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## Sunday, November 04, 2018 ... //

### Scholze, Stix don't have the magic power to veto arbitrary proofs

The $abc$ conjecture is a proposition in number theory somewhat analogous to Fermat's Last Theorem. If three relatively prime (possibly negative) integers obey $a+b=c$, then some inequality holds$\Large \max (\abs a, \abs b, \abs c) \leq C_\epsilon \prod_{p|(abc)} p^{1+\epsilon}.$ In 2012, Šiniči Močizuki (this website is written in Czech English, some of you appreciate it) presented his alleged proof and now, over six years later, the validity of the proof remains disputed and its status is therefore uncertain.

I find the inequality above rather contrived and uninteresting – which very well may be just because I haven't studied those corners (and most corners) of number theory intensely enough – but Močizuki claims to have a whole profound theory, Inter-universal Teichmüller (IUT) theory [=arithmetic deformation theory], which can generate proofs of many number-theoretical propositions. I feel that the broader theory attracts me more than the $abc$ conjecture itself.

The Japanese guy's discovery would be rather profound if his claims are true. His proposed theory is grand, needs a lot of time to be understood, builds on some Grothendieck's spirit that remains incomprehensible to most, extends the Teichmüller theory of moduli spaces into arithmetics, and returns topological groups to the analyses of number-theoretical problems. The theory depends on anabelian arithmetic geometry recently developed in Japan (1990-2014).

The Hodge theaters, miniature representations of schemes or rings in which the number field is separated from the multiplicative structures, play an important role in the theory. All such theaters are isomorphic to each other but they may also be mapped to each other in a different way. All pre-Mochizuki mathematics may be said to take place in a single theater. But in his theory, one may move in between the theaters (with "translation", "reconstruction", and "quantification of distortion" which is required to be small in some precise sense) which adds some seemingly useless redundancy, but because one may return to the original theory after a detour, it's possible to find out new facts about objects in a single theater.

I do feel that the theaters are morally analogous to different observers' interpretations of some processes in quantum mechanics and some people's inability to even consider the numerous theaters as a useful tool (and their obsessive instinct to get rid of this "redundancy" from the beginning) is analogous to the inability of many people to even consider the basics of quantum mechanics and its refutation of the classical objective existence of observables.

There are various reasons why not too many mathematicians have tried to absorb everything that Močizuki wants to say. It's too new, the papers are too large, depend on many things, the spirit disagrees with the toolkit that (not only) junior people in the field are supposed to repeat all the time. In particular, while Robert Langlands has expressed his desire to understand Močizuki's theory, the "Langlands program" is a much more popular paradigm outside Asia and could be interpreted as a competitor to Močizuki's IUT. To some extent, these regional preferences may be compatible with the health of the world's mathematics community. But some of the reasons for this calm reception might be rather political and pathological.

Even though I am not a clear fan of Močizuki or his ideas, I was shocked when I saw the top Quora answer to a straightforward question:

Did Peter Scholze and Jakob Stix really find a serious flaw in Shinichi Mochizuki's proof of $abc$ conjecture?
OK, Scholze and Stix made these negative claims, before and after some very tense March 2018 meetings with Močizuki. Especially Scholze is sort of a darling of certain community and the most often talked about fresh winner of the Fields Medal. Let me admit that my guess is that Močizuki is a far deeper mathematician than Scholze. So are the Europeans' criticisms valid?

What I've heard and seen from these criticisms seems shallow to me. It may be right but it doesn't look persuasive. People on Močizuki's side – who basically say to have verified the proof – may have been less famous with the media and they're less independent of Močizuki. But it seems to me that they have done much more genuine work than Scholze and Stix.

And I seem to have lots of empathy for the kind of feelings that Močizuki had while reading the criticisms, e.g.:
I can only say that it is a very challenging task to document the depth of my astonishment when I first read this Remark! This Remark may be described as a breath-takingly (melo?)dramatic self-declaration, on the part of SS, of their profound ignorance of the elementary theory of heights, at the advanced undergraduate/beginning graduate level.
Well, I don't really understand IUT but the sentiment above does indicate that he is fighting against a similar kind of pop-science and P.R.-driven superficial nonsense – and reacts very similarly as I react to kilotons of breathtaking melodramatic crackpots who often fail to understand advanced undergraduate or rather rudimentary graduate school material.

(Incidentally, if you care about the skeleton of the SS-vs-Močizuki disagreement, it's all about the SS' additional assumptions under which Močizuki's strategy yields vacuous results. He agrees but says it's really a straw man argument because if the additional assumptions aren't made, the results of his arguments are non-trivial, which is at least logically possible, and SS don't seem to have anything to counter this Močizuki's criticism of their argument. So unless I have overlooked some other part of the SS' criticism somewhere, their claim to have found a gap in the Močizuki proof is just a feeling combined with faith, not a fact.)

You may dislike any "personally sounding" comments like Močizuki's but at some moment, they simply are absolutely appropriate and needed. It's when a discussion becomes completely meaningless – lowered to a much lower level than the level where the actual scientific work was being done. And maybe it is the case here. Sometimes, we only scream that the emperor has no clothes and what the other side generates – whether it's polite or not – is just rubbish.

I am not able to settle the question whether Močizuki is right. But even with this uncertainty, I think that certain societal phenomena that are happening around mathematics – and it's obviously analogous to many things happening around physics and other fields – are certainly incredibly pathological.

OK, the Quora question about the validity of Scholze's and Stix's criticism of Močizuki's proof has two answers now. One, hugely read, upvoted, and insane one; and one, almost invisible and even less upvoted, sensible one.

The invisible answer is by Xiaowei Xu, one young mathematician, and points out that the fate of the proof and the possible new field of number theory is being crippled by street rumors spread by people who demonstrably haven't digested everything they need to digest in order to have a justified opinion. Xu, who links to a very wise and similarly frustrated essay by Ivan Fesenko (II), is disappointed by the rotten conditions in the field. After all, according to some rather standard rules that once worked, the proof could have been "officially validated" by a limited number of reviewers who give an approval to Močizuki's paper – and this has arguably happened. Xu's answer was read by less than 300 people and only has 4 upvotes including mine.

The much more widely read answer is by Senia Sheydvasser, a postdoc at CUNY who seems rather invisible to me in the world of actual papers. He is clearly one of the guys who became extremely influential on Quora – perhaps in some way analogously to the physics crackpot Richard Muller. OK, Senia's answer has over 2,500 views and over 100 upvotes. Dr Senia, have they found a gap in the proof?

I would put it thus: they certainly found a serious flaw in the proof. Whether or not the flaw is the one that they actually identify is not entirely clear.
Scholze and Stix have "certainly" found a serious flaw in the proof but it may be a different flaw than the flaw they are actually talking about! ;-) Holy cow, how can you be "certain" that they have found a "serious flaw" if the serious flaw is something that they haven't even "identified"? Is it possible to find something without identifying it?

Senia is basically saying that even without saying the proposition of the form $XY$, we must be "certain" that Scholze and Stix have "found" the proposition of the form $XY$ and it is a very important one. Unbelievable. Not even Isaac Newton was able to "find" groundbreaking scientific results without talking or writing about them, without "identifying" what they are supposed to be. So I was curious how this Senia justified his extraordinary claims:
What has happened with the $abc$ conjecture is that Mochizuki has a manuscript that he claims is a proof—however, no one (other than maybe some of his students) seems to be able to understand how it is a proof. This is a problem: [...]
The inability of most people including the bulk of mathematicians to understand Močizuki's proof may be considered a "problem" in some emotional or subjective way. But it is not a proof that Močizuki's proof is wrong. All state-of-the-art proofs of complex enough propositions in mathematics are incomprehensible to most people, including mathematics PhDs, especially to those who haven't dedicated an appropriate amount of time, aren't they? Senia seems to be aware of similar things:
[...] in practice, nobody ever writes out the whole proof with all of its gory logical details (as it would make the whole thing completely unreadable—imagine trying to understand how a computer program worked by reading machine code, or if you want to get really sadistic, binary). However, it should include enough information that any competent mathematician working in the area can reconstruct the full proof if needed.
Senia admits that proofs by professional mathematicians aren't supposed to be pedagogically perfect textbooks capable of teaching cutting-edge mathematics to the most retarded schoolkids. It clearly isn't even possible. It isn't even possible to guarantee that a proof of a complex enough proposition will be understandable to most mathematics PhDs.

But according to Senia, proofs are obliged to contain enough detail that any competent mathematician in the area may reconstruct the full proof if needed.

In reality, the history makes it totally clear that deep enough game-changing results in mathematics – but also in physics and all natural sciences – often face the violent misunderstanding not only by "some" professionals in the area but, rather frequently, "most" professionals in the area if not an overwhelming majority. So the condition Senia "demands" would have made many – if not most – of the truly important advances in mathematics and science impossible.

But let us ignore the fact that true revolutionaries were often "misunderstood" by the "peers".

There is an even deeper problem with Senia's condition. Even if you assume that the condition is legitimate – that the proofs should be reconstructable by specialized experts if needed – it isn't demonstrably relevant to the present situation and most other situations. Why? Because there's no evidence that Scholze and Stix really "needed" to reconstruct Močizuki's proof.

Instead, there is quite some evidence that they haven't tried hard enough to read all the texts and reconstruct the proofs. It seems rather likely that they're not even motivated to do so. As we can see, their simply saying that the proof is wrong is just fine for their personal careers and even fame. "Maybe" that's enough for what they care. And they will always find lots of people who trust them. So why should they reconstruct it?

Senia gets really explicit about his philosophy of a mindless belief in some chosen practitioners in his third paragraph:
Scholze and Stix are competent mathematicians, and have spent a protracted amount of time studying the proof and the underlying mathematics. If they don’t understand the proof, there is no proof.
Oh, really? All the world's hyped and celebrated geologists have spent a protracted amount of time – some 30 years – by studying Alfred Wegener's proof of the existence of continental drift. They said they didn't understand it and they said much worse things about it. Nevertheless, all sane people agree that what Wegener had was a proof of continental drift. We use more or less the same one when teaching the stuff today.

The Schutzstaffel guys are just two people. Four main U.S. mathematical societies have 35,800 numbers in total. Clearly, most of them are less competent in this branch of number theory than Scholze and Stix. But the number of those who are competent is still much higher than two. To say that "there is no proof because two particular people don't see a proof" is just plain insane. Even if a great majority of people similar to Scholze and Stix failed to get the proof, it would still fail to show that "there is no proof". It would still not be a solid reason for a professional number theorist not to study Močizuki's papers.

Senia's last paragraph starts as follows:
Now, Mochizuki’s objection is that Scholze and Stix are misunderstanding his proof. That might be true. However, if this is true, Mochizuki or one of his students could remedy the situation by writing out some of the details he is leaving out. [...]
As you can see, Senia – and those 100+ idiots who upvoted his answer – would like to turn mathematics (and probably also science and many other fields of human activity) to a platform of sociological rituals enforcing group think. After the group think is enforced, there is a vote, you know what the outcome of the vote is in advance, and that's how the truth is established. They don't want the substance to matter at all. Instead, what they want to be decisive is what someone says about the substance, and that someone should better be powerful and your ally. By having such allies, you don't need to read any "heretical" papers at all.

In fact, one commenter under Senia's answer makes this point rather clear. He celebrates that mathematics should turn into a social science – he praises Senia for highlighting the sociological mechanisms to decide about the truth value of proofs. Well, it isn't even a social science. It would be a mindless irrational cult.

We may describe this sick attitude to the truth as a generalization of the standard left-wingers' fight against hard and exact sciences, against the independent thinking, and for mindless group think revolving around grievance studies and adjacent social pseudosciences. But we may arguably pick much more specific analogies for this Senia-style thinking. The paragraph I quoted seconds ago is analogous to #MeToo. Why? Because the accuser is always claimed to be right.

If a disgusting and untrustworthy witch or bitch claims to have been raped by a successful man, everyone is bullied into believing these mostly ludicrous and insulting claims – in many cases, the successful man wouldn't even touch such an individual. Similarly, if a Scholze and a Stix say something bad about a proof by a Japanese man, they must be trusted. There is no proof!

This is not how mathematics can work, this is not how physics may work, this is not how courts may work, this is not how a civilized society may work. The accused person may also be right and this possibility mustn't be eliminated at the very beginning by some bizarre screaming and demagogic sociological would-be arguments. And criticized proofs may be solid, too. When a Scholze and a Stix fail to understand a proof, it doesn't have to imply a flaw in the proof. It may be a sign of their flaw, too. Even in the absence of dishonesty or laziness, a simple reason why they may fail to understand Močizuki's papers may be simply because they're far weaker, less creative, less imaginative mathematicians than he is – who simply can't learn anything truly new that is outside their mental boxes. If people are threatened so that they are even afraid to make this trivial point, it means that there are powerful pressures that want to eliminate all meritocracy and rational thinking from the process.

What we're seeing is that our institutions are being filled by dishonest and incompetent NPCs who claim to be competent but who always prefer to politically endorse an opinion that says that "there is even no proof to be read". It's so much easier for this lazy dishonest scum – a scum which praises itself, however – to join such a bandwagon than to study what they should study if they were actually fulfilling the moral duties that a mathematician really has if he or she is a real mathematician.

Needless to say, the critics of string theory, quantum mechanics, or natural sciences are analogous to one extent or another. They're lazy mediocre pseudointellectuals but because of the dropping standards and also affirmative action, they got to the system and they simply defend their political interests instead of doing honest scholarly work. They are building an infrastructure that makes sure that melodramatic self-declarations aligned with the collective interests of this lazy majority can politically beat genuine scientific arguments, including the most solid and profound ones. Incidentally, I think it's no coincidence that the target (Močizuki) is Asian – Asians (including people with names similar to Motlzuki and Nakamotl) are generally the main victims of affirmative action which is largely a new institutionalized racism. And it just happens that all the anti-Močizuki criticisms comes from the U.S. and Germany, two top politically correct countries with minimal experience in arithmetic anabelic geometry.

There is a controversy about Močizuki's proof and the "general recipes" proposed by Senia how to resolve such situations are clearly unacceptable and incompatible with any progress in mathematics or any discipline requiring mental powers. Scholze and Stix say that Močizuki doesn't have a proof of the $abc$ conjecture, Močizuki says that they don't have any proof that there is a mistake or gap in his proof. The situation is really symmetric and whoever claims to be "certain" that Scholze and Stix are right is simply a dishonest charlatan who places irrationality, group think, fallacies of sociological arguments, and laziness above the merit.

Sadly, the number of such people who completely suck has increased dramatically, even in the Academia – and maybe especially in the Academia.


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1. Math
2. Calculus
3. solve the following equation giving the exact solutions which lie...

# Question: solve the following equation giving the exact solutions which lie...

###### Question details

Solve the following equation, giving the exact solutions which lie in [0, 2π). (Enter your answers as a comma-separated list.)

2 tan(x) = 1 − tan2(x)

format: plain


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Difference between revisions of "Reach for the Stars"

Reach for the Stars
This event is an event held in the current season.

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Division B Results
 1st Daniel Wright Junior High School 2nd 3rd

Reach for the Stars is a Division B event for the 2019-2020 season. This event rotates every two years with Solar System. It was previously an event during the 2011-2012 season, 2012-2013 season, 2015-2016 season, and 2016-2017 season.

2016-2017 Rules

Each team is allowed to bring 2 double-sided 8.5" x 11" sheets of notes, and may be asked to bring a clipboard and red filtered flashlight. You are allowed to put anything on this paper, such as text, illustrations, tables, and pictures. Calculators are also allowed unless told otherwise by event supervisors.

Part I

In Part I, students are asked to identify a specified list of stars, constellations, and deep sky objects (DSOs), which may appear on star charts, HR diagrams, planetariums, or other forms of display. Teams must also be knowledgeable about the evolutionary stages of the stars and deep sky objects on the list.

For lists of this year's and past years' stars and objects, please see the Star and DSO list.

Star Charts

During competitions, the star charts given can be from any location on Earth, in any season and at any time of the night. Therefore, it is crucial to be able to recognize stars and constellations in any orientation.

Below are three star charts that together cover all Stars and Deep Sky Objects from the 2017 Reach for the Stars list. Mizar and Alcor are binary systems, while 30 Doradus is in the Large Magellanic Cloud.

 NYC, Summer Solstice
 NYC, Winter Solstice
 Sydney, Summer Solstice

Identification Tips

• Go outside and look at the night sky. This is a great way to learn the constellations. Look up into the sky and use a star chart to find constellations and stars. Doing this even a few times a month really pays off.
• Flash cards with the constellation/star/DSO on the front and its name on the back can really help. Make different flash cards with different information on the front page, such as constellation shape, location on a star chart, or pictures.
• Another tip is to use Quizlet, which is great for studying constellations, stars, and DSO's, and may show images that could be seen on a test.
• When working with star charts or looking at the night sky, many have found it very helpful to relate easy-to-find constellations such as Orion or Ursa Major (Big Dipper) to the constellations around them. This guides you to the constellation via others, rather than having to rely only on the shape. It may be helpful to include a section on the reference sheet about finding constellation you have trouble with on a sky chart. Common "pathways" include:
• Using the handle of the Big Dipper asterism to find Bootes and Arcturus, and Virgo and Spica, and using the cup to find Ursa Minor and Polaris.
• Using Orion to find Taurus, then Auriga, and then Gemini, Perseus, and Andromeda. Cassiopeia is also good for finding Perseus and Andromeda.
• Looking for the Winter Triangle (Betelgeuse, Sirius, Procyon) and Summer Triangle (Vega, Altair, Deneb) to find those stars and constellations.
• Using the Zodiac constellations, which are close to each other around the Ecliptic. This helps with Leo, Virgo, Scorpius, Sagittarius, and Ophiuchus (along with Serpens) especially, but also Gemini and Taurus.
• Sometimes, the test will use a StarLab or planetarium for the identification portion. Put some time in to familiarize yourself with how the skies look on it. This will help reduce confusion on the identifications and reduce the amount of time spent on those questions.
• There is always a chance that a bad star map may be used, so make sure to get accustomed to using less-than-clear maps and images.
• In general, the brightest star (lowest apparent magnitude) in a constellation is denoted Alpha, and second brightest Beta, and so on. After the Greek alphabet has been used up, numbers are used - 1 is dimmer than Omega but brighter than 2. However, there are exceptions: Betelgeuse (Alpha Orionis) usually appears dimmer than Rigel (Beta Orionis), and Castor (Alpha Geminorum) appears to be dimmer than Pollux (Beta Geminorum).
• While identification is not the only part of this event, it is a good way to begin preparing. For the rest of the event, see Part II. A good resource is Astronomy Today.

Part II

In Part II, students are asked to complete tasks relating to a topics in Astronomy. Students should know about the general characteristics of stars, galaxies, star clusters, etc. Furthermore, students should be able to find a star's spectral class, surface temperature, and evolutionary stage (protostar, main sequence, giant, supergiant, white dwarf) given its position on the Hertzsprung-Russell diagram, or place the star on the diagram given these information. Finally, students should be familiar with basic Astrophysics knowledge such as Luminosity scales and relationships, temperature relationships, flux, and distance measures. For a more in-depth study of astrophysics, please see the Astronomy page, but Reach for the Stars is unlikely to reach the level of complexity of the C Division event.

Stellar Evolution

This section gives a brief overview of Stellar Evolution. A more detailed discussion of stellar evolution may be found on the Astronomy/Stellar Evolution page, and other associated pages.

Stellar Evolution for stars with different masses.

Formation

Star formation starts in a dense interstellar cloud - a dark dust cloud or a molecular cloud. After some instability or disturbance, the cloud collapses under its own gravity, and form clumps of matter. As the clump contracts further, its density grows, its temperature rises, and the clump becomes a protostar.

The protostar contracts further and evolves onto the main sequence following the Hayashi track. It enters main sequence when nuclear fusion begins.

Main Sequence

Stars spend the majority of their lives on the main sequence. Heavier stars spend significantly less time on the main sequence because their rate of fusion is much higher. The radiation pressure from fusion balances with gravity, and the star is in hydrostatic equilibrium.

Lower-mass stars fuse Hydrogen into Helium mostly using the proton-proton (p-p) chain. This process starts at around $4\cdot10^6$ Kelvins.

Higher-mass stars fuse Hydrogen into Helium mostly using the CNO Cycle. The CNO cycle uses Carbon, Nitrogen and Oxygen as catalysts, and is dominant in temperatures higher than $17\cdot10^6$ Kelvins.

Post-Main Sequence, Low Mass

When the core of a low-mass star is depleted of Hydrogen, nuclear fusion subsides because Helium fusion occurs at much higher temperature. The core contracts, and the heat generated by the contraction heats up the outer layers, creating a Hydrogen-burning shell. The star also expands and becomes a red giant.

The core then reaches high enough temperature for helium fusion. For a moment, helium is fused rapidly in a runaway condition known as the helium flash, but then subsides as the star enters the horizontal branch.

When a low-mass star runs out of helium, it does not heat up high enough to fuse Carbon into heavier elements, and fusion stops. The core compresses further into a white dwarf, when electron degeneracy pressure prevents it from compressing further, and the outer shell expands into a planetary nebula, heated up by the white dwarf core. The white dwarf slowly cools and becomes a black dwarf.

Post-Main Sequence, High Mass

When the core of a high-mass star is depleted of Hydrogen, the star expands into a red supergiant. Its luminosity stays roughly the same because the temperature of the outer shell decreases. Helium starts fusing into carbon without a runaway process, so there is no helium flash.

A high-mass star is able to fuse elements up to iron, at which point further fusion consumes energy instead. The iron core contracts further. The star explodes in a bright core-collapse supernova. Then, either electrons combine with protons and the core is made of neutron degenerate matter - a neutron star - or the core contracts further into a black hole. Neutron stars rotate very rapidly and have very strong magnetic fields. Some neutron stars are called pulsars and magnetars for their additional features.

Spectral Classification

Harvard Spectral Classification

There are 7 spectral Classes (O,B,A,F,G,K,M). This order is based on decreasing surface temperature. A Class stars have the strongest Hydrogen lines, while M-Class stars have the weakest hydrogen lines. Each class is then subdivided into 10 subdivisions (0-9).

The following is a table with properties of each of the spectral classes.

Spectral Class Properties
Type Temperature (Kelvin) Color Hydrogen
O 30,000-60,000 Blue Weak
B 10,000-30,000 Blue-White Medium
A 7,500-10,000 White Strong
F 6,000-7,500 White Medium
G 5,000-6,000 Yellow Weak
K 3,500-5,000 Yellow-Orange Very Weak
M 2,000-3,500 Red Very Weak

The following is the class of each of the stars on the list:

Class O: Zeta Ophiuchi
Class B: Rigel, Spica, Regulus, and Algol
Class A: Vega, Sirius A, Deneb, Altair, Castor, Mizar, and, Alcor
Class F: Procyon, and Polaris
Class G: Capella
Class K: Arcturus, Aldebaran, and Pollux,
Class M: Betelgeuse, and Antares

There are also S, N, and Y for brown dwarfs, which are generally not considered stars.

Yerkes Spectral Classification

The Yerkes Spectral Classification is based on luminosity and temperature. It is also known as luminosity classes. There are seven main luminosity classes:

Type Ia: Bright Supergiants
Type Ib: Normal Supergiants
Type II: Bright Giant
Type III: Normal Giant
Type IV: Sub-Giants
Type V: Main Sequence
Type VI: Sub-Dwarf
Type VII: White Dwarf

There is also Type 0, for hypergiants. However, these are exceedingly rare; examples include VY Canis Majoris, the Pistol Star, and R136a1.

NOTE: This and the following section contain some algebra. If you are not yet comfortable with algebra, you can still read these sections for the theoretical concepts.

The radiation laws show relationships between stellar temperature, radius, and luminosity. All three laws are regarding black bodies, ideal objects that absorbs all incoming radiation. Stars, with little incoming radiation, are often approximated as black bodies to simplify calculations.

Both Wien's Law and Stefan's Law are proportionality statements, that a change in one quantity is always accompanied by change in other. These can be turned into equations by introducing a constant known as a proportionality constant. The proportionality statement $y\propto x$ denotes that if $x$ changes by a factor $k$ (here, "k" is just an arbitrary variable), $y$ also changes by $k$. However, it does not mean that the values are equal to each other: a proportionality constant needs to be added.

At Division B it is unlikely that one will perform calculations with these laws, but general questions regarding these laws, such as the proportionality, may be asked. For details about calculations with these laws, visit the Astronomy page.

Wien's Law

Wien's displacement law states that the wavelength where a black-body emits most of its radiation is inversely proportional to the temperature. In other words, as the temperature of a star increases, the wavelength at which the star emits most of its radiation decreases. As a proportionality statement,

$\lambda_{max}\propto\frac1T$,

where ${\lambda}_{max}$ is the wavelength of maximum output of radiation from an object and $T$ is Temperature. For example, if the temperature of a star is multiplied by 2, the wavelength of maximal radiation would be divided by 2.

In order to use this as a normal equation, a proportional constant needs to be added. In this case, $b=2900\mu m\cdot K$ known as the Wien's displacement constant, is used. Then,

$\lambda_{max}=\frac{b}{T},$

where $\lambda_{max}$ is in micrometers and $T$ is in Kelvin. The equation form of the law is more common in Division C, and is unlikely to appear in this event.

Stefan-Boltzmann's Law

The Stefan-Boltzmann Law states that the total energy emitted from a black-body per unit surface area is proportional to the fourth power of its temperature. In equations,

$j^*\propto T^4$

where $j^*$ is the total energy emitted per unit area and $T$ is Temperature. For example, if the temperature of a star is multiplied by 2, the total energy emitted per unit surface area would be multiplied by $2^4=16$.

The proportional constant for this equation is the Stefan–Boltzmann constant $\sigma=5.67\cdot 10^{-8}\mathrm{W/m}^2\mathrm{K}^4$. Therefore, $j^*=\sigma T^4$, where $j^*$ is in Watts per square meter, and $T$ is in Kelvin. Again, the equation form is unlikely to appear.

Since all black-bodies we encounter are considered to be spheres, they have surface area $A=4\pi R^2$, where $R$ is the radius of the object. The luminosity ($L$) of a star, which is the total energy emitted from the star, is the product of $j^*$ and $A$. Therefore, since $A\propto R^2$ and $j^*\propto T^4$, we end up with the following relationship:

$L\propto R^2 T^4.$

Stefan's Law relates three different quantities of a star: its luminosity, temperature and radius. Notice that an increase in temperature will have much more of an effect on luminosity than an increase of the same factor in radius, since temperature is raised to the fourth power.

Putting all of the constants back into the relationship to make it an equation again, $L=4\pi {R}^{2}\sigma {T}^{4}$, with $L$ in Watts, $R$ in meters, and $T$ in Kelvin. Again, questions requiring the use of this specific equation are beyond the scope of this event.

Planck's Law

Planck's Law states that a hotter blackbody emits more energy at every frequency than a cooler blackbody. The equation form of the law is complicated, while on a radiance vs. temperature graph the law states that the curve for a hotter blackbody never dips below that of a cooler one.

Magnitude and Luminosity Scales

The luminosity of a celestial object refers to how much radiation (visible light, infrared, x-ray, etc.) it emits per unit time. Luminosity is measured in Joules per second or Watts. The luminosity, of the sun, for example, is $L_\odot=3.846\cdot 10^{26}$ watts. Magnitude scales are different methods to express luminosity.

Apparent Magnitude

The apparent magnitude, denoted by $m$, denotes the brightness of a celestial object as seen by an observer on Earth. The brighter an object appears, the lower its apparent magnitude. It is a logarithmic scale, not a linear scale, which means that a small decrease in magnitude results in a much greater increase in luminosity. For example, an object with apparent magnitude 5 less than that another would seems 100 times more luminous. Logarithms are very advanced for Division B, but at the very least, it is important to know that a small change in magnitude represents a much larger change in luminosity.

For example, the sun has apparent magnitude of -26.74, while Deneb has apparent magnitude of 1.2. Because of this, the sun seems $100^{(1.2+26.74)/5}\approx 150 \text{ billion}$ times brighter than Deneb! Apparent magnitude depends on both the luminosity of the object and its distance from Earth: while Deneb is more luminous than the sun, to an observer on Earth it is dimmer because it is farther away.

The system of apparent magnitude originated from Greece, where the brightest stars in the night sky were of first magnitude ($m=1$), while the faintest to the naked eye were of sixth magnitude ($m=6$). The system was formalized and extended beyond 1 to 6 in 1856 by N. R. Pogson.

Absolute Magnitude

The absolute magnitude, denoted by $M$, denotes the brightness of a celestial object as seen by an observer 10 parsecs (about 32.6 light years) away from the object. Similarly, an object with absolute magnitude 5 less that of another would be 100 times more luminous. The absolute magnitude is basically another way of expressing the luminosity of the object. Scientists often consider the absolute bolometric magnitude $M_b$ of an object, meaning that its radiation is being measured across all wavelengths.

For example, the sun has absolute magnitude 4.83, while Deneb has absolute magnitude -8.38. This means Deneb is $100^{(4.83+8.38)/5}\approx 192\text{ thousand}$ times more luminous than the sun. A typical Type Ia supernova has an absolute magnitude of about -19.3.

Inverse Square Law

The inverse square law says that a certain quantity is inversely proportional to the square of the distance relating to that quantity. In this case, "inversely proportional" means that an increase of one number causes a decrease in the other number. For example, suppose an astronomer measures a star of some intensity ($I_1$) at a certain distance ($D$) from the source. By the inverse square law, we have the following proportion:

$I_1\propto \frac{1}{D^2}$.

This law also applies to Newton's Law of Gravitation. The law states that:

$F=\frac{Gm_1 m_2}{r^2}$

where $m_1$ and $m_2$ are the masses of two objects, $r$ is the distance between the two objects, and $G$ is a special constant called Newton's gravitational constant. Since most objects in space are very far away from each other, the bottom part of the fraction is much larger than the top part of the fraction, so the law can be approximated for most far-apart objects by $F\approx\frac{1}{r^2}$.

The law also applies to the electrostatic force and the intensity of sound wave in a gas.

Distance Modulus

Distance modulus is a way to relate the absolute and apparent magnitudes of objects with the distance between them. The distance modulus equation is as follows:

$m-M = 5\log_{10} (d) - 5,$

where $m$ is apparent magnitude, $M$ is the absolute magnitude, and $d$ being the distance to the object in parsecs. This equation uses a logarithm, which will more often than not be outside the scope of the Division B event, but for a brief tutorial on logarithms, please see this link. A different way of expressing this equation is:

$d=10^{\frac{m-M+5}{5}}$.

For example, the Supernova SN 2011fe had peak apparent magnitude of $m=+9.9$, while its absolute magnitude is about $M=-19.3$. Therefore, the supernova is

$d=10^{\frac{9.9+19.3+5}{5}}\approx 7\text{ Mpc},$

away from Earth, close to the experimental value of $6.4\pm0.5\text{ Mpc}$.

This relationship can be found by using the laws that were discussed earlier. By the inverse square law, an observer 10 times as far as another from the same object would see the object as being 100 times less bright, and so they would mark it as having an apparent magnitude of 5 more than the other observer. Since absolute magnitude is the "apparent magnitude" of an observer 10 parsecs away, the first distance equation can be found by relating all of these factors. This is helpful to know in case theoretical questions are asked about the distance modulus relationship, but tests are unlikely to ask about the fine details for Reach for the Stars.

Galaxies

There are three main types of galaxies: Spiral, Elliptical, and Irregular. However, in the 2013 rules, there are no galaxies on the list. Nevertheless, galaxies are an important part of astronomy, so here is a brief background on the types of galaxies.

Spiral Galaxies

An example of a spiral galaxy: (M31 Andromeda Galaxy)

Spiral Galaxies are named so because they have prominent spiral arms and a central "galactic nucleus" or central bulge.

An example of a Barred-Spiral Galaxy: (NGC 1300)

Spiral Galaxies also have a very large rate of star formation in the spiral arms of the galaxy. Also, almost all spiral galaxies have a galactic halo that surrounds the galaxy. These halos contain stray stars and globular clusters. It is also theorized that many spiral galaxies have supermassive black holes at the center of the galaxy. Our own galaxy, The Milky Way, is a spiral galaxy, and is also theorized to have a supermassive black hole at its center, called Sgr A*. There is also a sub-division of spiral galaxies, known as barred-spiral galaxies. Barred-spirals have a central bar, and then have spiral arms shooting off at each end of the bar.

Spirals are classified by presence of a central bar and how tightly the rings are wound.

Sa tightly wound, no central bar moderately tightly wound, no central bar loosely wound, no central bar very loosely wound, no central bar tightly wound, central bar moderately tightly wound, central bar loosely wound, central bar very loosely wound, central bar

The spiral galaxies on the list for 2020 are:

• M31 Andromeda Galaxy (in Andromeda)
• M101 Pinwheel Galaxy (in Ursa Major)
• NGC 4676 Mice Galaxies (in Coma Berenices)
• NGC 4038/4039 Antennae Galaxies (in Corvus)

Lenticular Galaxy

Lenticular Galaxies are intermediate between spiral and elliptical Galaxies, they contain a large scale disk, but do not have spiral arms. Like Elliptical Galaxies, they contain older Stars and have a low rate of star formation.

The Lenticular Galaxies on the list for 2020 are:

• M104 Sombrero Galaxy (in Virgo)
• NGC5128 Centaurus A (in Centaurus)
An example of an Elliptical Galaxy:(M84)

Elliptical Galaxy

Elliptical Galaxies appear just like they sound- they are elliptical/ spherical. Elliptical Galaxies contain mostly old Population II stars, and also, they have a very low rate of star formation because there is barely any interstellar matter in elliptical galaxies. There is the least amount of Elliptical Galaxies in the known Universe. Also, they are classified by how spherical they are, with E followed by a number from zero to seven. Zero indicates perfectly spherical; seven indicates the extremely elongated and cigar-shaped.

The Elliptical Galaxies on the list for 2020 are:

• M60 (in Virgo)
• NGC 4555 (in Coma Berenices)
An example of an Irregular Galaxy

Irregular Galaxies

Irregular also appear just how they sound- they are without a definite shape. They are normally formed by Spiral or Elliptical Galaxies that have been deformed by different forces- such as gravity. They contain a lot of interstellar matter. There are distinctions between "normal" irregular galaxies - with no hint of shape - and peculiar galaxies, that have some hint of form - usually, they were bent out of shape by outside forces or became violently active.

The Irregular Galaxies on the list for 2020 are:

• Large Magellanic Cloud (in Dorado and Mensa)
• Small Magellanic Cloud (in Tucana)

Sample Tests

Identification practice: Reach for the Stars Test (2009)
RFTS Test and Pic Sheet for the test
Also be sure to check out the Reach for the Stars Test Exchange.


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## Electromagnetic corrections to the hadronic vacuum polarization of the photon within QED_L and QED_M

Preprint number: CP3-Origins-2017-41 DNRF90
Authors: Andrea Bussone (CP3-Origins & DIAS), Tadeuz Janowski (CP3-Origins & DIAS), and Michele Della Morte (CP3-Origins & DIAS)
We compute the leading QED corrections to the hadronic vacuum polarization (HVP) of the photon, relevant for the determination of leptonic anomalous magnetic moments, $$a_\ell$$. We work in the electroquenched approximation and use dynamical QCD configurations generated by the CLS initiative with two degenerate flavors of non-perturbatively O($a$)-improved Wilson fermions. We consider QED$$_{\rm L}$$ and QED$_{\rm M}$ to deal with the finite-volume zero modes. We compare results for the Wilson loops with exact analytical determinations. In addition we make sure that the volumes and photon masses used in QED$$_{\rm M}$$ are such that the correct dispersion relation is reproduced by the energy levels extracted from the charged pions two-point functions. Finally we compare results for pion masses and the HVP between QED$$_{\rm L}$$ and QED$$_{\rm M}$$. For the vacuum polarization, corrections with respect to the pure QCD case, at fixed pion masses, turn out to be at the percent level.


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# Latex reference sheet for usage with Word 2007

Anyone knows where one could find a Latex reference sheet for usage in Word 2007? I'm interested in some which mostly deals with writing math. Formulas, text formatting and page layout are of lesser importance.

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You either use Word or you use LaTeX. You can't do both at the same time.

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The Microsoft Equation Editor is not LaTeX. It has some similarities of syntax, but it would be no use having a LaTeX reference sheet as there are too many differences. What you want is a MS-Word equation editor reference sheet. – Rob Hyndman Dec 20 '09 at 12:47
... okey, fair enough ... I did specify "for usage with word'07" ... so that should've given something away ... – Rook Dec 20 '09 at 18:08
... in any case, every formula that I had so far in a latex document worked without question ... – Rook Dec 20 '09 at 18:09
You can edit Latex source in Word. In fact, that can be a very useful thing to do sometimes; I've worked this way with some clients. – Charles Stewart Jan 17 '10 at 2:04

If you're using MS Office, you can't access LaTeX. It's like driving bus and car in one time...

As you commented Rob's post, Word 2007

For Microsoft Office 2007, Equation Editor is included as is, however some Office 2007 applications also include a reengineered equation editor with support for a TeX-like linear input/edit language called "Office Math Markup Language" (Office MathML or OMML) in addition to its WYSIWYG interface.

For me that means, that programmers of MS created GUI for using TeX for typesetting math. So that TeX is used only in box with math text, rest of the document is created by Word.

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Tex-like means they’re actually not using Tex but something which is, well, Tex-like. – Debilski Dec 20 '09 at 13:52
@Crowley - ... I thought I mentioned exactly that in my question ... – Rook Dec 20 '09 at 18:10


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# Without adding, find the sum: i) 1+3+5+7+9\  ii) \ 1+3+5+7+9+11+13+15+17+19 iii) 1+3+5+7+9+11+13+15+17+19+21+23

Updated On: 27-06-2022

Text Solution

Solution : (i) In the first series we are given 5 terms having difference 2 between two consecutive numbers.
Therefore, their sum will be:
5^2=25

(ii)In the second series we are given 10 terms having difference 2 between two consecutive numbers.
Therefore, their sum will be:
10^2=100

(iii)In the third series we are given 12 terms having difference 2 between two consecutive numbers.
Therefore, their sum will be:
12^2=144.


text:
Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp019z903232z
 Title: Development of miniaturized, spectroscopically assisted Penning gauges for fractional helium and hydrogen neutral pressure measurements Contributors: Flesch, K.Kremeyer, T.Schmitz, O.Soukhanovskii, V.Wenzel, U.U. S. Department of Energy contract number DE-AC02-09CH11466DE-SC0012315DE-SC0014210DE-AC52-07NA27344 Keywords: Penning gaugePartial pressureHelium spectroscopy Issue Date: Nov-2016 Publisher: Princeton Plasma Physics Laboratory, Princeton University Related Publication: Review of Scientific Instruments, vol 87, p. 11E529 (November 2016) Abstract: Direct measurements of the helium (He) fractional neutral pressure in the neutral gas around fusion devices is challenging because of the small mass difference between the abundant D2 molecules and the He atoms which will be produced by D-T fusion. To study He exhaust, Penning gauges were used to measure total neutral pressure assisted by spectroscopy to resolve the D and He partial pressures. In this contribution, initial results are shown from developing this technique into a miniaturized configuration for direct in-situ measurements in the divertor of fusion devices. The configuration is based off a gauge originally designed for the National Spherical Tokamak Experiment-Upgrade (NSTX-U). The goal of this new miniaturized design it to reduce the space required by the gauge on the device and use of the inherent magnetic field of the machine rather than permanent magnets inside the gauge, enabling it to be adapted into a system that can be extended directly into the divertor region. The feasibility test of the method for NSTX-U and the Wendelstein 7-X (W7-X) stellarator are surveyed. For W7-X, a commercial Penning Gauge has been installed on an outboard vacuum flange as a generic feasibility test in the neutral gas environment of a stellarator. At an integration time of 25s, helium lines can be seen down to 10^-5 mbar and H-alpha lines down to 10^-6 mbar. Successful measurement of the total as well as the fractional neutral pressures of He and H has been shown. A first prototype of the miniature Penning gauge has been tested in Madison and shows a near linear power law scaling between current and pressure: I = C*P^n with n = 1.0 - 1.2. Pressure measurements were achieved starting at 10^-3 mbar and down to 10^-6 mbar. A modular gauge is being assembled, which allows easy interchangeability of the anode to test new anode geometries, in order to improve optical access and increase spectroscopic sensitivity. This shall enable an increase of the time resolution of the spectroscopically assisted fractional neutral pressure measurements to up to 1kHz. URI: http://arks.princeton.edu/ark:/88435/dsp019z903232z Referenced By: http://dx.doi.org/10.1063/1.4960815 Appears in Collections: NSTX-U

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# Lanthanides and actinides resemble in

$\begin{array}{1 1}(a)\;\text{Electronic configuration}&(b)\;\text{Ionization energy}\\(c)\;\text{Oxidation state}&(d)\;\text{Formation of complexes}\end{array}$

The oxidation state in both (lanthanide and actinide) is +3. The property of actinide are very similar to those of lanthanide when both are in +3 state
Hence (C) is the correct answer.
edited Oct 2

Oxidation state


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# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$

### 3.2.2 The Homotopy Groups of a Kan Complex

Let $X$ be a topological space and let $x \in X$ be a point. For every positive integer $n$, we let $\pi _{n}(X,x)$ denote the set of homotopy classes of pointed maps $( S^ n, x_0) \rightarrow (X,x)$, where $S^ n$ denotes a sphere of dimension $n$ and $x_0 \in S^ n$ is a chosen base point. The set $\pi _{n}(X,x)$ can be endowed with the structure of a group, which we refer to as the $n$th homotopy group of $X$ (with respect to the base point $x$). Note that the sphere $S^ n$ can be realized as the quotient space $| \Delta ^ n | / | \operatorname{\partial \Delta }^ n |$, obtained from the topological simplex $| \Delta ^ n |$ by collapsing its boundary to the point $q$. We can therefore identify pointed maps $(S^ n, x_0) \rightarrow (X,x)$ with maps of simplicial sets $f: \Delta ^ n \rightarrow \operatorname{Sing}_{\bullet }(X)$ which carry the boundary $\operatorname{\partial \Delta }^ n$ to the simplicial subset $\{ x\} \subseteq \operatorname{Sing}_{\bullet }(X)$. In , Kan elaborated on this observation to give a direct construction of the homotopy group $\pi _{n}(X,x)$ in terms of the simplicial set $\operatorname{Sing}_{\bullet }(X)$ (and the vertex $x$). Moreover, his construction can be applied directly to any Kan complex.

Notation 3.2.2.1. Let $B$ be a simplicial set and let $A \subseteq B$ be a simplicial subset. We let $B/A$ denote the pushout $B \coprod _{A} \{ q_0\}$, formed in the category of simplicial sets. We regard $B/A$ as a pointed simplicial set, with base point given by the vertex $q_0$.

Remark 3.2.2.2. Let $B$ be a simplicial set and let $A$ be a simplicial subset. Then the simplicial set $B/A$ can be described more informally as follows: it is obtained from $B$ by collapsing the simplicial subset $A \subseteq B$ to a single vertex $q_0$. Beware that this informal description is a bit misleading when $A = \emptyset$: in this case, the natural map $B \rightarrow B/A$ is not surjective (instead, $B/A$ can be described as the coproduct $B_{+} = B \coprod \{ q_0 \}$, obtained from $B$ by adding a new base point).

Example 3.2.2.3. For $n \geq 0$, the geometric realization $| \Delta ^ n / \operatorname{\partial \Delta }^ n |$ can be obtained from the topological $n$-simplex $| \Delta ^ n |$ by collapsing the boundary $| \operatorname{\partial \Delta }^ n |$ to a point (or by adding a new base point, in the degenerate case $n = 0$). It follows that $| \Delta ^ n / \operatorname{\partial \Delta }^ n |$ is homeomorphic to a sphere of dimension $n$.

Construction 3.2.2.4. Let $(X,x)$ be a pointed Kan complex and let $n$ be a nonnegative integer. We let $\pi _{n}(X,x)$ denote the set $[ \Delta ^ n / \operatorname{\partial \Delta }^ n, X]_{\ast }$ of pointed homotopy classes of maps from $\Delta ^ n / \operatorname{\partial \Delta }^ n$ to $X$ (Notation 3.2.1.8). For $n > 0$, we will refer to $\pi _{n}(X,x)$ as the $n$th homotopy group of $X$ with respect to the base point $x$ (see Theorem 3.2.2.10 below). In the special case $n = 1$, we refer to $\pi _{1}(X,x)$ as the fundamental group of $X$ with respect to the base point $x$.

Notation 3.2.2.5. Let $(X,x)$ be a pointed Kan complex and let $n$ be a nonnegative integer. Then the set of pointed morphisms $\Delta ^ n / \operatorname{\partial \Delta }^ n \rightarrow X$ can be identified with the set of $n$-simplices $\sigma : \Delta ^ n \rightarrow X$ having the property that $\sigma |_{ \operatorname{\partial \Delta }^ n}$ is equal to the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \subseteq X$. In this case, we write $[\sigma ]$ for the image of $\sigma$ in the set $\pi _{n}(X,x)$. Note that, if $\tau$ is another $n$-simplex of $X$ for which $\tau |_{ \operatorname{\partial \Delta }^ n}$ is the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \subseteq X$, then the equality $[\sigma ] = [\tau ]$ holds in $\pi _{n}(X,x)$ if and only if there exists a homotopy $h: \Delta ^1 \times \Delta ^ n \rightarrow X$ satisfying $\sigma = h|_{ \{ 0\} \times \Delta ^ n }$, $\sigma ' = h|_{ \{ 1\} \times \Delta ^ n }$, and $h|_{ \Delta ^1 \times \operatorname{\partial \Delta }^ n }$ is the constant map taking the value $x$.

Example 3.2.2.6. Let $(X,x)$ be a pointed Kan complex. Then $\pi _{0}(X,x)$ can be identified with the set $\pi _0(X)$ of connected components of $X$ (Definition 1.1.6.8). Beware that, unlike the higher homotopy groups $\{ \pi _{n}(X,x) \} _{n \geq 1}$, there is no naturally defined group structure on $\pi _{0}(X,x)$.

Example 3.2.2.7. Let $X$ be a topological space and let $x \in X$ be a base point, which we identify with a vertex of the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$. For every positive integer $n$, we can identify $\pi _{n}( \operatorname{Sing}_{\bullet }(X), x)$ with the set $\pi _{n}(X,x)$ of (pointed) homotopy classes of maps from the sphere $S^ n \simeq | \Delta ^ n / \operatorname{\partial \Delta }^ n |$ into $X$.

Example 3.2.2.8. Let $X$ be a Kan complex, let $x$ be a vertex of $X$, and let $e,e': x \rightarrow x$ be edges of $X$ which begin and end at the vertex $x$. Then the equality $[e] = [e']$ holds in the fundamental group $\pi _{1}(X,x)$ if and only if $e$ is homotopic to $e'$ as a morphism in the $\infty$-category $X$ (in the sense of Definition 1.3.3.1); see Corollary 1.3.3.7.

Remark 3.2.2.9. Let $n$ be a nonnegative integer. By virtue of Corollary 3.1.7.2, there exists an anodyne morphism $f: \Delta ^ n / \operatorname{\partial \Delta }^ n \rightarrow Q$, where $Q$ is a Kan complex. Let $q \in Q$ denote the image of the base point $q_0$ of $\Delta ^ n / \operatorname{\partial \Delta }^ n$. If $(X,x)$ is a pointed Kan complex, then precomposition with $f$ induces a trivial Kan fibration $\operatorname{Fun}( Q, X ) \rightarrow \operatorname{Fun}( \Delta ^ n / \operatorname{\partial \Delta }^ n, X)$ (Proposition 3.1.6.10), hence also a trivial Kan fibration

$\operatorname{Fun}(Q,X) \times _{ \operatorname{Fun}( \{ q\} , X) } \{ x\} \rightarrow \operatorname{Fun}( \Delta ^ n / \operatorname{\partial \Delta }^ n ) \times _{ \operatorname{Fun}( \{ q_0 \} , X) } \{ x\} .$

Passing to connected components, we see that $f$ induces a bijection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }}( Q, X ) \simeq \pi _{n}(X,x)$. In other words, the functor $(X,x) \mapsto \pi _{n}(X,x)$ is corepresentable (in the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$) by the pointed Kan complex $(Q,q)$ (which can be regarded as a combinatorial incarnation of the $n$-sphere).

Theorem 3.2.2.10. Let $(X,x)$ be a pointed Kan complex and let $n$ be a positive integer. Then there is a unique group structure on the set $\pi _{n}(X,x)$ with the following properties:

$(a)$

Let $e: \Delta ^{n} \rightarrow \{ x\} \rightarrow X$ be the constant map. Then the homotopy class $[e]$ is the identity element of $\pi _{n}(X,x)$.

$(b)$

Let $f: \operatorname{\partial \Delta }^{n+1} \rightarrow X$ be a morphism of simplicial sets, corresponding to a tuple $(\sigma _0, \sigma _1, \ldots , \sigma _{n+1})$ of $n$-simplices of $X$ (see Exercise 1.1.2.8). Assume that each restriction $\sigma _{i}|_{ \operatorname{\partial \Delta }^ n }$ is equal to the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \subseteq X$. Then $f$ extends to a map $\Delta ^{n+1} \rightarrow X$ if and only if the product

$[ \sigma _0 ]^{-1} [ \sigma _1 ] [ \sigma _2 ]^{-1} [ \sigma _3 ] \cdots [ \sigma _{n+1} ]^{ (-1)^{n} }$

is equal to the identity element of $\pi _{n}(X,x)$.

Moreover, if $n \geq 2$, then the group $\pi _{n}(X,x)$ is abelian.

We will give the proof of Theorem 3.2.2.10 in §3.2.3.

Exercise 3.2.2.11. Show that when $n > 0$ is odd, condition $(a)$ of Theorem 3.2.2.10 follows from condition $(b)$ (beware that this is not true when $n$ is even).

Example 3.2.2.12. In the special case $n=1$, we can rewrite condition $(b)$ of Theorem 3.2.2.10 as follows:

• Let $f$, $g$, and $h$ be edges of $X$ which begin and end at the vertex $x$. Then the equality $[h] = [g] [f]$ holds (in the fundamental group $\pi _{1}(X,x)$) if and only if there exists a $2$-simplex $\sigma$ of $X$ which witnesses $h$ as a composition of $f$ and $g$ (in the sense of Definition 1.3.4.1), as indicated in the diagram

$\xymatrix@R =50pt@C=50pt{ & x \ar [dr]^{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & x. }$

It follows that the fundamental group $\pi _{1}(X,x)$ can be identified with the automorphism group of $x$ as an object of the fundamental groupoid $\pi _{\leq 1}(X) = \mathrm{h} \mathit{X}$.

Warning 3.2.2.13. Let $(X,x)$ be a pointed Kan complex, so that $\pi _{1}(X,x)$ can be identified with the set $\operatorname{Hom}_{\pi _{\leq 1}(X) }(x,x)$ of homotopy classes of paths from $x$ to itself. We have adopted the convention that the multiplication on $\pi _{1}(X,x)$ is given by composition in the homotopy category $\mathrm{h} \mathit{X}$. In other words, if $f,g: x \rightarrow x$ are edges which begin and end at $x$, then the product $[g] [f] \in \pi _{1}(X,x)$ is the homotopy class of a path which can be described informally as traversing the path $f$ first, followed by the path $g$. Beware that the opposite convention is also common in the literature (note that his issue is irrelevant for the higher homotopy groups $\{ \pi _{n}(X,x) \} _{n \geq 2}$, since they are abelian).

Remark 3.2.2.14. Let $(X,x)$ be a pointed Kan complex. For $n \geq 2$, the homotopy group $\pi _{n}(X,x)$ is abelian. We will generally emphasize this by using additive notation for the group structure on $\pi _{n}(X,x)$: that is, we denote the group law by

$+: \pi _{n}(X,x) \times \pi _{n}(X,x) \rightarrow \pi _{n}(X,x) \quad \quad (\xi , \xi ') \mapsto \xi + \xi '.$

With this convention, we can restate property $(b)$ of Theorem 3.2.2.10 as follows:

$(b)$

Let $f: \operatorname{\partial \Delta }^{n+1} \rightarrow X$ be a morphism of simplicial sets, corresponding to a tuple $(\sigma _0, \sigma _1, \ldots , \sigma _{n+1})$ of $n$-simplices of $X$. Then $f$ extends to an $(n+1)$-simplex of $X$ if and only if the sum $\sum _{i = 0}^{n+1} (-1)^{i} [ \sigma _ i ]$ vanishes in $\pi _{n}(X,x)$.

Remark 3.2.2.15 (Functoriality). Let $f: X \rightarrow Y$ be a morphism of Kan complexes, let $x$ be a vertex of $X$, and set $y = f(x)$. For each $n \geq 1$, the morphism $f$ induces a homomorphism $\pi _{n}(f): \pi _{n}(X,x) \rightarrow \pi _{n}(Y,y)$, characterized by the formula $\pi _{n}(f)( [\sigma ] ) = [ f(\sigma ) ]$ for each $n$-simplex $\sigma$ of $X$ for which $\sigma |_{ \operatorname{\partial \Delta }^ n}$ is the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \hookrightarrow X$.

Remark 3.2.2.16. Let $X$ be a Kan complex and let $x$ be a vertex of $X$. Then $x$ can also be regarded as a vertex of the opposite simplicial set $X^{\operatorname{op}}$, which is also a Kan complex. For $n \geq 1$, we have an evident bijection $\varphi : \pi _{n}(X,x) \simeq \pi _{n}(X^{\operatorname{op}}, x)$. If $n \geq 2$, then this bijection is an isomorphism of abelian groups. Beware that, in the case $n=1$, it is generally not an isomorphism of groups: instead, it is an anti-isomorphism (that is, it satisfies the identity $\varphi ( \xi \xi ' ) = \varphi (\xi ') \varphi (\xi )$ for $\xi , \xi ' \in \pi _{1}(X,x)$; see Warning 3.2.2.13 above).

Remark 3.2.2.17. Let $(X,x)$ be a pointed Kan complex and let $n$ be a positive integer. Suppose that $\sigma , \sigma ': \Delta ^{n} \rightarrow X$ are $n$-simplices of $X$ for which $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ and $\sigma '|_{ \operatorname{\partial \Delta }^{n} }$ are equal to the constant map $\operatorname{\partial \Delta }^{n} \rightarrow \{ x\} \subseteq X$. It follows from Theorem 3.2.2.10 that the equality $[ \sigma ] = [\sigma ']$ holds (in the homotopy group $\pi _{n}(X,x)$) if and only if there exists an $(n+1)$-simplex $\tau$ of $X$ such that $d_0(\tau ) = \sigma$, $d_1(\tau ) = \sigma '$, and $d_{i}(\tau )$ is the constant map $\Delta ^{n} \rightarrow \{ x\} \subseteq X$ for $2 \leq i \leq n+1$.

Exercise 3.2.2.18 (Homotopy of Eilenberg-MacLane Spaces). Let $M_{\ast }$ be a chain complex of abelian groups and let $X = \mathrm{K}(M_{\ast })$ be the associated Eilenberg-MacLane space (Construction 2.5.6.3). Let $x \in X$ be the vertex corresponding to the zero element, and let $n$ be a positive integer. Note that a pointed map from $\Delta ^ n / \operatorname{\partial \Delta }^ n$ to $X$ can be identified with a map of chain complexes $\mathrm{N}_{\ast }( \Delta ^ n, \operatorname{\partial \Delta }^ n; \operatorname{\mathbf{Z}}) \simeq \operatorname{\mathbf{Z}}[n] \rightarrow M_{\ast }$: in other words, it can be identified with an $n$-cycle of the chain complex $M_{\ast }$, which we will denote by $\overline{\sigma }$.

$(1)$

Let $\sigma , \sigma ': \Delta ^ n \rightarrow X$ be $n$-simplices whose restriction to $\operatorname{\partial \Delta }^ n$ is equal to the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \hookrightarrow X$. Show that $[\sigma ] = [\sigma ']$ in $\pi _{n}(X,x)$ if and only if $\overline{\sigma }$ and $\overline{\sigma }'$ are homologous as $n$-cycles of $M_{\ast }$ (use Remark 3.2.2.17).

$(2)$

Show that the $[\sigma ] \mapsto [\overline{\sigma }]$ induces an isomorphism from $\pi _{n}(X,x)$ to the homology group $\mathrm{H}_{n}(M)$.

In particular, if $A$ is an abelian group and $m \geq 0$ is an integer, then the homotopy groups of the Eilenberg-MacLane space $X = \mathrm{K}(A,m)$ are given by

$\pi _{n}(X,x) = \begin{cases} A & \text{ if } n=m \\ 0 & \text{ otherwise. } \end{cases}$


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Statistical physics

Statistical mechanics

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Kinetic theory of gases

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Theory that gases are made up of a bunch of small billiard balls that don't interact with each other.
This theory attempts to deduce/explain properties of matter such as the equation of state in terms of classical mechanics.

Maxwell-Boltzmann vs Bose-Einstein vs Fermi-Diract statisics

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A good conceptual starting point is to like the example that is mentioned at The Harvest of a Century by Siegmund Brandt (2008).
Consider a system with 2 particles and 3 states. Remember that:
Therefore, all the possible way to put those two particles in three states are for:
• Maxwell-Boltzmann distribution: both A and B can go anywhere:
State 1 State 2 State 3
AB
AB
AB
A B
B A
A B
B A
A B
B A
• Bose-Einstein statistics: because A and B are indistinguishable, there is now only 1 possibility for the states where A and B would be in different states.
State 1 State 2 State 3
AA
AA
AA
A A
A A
A A
• Fermi-Dirac statistics: now states with two particles in the same state are not possible anymore:
State 1 State 2 State 3
A A
A A
A A
Both Bose-Einstein statistics and Fermi-Dirac statistics tend to the Maxwell-Boltzmann distribution in the limit of either:
TODO: show on forumulas. TODO experimental data showing this. Please.....

Maxwell-Boltzmann distribution (MB distribution)

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Experimental verification of the Maxwell-Boltzmann distribution

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Most applications of the Maxwell-Boltzmann distribution confirm the theory, but don't give a very direct proof of its curve.
Here we will try to gather some that do.
Zartman Ko experiment
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Stern-Zartman experiment (1920)
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Is it the same as Zartman Ko experiment? TODO find the relevant papers.
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Thermodynamics

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Boltzmann constant (, )

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This is not a truly "fundamental" constant of nature like say the speed of light or the Planck constant.
Rather, it is just a definition of our Kelvin temperature scale, linking average microscopic energy to our macroscopic temperature scale.
The way to think about that link is, at 1 Kelvin, each particle has average energy: $$1/2kT (1)$$ per degree of freedom.
This is why the units of the Boltzmann constant are Joules per Kelvin.
For an ideal monatomic gas, say helium, there are 3 degrees of freedom. so each helium atom has average energy: $$3/2kBT (2)$$
If we have 2 atoms at 1 K, they will have average energy , and so on.
Another conclusion is that this defines temperature as being proportional to the total energy. E.g. if we had 1 helium atom at 2 K then we would have about energy, 3 K and so on.
This energy is of course just an average: some particles have more, and others less, following the Maxwell-Boltzmann distribution.

Thermodynamic potential

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chemistry.stackexchange.com/questions/7696/how-do-i-distinguish-between-internal-energy-and-enthalpy/7700#7700 has a good insight:
To summarize, internal energy and enthalpy are used to estimate the thermodynamic potential of the system. There are other such estimates, like the Gibbs free energy G. Which one you choose is determined by the conditions and how easy it is to determine pressure and volume changes.

Enthalpy ()

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Adds up chemical energy and kinetic energy.
Wikipedia mentions however that the kinetic energy is often negligible, even for gases.
The sum is of interest when thinking about reactions because chemical reactions can change the number of molecules involved, and therefore the pressure.
To predict if a reaction is spontaneous or not, negative enthalpy is not enough, we must also consider entropy via Gibbs free energy.

Gibbs free energy ()

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TODO understand more intuitively how that determines if a reaction happens or not.
$$ΔG=ΔH−TΔS (3)$$
At least from the formula we see that:
• the more exothermic, the more likely it is to occur
• if the entropy increases, the higher the temperature, the more likely it is to occur
• otherwise, the lower the temperature the more likely it is to occur
A prototypical example of reaction that is exothermic but does not happen at any temperature is combustion.
Reversible reaction
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I think these are the ones where , i.e. enthalpy and entropy push the reaction in different directions. And so we can use temperature to move the Chemical equilibrium back and forward.

Entropy

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OK, can someone please just stop the philosophy and give numerical predictions of how entropy helps you predict the future?
The original notion of entropy, and the first one you should study, is the Clausius entropy.
For entropy in chemistry see: entropy of a chemical reaction.

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Second law of thermodynamics

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Subtle is the Lord by Abraham Pais (1982) chapter 4 "Entropy and Probability" mentions well how Boltzmann first thought that the second law was an actual base physical law of the universe while he was calculating numerical stuff for it, including as late as 1872.
But then he saw an argument by Johann Joseph Loschmidt that given the time reversibility of classical mechanics, and because they were thinking of atoms as classical balls as in the kinetic theory of gases, then there always exist a valid physical state where entropy decreases, by just reversing the direction of time and all particle speeds.
So from this he understood that the second law can only be probabilistic, and not a fundamental law of physics, which he published clearly in 1877.
Time reversibility
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Time reversibility of classical mechanics
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Considering e.g. Newton's laws of motion, you take a system that is a function of time , e.g. the position of many point particles, and then you reverse the speeds of all particles, then is a solution to that.
Time reversibility of gravity
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I guess you also have to change the sign of the gravitational constant?

Phase transition

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TODO can anything interesting and deep be said about "why phase transition happens?" physics.stackexchange.com/questions/29128/what-causes-a-phase-transition on Physics Stack Exchange

Second-order phase transition

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The more familiar transitions we are familiar with like liquid water into solid water happen at constant temperature.
However, other types of phase transitions we are less familiar in our daily lives happen across a continuum of such "state variables", notably

Refrigerator

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Dilution refrigerator

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Used for example in some times of quantum computers, notably superconducting quantum computers. As mentioned at: youtu.be/uPw9nkJAwDY?t=487, in that case we need to go so low to reduce thermal noise.
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Temperature

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For scales from absolute 0 like Kelvin, is proportional to the total kinetic energy of the material.
The Boltzmann constant tells us how much energy that is, i.e. gives the slope.

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anonymous 5 years ago I can't seem to solve for y in the following equation. I can get the answer from wolfram alpha, but I want to see the steps. Can somebody provide a step by step to solve for y? Thanks! x^2 + y^2 = cy

1. anonymous

Use the quadratic equation, with A = 1, B = c, and C = -x^2

2. anonymous

$(-c \pm \sqrt{c^2+4x^2})/2$

3. anonymous

My daughter and I are working through it now... thanks for the helpful hint!

Find more explanations on OpenStudy


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# Worked example: Evaluating functions from equation

Common Core Math:

## Video transcript

The function f of x is defined as f of x is equal to 49 minus x squared. Find the value of f of 5. So whenever you're dealing with a function, you take your input. In this case, our input is going to be our 5. We input it into our little function box, and we need to get our output. And they defined the function box here as whatever your input is, take that, square it, and then subtract it from 49. So f of 5, every time I see an x here, since f of x is equal to this, every time I see an x, I would replace it with the input. So f of 5 is going to be equal to 49 minus-- instead of writing x squared, I would write 5 squared. So this is equal to 49 minus 25. And 49 minus 25 is equal to 24. And we are done.