MathWorks toolbox, have been included in this book; they can be found in Appendix B. This book has been written as a collection of lessons. It is meant to be an introduction to the general field of estimation theory and, as such, is not encyclopedic in content or in references. The supplementary material, which has been included at the end of many lessons, provides additional breadth or depth to those lessons. This book can be used for self-study or in a one-semester course. Each lesson begins with a summary that describes the main points of the lesson and also lets the reader know exactly what he or she will be able to do as a result of completing the lesson. Each lesson also includes a small collection of multiple-choice summary questions, which are meant to test the reader on whether or not he or she has grasped the lesson's key points. Many of the lessons include a section entitledr Computation.'' When I decided to include material about computation, it was not clear to me whether such material should be collected together in one place, say at the rear of the book in an appendix, or whether it should appear at the end of each lesson, on demand so to speak. I sent letters to more than 50 colleagues and former students asking them what their preference would be. The overwhelming majority recommended having discussions about computation at the end of each lesson. I would like to thank the following for helping me to make this decision: Chong-Yung Chi, Keith Chugg, Georgios B. Giannakis, John Goutsias, Ioannis Katsavounidis, Bart Kosko, Li-Chien Lin, David Long, George Papavassilopoulos, Michael Safonov, Mostafa Shiva, Robert Scholtz, Ananthram Swami, Charles Weber, and Lloyd Welch. Approximately one-half of the book is devoted to parameter estimation and the other half to state estimation. For many years there has been a tendency to treat state estimation, especially Kalman filtering, as a stand-alone subject and even to treat parameter estimation as a special case of state estimation. Historically, this is incorrect. In the musical Fiddler on the Roof, Tevye argues on behalf of Tradition!'' Estimation theory also has its tradition, and it begins with Gauss and parameter estimation. In Lesson 2 we show that state estimation is a special case of parameter estimation; i.e., it is the problem of estimating random parameters when these parameters change from one time instant to the next. Consequently, the subject of state estimation flows quite naturally from the subject of parameter estimation. There are four supplemental lessons. Lesson A is on sufficient statistics and statistical estimation of parameters and has been written by Professor Rama Chellappa. Lessons B and C are on higher-order statistics. These three lessons are on parameter estimation topics. Lesson D is a review of state-variable models. It has been included because I have found that some people who take a course on estimation theory are not as well versed as they need to be about state-variable models in order to understand state estimation. This book is an outgrowth of a one-semester course on estimation theory taught at the University of Southern California since 1978, where we cover all its contents at the rate of two lessons a week. We have been doing this since 1978. I wish to thank Mostafa Shiva, Alan Laub, George Papavassilopoulos, and Rama Chellappa for encouraging me to convert the course lecture notes into a book. The result was the first version of this book, which was published in 1987 as {\it Lessons in Digital Estimation Theory}. Since that time the course has been taught many times and additional materials have been included. Very little has been deleted. The result is this new edition. Most of the book's important results are summarized in theorems and corollaries. In order to guide the reader to these results, they have been summarized for easy reference in Appendix A. Problems are included for all the lessons (except Lesson 1, which is the Introduction), because this is a textbook. The problems fall into three groups. The first group contains problems that ask the reader to fill in details, which have been left to the reader as an exercise.'' The second group contains problems that are related to the material in the lesson. They range from theoretical to easy computational problems, easy in the sense that the computations can be carried out by hand. The third group contains computational problems that can only be carried out using a computer. Many of the problems were developed by students in my Fall 1991 and Spring 1992 classes at USC on Estimation Theory. For these problems, the name(s) of the problem developer(s) appears in parentheses at the beginning of each problem. The author wishes to thank all the problem developers. While writing the first edition of the book, the author had the benefit of comments and suggestions from many of his colleagues and students. I especially want to acknowledge the help of Georgios B. Giannakis, Guan-Zhong Dai, Chong-Yung Chi, Phil Burns, Youngby Kim, Chung-Chin Lu, and Tom Hebert. While writing the second edition of the book, the author had the benefit of comments and suggestions from Georgios B. Giannakis, Mithat C. Dogan, Don Specht, Tom Hebert, Ted Harris, and Egemen Gonen. Special thanks to Mitsuru Nakamura for writing the estimation algorithm M-files that appear in Appendix B; to Ananthram Swami for generating Figures B 4, B 5, and B 7; and to Gent Paparisto for helping with the editing of the galley proofs. Additionally, the author wishes to thank Marcel Dekker, Inc., for permitting him to include material from J. M. Mendel, Discrete Techniques of Parameter Estimation: The Equation Error Formulation}, 1973, in Lessons 1--3, 5--9, 11, 18, and 23; Academic Press, Inc., for permitting him to include material from J. M. Mendel, Optimal Seismic Deconvolution: An Estimation-based Approach}, copyright\,\copyright\,1983 by Academic Press, Inc., in Lessons 11--17, 19--21, and 25; and the Institute of Electrical and Electronic Engineers (IEEE) for permitting him to include material from J. M. Mendel, Kalman Filtering and Other Digital Estimation Techniques: Study Guide}, copyright, 1987 IEEE, in Lessons 1--3, 5--26, and D. I hope that the readers do not find it too distracting when I reference myself for an item such as a proof (e.g., the proof of Theorem 17-1). This is done only when I have taken material from one of my former publications (e.g., any one of the preceding three), to comply with copyright law, and is in no way meant to imply that a particular result is necessarily my own. I am very grateful to my editor Karen Gettman and to Jane Bonnell and other staff members at Prentice Hall for their help in the production of this book. Finally, I want to thank my wife, Letty, to whom this book is dedicated, for providing me, for more than 30 years, with a wonderful environment that has made this book possible.

1. Introduction, Coverage, Philosophy, and Computation.

2. The Linear Model.

3. Least-Squares Estimation: Batch Processing.

4. Least-Squares Estimation: Singular-Value Decomposition.

5. Least-Squares Estimation: Recursive Processing.

6. Small Sample Properties of Estimators.

7. Large Sample Properties of Estimators.

8. Properties of Least-Squares Estimators.

9. Best Linear Unbiased Estimation.

10. Likelihood.

11. Maximum-Likelihood Estimation.

12. Multivariate Gaussian Random Variables.

13. Mean-Squared Estimation of Random Parameters.

14. Maximum A Posteriori Estimation of Random Parameters.

15. Elements of Discrete-Time Gauss-Markov Random Sequences.

16. State Estimation: Prediction.

17. State Estimation: Filtering (The Kalman Filter).

18. State Estimation: Filtering Examples.

19. State Estimation: Steady-State Kalman Filter and Its Relationships to a Digital Wiener Filter.

20. State Estimation: Smoothing.

21. State Estimation: Smoothing (General Results).

22. State Estimation for the Not-So-Basic State-Variable Model.

23. Linearization and Discretization of Nonlinear Systems.

24. Iterated Least Squares and Extended Kalman Filtering.

25. Maximum-Likelihood State and Parameter Estimation.

26. Kalman-Bucy Filtering.

A. Sufficient Statistics and Statistical Estimation of Parameters.

B. Introduction to Higher-Order Statistics.

C. Estimation and Applications of Higher-Order Statistics.

D. Introduction to State-Variable Models and Methods.

Appendix A: Glossary of Major Results.

Appendix B: Estimation of Algorithm M-Files.

References.

Index.


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# zbMATH — the first resource for mathematics

Residually finite groups with the same finite images. (English) Zbl 0309.20007

##### MSC:
 20D15 Finite nilpotent groups, $$p$$-groups 20K35 Extensions of abelian groups
Full Text:
##### References:
 [1] R.C. Brigham : On the isomorphism problem for just-infinite groups . Comm. Pure and Applied Math. XXIV. (1971) 789-796. · Zbl 0217.07703 [2] L.L. Dyer : On the isomorphism problem for polycyclic groups . Math. Zeitsch. 112 (1969) 145-153. · Zbl 0186.31901 [3] G.H. Hardy and E.M. Wright : An introduction to the theory of numbers . Clarendon Press, Oxford. · Zbl 0020.29201 [4] G. Higman : Unpublished. [5] K.A. Hirsch : On infinite solvable groups. II . Proc. London Math. Soc. (2) 44 (1938) 336-344. · Zbl 0019.15602 [6] R. Hirshon : On cancellation in groups . American Math. Monthly 76 (1969), 1037-1039. · Zbl 0186.03801 [7] H. Neumann : Varieties of groups , Ergebnisse der Math. und ihrer Grenzgebiete, Band 37. Springer-Verlag, New York, 1967, MR. 35 # 6734. · Zbl 0251.20001 [8] R. Remak : Über die Zerlegung der endlichen Gruppen in direkte unzerlegbare Faktoren . J. reine angew. Math. 139 (1911) 293-308. · JFM 42.0156.01 [9] V.N. Remeslennikov : Conjugacy of subgroups in nilpotent groups . Algebra i Logika Sem. 6 (1967) no. 2 61-76. (Russian) MR. 36 # 1545. · Zbl 0189.30902 [10] E.A. Walker : Cancellation in direct sums of groups . Contributions to the theory of groups (1956) 101-129. University of Kansas. · Zbl 0071.25203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# TimeReverse¶

class acoular.tprocess.TimeReverse

Calculates the time-reversed signal of a source.

result(num)

Python generator that yields the output block-wise.

Parameters
numinteger

This parameter defines the size of the blocks to be yielded (i.e. the number of samples per block).

Returns
Yields samples in blocks of shape (num, numchannels).

Time-reversed output of source. The last block may be shorter than num.