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# How is this expression for a GHZ state obtained in the nature paper by Pan et al. (2000)?

Can someone tell me how the authors of the paper "Experimental test of quantum nonlocality" (Nature link to abstract) have rewritten their equation 1 in terms of equation 2 and 3?

• web.physics.ucsb.edu/~quopt/ghz.pdf Jun 19 at 19:54
• To clarify: are you asking how $(4)$ comes from $(1)$, $(2)$ and $(3)$? If so, then you can probably just substitute inverted $(2)$ and inverted $(3)$ into $(1)$... If not, then could you clarify what the question is? Jun 19 at 22:02
• Yes, that was my question. How can I substitute into 1 when equations 2 and 3 are V' and H' ??? Jun 19 at 22:05
• Do you mean expressing H and V in terms of H' and V' from 2 and 3 and then inserting them into 1?? Jun 19 at 22:07

First, in order to express $$|H'\rangle$$, $$|V'\rangle$$, $$|R\rangle$$ and $$|L\rangle$$ in terms of $$|H\rangle$$ and $$|V\rangle$$, add and subtract the pair of equations $$(2)$$ and add and subtract the pair $$(3)$$, to get

$$|H'\rangle + |V'\rangle = \sqrt2|H\rangle \\ |H'\rangle - |V'\rangle = \sqrt2|V\rangle \\ |R\rangle + |L\rangle = \sqrt2|H\rangle \\ |R\rangle - |L\rangle = i\sqrt2|V\rangle$$

which means that

$$|H\rangle = \frac{1}{\sqrt{2}}\left(|H'\rangle + |V'\rangle\right) = \frac{1}{\sqrt{2}}\left(|R\rangle + |L\rangle\right) \\ |V\rangle = \frac{1}{\sqrt{2}}\left(|H'\rangle - |V'\rangle\right) = \frac{1}{i\sqrt{2}}\left(|R\rangle - |L\rangle\right).$$

Substituting the last equations into $$(1)$$ according to the choices of polarization in text, we obtain

\begin{align} |\Psi\rangle &= \frac{|H\rangle|H\rangle|H\rangle + |V\rangle|V\rangle|V\rangle}{\sqrt{2}} \\ & = \frac{(|R\rangle + |L\rangle)(|R\rangle + |L\rangle)(|H'\rangle + |V'\rangle)-(|R\rangle - |L\rangle)(|R\rangle - |L\rangle)(|H'\rangle - |V'\rangle)}{4}. \end{align}

Notice that the only terms that do not cancel are those with odd number of kets in $$\{|L\rangle,|V'\rangle\}$$, i.e.

$$|\Psi\rangle = \frac{1}{2}\left(|L\rangle|R\rangle|H'\rangle + |R\rangle|L\rangle|H'\rangle + |R\rangle|R\rangle|V'\rangle + |L\rangle|L\rangle|V'\rangle\right)$$

in agreement with $$(4)$$.

• Thanks a lot!!!! Jun 20 at 8:58
• Hi @HaxhiPantina and welcome to QCSE! If an answer solved your doubts, consider accepting it by clicking the check mark. This helps keep focus on questions that haven’t been solved. Jun 20 at 12:34


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maths > fractions

Reciprocal of a fraction

what you'll learn...

overview

"Reciprocal" or "multiplicative inverse" of a fraction is another fraction that results in $1$$1$ when the two fractions are multiplied.

invert to reverse

The word "reciprocal" means: backwards; in reverse.

Consider the multiplication $4×3$$4 \times 3$

•  $4$$4$ is the multiplicand

•  $3$$3$ is the multiplier

•  $12$$12$ is the product

The multiplier has modified the multiplicand.
How to reverse or take backward the multiplication?
The reverse of the multiplier $3$$3$ is the reciprocal $\frac{1}{3}$$\frac{1}{3}$
Because the product $12$$12$, when multiplied by the reciprocal $\frac{1}{3}$$\frac{1}{3}$ gives the multiplicand $4$$4$.

Reciprocal is also called the multiplicative inverse.

Consider the multiplication $\frac{2}{3}×\frac{5}{8}=\frac{5}{12}$$\frac{2}{3} \times \frac{5}{8} = \frac{5}{12}$

•  $\frac{2}{3}$$\frac{2}{3}$ is the multiplicand

•  $\frac{5}{8}$$\frac{5}{8}$ is the multiplier

•  $\frac{5}{12}$$\frac{5}{12}$ is the product

The reverse of the multiplier $\frac{5}{8}$$\frac{5}{8}$ is the reciprocal $\frac{8}{5}$$\frac{8}{5}$
The product $\frac{5}{12}$$\frac{5}{12}$, when multiplied by the reciprocal $\frac{8}{5}$$\frac{8}{5}$ gives the multiplicand $\frac{2}{3}$$\frac{2}{3}$.

Reciprocal is also called the multiplicative inverse.

"Reciprocal" or "multiplicative inverse" of a fraction is the fraction that inverses the number to a product result $1$$1$.

Reciprocal of a Fraction: For a fraction $\frac{p}{q}$$\frac{p}{q}$, the "Reciprocal" or "multiplicative inverse" is $\frac{q}{p}$$\frac{q}{p}$ as $\frac{p}{q}×\frac{q}{p}=1$$\frac{p}{q} \times \frac{q}{p} = 1$.

examples

What is the reciprocal of $\frac{14}{18}$$\frac{14}{18}$?
The answer is '$\frac{18}{14}$$\frac{18}{14}$'

What is the reciprocal of $1\frac{3}{4}$$1 \frac{3}{4}$
The answer is '$\frac{4}{7}$$\frac{4}{7}$'. The mixed fraction $1\frac{3}{4}$$1 \frac{3}{4}$has to be converted to equivalent improper fraction $\frac{7}{4}$$\frac{7}{4}$ before finding the reciprocal $\frac{4}{7}$$\frac{4}{7}$

summary

»  Multiplicative inverse
reciprocal means "backwards; in reverse"
→  reciprocal of $\frac{a}{b}$$\frac{a}{b}$ is $\frac{b}{a}$$\frac{b}{a}$

Outline

format: plain


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Bear plus snowflake equals polar bear 384 points by soopurman 45 days ago | hide | past | favorite | 122 comments

 This is definitely an example where "character" was the wrong word. We start out OK with bytes being distinct, but by the end we're talking about how a character is made out of several characters. If we think in terms of code points (Rust's native char type actually provides here a slightly different thing, a Unicode Scalar Value, which certain types of code point are not, but close enough) clearly a code point isn't made out of several code points, so we needed a different word.I like squiggle, if you're a text rendering engine you might want to use "glyph" although you might already need that word for something else. But try to avoid character because that word already has far too many meanings, most of which won't be what you wanted.
 I think the term you are looking for is grapheme cluster.
 We're so lazy that we can't even be bothered to type "character" half the time, so "grapheme cluster" has no chance to catch on unless we can think of a "char"-style abbreviation. I'll start the bikeshed with "gracl", but ambitious folks may wish to argue for "clog" ("cluster of graphemes").
 How about graph? One character longer than char and probably not already in use in the area of computer science.
 Is this sarcasm?
 Almost certainly a joke.
 I laughed for a good two minutes. Still laughing cause I can't tell whether he's talking about nodes or vertices
 Node because that's not a major software term yet.
 It's really an Extended Grapheme Cluster. In our codebase (Darklang) we abbreviate it to EGC.
 Elixir uses the term grapheme already:
 Is that what Go uses "rune" for?
 No. A "rune" is one codepoint, Golang just decided that they can't use the same name as everyone else (as is tradition in Go).
 No, because a grapheme cluster is à where the letter is combined by a and  which are two Unicode Scalar Values.A Go rune is more like one Unicode Scalar Value.However, be careful, terminology is subtle: what is a Code Point, what a Scalar Value, etc.
 clugr 'sounds' better than clog though (unless you want to evoke pipe somehow)...clugr evokes more Freddy to me
 character…grapheme cluster…Charster!
 Unicode uses the term grapheme cluster to emphasize the fact that it can be composed of a cluster of codepoints, and perhaps also to distinguish it from the linguistic term grapheme. But ultimately in the context of Unicode I think you could just get away with using grapheme. That's what Perl did when it invented Normalization Form Grapheme (NFG).
 You can refer to the definitions of these terms. [1] It's best not to reinvent how these terms are used. There has been far too much of that, already.PS Be prepared for a long read.
 > I like squiggleGo uses "Runes" which is a pretty unambiguous and memorable term.Though, in Go's case they don't include use of ZWJ, as they're limited to 32-bits.
 It's somewhat ambiguous because Unicode has a code block for actual runes which are not the same as the "runes" you're talking about
 I believe this name is inherited from Plan 9 (which also happens to be the origin of UTF-8).
 A Go "rune" is not the same as a "single rendered/visible character" though, a grapheme cluster/glyph/squiggle is from what I understand.
 I think, zwjchar has a chance: it is unambiguous, so when in future standard evolves we could distinguish it. Besides, technical people like precision and cryptic names.
 This reminds me of a blog post that I can no longer remember that discusses how Chinese is a lot of arranging several related kanji into a single character to express a new idea (please link it if you can find it).I would love to see unicode characters to allow for arbitrary combinations beyond those defined using just ZWJ to allow more flexibility (e.g. blizzard could be created by adding like "slowflake x 5" which creates a single character with five snowflakes, without having to create an entirely new character representing blizzard from snowflake + ZWJ + snowflake).As an aside, my favorite ZWJ magic is black flag + ZWJ + skull and crossbones = pirate flag.Also see https://en.wikipedia.org/wiki/Blissymbols for more symbol language fun.
 The notion that Chinese characters describe ideas that are formed from smaller characters that also describe ideas is a nice story, but it's not really the case for the majority of Chinese characters. See the Wikipedia page, Principles of formation section[0].Many Chinese characters are constructed with two parts of smaller characters, one of which can indicate a general concept while the other provides a vague pronunciation hint. This isn't really good enough to provide you with enough information to guess at either the meaning or the pronunciation if you don't already know the word, but if you do already know the word from spoken language, then the hints might be enough to recognize the character when you read it too.
 If you'd like to decompose Chinese characters (e.g. if you're trying to use handwriting recognition and it's /almost/ right), try the Pingtype keyboard. Enter the character into the lower text box, and click Search. It'll decompose, and recompose characters based on the parts. I agree that the meaning is often hard to grasp from the glyphs, and can even be misleading.
 Right, but there are still some nice examples, like:木 = tree森林 = forestAnd:田 = field力 = strength男 = man (because he labours in the fields)
 This breaks down very quickly as you get more abstract.人 = person "rén"门 = gate "mén"们 = plurality marker "mėn"也 = also "yé"他 = he "tā"他们 = them "tāmėn"One can come up with a story for why person + gate is plural, but the inclusion of gate in the character for plurality is really more about the sound of gate.
 This is even more interesting because prior to the 1920s 他 was a gender neutral third person pronoun. And in ancient Chinese 也 was probably pronounced differently to today's yě, perhaps more like the ā of tā. A great resource for looking up the history of individual characters is Wiktionary:https://en.wiktionary.org/wiki/%E4%BB%96https://en.wiktionary.org/wiki/%E4%B9%9FWiktionary is neat because it also shows you pronunciations in several different Chinese languages (Mandarin, Cantonese, Minnan, Hakka etc) and the relationship to Japanese, Korean and Vietnamese, if it's there.
 I think 他 is still gender neural, I put down "he" out of laziness, since the feminine-only form is 她... but you make good points.
 Yes totally. I wasn't contradicting GP. The pattern where part of the character indicates the sound is so common that it helps me a great deal when reading Chinese. My speaking/hearing is better than my reading. I might not know how to write a particular character but, if I can guess the sound, then that plus the surrounding context are enough for me to know what the word is.
 Yes! Thanks for the link :D
 For the next Unicode emoji extension, we might as well just take a word vector model and PCA project it into the Unicode 1 dimensional space. Somebody who is better at linear algebra should comment and tell me how wrong I am.
 SignWriting, a writing system for Signed Languages, would need something like this, because rotation and location of the graphemes matter. Imagine the letter a being rotated by 45 degrees and moved to the upper left corner of the drawing area. How would you encode this?There's a symbol for the torso, for the fist, for touch and for the strongness of touch. If you hit your own shoulder with a fist with the knuckles facing upwards, you rotate the symbol for the fist, paint the torso and the fist next to each other and put the touch symbol approximately where the shoulder would be.
 It's brilliant, but horrible, since you need somewhere between 300 and 1000 dimensions of floating point word vector semantic space to get good separation. Maybe it's shrunk since I last checked.Even with less granularity, that's still like 256 bytes per grapheme.I think this would be great for machine translation though.
 It's also similar to forming words from multiple kanji. E.g. like Chinese "penguin" is combined from "business" and "goose" (not completely accurate translation).Imagine poor kids 100 years from now having to memorize all those emojis and their combinations at school. Or having to know 3000 basic emojis in order to be able to read news.
 >Imagine poor kids 100 years from now having to memorize all those emojis and their combinations at school.I mean, we all know what a "polar bear" is, right? If those kids know how to say "penguin", and they know how to write the parts, then they know how to write "penguin" in Chinese.Meanwhile, the parts in "penguin" aren't used in any other word. The "pen" isn't a writing utensil. And what's a "guin"? If you didn't learn how to spell beforehand and were suddenly asked to write "penguin", maybe you'd write "pengwin" or "pengwen", or mishear it as "pengwing" (because birds have wings so it makes sense, right?) English writing is brute-force memorization of spelling with some patterns that often don't hold, just like Chinese is brute force memorization with some patterns regarding sound or meaning that often don't hold.And we can see that all English speakers still struggle to spell some words that they don't encounter often, and some even struggle to spell words they do use often.
 No, the "pen" in "penguin" means "head" - it's from Welsh. English spelling is a lot easier if you know the languages it was derived from or influenced by.An advantage of non-phonetic spelling is that it doesn't privilege any one accent over any other, so allowing a polycentric acrolect with each variety picking up accent and vocabulary from the local dialect, but maintaining mutual comprehension in writing.
 >English spelling is a lot easier if you know the languages it was derived from or influenced by.Knowing a bunch of languages in order to have a proper context for English spelling is surely a higher threshold than memorizing the characters in your own language.
 It’s still debated if that is the correct etymology. A different hypothesis has “penguin” deriving from the Latin “pinguis” (fat).
 Certainly in Czech the two are related:tučňák = noun, penguintučný = adjective, fatI don't know if that helps determine the origin of the English word, but it's definitely funny to think that there’s a nation which calls penguins “fatties” :D
 The Chinese character "企" has multiple meanings. One is "to stand on tiptoe".[1] So "企鹅 (penguin)" means "standing goose".
 The character 企 also looks like a penguin by itself.
 I wish it was that simple. Though some characters are, the majority have evolved through usage along with pronunciation. This makes similar characters mean very different things and are pronounced differently. Mastering Chinese is about memorizing the 2000-3000 most used characters.
 This reminds me of a game I made! Matching emojis to produce the given emoji prompt. I didn't use any complex math tho, and my list of combinations is arbitrarily made by hand.
 Haha, juice box = apple + hammer.dang, it would be cool if mods could mark a post as "allow emoji here".
 I think the HN codebase is mostly frozen.
 Dang has alluded it's being worked on so long threads wont need to be paginated anymore.
 Only tangentially related, but it's a question that have bothered me for a while: Is there a reason why composite Math symbol aren't part of Unicode? Things like series, limits or integrals look like a really good fit for this kind of composition, and you could get rid of mathjax for 80% of its usage.
 I gotta think it's because all the math you mention cannot render within a normal line height. That's not the end of the world for rendering, but maybe that counts as a bridge too far in Unicode? Or maybe it just boils down to "because it doesn't yet" and as soon as someone makes a good proposal it'll happen
 Unicode already has a bunch of things that extend the height like those "corrupted text" generators.
 Those are a ton of combining accents put together. Crucially: they don't make the line any taller—that's what makes them so hard to read! An integral should be a little taller than the line height but you still want the integral's limits to be legible. Maybe I'm being too prescriptive; ∫₀¹𝑥 𝑑𝑥 would read ok if that 1 were stacked on top of the 0 without any other changes.My apologies if HN's formatter strips the special characters (edit: it didn't yay!): I wrote integral sign, subscript 0, superscript 1, math italic x, thin space, math italic d, math italic x. And maybe the idea is we should use words like "the integral from zero to one of x with respect to x." IDK.
 There is a proposal by the Julia language team
 Everybody, if you're a somebody in your field, please go sign this.
 I guess the question is how much typesetting do you want to stick into unicode? It works OK for emoji because the results are always simple: it's just another individual emoji, with the same size & characteristics as the original.That said unicode is not free from typesetting weirdness, see the character ﷽.
 consider the following:𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫𒐫
 As noted, it's a bit more complicated than just putting things above or below a symbol. Also, if you look closer at the output from MathJax, you'll notice things like the setting of x+y=z has slightly different spacing around the + than the =. For that matter, properly typeset mathematics will also the spaces around the colons differently in, e.g., f: X → { y ∊ *R* : |y| < 1 }. Math typesetting is not a simple matter. I'd also note that the existence of, e.g., ² or ₆ in Unicode is more a result of allowing the mapping of legacy encodings and that there is a preference in Unicode that in general, superscripts and subscripts not be handled through encodings but rather through the layout engine of the application which is why there is no superscripted decimal point or other such characters.
 1) Combining emoji compose into a finite and reasonably small set of combined symbols. The set of mathematical formulas isn't finite in the same kind of way.2) Layout of mathematical formulas is reasonably complicated. It doesn't make sense to force that complexity to be included in every text layout engine.
 There is UnicodeMath[1], which was developed by Microsoft and is the default representation used by the Word equation editor.It looks like this:e=lim┬(n→∞)(1+1/n)^nf(x)=∑_(k=-∞)^∞〖c_n e^(ⅈkx)〗∫_(-∞)^(+∞)〖exp(-a/2 x^2) ⅆx〗=√(2π/a)I find it quite readable, even for quite complicated formulae like the above. You can also replace the unicode symbols with Latex-style escape strings, like \sum or \below.
 Thank \$diety for emojis making sure most user facing code ends up handling extended code points correctly, and teaching devs about the multibyte encodings.I imagine in a world without them existing a lot of the non-ascii paths would not be regularly used.
 It is so stupid that Emojis render differently on different platforms. In particular there are subtle differences among the facial expression emojis that make them pretty useless if you’re not sure the recipient is on the same platform that you’re choosing one on.
 It's a reminder that these are glyphs, on the same plane as words, aimed at portability.I think what you are looking for is stamps and png/gifs, which are also supported in most relevant chat platforms these days.
 An encoding for words that allowed rendering as different words on different platforms would also be stupid. MacOS: grimace Windows: frown Android: scowl Samsung: growl Firefox: angry face ` That’s pretty much what the emoji situation is today. I don’t understand what the actual use case is for the requirement that emojis differ across platforms.Yes, I realize they are widely used, but they are widely used despite this stupidity, not because of it.
 Do you have any examples of current emojis that clearly have a very different meaning on different platforms? What you're saying might have been true 5+ years ago, but over time emojis have become more and more similar.Here is just one random article from Emojipedia about the history of the "folded hands" emoji: https://blog.emojipedia.org/emojiology-folded-hands/ There are many more examples.In addition to that, almost all emoji keyboards now autocomplete the emoji based on standard names, so if you search for "disappointed" on most any emoji keyboard, you will get the same face.For reference, here is the current official emoji set, including the standard names and images showing how they render on different platforms: https://unicode.org/emoji/charts/full-emoji-list.html
 The famous example is the pistol emoji, which is rendered as either a real gun or a water gun depending on platform. Lots of scope for misunderstanding there!
 This has not been true since early 2018. See: https://blog.emojipedia.org/all-major-vendors-commit-to-gun-... There was only a couple of years when vendors rendered it differently, and they worked toward a consensus, as they have for many other emojis.Even if there was no vendor consensus, I'm not sure what the misunderstanding could be with this particular emoji. There is only one pistol emoji, and regardless of whether it is rendered as a water pistol or a revolver, it still is used to represent the concept of a pistol. There are better examples of emojis that used to be displayed with a facial expression or hand gesture that had a relatively different meaning depending on the platform. For example face with rolling eyes, person tipping hand (information desk person) and so on.
 It still rendered as a revolver on the phone I used to write that comment. Guess I'm still on 2017 Android. Please do not invite me to any water gun parties.
 > Do you have any examples of current emojis that clearly have a very different meaning on different platforms? What you're saying might have been true 5+ years ago, but over time emojis have become more and more similar.Well... yes? That's because avalys is obviously correct. The "solution" defined by the Unicode consortium is so spectacularly stupid that everyone has unanimously agreed to move away from it by synchronizing their images, because it makes no sense to send an image unless you know what it will look like.
 Rendering differently on an older platform is also an important difference (consider those who can't afford the latest phones). E.g. the gun one has a very different meaning on (some) older platforms from newer ones.
 I think this is basically what's happening on the JCK plane with glyphs that have the same code, but slightly different representations depending on the font (e.g. Chinese or Japanese), with potentially wildly different meanings in the language they are used.To your point, this is an aspect where integrating and regularily adding emojis to unicode pushed a very technical system under popular attention, and attracted a lot of people from unrelated backgrounds who now have to understand how it works and what its goals in the first place.
 > I think this is basically what's happening on the JCK plane with glyphs that have the same code, but slightly different representations depending on the font (e.g. Chinese or Japanese), with potentially wildly different meanings in the language they are used.It was even more dumb when they did it for CJK. Unicode is now neither fish nor fowl; you can't rely on things to look the same in different places, but you can't rely on them to mean the same either; there's no proper separation of concerns because they decided it's fine for the meaning of a character to change based on the font you're using.
 Yes, it's not an ideal solution on many respects.I kind of get why we got here, as encodings were a quagmire for a long time, and coming to a real clean, everybody's happy solution with unicode looked completely unrealistic.We now have way better compatibility, got to settle for one encoding in most cases, and the annoyances are for now somewhat manageable (if/when China opens more to the global net, it might be a different story). Installing fonts is still easier than adding encoding compatibilities.
 On a serious note: Yes, I agree. On a comical note: Reminds me of the great "Cheeseburger-gate" scandal of 2017, in which Google put the cheese under the burger patty instead of on top of. https://www.theverge.com/2017/10/30/16569346/burgergate-emoj...
 Problem is that each tech company uses a proprietary set of emoji so they couldn't standardize even if they wanted. Many instant messaging apps work around this by using their own emoji set for all users but this is also often annoying especially when the emoji are ugly.
 The solution is to allow users to use their own svg libraries when sending messages, and the ability to manage these libraries by copying inline symbols from messages sent by others.This would promote an open culture of creating inline symbols. E.g. where is our "covid" symbol, and why do we have to wait for the Unicode consortium to define it for us, and app makers to implement it??
 I like telegram's sticker system best. User defined images which you can organize and store in the UI for easy access.
 Please not. That would be the tower to Babel. Afterwards we are all just going bla bla and don't understand each other at all, because everyone invents its own funny secret emoji language.
 > Afterwards we are all just going bla bla and don't understand each other at all, because everyone invents its own funny secret emoji language.Just like we all go "bla bla" on the phone because we can choose our own sounds?
 Yes, we could have wide support for inline svg by now.
 Relatedly, I’ve always rather enjoyed the talk “Love Hotels & Unicode”[1]
 If you want to see how an emoji will render on various platforms, or generally search what's available, Emojipedia is good for that: https://emojipedia.org/kiss-woman-woman-medium-dark-skin-ton...
 Is there a good reason these composite emojis aren't all single code points? They have to at least mostly be separate images to be rendered, so it's not like it's an impossible combinatorial explosion. I remember there are only on the order of 1000 valid combinations.
 I would assume for backward compatibility. If you don’t have the font with the updated emojis you can still get the reference.
 There was a flap when people realized that you can combine the zero-width joiner with the "Prohibited" symbol to put that on top of anything you don't like.
 These kind of remind me of ligatures. Not exactly the same since (as far as I'm aware) ligatures are always based on a direct blending/combination of the two character's shapes, but still similar.
 I think programming ligatures kind of break this assumption - particularly != becoming the not equals sign.
 I'm working on a game for younger kids using pictures/emojis for "universal" conversation rather than text, so this gave me some good ideas. Thanks!
 I've had success playing Concept with my 5yo. https://boardgamegeek.com/boardgame/147151/concept
 Seconding the suggestion, concept is a great game for people of all ages learning a language. It's also a good game overall :)
 Yes, I should note that I have also had substantial success playing Concept with friends, other family, and co-workers.Although we've never actually kept score, through any of that.
 But as a software developer, it’s always fun to think about edge cases, and squeezing almost 5KB into a 280-“character” tweet is fun This makes me wonder if anyone has created a version of base64 that uses the vast, sprawling space of unicode to take advantage of these glyph-count-based restrictions.If they have, I hope they called it uuuniencode.
 There are several of these. base65536 is the one that seems to pop up the most often on HN, although base2048 is more useful for Twitter. On the GitHub page the dev helpfully links to the various implementations: https://github.com/qntm/base2048
 This is what you're looking for:https://github.com/qntm/base2048It can store 385 bytes per tweet. This link includes a bit more technical explanation of how Twitter counts characters towards the limit. Apparently, using the entire range of unicode characters does not improve compression because of the double weighting of emojis and other characters as described in TFA. It links to a base131072 encoding which can only store 297 bytes per tweet.
 Whoah, it's a crafting game.
 Emojis need more serious attention. A lot of online conflicts start from miscommunication due to lack of facial expressions. To the point thats its driving a social crisis and division in the real world.
 As a recent observer of a discussion wherein the dramatic differences between single, double, and triple-dot sentence terminators were argued, I think concerns about emoji facial expressions might be valid, but resolving them would still leave us in a world of crisis and division..
 I can't stand ".." being used in communications. It's not a thing, and because of that it's incredibly ambiguous!
 What was the outcome of that argument?
 I think the main takeaway for the majority of participants was that I'm old and don't know how to talk on the Internet.
 Humans came up with a way to express musical notes on paper. To me that seems much more challenging. The list of common emotions is not that numerous. We need some shortcut, compact way of expressing them in writing. It would go a long way to score, or markup our words with some sort of emotion meter. Perhaps thats what music is to singing. Just maybe a visual way of doing that. Emojis are hard to use.
 I'm not disagreeing, but note that humans have not come up with a way to express musical notes in Unicode or Emoji. The second vertical dimension is critical for music notation.I'd suggest that the same idea applies to nuanced facial expressions. You could certainly devise a set of glyphs to stand in, but they would have to be learned by everyone.The failures we see today are attempts at rendering expressions on drawn faces -- and to be fair, even a high res photographic still image of a real human making an expression could be easy to misinterpret. Especially across cultures and subcultures. My 15-year old niece has very strong opinions on how many periods I use to end sentences..I think "fixing" emoji is probably a lost cause. But I am biased since I don't care at all. The most interesting and amusing thing I know about emoji is that when Apple changed the "gun" from a realistic-looking Glock/etc to a toy squirt gun, all the ingrates who used the image in a threatening manner ended up looking silly to iOS users. Android followed quickly.I support the same sillification of negative and angry facial expression emoji as well, FWIW, and I think it's probably hopeless to try to cram nuance into any of them. Fortunately, we still have words.
 compare, in order of relative "uh-oh!":OK.Ok.Okok
 I also find (thumbs up emoji) and (ok emoji) to be somewhat ambiguous. I see them as potentially sarcastic. Really, sarcasm is the biggest barrier to effective online communication.Apparently you cannot use emojis on HN.
 Don't forget:kkkk...et cetera et alii ad nauseam
 This seems like much too large a problem to tackle with emoji.Even as complex as emoji are becoming, they really still don't address the issues behind online miscommunication. And I really doubt they ever could.
 Maybe people need to learn how to express themselves preciselyon the other hand people still for some reason believe that "lol" only means "laughing out loud" which's lol itself.so maybe it's impossible?
 The Emoji department of the unicode consortium is really just three letter agencies making sure text parsing and font rendering libs will continue to have exploitable bugs. ;-)
 I wonder where is the "substitution" of the codepoint sequence (from a sequence to a single codepoint) done? Very concretely and practically: is it the font doing the substitution? What else if not? How do they decide that a sequence should be substituted?
 Even our character encodings have turned into bloatware.
 Conspiracy theory:1. specification authors want to make sure the extended grapheme cluster algorithms are widely adopted so that implementations can correctly deal with devanagari 2. they notice no one gives a shit about brown people and their writing systems 3. combining emojis requiring the use of the same underlying algorithms were popularised in order to push the adoption
 I'm on Linux right now and, sadly, couldn't see the magic character properly. :(Anyone know if I'm missing anything, or is there no support for 13.1 yet? My standard routine is to just install every noto font I can find (noto-fonts noto-fonts-cjk noto-fonts-emoji noto-fonts-extra).
 It works for me with Firefox on NixOS using Noto Color Emoji, from, I'm assuming, noto-fonts-emoji.
 Works here on a fresh (yesterday) Linux Mint install.
 Same on fresh, stock openSUSE Tumbleweed with Gnome 40
 Really enjoyed, thank you for sharing. It was interesting to learn how emojis are composed from component parts. I'm wondering where that will lead to in the future? Will we have a vast built-in library of emojis of ever increasing complexity?
 From the title I thought this is similar to word arithmetic based on Word2Vec representations, where for example we approximately have:King - Man + Woman = Queen Or Brother - Man + Woman = SisterAnd some sexist variants of these too. But the article is on a different topic.
 This reminds me of a fun iOS pizzle game from a few years back called "Alchemy" - you started with icons for earth, air, fire, and water and then combined them in a similar way to the title example to create others.
 Yes it reminded me of that too. I thought it was a repost - pretty sure its original web version was posted on HN. I spent a lot of toilet visits on that game... became pretty unfun once the board is super full, though.
 I thought this was going to be about word embeddings!
 That Twitter limit is interesting. I wonder how much data can fit in 140 emojis, if all existing and valid combinations are allowed?
 reminds me of https://www.figma.com/blog/when-fonts-fall/

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# P-T graph of water

375

### Staff: Mentor

It explains it in the link a bit further down.

3. ### Outrageous

375
So can I say at higher pressure what is more likely to form will be denser?
One thing I don't understand is let's say carbon dioxide, when we apply higher pressure, the freezing curve of CO2 slopes positively to the right , that means higher the pressure ,higher the freezing point, more difficult to form solid. Is this correct?
But I always think that we need high pressure so that we can easily change from gas to liquid then to solid.

4. ### K^2

2,470
Not exactly. Try to think of it in terms of Le Chatelier's principle. Imagine I take a block of ice just bellow melting point. Then I compress it. In other words, I raise pressure. Le Chatelier's principle tells me that system will try to resist the change. In other words, it will try to do something to reduce pressure. For example, it can go into a denser state if one is available. You can also see what happens based on PV diagram. If I increase pressure, temperature just bellow freezing becomes just above freezing. Ice melts. Combined with above it tells me that substance found a denser state, which happens to be liquid.

Higher freezing point means solid will form at higher temperature. It will be easier to make solid CO2 at higher pressure. What's more important is that if you start with a block of frozen CO2 at just bellow the freezing point and compress it, increasing pressure, you stay bellow freezing point. On the other hand, if you start with liquid CO2 just above freezing point, compressing it will put you bellow freezing point, meaning that in this case the solid is denser.


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# The TOC tree

Since reST does not have facilities to interconnect several documents, or split documents into multiple output files, Sphinx uses a custom directive to add relations between the single files the documentation is made of, as well as tables of contents. The toctree directive is the central element.

Note

Simple "inclusion" of one file in another can be done with the :dudir:include directive.

## Special names

Sphinx reserves some document names for its own use; you should not try to create documents with these names -- it will cause problems.

The special document names (and pages generated for them) are:

• genindex, modindex, search

These are used for the general index, the Python module index, and the search page, respectively.

The general index is populated with entries from modules, all index-generating :ref:object descriptions <basic-domain-markup>, and from :rst:dir:index directives.

The Python module index contains one entry per :rst:dir:py:module directive.

The search page contains a form that uses the generated JSON search index and JavaScript to full-text search the generated documents for search words; it should work on every major browser that supports modern JavaScript.

• every name beginning with _

Though only few such names are currently used by Sphinx, you should not create documents or document-containing directories with such names. (Using _ as a prefix for a custom template directory is fine.)

Footnotes

 [1] The maxdepth option does not apply to the LaTeX writer, where the whole table of contents will always be presented at the begin of the document, and its depth is controlled by the tocdepth counter, which you can reset in your :confval:latex_preamble config value using e.g. \setcounter{tocdepth}{2}.
 [2] A note on available globbing syntax: you can use the standard shell constructs *, ?, [...] and [!...] with the feature that these all don't match slashes. A double star ** can be used to match any sequence of characters including slashes.


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# 9.E: Separation, Purification, & Identification of Organic Compounds (Exercises)

Exercise 9-1 Suppose you are standing on the end of a pier watching the waves and, between your position and a buoy $$200 \: \text{m}$$ straight out, you count 15 wave crests. Further, suppose a wave crest comes by every 15 seconds. Calculate $$\nu$$ in $$\text{Hz}$$, $$\lambda$$ in $$\text{m}$$, $$c$$ in $$\text{m sec}^{-1}$$, and $$\bar{\nu}$$ in $$\text{km}^{-1}$$.

Exercise 9-2 Blue light has $$\bar{\nu} = 20,800 \: \text{cm}^{-1}$$. Calculate $$\nu$$ in $$\text{Hz}$$ and $$\lambda$$ in $$\text{nm}$$.

Exercise 9-3 Calculate the energy in $$\text{kcal mol}^{-1}$$ that corresponds to the absorption of 1 einstein of light of $$589.3 \: \text{nm}$$ (sodium $$D$$ line) by sodium vapor. Explain how this absorption of light by sodium vapor may have chemical utility.

Exercise 9-4

a. Use Equations 9-1 and 9-2 to calculate the wavelength in $$\text{nm}$$ and energy in $$\text{kcal}$$ of an einstein of radiation of radio-frequency energy in the broadcast band having $$\nu = 1 \: \text{MHz}$$ (1 megahertz) $$= 10^6 \: \text{sec}^{-1}$$ and knowing that the velocity of light is approximately $$3 \times 10^8 \: \text{m sec}^{-1}$$.

b. In photoelectron spectroscopy, x-rays with energies of approximately $$1250 \: \text{eV}$$ are used ($$1 \: \text{eV}$$ (electron volt) $$\text{mol}^{-1} = 23.05 \: \text{kcal}$$). What would $$\lambda$$ (in $$\text{nm}$$) be for such x-rays?

Exercise 9-5 The microwave spectrum of pure trans-2-butenoic acid $$\left( \ce{CH_3CH=CHCO_2H} \right)$$ shows patterns exactly like those of Figure 9-8, which indicate the presence of two different conformations. What are these conformations, and why are there only two of them? (You may be helped by reviewing Section 6-5.)

Exercise 9-6 Use Equation 9-3 and any other pertinent data to predict which compound in each group would absorb in the infrared at the highest frequency for the changes in the stretching vibration of the specified bond. Give your reasoning.

a. $$\ce{R-Cl}$$, $$\ce{R-Br}$$, $$\ce{R-F}$$ (carbon-halogen)
b. $$\ce{CH_3-NH_2}$$, $$\ce{CH_2=NH}$$, $$\ce{HC \equiv N}$$ (carbon-nitrogen)

Exercise 9-7 Which compound in each group would have the most intense infrared absorption band corresponding to stretching vibrations of the bonds indicated? Give your reasoning.

a. $$\ce{(CH_3)_2C=O}$$, $$\ce{(CH_3)_2C=CH_2}$$ (multiple bond)
b. $$\ce{CH_3-CH_3}$$, $$\ce{CH_3-O-CH_3}$$ ($$\ce{C-C}$$ vs. $$\ce{C-O}$$)
c. $$\ce{CH_3C \equiv CH}$$, $$\ce{CH_3C \equiv CCH_3}$$ (multiple bond)
d. $$\ce{H-Cl}$$, $$\ce{Cl-Cl}$$

Exercise 9-8* How many vibrational modes are possible for (a) $$\ce{CS_2}$$ (linear), (b) $$\ce{BeCl_2}$$ (linear), and (c) $$\ce{SO_2}$$ (angular)? Show your reasoning.

Exercise 9-9* Suppose an infrared absorption occurs at $$3000 \: \text{cm}^{-1}$$. Calculate the corresponding frequency $$\nu$$ in $$\text{sec}^{-1}$$; $$\lambda$$ in $$\text{nm}$$, angstroms, and microns, and energy change in $$\text{kcal mol}^{-1}$$. Using Equation 4-2 and neglecting $$\Delta S$$, calculate the fraction of the molecules that would be in the ground state and in the first vibrational excited state (above ground state by $$3000 \: \text{cm}^{-1}$$) at $$298^\text{o} \text{K}$$.

Exercise 9-10 Use Table 9-2 to map the approximate positions and intensities expected for the characteristic infrared bands corresponding to the stretching vibrations of the various kinds of bonds in the following molecules:

a. 1,1,1-trideuteriopropanone (trideuterioacetone)

b. propyne

c. ethyl ethanoate (ethyl acetate)

d. propanenitrile (acrylonitrile)

e. 2-oxopropanoic acid (pyruvic acid)

f. ethanol (ethyl alcohol) (both as pure liquid and as a dilute solution in $$\ce{CCl_4}$$)

Exercise 9-11 The infrared spectra shown in Figure 9-14 are for compounds of formula $$\ce{C_3H_O}$$ and $$\ce{C_3H_6O_2}$$. Use the data in Table 9-2 and the molecular formulas to deduce a structure for each of these substances from its infrared spectrum. Indicate clearly which lines in the spectra you identify with the groups in your structures.

Figure 9-14: Infrared spectra for Exercise 9-11. Spectrum (a) corresponds to $$\ce{C_3H_6O}$$ and Spectrum (b) to $$\ce{C_3H_6O_2}$$.

Exercise 9-12* Classify the following molecules according to the general characteristics expected for their infrared and Raman spectra:

a. $$\ce{HC \equiv CH}$$
b. $$\ce{ICl}$$
c. $$\ce{CO}$$
d. $$\ce{CF_2=CH_2}$$ (double-bond stretch only)
e. $$\ce{(CH_3)_2C=CH_2}$$
f. $$\ce{CH_3CH=CHCH_3}$$

Exercise 9-13* Carbon dioxide gives two infrared absorption bands but only one Raman line. This Raman line corresponds to a different vibration than the infrared absorptions. Decide which vibrational modes are infrared active (i.e., make the molecule electrically unsymmetrical during at least part of the vibration) and which is Raman active (i.e., occurs so the molecule is electrically symmetrical at all times during the vibration, see Section 9-7A).

Exercise 9-14 List the kinds of electronic transitions that would be expected for azaethene (methyleneimine), $$\ce{CH_2=NH}$$, in order of increasing energy. Use the data in Table 9-3 to predict approximately the wavelengths at which the three lowest-energy transitions should occur.

Exercise 9-15 Calculate the percentage of the incident light that would be absorbed by an $$0.010 \: \text{M}$$ solution of 2-propanone (acetone) in cyclohexane contained in a quartz cell $$0.1 \: \text{cm}$$ long at $$280 \: \text{nm}$$ and at $$190 \: \text{nm}$$ (see footnote $$a$$ of Table 9-3).

Exercise 9-16 Explain why the absorption band at $$227.3 \: \text{nm}$$ for trimethylamine, $$\ce{(CH_3)_3N}$$, disappears in acid solution.

Exercise 9-17 A compound of formula $$\ce{C_4H_6O}$$ has two absorption bands in the ultraviolet: $$\lambda = 320 \: \text{nm}$$, $$\epsilon = 30$$ and $$\lambda = 218 \: \text{nm}$$, $$\epsilon = 18,000$$ in ethanol solution. Draw three possible structures that are consistent with this information.

Exercise 9-18 2,4-Pentadione exists in equilibrium with 4-hydroxy-3-penten-2-one:

The infrared spectrum of the liquid mixture shows a broad absorption band at $$3000$$-$$2700 \: \text{cm}^{-1}$$ and an intense absorption band at $$1613 \: \text{cm}^{-1}$$. In cyclohexane solution, the substances has $$\lambda_\text{max}$$ at $$272 \: \text{nm}$$ with $$\epsilon_\text{max} = 12,000$$.

a. What can you conclude from this data as to the magnitude of $$K$$, the equilibrium constant for the interconversion of the two forms?

b. What can you deduce from the fact that the absorption at $$272 \: \text{nm}$$ is much weaker in aqueous solution (pH 7) than it is in cyclohexane?

Exercise 9-19* The electronic absorption spectrum of 2-nitrobenzenol has $$\lambda_\text{max}$$ in $$0.1 \: \text{M} \: \ce{HCl}$$ at $$350 \: \text{nm}$$. In $$0.1 \: \text{M} \: \ce{NaOH}$$, the benzenol is largely converted to its anion, and $$\lambda_\text{max}$$ shifts to $$415 \: \text{nm}$$.

The ground-state resonance forms of 2-nitrobenzenol and its anion include

Explain how the relative importance of these resonance forms to the ground and excited states of 2-nitrobenzenol and its anion can account for the fact that the anion absorbs at longer wavelengths than does 2-nitrobenzenol. (Review Section 6-5B)

Exercise 9-20* A solution containing the two forms of the important coenzyme nicotinamide adenine dinucleotide (abbreviated $$\ce{NAD}^\oplus$$ and $$\ce{NADH}$$; see Section 15-6C for structures) has an absorbance in a $$1$$-$$\text{cm}$$ cell of 0.311 at $$340 \: \text{nm}$$ and 1.2 at $$260 \: \text{nm}$$. Both $$\ce{NAD}^\oplus$$ and $$\ce{NADH}$$ absorb at $$260 \: \text{nm}$$, but only $$\ce{NADH}$$ absorbs at $$340 \: \text{nm}$$. The molar extinction coefficients are

$\begin{array}{lll} \underline{\text{Compound}} & \underline{260 \: \text{nm}} & \underline{340 \: \text{nm}} \\ \ce{NAD}^\oplus & 18,000 & \sim 0 \\ \ce{NADH} & 15,000 & 6220 \end{array}$

Calculate the proportions of $$\ce{NAD}^\oplus$$ and $$\ce{NADH}$$ in the mixture.

Exercise 9-21 Use Figure 9-24 to map the nmr spectrum you would expect for $$\ce{^{13}CCl_3 \: ^1H}$$ in a field-sweep spectrometer in which the transmitter frequency is kept constant at $$30 \: \text{MHz}$$ and the magnetic field is swept from 0 to 30,000 gauss. Do the same for a frequency-sweep spectrometer when the magnetic field is kept constant at 10,000 gauss and the frequency is swept from $$0$$ to $$100 \: \text{MHz}$$. (For various reasons, practical spectrometers do not sweep over such wide ranges of field or frequency.)

Exercise 9-22* In nmr experiments, structural inferences sometimes are drawn from differences in resonance frequencies as small as $$1 \: \text{Hz}$$. What difference in energy in $$\text{kcal mol}^{-1}$$ does $$1 \: \text{Hz}$$ represent?

Exercise 9-23* The intensity of nmr signals normally increases markedly with decreasing temperature because more magnetic nuclei are in the $$+\frac{1}{2}$$ state. Calculate the equilibrium constant at $$-90^\text{o}$$ for the $$+\frac{1}{2}$$ and $$-\frac{1}{2}$$ states of $$\ce{^1H}$$ in a magnetic field of 42,300 gauss when the resonance frequency is $$180 \: \text{MHz}$$.

Exercise 9-24

a. Identify the protons with different chemical shifts in each of the structures shown. Use letter subscripts $$\ce{H}_A$$, $$\ce{H}_B$$, and so on, to designate nonequivalent protons. Use models if necessary.

(i) cis- and trans-2-butene
(iii) 1-chloro-2,2-dimethylbutane
(iv) 2-butanol
(v) trans-1,2-dibromocyclopropane

b.* Why does 3-methyl-2-butanol have three methyl resonances with different chemical shifts in its proton nmr spectrum?

c.* For the compounds in Part a designated those protons (if any) that are enantiotopic or diastereotopic.

Exercise 9-25 Use Equation 9-4 to calculate the chemical shift of the $$\ce{-CH_2}-$$ protons on

a. $$\ce{CH_2Cl_2}$$
b. $$\ce{ClCH_2OCH_3}$$
c. $$\ce{C_6H_5CH_2CO_2H}$$

Exercise 9-26 If the $$\ce{-NH_2}$$ protons of 2-aminoethanol, $$\ce{NH_2CH_2CH_2OH}$$, have a shift of $$1.1 \: \text{ppm}$$ and the $$\ce{-OH}$$ proton has a shift of $$3.2 \: \text{ppm}$$, what will be the observed average ($$\ce{-NH_2}$$, $$\ce{-OH}$$) proton shift if exchange is very fast?

Exercise 9-27 In reasonably concentrated solution in water, ethanoic acid (acetic acid) acts as a weak acid (less than $$1\%$$ dissociated). Ethanoic acid gives two proton nmr resonance lines at $$2$$ and $$11 \: \text{ppm}$$, relative to TMS, whereas water gives a line at $$5 \: \text{ppm}$$. Nonetheless, mixtures of ethanoic acid and water are found to give only two lines. The position of one of these lines depends on the ethanoic acid concentration, whereas the other one does not. Explain how you would expect the position of the concentration-dependent line to change over the range of ethanoic acid concentrations from $$0$$-$$100\%$$.

Exercise 9-28 The proton nmr spectrum of a compound of formula $$\ce{C_6H_{12}O_2}$$, is shown in Figure 9-31. The signals are shown relative to TMS as the standard, and the stepped line is the integral of the area under the peaks from left to right. The infrared spectrum of the same compound shows a broad band at $$3300 \: \text{cm}^{-1}$$ and a strong band at $$1700 \: \text{cm}^{-1}$$. Deduce the structure of the compound and name it by the IUPAC system.

Figure 9-31: Proton nmr spectrum of a compound, $$\ce{C_6H_{12}O_2}$$, at $$60 \: \text{MHz}$$ relative to TMS at $$0.00 \: \text{ppm}$$. The stepped line is the integral running from left to right. See Exercise 9-28.

Exercise 9-29 Sketch the proton chemical shifts in $$\text{ppm}$$ and $$\text{Hz}$$ as well as the integral you would expect for each of the following substances at $$60 \: \text{MHz}$$. (The spin-spin splitting of the resonance lines evident in Figures 9-23 and 9-27, but not seen in Figure 9-31, can be safely neglected with all of the compounds listed.)

a. $$\ce{(CH_3)_3CCH_2OCH_3}$$
b. $$\ce{CH_2COC(CH_3)_3}$$
c. $$\ce{HCOC(CH_3)_2CHO}$$
d.
e. $$\ce{(CH_3)_2C=CCl_2}$$
f. $$\ce{(CH_3)_3COC \equiv CH}$$
g. $$\ce{(CH_2Cl)_3CCO_2H}$$
h.* cis-1-methyl-4-tert-butyl-1,2,2,3,3,4,5,5,6,6-decachlorocyclohexane

Exercise 9-30 Write structures for compounds with the following descriptions (There may be more than one correct answer, but only one answer is required.)

a. $$\ce{C_2H_6O}$$ with one proton nmr shift
b. $$\ce{C_6H_{12}}$$ with one proton nmr shift
c. $$\ce{C_5H_{12}}$$ with one proton nmr shift
d. $$\ce{C_4H_8O}$$ with two different proton nmr shifts
e. $$\ce{C_4H_8O_2}$$ with three different proton nmr shifts
f. $$\ce{C_4Cl_8}$$ with two different $$\ce{^{13}C}$$ nmr shifts

Exercise 9-31 Sketch the proton nmr spectrum and integral expected at $$60 \: \text{MHz}$$, with TMS as standard, for the following substances. Show the line positions in $$\ce{Hz}$$; neglect spin-spin couplings smaller than $$1$$ to $$2 \: \text{Hz}$$ and all second-order effects. Remember that chlorine, bromine, and iodine (but not fluorine) act as nonmagnetic nuclei.

a. $$\ce{CH_3Cl}$$
b. $$\ce{CH_3CH_2Cl}$$
c. $$\ce{(CH_3)_2CHCl}$$
d. $$\ce{CH_3CCl_2CH_2Cl}$$
e. $$\ce{(CH_3)_3CCl}$$
f. $$\ce{CHCl_2CHBr_2}$$
g. $$\ce{CH_3CHClCOCH_3}$$
h. $$\ce{CH_3CH_2CO_2CH_2CH_3}$$
i. $$\ce{ClCH_2CH_2CH_2I}$$
j. $$\ce{(ClCH_2)_3CH}$$

Exercise 9-32* The proton-proton coupling in 1,1,2,2-tetrachloroethane cannot be observed directly because the chemical shift is zero. However, measurements of the splittings in $$\ce{^{13}CCl_2H-^{12}CCl_2H}$$ show that the proton-proton coupling in $$\ce{CHCl_2CHCl_2}$$ is $$3.1 \: \text{Hz}$$. Explain how you can use this information to deduce the favored conformation of $$\ce{CHCl_2CHCl_2}$$. Draw a sawhorse representation of the preferred conformation.

Exercise 9-33 The proton-proton coupling in meso-2,3-dibromobutanedioic acid (determined by the same procedure as for 1,1,2,2-tetrachloroethane, see Exercise 9-32) is $$11.9 \: \text{Hz}$$. Write a sawhorse structure for the preferred conformation of this molecule.

Exercise 9-34*

a. Show how the assignment of $$J_{AB} = J_{BC} = 2 J_{AC}$$ leads to the prediction of four equally spaced and equally intense lines for the methyl resonance of 2-phenylpropene.

b. What would the splittings of the alkenic and methyl protons look like for trans-1-phenylpropene if $$J_{AB} = 16 \: \text{Hz}$$, $$J_{AC} = 4 \: \text{Hz}$$, and $$J_{BC} = 0 \: \text{Hz}$$?

Exercise 9-35 Interpret fully each of the proton nmr spectra shown in Figure 9-40 in terms of the given structures. For spin-spin splittings, explain how the patterns arise and predict the intensities expected from simple theory.

Figure 9-40: Proton nmr spectra at $$60 \: \text{MHz}$$ relative to TMS $$= 0.00 \: \text{ppm}$$. See Exercise 9-35.

Exercise 9-36 Figure 9-41 shows proton nmr spectra and integrals at $$60 \: \text{MHz}$$ for three simple organic compounds. Write a structure for each substance that is in accord with both its molecular formula and nmr spectrum. Explain how you assign each of the lines in the nmr spectrum.

Figure 9-41: Proton nmr spectra and integrals for some simple organic compounds at $$60 \: \text{MHz}$$ relative to TMS, $$0.00 \: \text{ppm}$$. See Exercise 9-36.

Exercise 9-37 Figure 9-42 shows the proton nmr spectrum of a compound, $$\ce{C_5H_8O_2}$$. Which of the following structures fits the spectrum best? Explain. Remember that the protons of

are expected to be nonequivalent; that is, they have different chemical shifts if $$\ce{R}$$ and $$\ce{R'}$$ are different groups.

Figure 9-42: Proton spectrum of a compound, $$\ce{C_5H_8O_2}$$, at $$60 \: \text{MHz}$$ relative to TMS as standard. See Exercise 9-37.

Exercise 9-38 Suppose that you had six unlabeled bottles containing caffeine, hexachlorophene, phenacetin, DDT, 1,3-dimethyluracil, and 1-phenylethanamine. The nmr spectrum of each of these compounds is shown in Figure 9-43. Match the lettered spectra with the appropriate individual structures so the bottles can be labeled properly. Give your reasoning.

Figure 9-43: Proton nmr spectra of compounds at $$60 \: \text{MHz}$$. See Exercise 9-38.

Exercise 9-39 Show how one can use the asymmetry of the line intensities of the $$60$$-$$\text{MHz}$$ proton spectrum in Figure 9-45 to show which groups of lines are interconnected by spin-spin coupling. Write structural formulas for the compounds involved that fit the observed splitting patterns and chemical shifts.

Exercise 9-40 Explain why it is correct to characterize $$16$$ and $$17$$ as diastereomers and not enantiomers.

Exercise 9-41* When one takes the proton nmr spectrum of ordinary trichloromethane (chloroform, $$\ce{CHCl_3}$$) under high gain, the spectrum shown in Figure 9-49 is obtained. The weak outside peaks are separated by $$210 \: \text{Hz}$$ and together have an integrated intensity of slightly over $$1\%$$ of the main peak. Explain how these weak proton signals arise.

Figure 9-49: Proton nmr spectrum at $$60 \: \text{MHz}$$ of trichloromethane taken with high-detection sensitivity. See Exercise 9-41.

Exercise 9-42* With reference to the data summarized in Figure 9-47 and the discussion in Section 9-10L, sketch qualitatively the proton-decoupled $$\ce{^{13}C}$$ spectra you would expect for

a. $$\ce{(CH_3)_3CCH_2OH}$$
b. $$\ce{CCl_3CH_2OCOCH_3}$$

Exercise 9-43* Figure 9-50 shows the $$\ce{^1H}$$ and $$\ce{^{13}C}$$ nmr spectra of a compound $$\ce{C_6H_{10}O}$$. With the aid of these spectra, deduce the structure of $$\ce{C_6H_{10}O}$$. It will be seen that the $$\ce{^{13}C}$$ spectrum is quite simple, even though the proton spectrum is complex and difficult to interpret.

Figure 9-50: (a) Proton and (b) $$\ce{^{13}C}$$ spectra of a compound $$\ce{C_6H_{10}O}$$ taken at $$60 \: \text{MHz}$$ and $$15.1 \: \text{MHz}$$, respectively. Because of the special way the $$\ce{^{13}C}$$ spectrum was determined, the peak at $$209 \: \text{ppm}$$ is smaller than it should be. The intensity of this peak is, correctly, the same as the peak at $$25.5 \: \text{ppm}$$. See Exercise 9-43.

Exercise 9-44 Explain how a mass spectrometer, capable of distinguishing between ions with $$m/e$$ values differing by one part in 50,000, could be used to tell whether an ion of mass 29 is $$\ce{C_2H_5^+}$$ or $$\ce{CHO^+}$$.

Exercise 9-45

a. Calculate the relative intensities of the $$\left( \ce{M} + 1 \right)^+$$ and $$\left( \ce{M} + 2 \right)^+$$ ions for a molecule of elemental composition $$\ce{C_3H_7NO_2}$$.

b. The $$\ce{M}^+$$, $$\left( \ce{M} + 1 \right)^+$$, and $$\left( \ce{M} + 2 \right)^+$$ ion intensities were measured as 100, 8.84, and 0.54 respectively, and the molecular weight as 120. What is the molecular formula of the compound?

c. In our example of how natural $$\ce{^{13}C}$$ can be used to determine the number of carbon atoms in a compound with $$\ce{M}^+ = 86$$ and a $$\left( \ce{M} + 1 \right)^+/\ce{M}^+$$ ratio of 6.6/100, we neglected the possible contribution to the $$\left( \ce{M} + 1 \right)^+$$ peak of the hydrogen isotope of mass 2 (deuterium). The natural abundance of deuterium is $$0.015\%$$. For a compound of composition $$\ce{C_6H_{14}}$$, how much do you expect the deuterium to contribute to the intensity of the $$\left( \ce{M} + 1 \right)^+$$ peak relative to the $$\ce{M}^+$$ peak?

Exercise 9-46 Show how the molecular weights of 2-propanone, propanal, and 2-butanone can be estimated from the mass spectra in Figure 9-52. Suggest a possible origin for the strong peaks of mass 57 in the spectra of propanal and 2-butanone, which is essentially absent in 2-propanone, although 2-propanone (and 2-butanone) show strong peaks at mass 43.

Exercise 9-47 The mass spectrum of propylbenzene has a prominent peak at mass number 92. With (3,3,3-trideuteriopropyl)benzene, this peak shifts to 93. Write a likely mechanism for breakdown of propylbenzene to give a fragment of mass number 92.

Exercise 9-48 The mass spectra of alcohols usually show peaks of $$\left( \ce{M} - 18 \right)$$, which correspond to loss of water. What kind of mechanisms can explain the formation of $$\left( \ce{M} - 18 \right)$$ peaks, and no $$\left( \ce{M} - 19 \right)$$ peaks, from 1,1-dideuterioethanol and 1,1,1,3,3-pentadeuterio-2-butanol?

Exercise 9-49 Explain how the postulated rearrangement of the $$\ce{M}^+$$ ion of ethyl butanoate (Section 9-11) is supported by the fact that the 2,2-dideuterio compound gives a peak with $$m/e =$$ 90, the 3,3-dideuterio isomer gives a $$m/e$$ 88 peak, while the 4,4,4-trideuterio isomer gives a $$m/e$$ 89 peak.

Exercise 9-50 What is the likely structure for the major fragment ion with $$m/e =$$ 45 derived from methoxyethane (methyl ethyl ether) on electron impact?

Exercise 9-51 A certain halogen compound gave a mass spectrum with molecular ion peaks at $$m/e$$ 136 and 138 in about equal intensities. The nmr spectrum of this compound gave only a single resonance around $$1.2 \: \text{ppm}$$. What is the structure of the compound? Give your reasoning.

Exercise 9-52* The mass spectra of three compounds, A, B, and C, are given below in tabular form. Only the peaks of significant intensity are reported.

a. Compound A is $$\ce{CH_3CH_2CH_2COCH_3}$$. Show how this material can fragment to give the peaks marked with an asterisk and, where possible, how the isotope peaks help establish your assignments.

b. Determine the molecular weight and the molecular formula of Compounds B and C from the spectral data. Suggest a likely structure for each peak marked with an asterisk.

## Contributors

• John D. Robert and Marjorie C. Caserio (1977) Basic Principles of Organic Chemistry, second edition. W. A. Benjamin, Inc. , Menlo Park, CA. ISBN 0-8053-8329-8. This content is copyrighted under the following conditions, "You are granted permission for individual, educational, research and non-commercial reproduction, distribution, display and performance of this work in any format."


text:
SEARCH HOME
 Math Central Quandaries & Queries
 Hi! My question is: Determine how the area of the Reuleaux Triangle of width h compared to the area of a circle of width h? Thanks! Dani

Hi Dani,

Start with an equilateral triangle ABC with each side of length h. With the vertex A as centre, draw an arc of the circle of radius h that passes through the other two vertices. Repeat this construction at the vertices B and C. The resulting plane figure ABC is the Reuleaux Triangle of width h.

Consider he plane figure bounded by the line segments AB and CA and the arc BC. This is a sector of a circle with centre A, radius h and centre angle 60 degrees or /3 radians. Thus the area of this sector is

1/2 h2 /3 or 1/6 h2

The similar sectors with centres B and C each have the same area and hence the sum of these three areas is

3(1/6 h2 ) = 1/2 h2

Looking back at the diagram you can see that this calculation included the equilateral triangle ABC three times and hence the area of the Reuleaux Triangle ABC is

1/2 h2 - twice the area of the equilateral triangle ABC.

Find the area of the equilateral triangle and simplify this expression. Compare this area to the area of the circle of width h. The circle of width h has radius h/2 and hence its area is

h2/4

I hope this helps,
Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.


text:
# Are topologies induced by following families of seminorms same?

Let $$G$$ be an open subset of $$\mathbb R ^n$$ and $$D(G)$$ denotes the set of smooth functions with compact support in $$G$$.

Consider following families of seminorms,

1. For $$f\in D(G)$$

$$||f||_N = \sup \{ | D^{\alpha}f(x)|: x\in G,|\alpha| \leq N \}$$ for $$N\in \mathbb N$$.

1. Let $$K_n$$ be a nested sequence of compact sets which exhaust $$G$$ then for $$f \in D(G)$$ define

$$\nu_N (f) = \sup \{ | D^{\alpha}f(x)|: x\in K_N,|\alpha| \leq N \}$$ for $$N\in\mathbb N$$.

where $$\alpha$$ is a multi-index.

Are topologies induced on $$D(G)$$ by above two families same?

My guess is NO but I am unable to prove it.

EDIT- I am interested in this question because on $$C^\infty(G)$$ we give topology induced by family 2. In chapter 6 of Rudin’s Functional Analysis, it says family 1 doesn’t give good topology on $$D(G)$$ (it’s not complete). But Rudin didn’t talk about family 2 even though it’s subspace topology on $$D(G)$$ as a subset of $$C^\infty(G)$$.

• In the definition of $\nu_N$, you wrote $x\in K_N$. Did you mean $x\in K_m$? Since $N$ is already used as the size of the maximal multi index? Dec 26 '18 at 7:33
• It’s $x\in K_N$. In fact it’s a topology which we give on $C^\infty (G)$. Dec 26 '18 at 8:07
• Can you define the natural basis of neighborhoods of $0$ in both topology ? (the translates of those neighborhoods are a basis of the topology). Will a neighborhood in the 1st topology be open in the 2nd topology, or is it containing some open set of the 2nd topology ? Dec 27 '18 at 6:13
• @reuns I tried that but couldn’t figure out! Dec 27 '18 at 6:39
• Let $\|f\|_{K_n,n}= \sup_{x \in K_n, |\alpha| \le n}|D^\alpha f(x)|$ and $U_1(r,N) = \{ f \in C^\infty_c(G), \forall n \le N, \|f\|_{G,n}<r_n \}$. Is it open in the 1st topology ? Does it contain an open set in the 2nd topology ? (let $T(C^\infty_c(G), \|.\|_{K_n,n})$ be the set of open sets of the topological vector space $C^\infty_c(G), \|.\|_{K_n,n}$, then $T_2 = \bigcap_n T(C^\infty_c(G), \|.\|_{K_n,n})$) Dec 27 '18 at 12:19

With the topology from 2, for functions in $$C^\infty(\Omega)$$, $$\lim_n f_n = f$$ can be thought as of $$D^\alpha f_n \rightarrow D^\alpha f$$ uniformly on any compact subsets, and for each $$|\alpha| < \infty$$.

One can construct $$f_n\in \mathcal{D}(\Omega)$$ and yet the limit $$f\in C^\infty(\Omega)$$ does not have compact support.

2 gives a weaker topology than 1 on $$\mathcal{D}(\Omega)$$, since 1 is "converge uniformly" and 2 is "converge locally uniformly". For example, let $$\phi$$ have support in $$[0,1]$$, define $$\psi_m(x) = \phi(x) + \phi(x-1) + \cdots + \phi(x-m)$$ this sequence is Cauchy in 2, but not Cauchy in 1. This also shows that $$\mathcal{D}(\Omega)$$ with the topology induced by 2 is not complete. (Similar to Rudin's example on page 151.)

• So we are not interested in topology 2 because it also is not complate. But how to justify the line “2 gives a weaker topology than 1 on $D(\Omega)$”? Dec 28 '18 at 15:37
• Two families of semi-norms generate the same topology if and only if any semi-norm from one family can be bounded by some semi-norm of the other family. In this case $\nu_N \leq C\|\cdot \|_N$ is clear, and I don't see a way to have the bound the other way around.
– Xiao
Dec 28 '18 at 16:32
• Maybe the more precise statement is given two families of semi-norms $\mathcal P$ and $\mathcal Q$, $\tau_P$ is finer, i.e. $\tau_Q \subset \tau_p$ if for each $q\in \mathcal {Q}$, there exist $n$ semi-norms $p_1, \cdots, p_n \in \mathcal {P}$ and a constant $C$ such that $$q\leq C \max \{ p_1, \cdots p_n\}.$$
– Xiao
Dec 28 '18 at 16:43
• Could you give reference for statement in your last comment?. Thanks a lot for your help. Dec 28 '18 at 17:08
• @MayureshL It should be easy to see with the metric explicitly. For $k,l\geq M$, $|\psi_k -\psi_l|$ will vanish on $[0, M]$. This means $\nu_N (|\psi_k -\psi_j|)$ will all be zero for $N=1, 2,\cdots, M'$.
– Xiao
Dec 31 '18 at 17:58


text:
# 1.6. Instructions for Developers#

## 1.6.1. Getting started#

2. Install required packages with e.g. conda/anaconda

conda install jupyter-book

3. Many chapters require further packages, which you should install if you plan to compile the whole book (not always required).

conda install numpy scipy matplotlib ipython ipywidgets jupyterlab
pip install itikz

4. The typical use case is that markdown (.md) and notebooks (.ipynb) are written with JupyterLab, which can be started (in the folder of the project) by

jupyter lab &

The easiest way to start is to write static Markdown files. This disadvantage is that then you cannot use live code-examples in your document. Upgrading from markdown to notebooks is however easy.

5. Once you are reasonably happy with your new content, compile the html-book (in the folder of the project) by

jupyter-book build .

Verify that the html-book looks like intended - some elements might work slightly differently.

## 1.6.2. Tips and tricks#

### 1.6.2.1. Audio#

Recording with python/jupyter is a nightmare, but playing audio works like a dream when using IPython and ipywidgets. See Vocoder for an example.

### 1.6.2.2. Interactive elements#

Some demonstrations are best when readers get to modify parameters themselves and to observe the output. IPython and ipywidgets allows doing that. Unfortunately, some of the more advanced interactive elements currently require a running jupyter/python kernel (i.e. have to run the notebook on a server). For an example, check out Representations/Spectrogram and the STFT.

### 1.6.2.3. Flow graphs and other static visualizations#

Tikz and PGF is a combination powerful tool for visualizations inside LaTeX documents. These can be incorporated in jupyter notebooks through itikz. For examples, see Security and Privacy.

Tip: In some scenarios, characters will get messed up when having multiple itikz-elements in a single document. If you’re having this issue, please see the solution at the discussion on Itikz-github.

### 1.6.2.4. Hidden and removed codeblocks in Jupyter notebooks#

Codeblocks can be used as pedagogical elements, but often there also codeblocks which are just to plot stuff, which is not very interesting for a reader. It is then recommended to hide the codeblock to avoid visual clutter. A button “click to show” button then appears where the reader can see how the backend works.

An alternative is to remove the codeblock from the output entirely. This is best in cases where the codeblock does not have any pedagogic purpose, like when creating flowgraphs with itikz (see above).

Both functions can be achieved with “tags” in the jupyter notebook. This can be found by clicking on the “Property inspector” logo at the far right side of the jupyter notebook. There it is possible to add either “hide-input” or “remove-input” for respective functions. For more information see Jupyter-book/Hide or remove content.

For an example of the use of both hidden and removed codeblocks, see Enhancement/Noise attenuation.

### 1.6.2.5. References and citations#

We use a bibtex based approach stored at references.bib. For instructions, see e.g. Jupyter-book/citations and references. For an example, see Computational_models_of_human_language_processing.

### 1.6.2.6. Executing notebooks / MyBinder badge#

To allow for the user to run the code in notebooks, there needs to be a jupyter server. Such a server can be installed locally (see Using this document), but importantly, there are public servers which can be used as well. The simplest approach is to use mybinder.org. To make it simple to use, please place a badge on top of the notebook, which directly runs the notebook on the server with the following steps:

1. Find the github url, folder and filename.

2. Enter the above in the form at mybinder.org.

3. This autogenerates a command to generate the badge; copy that code to the top of your jupyter notebook.

4. Be sure to test it by clicking the badge. For an example of use, see the page Noise attenuation.


text:
# Is there a source for a diagrammatic description of the induction functor C->Z(C)?

Suppose that C is a fusion category (over the complex numbers) and that Z(C) is its Drinfel'd center. By definition an object in Z(C) consists of an object V in C together with a collection of half-braidings $V \otimes W \rightarrow W \otimes V$ for every object W in C satisfying some naturality conditions. Hence there is a restriction functor R:Z(C)->C given by forgetting the half-braiding. Adjoint to this is an induction functor I:C->Z(C).

A Theorem of Etingof-Nikshych-Ostrik says that $R(I(V)) = \bigoplus_X X \otimes V \otimes X^{\*}$. In particular, we see that I(V) is $\bigoplus_X X \otimes V \otimes X^{\*}$ (where X ranges over simple objects up to isomorphism) together with some particular choice of half-braidings. I'm pretty sure I know what those half-braidings are. In particular there's a nice picture (the X,Y summand of the half-braiding with W is a sum over diagrams with a trivalent vertex connecting W to X* and Y* and another trivalent vertex connecting X and Y to W, where the two vertices range over dual bases).

What I really would like is a reference that explains this so that I don't have to write it up myself. The only description I know is in ENO's "On Fusion Categories" where it's written in terms of weak Hopf algebras.

The motivation is removing any mention of weak Hopf algebras from the construction in Section 5 (about cyclotomicity of certain Drinfel'd centers) in Scott and my Noncyclotomic Fusion Categories. It turns out that the diagram description above can be slightly modified (in a way suggested to me by Ben Webster) in order to give a description of I(V) where V is an object in a non-split fusion category over an arbitrary field.

-
Hi Noah. I'm a bit confused about a few sentences in your post. (1) "By definition Z(C) consists of an object V in C together with a collection of half-braidings..." Certainly you mean "an object of Z(C) is an object V in C together with a collection of half-braidings...", right? (2) I don't know what the sum $\bigoplus_X$ is supposed to be over. More generally, I guess I should read your paper with Scott to figure out what the Y and W are, and what are your trivalent vertices :) Which is all to say: I'm no help for a reference request. –  Theo Johnson-Freyd Mar 9 '10 at 23:11
I rewrote the question to address your excellent complaints. I wasn't sure what to say to clarify what Y is... The point is that maps between a direct sum and itself are indexed by two things, the summand in the source and the summand in the target. –  Noah Snyder Mar 9 '10 at 23:32
None of this is in the paper with Scott yet. By "trivalent vertex" I just mean an element of a space of the form Hom(A, B \otimes C). –  Noah Snyder Mar 9 '10 at 23:35
In the meanwhile a reference has appeared: Theorem 2.3 in arxiv.org/abs/1004.1533 –  pasquale zito Apr 13 '10 at 14:10
Thanks! If you make that an answer instead of a comment I'll accept it. –  Noah Snyder Apr 13 '10 at 16:25

In the meanwhile a reference has appeared: Theorem 2.3 in http://arxiv.org/abs/1004.1533

-

I suggest trying this paper of Mueger: http://arxiv.org/abs/math/0111205. In section 3.3 he defines a map $Hom_{C}(X,Y) \rightarrow Hom_{Z(C)}((X,e_X),(Y,e_Y))$ (actually a trace preserving conditional expectation) which, as far as I understand, is what you are looking for. It implies $R(I(V)) = \oplus_X X \otimes V \otimes X^*$ almost tautologically, I would say.

If I remember well, Mueger assumed $dim C$ to be non-zero, but this turned out to be always true (proven in the paper of Etingof-Nykshych-Ostrik).

Edit I read your question again, and now I see which half-braiding you mean. Proof and diagrams are almost the same as the ones used to show that $\oplus_X X \boxtimes X^{op}$ is a Frobenius algebra in $C \boxtimes C^{op}$. I doubt it has been written in detail anywhere.

Concerning the theorem, I don't know if the following is helpful. Denote $Q = \oplus_X X \boxtimes X^{op}$. $Z(C)$ is equivalent to the tensor category of $Q-Q$-bimodules. For $V \in C$, $I(V)$ corresponds to $Q \otimes (V \boxtimes 1) \otimes Q$. Notice that in general $(V^* \boxtimes 1) \otimes Q \sim (1 \boxtimes V^{op}) \otimes Q$ and there is a canonical choice for this isomorphism.

For $(Y,e_Y) \in Z(C)$, $(Y \boxtimes 1) \otimes Q$ has the structure of a $Q-Q$-bimodule: let $Q$ act on the right the obvious way, and use the half braiding $e_y$ to let $Q$ act from the left as well. All $Q-Q$ bimodules are of this form (up to isomorphism). Thus the restriction functor sends $(Y \boxtimes 1) \otimes Q$ to $Y$.

As you mentioned, $\oplus_X X \otimes V \otimes X^*$ has a natural choice of half braiding: choose a basis for each $Hom_C(X \otimes Y, Z)$, where $X,Y,Z$ span a complete set of irreducibles, and use them (together with the rigidity structure) to build an isomorphism $(\oplus_X X \otimes V \otimes X^*) \otimes Y \sim Y \otimes (\oplus_Z Z \otimes V \otimes Z^*)$ satisfying the necessary conditions.

Or define it through this sequence of isomorphims (so maybe you can avoid diagrams, in the end): $$(Y \boxtimes 1) \otimes ((\oplus_X X \otimes V \otimes X^*) \boxtimes 1) \otimes Q \sim (Y \boxtimes 1) \otimes (\oplus_X (X \otimes V ) \boxtimes X^{op}) \otimes Q$$

$$\sim (Y \boxtimes 1) \otimes Q \otimes ( V \boxtimes 1) \otimes Q \sim (1 \boxtimes Y^{op *}) \otimes ((\oplus_X X \otimes V \otimes X^*) \boxtimes 1) \otimes Q$$

$$\sim ((\oplus_X X \otimes V \otimes X^*) \boxtimes 1) \otimes ( 1 \boxtimes Y^{op *}) \otimes Q \sim ((\oplus_X X \otimes V \otimes X^*) \boxtimes 1) \otimes (Y \boxtimes 1) \otimes Q.$$

Thus $((\oplus_X X \otimes V \otimes X^*) \boxtimes 1) \otimes Q \sim Q \otimes (V \boxtimes 1) \otimes Q$ not only as objects in $C \boxtimes C^{op}$, but also as $Q-Q$-bimodules (and $(R(I(V)) \sim \oplus_X X \otimes V \otimes X^*$).

-
Could you clarify the tautological bit? I'm guessing you want to do something like Hom_C(V,R(Y)) = Hom_Z(C) (I(V),Y) = Im \pi: Hom_C(R(I(V)),Y)->Hom_Z(C)(I(V),Y), and then let Y vary over simples... But I'm not seeing the whole argument. –  Noah Snyder Mar 10 '10 at 2:02
Sorry, it's not so straightforward as I thought... The isomorphism between Hom spaces you were mentionig is Lemma 5.2. - unfortunately stated not directly in terms of Z(C), but of an equivalent category. –  pasquale zito Mar 10 '10 at 2:51
Added some stuff, and removed a false statement in an intermediate version. –  pasquale zito Mar 10 '10 at 13:31
That's actually really helpful. I knew the "dual over a module category" definition of the center, and I knew that any dual over a module category is the category of bimodules over an appropriate algebra object. But I hadn't realized that the algebra object was this one. That may actually allow me to avoid the whole question above and instead just build a version of that algebra object over the smaller field. –  Noah Snyder Mar 10 '10 at 16:37


text:
News & Views | Published:

MANY-BODY PHYSICS

# Escape the thermal fate

Nature Physicsvolume 14pages637638 (2018) | Download Citation

### Subjects

Many-body quantum systems fail to reach thermalization only under specific circumstances. An analysis now reveals a new, different kind of non-equilibrating dynamics based on the many-body analogue of quantum scars in single-particle quantum chaos.

The state of thermal equilibrium is the destiny of any sufficiently large isolated system after a sufficiently long time. Until now we knew only one way to avoid this fate: the presence of a set of independent conserved quantities, where the size of the set grows proportionally to the size of the system. This is the case for integrable systems, or if quenched disorder gives rise to many-body localization1. Writing in Nature Physics, Christopher J. Turner and co-workers2 have now introduced a substantially different way to avoid thermalization: owing to the presence of a limited number of special, atypical energy eigenstates, a system can exhibit persistent oscillations of observable quantities.

To understand what happens, let us review how systems thermalize. The key is the eigenstate thermalization hypothesis (ETH)3,4. Despite its name, related to historical reasons, the ETH is now believed to underlie thermalization in essentially every isolated quantum system that thermalizes1.

In any isolated quantum system, the time-dependent expectation value of the operator representing a certain observable can be written down as a sum of two terms. The first one is time-independent, and is given by the sum over all the diagonal elements of the operator in the basis formed by the energy eigenstates of the system, weighted by the occupation numbers — the modulus squared of the overlap between the system’s initial state and each energy eigenstate. All the off-diagonal elements of the operator are included in a second term, whose individual sub-terms oscillate in time. Now, if an equilibrium state — thermal or not1,5 — exists, the equilibrium value of the observable must be equal to the infinite time average of its time-dependent expectation value. This corresponds to the diagonal part alone, because the infinite-time average of each oscillating sub-term in the off-diagonal part is zero. This outcome is sometimes called the prediction of the diagonal ensemble.

On the other hand, according to statistical mechanics, the value of an observable at thermal equilibrium should follow the prediction of the microcanonical ensemble. This is given by an average over the same diagonal matrix elements as above, but without any weights and with the average running only over eigenstates that are very close in energy to the mean energy of the initial state of the system. And now, a conundrum: how can the predictions of the microcanonical and diagonal ensembles agree, when the former does not depend on the occupation numbers but the latter does? The only way they can agree for all initial states that are narrow in energy — a condition that is generically valid1 — is if the diagonal matrix elements have the same value and factor out of the sum in any energy window whose width is comparable to the energy width of the initial state. This is the core idea of the ETH.

It is at this point where systems with many additional conserved quantities behave differently: to obtain diagonal elements that are the same, one must consider not only an energy-narrow window, but additional windows that are narrow in all the conserved quantities. This is the microcanonical version of the generalized Gibbs ensemble1.

Furthermore, to have equilibration, the off-diagonal matrix elements should be exponentially small in the size of the system, which ensures that the expectation value will be close to the diagonal ensemble’s prediction for an overwhelming majority of the time.

When the ETH first became widely known, some researchers cautioned that even if most eigenstates of a generic system satisfy the ETH, embedded among them could be rare states whose diagonal matrix elements might be outliers6,7. The consensus nowadays is that this does not generically happen. Turner and colleagues now offer a vindication of this scenario: a system where rare eigenstates prevent thermalization of particular initial states. Additionally, this system does not even equilibrate, which means that some of the off-diagonal matrix elements are anomalously large, as well as some occupation numbers — leading to a violation of Riemann’s equilibration conditions5. Remarkably, though, this is an experimentally relevant scenario: indeed, explaining the recently observed persistent oscillations in a 51-Rydberg-atom quantum simulator8 was a motivation for the authors’ work.

Intriguingly, the authors suggested that these rare and special states, whose number grows only linearly with the system size, are many-body analogues of quantum scars in single-particle quantum chaos (Fig. 1). The argument goes as follows. The system considered by the authors is mostly chaotic, because the energy-level spacing approaches the Wigner–Dyson distribution, and most eigenstatesare are ergodically spread out over the Hilbert space, as measured by the participation ratio. However, the participation ratio of the special states is larger, meaning that they are more concentrated in certain parts of the Hilbert space, similarly to how quantum scars are more concentrated in certain parts of the real space. Just as quantum scars concentrate around classical periodic orbits, special states are found concentrated around certain product states. This gives rise, in an effective tight-binding model, to periodic motion connecting two product states.

The effect on the full system is persistent oscillations at the experimentally observed frequency in the time evolution of a particular initial state, labelled $$\left|{{\mathbb{Z}}}_{2}\right\rangle$$, which has high overlaps with the special states. Many-body localization can be ruled out, since the system is translationally invariant, is not disordered, and shows ballistic propagation of entanglement. Turner et al. suggested that what is behind the existence of special states is a dynamical constraint, as their system —a spin-1/2 chain — does not allow relaxation of several adjacent up-spins. Indeed, their $$\left|{{\mathbb{Z}}}_{2}\right\rangle$$ state is the alternated up–down spins forming a charge-density wave.

Numerous further questions suggest themselves. The relevant concept of many-body periodic trajectory — namely the analogy with the usual quantum scars — must be generalized and made more rigorous to enable us to understand and classify all initial states that result in persistent oscillations. We should also try to better understand what class of system exhibits this new kind of dynamics. On the basis of these results, one can eventually envision engineering systems that have states with very long coherence times, to be used in quantum information processing.

## References

1. 1.

D’Alessio, L., Kafri, Y., Polkovnikov, A. & Rigol, M. Adv. Phys. 65, 239–362 (2016).

2. 2.

Turner, C. J., Michailidis, A. A., Abanin, D. A., Serbyn, M. & Papić, Z. Nat. Phys. https://doi.org/10.1038/s41567-018-0137-5 (2018).

3. 3.

Deutsch, J. M. Phys. Rev. A 43, 2046–2049 (1991).

4. 4.

Srednicki, M. Phys. Rev. E 50, 888–901 (1994).

5. 5.

Reimann, P. Phys. Rev. Lett. 101, 190403 (2008).

6. 6.

Roux, G. Phys. Rev. A 81, 053604 (2010).

7. 7.

Biroli, G., Kollath, C. & Läuchli, A. M. Phys. Rev. Lett. 105, 250401 (2010).

8. 8.

Bernien, H. et al. Nature 551, 579–584 (2017).

Download references

## Author information

### Affiliations

1. #### Department of Physics, University of Massachusetts Boston, Boston, MA, USA

• Vanja Dunjko
•  & Maxim Olshanii

### Corresponding authors

Correspondence to Vanja Dunjko or Maxim Olshanii.

## About this article

### DOI

https://doi.org/10.1038/s41567-018-0157-1

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text:
Rust Simple Stats Program

I am brand new to Rust and want to know if there are any convention violations in my code and ways how I can make it more readable:

use std::io;

fn main() {
let mut vec: Vec<i32> = Vec::new();
loop {
println!("Enter any number, 0 to terminate");
let mut num = String::new();
io::stdin()
if num.trim().parse::<i32>().unwrap() == 0 {break;}
vec.push(num.trim().parse::<i32>().unwrap());
}
let sum: i32 = vec.iter().sum();
println!("Sum: {}", sum);
println!("Min: {}", vec.iter().min().unwrap());
println!("Max: {}", vec.iter().max().unwrap());
println!("Mean: {}", sum as f32 / vec.len() as f32);
}

io::stdin().read_line(&mut num);

Here you are ignoring the Result from read_line; you should have seen a compiler warning about this. Add .unwrap() or .expect() to check the error and panic if there is a problem reading:

io::stdin().read_line(&mut num).expect("Read error");

(Below I'll be talking about handling errors instead of panicking. This case is different because an IO error during reading shouldn't occur in normal usage and there isn't really any way to recover from it.)

        if num.trim().parse::<i32>().unwrap() == 0 {break;}
vec.push(num.trim().parse::<i32>().unwrap());

In these two lines you are parsing the number twice. Put the parse result in a variable instead.

        let num = num.trim().parse::<i32>().unwrap();
if num == 0 {break;}
vec.push(num);

If the user makes a typo and enters something that does not parse as i32 (e.g. a letter) then the program will panic since you are .unwrap()ing the parse result. It probably makes sense to tell the user and let them reenter instead. We can combine that with the handling of 0 inputs:

        match num.trim().parse::<i32>() {
Err(_) => {
println!("{:?} wasn't an integer. Try again.", num);
}
Ok(0) => {
break;
}
Ok(x) => {
vec.push(x);
}
}

Finally, you used .unwrap() for the min and max, which means that it will panic if the user doesn't enter any numbers. It's generally better for a program to not panic on any input. We can eliminate the unwrap from a single print like this:

    if let Some(min) = vec.iter().min() {
println!("Min: {}", min);
}

But, since three of the four outputs don't make sense for zero items, I think it makes more sense to put them inside a single condition, even if that means keeping the unwraps:

    println!("Sum: {}", sum);
if !vec.is_empty() {
println!("Min: {}", vec.iter().min().unwrap());
println!("Max: {}", vec.iter().max().unwrap());
println!("Mean: {}", sum as f32 / vec.len() as f32);
}

One more thing. If the input to the program reaches end-of-file without containing 0, it will loop infinitely, reading no characters and producing error messages. (Well, your initial version didn't, because it panicked on a parse error instead). The right thing to do is to check whether read_line met end of file; this is reported in the return value but also detectable by the fact that num will be the empty string (without any newline).

        io::stdin().read_line(&mut num).unwrap();
if num == "" {
break;
}

Putting together all my suggested changes:

use std::io;

fn main() {
let mut vec: Vec<i32> = Vec::new();
loop {
println!("Enter any number, 0 to terminate");
let mut num = String::new();
if num == "" {
break;
}
match num.trim().parse::<i32>() {
Err(_) => {
println!("{:?} wasn't an integer. Try again.", num);
}
Ok(0) => {
break;
}
Ok(x) => {
vec.push(x);
}
}
}
let sum: i32 = vec.iter().sum();
println!("Sum: {}", sum);
if !vec.is_empty() {
println!("Min: {}", vec.iter().min().unwrap());
println!("Max: {}", vec.iter().max().unwrap());
println!("Mean: {}", sum as f32 / vec.len() as f32);
}
}

1. If you really intended the app to be used, you would probably want to use StructOpt to give help, etc.
2. There are several places where the program looks like it would crash if you just immediately gave it a 0 without any other values: the .unwrap()s and the division at the end. You would probably want to check for that up front.
3. Rather than doing unwrap()s like that you could use the anyhow crate and make main return a Result<()>. Then your .unwrap()s could just be ?s.
4. You can change the way input is provided to make the program much more flexible AND make the code easier to read. Just read lines from stdin until EOF, flat_map them to the numbers they contain, and collect the results as a Vec -- all in one line (something like stdin.lock().lines().flat_map(...).collect()).
5. You could make the program able to handle arbitrarily large data sets by NOT collecting the results in a Vec, but rather fold()ing to get the sum, min, max, and count, and then using those to produce the output at the end.

Update: I decided to go ahead and implement 2/4/5, to see what it would look like; here it is in the playground. And actually, that unwrap() on the parse should really be an expect.


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# Is this particular group cyclic? [duplicate]

If we consider $(\mathbb{Z}/120\mathbb{Z})^{\times}$ ?

Using the CRT we have : $(\mathbb{Z}/120\mathbb{Z})^{\times} \simeq (\mathbb{Z}/2^3\mathbb{Z})^{\times} \ \times (\mathbb{Z}/3\mathbb{Z})^{\times} \ \times (\mathbb{Z}/5\mathbb{Z})^{\times}$

By other properties $(\mathbb{Z}/120\mathbb{Z})^{\times}\simeq \mathbb{Z}/2\mathbb{Z}\ \times\mathbb{Z}/2\mathbb{Z}\ \times \mathbb{Z}/2\mathbb{Z}\ \times \mathbb{Z}/4\mathbb{Z}$

But we know that $(\mathbb{Z}/2^3\mathbb{Z})^{\times}$ is not cyclic and moreover $\gcd(2,2,2,4)\neq 1$ so the product cannot be a cyclic group. Does that mean that $(\mathbb{Z}/120\mathbb{Z})^{\times}$ is not cyclic too ?

• For $n=120$ it is not cyclic, as you have shown. – Dietrich Burde Mar 6 '17 at 20:30
• @Student thank you in fact it's a trap because if the group is not cyclic you cannot apply a powerful result on the number of solutions of $x^k \equiv 1 \pmod{ n}$. You will have to use CRT and it will be longer. – Maman Mar 6 '17 at 20:36
• @Student There is a powerful lemma which says if $(Z/nZ)^{\times}$ is cyclic then the number of solutions of $x^k \equiv 1 \pmod n$ is $\gcd(k,\phi(n))$ – Maman Mar 6 '17 at 20:45


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# Calculate probability using inclusion-exclusion and deduce formula for binomial coefficicient

We choose uniformly a group of $$k$$ people selected from $$n$$. For $$m \leq k$$, calculate using inclusion-exclusion the probability that $$m$$ special people are in the group and then deduce that \begin{align} {n-m\choose k-m}=\sum_{j=0}^{m} (-1)^j {m\choose j} {n-j \choose k} \end{align}

With this problem the underlying poset consists of nodes $$Q\subseteq [m]$$ (the set of special people) which represent groups of $$k$$ people selected from the $$n$$ total where the elements of $$Q$$ are missing, or possibly more of the special people. The weights on the $$Q$$ are as usual $$(-1)^{|Q|}.$$ The cardinality of the groups represented at $$Q$$ is clearly $${n-|Q|\choose k}.$$ Counting the groups presented at all $$Q$$ multiplied by their weight yields the closed form

$$\sum_{Q\subseteq [m]} (-1)^{|Q|} {n-|Q|\choose k} = \sum_{q=0}^m {m\choose q} (-1)^q {n-q\choose k}.$$

On the other hand, counting by computing the total weight contributed by each of the groups we find that a group that has all the special people only appears at $$Q=\emptyset$$ with total weight $$(-1)^{|\emptyset|} = 1.$$ A group that has exactly $$P$$ of the special people missing where $$|P|\ge 1$$ appears in all $$Q\subseteq P$$ for a total weight of zero since

$$\sum_{Q\subseteq P} (-1)^{|Q|} = \sum_{q=0}^{|P|} {|P|\choose q} (-1)^q = 0.$$

We conclude from these weights that the above sum counts exactly those groups with none of the special people missing, these having weight one, and the others having weight zero. Therefore it is equal to

$${n-m\choose k-m}$$

Here we have selected all $$m$$ special people first and choose $$k-m$$ people from the remaining set. Note. When $$n-|Q| \lt k$$ the set of groups represented at $$Q$$ is empty, so that for $$m+k\gt n$$ the nodes of the top $$m+k-n$$ rows of the poset represent zero possible groups. This does not affect the calculation of the sum of the weights, however, since a group with the special people from some $$P$$ missing is represented by the nodes of the embedded poset spanned by the two nodes $$Q=P$$ and $$Q=\emptyset$$ (which is entirely below $$P$$ rather than above where the potentially empty nodes are). None of the nodes of this poset are empty if $$n-|P|\ge k$$ which is a precondition for the group to be realized in the first place.

As a remark, observe that the sum is not difficult to evaluate. We get

$$\sum_{q=0}^m {m\choose q} (-1)^q [z^k] (1+z)^{n-q} = [z^k] (1+z)^n \sum_{q=0}^m {m\choose q} (-1)^q (1+z)^{-q} \\ = [z^k] (1+z)^n \left(1-\frac{1}{1+z}\right)^m = [z^k] z^m (1+z)^{n-m} = [z^{k-m}] (1+z)^{n-m} \\ = {n-m\choose k-m}.$$


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# Definition:Probability Theory

## Definition

Probability theory is the branch of mathematics which studies probability spaces.

## Also see

• Results about probability theory can be found here.

## Historical Note

The discipline of probability theory is suggested by some sources to be a creation shared between Pierre de Fermat and Blaise Pascal.

Their intent was to describe certain games of chance and to calculate various probabilities.

They developed the fundamental principles in a series of letters in the year $1654$.

The field was developed significantly by Jacob Bernoulli, the results of his research appearing in his posthumous Ars Conjectandi of $1713$.

Modern probability theory is more than just calculating the chance of getting $m$ heads while tossing $n$ coins.

It is used to study problems in fields as diverse as economics, genetics, sociology, astronomy and physics.