Better "In-Place" Amplification of QMA

$\def\braket#1#2{\langle#1|#2\rangle}\def\bra#1{\langle#1|}\def\ket#1{|#1\rangle}$ In [MW05](https://arxiv.org/abs/cs/0506068) the authors demonstrate so-called "in-place" amplitude amplification for QMA, exhibiting a method for Arthur to amplify his success probability without requiring any increase in the size of Merlin's witness state. Call the original machine $M$ and the amplified machine $M'$.

Suppose that $x$ is the input, the original machine is $M(x,w)$ and the amplified machine is $M'(x,w)$. The approach of MW05 does not guarantee that a witness state that was in the accepting set for $M(x,\cdot)$ (i.e. that made $M$ accept input $x$ with probability exceeding, say, $\frac{2}{3}$) is still in the accepting set for $M'(x, \cdot)$ (i.e. those states that make $M'(x, \cdot)$ accept with amplified probability). I explain why I think that it's actually not possible to make their method do this below the line.

**Q: Is there a way to perform amplification of QMA machines without increasing the length of Merlin's witness and while also guaranteeing that all initially valid witnesses for a given input stay valid witnesses for this input?**

* * *

Without explaining too much context and using their notation, the issue that seems to arise is that MW05 first proves that their approach works for states that are eigenvectors of $Q$. This is good enough for completeness because if there's a state that accepts with probability $p \geq a$ then there's an eigenvector with success probability at least $p$. However, I don't think this can be used to show that any "good" witness also accepts w.h.p. after amplification.

In general a "good" witness $\ket{\psi}$. will be some linear combination of eigenvectors $\sum_j \alpha_j\ket{\psi_j}$, some of which are "good" and have acceptance probability $\geq a$ and others of which are "bad" and have acceptance probability $\leq b$. Let $S$ be the set of "good" witness eigenvectors. Then, once we amplify, our success probability is roughly (for $r$ large enough I guess)

$$\sum_{j \in S} |\alpha_j|^2(1-2^{-r}).$$

In general $\sum_{j \in S} |\alpha_j|^2$ isn't $1$ so we're in trouble. In the worst case of witness eigenvectors, we can even construct a "good" witness $\ket{\psi}$ that after $r$-rounds of amplification accepts with probability roughly $\frac{a-b}{1-b}(1-2^{-r}) \leq \frac{1}{2}$ for $(a,b) = (\frac{2}{3}, \frac{1}{3})$.


Qubit fidelity of DWAVE device

Since DWAVE quantum device is constructed using superconducting flux qubits, each qubit cannot be produced identically so that the fidelity of the qubit must be different. DWAVE only provides the information of their devices in terms of number of qubits, couplers, etc without any calibration data of each qubit on their [website](https://www.dwavesys.com/solutions-and-products/systems/).

I was wondering if we can find some information about the qubit fidelity of DWAVE machine. Since the number of qubits is quite huge, I assume it's time-consuming to calibrate the device and methods like randomized benchmarking are not applicable anymore. Is there any specific calibration method for that?


How much did Atari pay for GEM?

Atari licensed Digital Research's GEM graphic user interface for the ST. As <https://www.filfre.net/2015/04/the-68000-wars-part-2-jack-is-back/> puts it:

> And of course in the wake of the Macintosh the ST simply had to ship with a mouse and an operating system to support it.

> It was this latter that presented by far the biggest problem. While the fairly conservative hardware of the ST could be put together relatively quickly, writing a modern, GUI-based operating system for the new computer represented a herculean task. Apple, for instance, had spent years on the Macintosh’s operating system, and when the Mac was released it was still riddled with bugs and frustrations. This time around Tramiel wouldn’t be able to just slap an archaic-but-paid-for old PET BASIC ROM into the thing, as he had in the case of the Commodore 64. He needed a real operating system. Quickly. Where to get it?

> He found his solution in a very surprising place: at Digital Research, whose CP/M was busily losing its last bits of business-computing market-share to Microsoft’s juggernaut MS-DOS. Digital had adopted an if-you-can’t-beat-em-join-em mentality in response. They were hard at work developing a complete Mac-like window manager that could run on top of MS-DOS or CP/M. It was called GEM, the “Graphical Environment Manager.” GEM was merely one of a whole range of similar shells that were appearing by 1985, struggling with varying degrees of failure to bring that Mac magic to the bland beige world of the IBM clones. Also among them was Microsoft’s original Windows 1.0 — another product that Tramiel briefly considered licensing for the ST. Digital got the nod because they were willing to license both GEM and a CP/M layer to run underneath it fairly cheap, always music to Jack Tramiel’s ears.

Just how cheap was it? How much did Atari pay DR? And what sort of terms were involved?

@tofro - details of other similar deals (eg Microsoft's licensing of SCP's 86-DOS) are known. It's plausible that details of this deal have come to light via a similar route.

Sadly, I suspect that the intersection of “people who would have even known at the time” (C-suite, accountant) and “people who care enough about old technology enough to visit Retro.SO” is very, very small. Of the intersection, the chances that they will remember what the negotiated price was is probably zero. Engineers would have been told “we licensed GEM, take care of the integration” but they would not have been told the licensing cost because it wasn’t relevant, NDA would likely impose need-to-know basis, and it was none of the engineer’s business to know.

@tofro I would normally expect NDA's to have expired after three and a half decades with the companies no longer in business. If they are really for life in this case, that in itself would be an interesting historical fact.

@user there were several companies selling their home/soho machines with GEM. Think Philips Yes and alike.

Who do you think could answer this? Everyone who really could know is either dead or very probably still under a lifetime NDA...

Would be interesting to know what Amstrad paid for it too. It was included with the PC1512.


What was IBM's internal Specification Language of the 1980s?

Within IBM's internal Development community, there was a move in the 1980s to bring our skills up to date. As part of this, we were introduced to a specification language, independent of the programming languages of the time (PL/S and Basic Assembler for example), in which a specification would be implemented. This specification language, as I recall, embodied set theory and logic in a type-safe context.

In fact, in the IBM Hursley Lab we eventually chose to use the Z notation from the Programming Research Group at Oxford University. This was perhaps more strictly mathematical than the IBM language - part of a Formal Methods approach.

However, I spent quite a bit of time with the IBM specification language and I would very much like to know if anyone recalls what it was called and what happened to it.
It had a “specification checker” for type consistency etc and I remember it very fondly, before getting immersed in the Z notation.

Thanks @davidbak I’m not thinking of VDM which, like Z, is more mathematically formal than the IBM language I am searching for.

See if this article rings a bell - dl.acm.org/doi/abs/10.1145/356596.356598 (PDF is freely available here)

I'm thinking it was the "Vienna Definition Language"? foldoc.org/Vienna+Definition+Language - see also en.wikipedia.org/wiki/Vienna_Development_Method#History


Ограничения на поиск пользователей через users.search api vk

У api вконтакте есть ограничение - 3 запроса в секунду. Однако у `users.search` есть какое-то своё ограничение на количество вызовов.

> Помимо ограничений на частоту обращений, существуют и количественные ограничения на вызов однотипных методов. По понятным причинам, мы не предоставляем информацию о точных лимитах.

Нужно найти 1000 человек в соцсети (клиенты компании). Если api бы работало хорошо, это было бы 300 секунд. А что делать с тем, что на N-м запросе api выдаёт 0 найденных? Похоже, что N-порядка десятков.
Другие методы работают, а этот выдаёт 0. Причём даже если попытаться поискать людей просто в браузере - тоже 0 найденных.

Кто сталкивался, поясните, надолго ли происходит такое отключение поиска. И может есть какое-то время паузы, выше которого поиск всегда работает и возвращает результаты?

[Проверка через execute тут](https://vk.com/dev/execute "Проверка через execute тут")

    var found = API.users.search({"q":"Иванов Иван"});
    return found;

Запуск этого кода даёт `response: { count: 0, items: [] }`, когда `users.search` вдруг перестаёт работать.

Возможный дубликат вопроса: Ограничения VK-APi

Но нельзя не согласиться с тем, что эта мера с неизвестным лимитом сильно портит жизнь и тем, кто готов играть по правилам. Видимо, выбирая защиту от злоупотреблений, недовольством соблюдающих правила пользователей решили пренебречь.

Если допустимое кол-во запросов будет известно заранее, то ботописатели обойдут ограничение, путём использования в парсере неограниченного количества учётных записей, коих в сети можно покупать, хоть пачками. А тут получается, что возврат нулевого значения не гарантирует в любой момент времени, что поиск не обнаружил данные по запрашиваемой строке. Может они есть, а может аккаунт уже забанен. Это ведёт к тому, что нужно выполнять отдельный запрос с заранее проверенным на корректность результатом на каждый нулевой ответ. Это конечно позволит проверить достоверность ответа, но сильно снизит КПД.

@alexis031182 честно говоря, из-за этой неоднозначности не очень хочется с ними работать, не беру работу по их апи больше. Программирование и нечёткие правила несовместимы.

Похоже, что не только users.search подвержен ограничению, но и wall.search. Проверено, "банят" довольно быстро с теми же симптомами, что и в вопросе. К сожалению, не удалось вычислить момент, с какого по счёту запроса это началось. Вполне вероятно, что audio.search, а также video.search тоже весьма ограничены.


Максимально возможное заполнение в клеточном автомате "Life"

Меня интересует алгоритм вычисления максимально возможного количества "живых" клеток на поле размером 100х100. Предполагается, что это поле - развернутый тор. Стартовое и конечное положения меня не интересуют. Но это должно быть возможное игровое состояние - т.е. состояние, соответствующее правилам (а не, например, просто заполненный квадрат 100x100). Интересует именно максимум. Возможно, кто-то логически объяснит, что может быть именно такое-то число и не больше ни меньше.

Возможно, многие знают правила игры. Но я их повторю:

> 1) в пустой (мёртвой) клетке, рядом с которой ровно три живые клетки, зарождается жизнь;
>
> 2) если у живой клетки есть две или три живые соседки, то эта клетка продолжает жить;
>
> 3) в противном случае (если соседей меньше двух или больше трёх) клетка умирает («от одиночества» или «от перенаселённости»)

**Уточнение:** Дабы не было никому обидно (и были понятны комменты), правку вношу в виде уточнения, а не меняя непосредственно сам вопрос.

Мне необходимо реальное промежуточное состояние. То, которое может быть достигнуто в принципе. При этом, хотелось бы знать максимально возможное количество живых особей в популяции. Да, действительно, это состояние может быть конечными или цикличным. Но меня не интересует стартовое состояние, когда оно соответствует правилам старта, но, на самом деле, оно не может быть получено иным путем.

Даже скажу больше... возможно, это состояние (состояние максимума популяции для квадрата 100x100) может быть достигнуто из расчета, что первоначальный квадрат был больше. Хотя желательно, чтобы он был ограничен в размерах и был тором.

Отвечая на вопрос, что значит _"возможное игровое состояние"_ отмечу, что в моем понимании, стартовое состояние является как бы предигровым состоянием или стартовым состоянием, но не состоянием, полученным в процессе игры. _"Возможное игровое состояние"_ в данном случае мной понимается как состояние, полученное в процессе игры.

Возможно, кто-то и не согласен с моим пониманием игрового состояния, но такое оно и есть. Представьте, что гости собираются и рассаживаются за стол (даже уже сели). Но это не застолье. Вот как-то так.

Если предлагаемая популяция циклична, то, может быть, имеет смысл сократить размер поля до 10x10?

Комментарии не предназначены для расширенной дискуссии; разговор перемещен в чат.


Dry water using graphite

[Dry water](https://en.wikipedia.org/wiki/Dry_water) or "powdered water" is essentially a bunch of extremely small water droplets surrounded by $\ce{SiO2}$.

Some time ago, I was messing around and mixed some powdered graphite with water. (The graphite was obtained from a pencil, so it was not pure. Also I mixed the two substances with a stick, not with a high-speed blender.) When a droplet of this was splashed onto the table, it could be blown around with air and even held on the hand without breaking, until it was squished.

The important part of this is, that that droplet could be placed back into the original mixture of water and graphite powder, and it would roll around on the surface.

So does anyone know if it's possible to use powdered graphite rather than silica particles to make dry water? If is is, what properties are different from the ordinary powdered water?

Graphite for pencils is mixed with clay, which is largely composed of silicates. I don't know whether pure graphite powder would display the same behavior you describe.

I may end up doing something similar to this for a school project. If I decide to mix the graphite with water instead of ink, I will measure its conductance. I could post my results here, maybe.

@Ivan But the 'dry water' uses hydrophobic silica particles (surface modified). I think that nanographite would be a suitable substitute.

Silica is highly hydrophilic, graphite is not quite like  that.


Mathematical models of oscillatory chemical reactions

I am a mathematician working on real-life models in ordinary differential equations. I want to know if there are any models of oscillatory chemical reactions that consist of three ordinary differential equations where one variable is much slower than the other two. In particular I am interested in bursting behavior, which consists of alternating trains of fast oscillations with periods of rest.

I have been searching and I couldn't find what I am looking for. I found about the Belousov-Zhabotinsky reaction, but in this case we have one fast variable and two slow variables, while I'm looking for two fast variables and one slow variable.


Interaction of trifluoroacetates with acetylcholinesterase

There exists a substance called TMTFA, or 3-(_N,N,N_ -Trimethylammonio)-2,2,2-trifluoroacetophenone. It is known for being able to inhibit acetylcholinesterase at femtomolar concentrations. The TMTFA-AChE adduct resembles the tetrahedral transition state through which the hydrolysis of acetylcholine by the enzyme normally proceeds, and this adduct is stabilized by the electron-withdrawing trifluoromethyl group present in the TMTFA molecule. This inhibition is reversible.

While there have been studies regarding trifluoroacetophenones and their interactions with AChE, I haven't been able to find studies exploring the interactions between AChE and trifluoroacetates, such as 3-(_N,N,N_ -trimethylammonio)phenyl trifluoroacetate or 2,2,2-trifluoroacetylcholine. Would these, and other, esters of trifluoroacetic acid inhibit AChE or be hydrolyzed normally? If it's the former, then what are the typical inhibitory concentrations of such esters? What is the shape of the adduct of the ester with AChE? How long would they remain bound to the enzyme? Have any studies been done regarding the bioactivity of such esters?

### References

  1. Nair, H. K.; Seravalli, J.; Arbuckle, T.; Quinn, D. M. Molecular Recognition in Acetylcholinesterase Catalysis: Free-Energy Correlations for Substrate Turnover and Inhibition by Trifluoro Ketone Transition-State Analogs. _Biochemistry_ **1994** , _33(28)_ , 8566–8576. DOI: [10.1021/bi00194a023](https://doi.org/10.1021/bi00194a023).
  2. Harel, M.; Quinn, D. M.; Nair, H. K.; Silman, I.; Sussman, J. L. The X-Ray Structure of a Transition State Analog Complex Reveals the Molecular Origins of the Catalytic Power and Substrate Specificity of Acetylcholinesterase. _J. Am. Chem. Soc._ **1996** , _118(10)_ , 2340–2346. DOI: [10.1021/ja952232h](https://doi.org/10.1021/ja952232h).

According to Ref.2, TMTFA binds titly to AcE from a Pacific electric ray, Torpedo californica. It might not do the same to Human AcE. I believe your intention is for inhibitors of Human AcE.


Reaction of Rubidium with vacuum system viewport

I have a vacuum system ($<1\times10^{-7}$ mbar) filled with Rubidium ($^{87}$Rb) vapour, at room temperature.

We also have a viewport near the Rubidium source, made of a stainless steel flange and a glass substrate, connected with a [Kovar](https://en.wikipedia.org/wiki/Kovar) alloy.

We are seeing potential degradation of the seal, which we do not see on other parts of the system. The only difference is the Nickel and Cobalt based Kovar alloy, not present anywhere else.

What are the reaction rates of Nickel and/or Cobalt with Rubidium?

btw, thanks to ability of rubidium to form suboxides even tiniest amount of oxygen may bind a lot of rubidium.

Please, check the mark of the glass and if it is suited for work with rubidium in this particular settings. In general, glasses are oxides that can potentially react with alkali metals in right circumstances. I see no reason for them to react with nickel/cobalt alloy. However, rubidium might react with oxide layer on the surfaces of this alloy and/or any dirt left on the seal (and glass, and any dirt left on the glass) and any passing air if the seal is... not good enough.


Whence “homomorphism”, “homomorphic”?

The [kernel](https://hsm.stackexchange.com/questions/2149/what-came-first-the-kernel-from-vector-spaces-or-from-group-theory) question leads to another : Today, **_homomorphism_** (resp. _isomorphism_) means what Jordan ([1870](https://archive.org/stream/traitdessubsti00jorduoft#page/56)) had called _isomorphism_ (resp. _[holoedric](https://hsm.stackexchange.com/questions/3108/jordan-called-isomorphisms-iso-and-homomorphisms-iso-holoedriques-and-iso) isomorphism_). How did the switch happen?

  * “Homomorphic” appears e.g. in de Séguier ([1904](https://www.emis.de/cgi-bin/JFM-item?36.0187.02), [pp. 65–66](https://archive.org/stream/thoriedesgroup00sguoft#page/65)):

> (...) supposons entre les groupes $\mathrm A$ et $\mathrm B$ une correspondance telle qu’à chaque élément de $\mathrm A$ réponde un élément au moins de $\mathrm B$ et à chaque élément de $\mathrm B$ un élément au moins de $\mathrm A$ et que si $a_i$ de $\mathrm A$ et $b_i$ de $\mathrm B$ se correspondent, $a_ia_k$ et $b_ib_k$ se correspondent aussi. On dit que $\mathrm A$ et $\mathrm B$ sont _homomorphes_.

  * “Homomorphism” (as _property_ of being homomorphic) appears e.g. in Loewy ([1912](https://www.emis.de/cgi-bin/JFM-item?43.0199.02), [p. 198](https://archive.org/stream/festschriftheinr06webeuoft#page/198)):

> §1 (...) untersucht den Homomorphismus zweier abstrakter Gruppen (...).

  * “Homomorphism” (as _map_ between groups) appears e.g. in Hopf ([1928](https://www.emis.de/cgi-bin/JFM-item?54.0610.02), [p. 128](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002507641&physid=phys139)):

> 4) Unter einem Homomorphismus der Gruppe $\mathfrak G$ in die Gruppe $\mathfrak H$ verstehen wir eine eindeutige Abbildung $f$ von $\mathfrak G$ auf einen echten oder unechten Teil von $\mathfrak H$, bei der stets $f(x+y)=f(x)+f(y)$ ist.

Are these the earliest occurrences?

Homomorphism is etymologically correct in the sense that the structures of two groups exhibit a degree of similarity when one can map into the other (trivial cases apart). Isomorphism is then more accurately used to mean "congruent" structure. ("Isos" is more commonly used to mean equal, but Euclid also uses it for congruent.)

The following text notes the use of homomorphism by de Séguier in 1904, and then states that the term was introduced by Klein without giving a reference. books.google.ca/…


Who was the first to use the "does not exist" sign ∄? I'm aware that Giuseppe Peano originated serifed ∃ and, moreover that Whitehead and Russell repurposed Peano's serifed ∃; I'm also aware that Gerhard Gentzen introduced the ∃ sans serif (and used it like Russell and Whitehead rather than like Peano). Yet, I cannot find who was the first to use ∄ as ~∃ to mean "there does not exist". Any idea? Thank you.


Who first defined polynomials as sequences?

> **Question 1.** When did the modern definition of a **polynomial** (as a sequence of coefficients, with multiplication defined by $\left(ab\right)_n = \sum\limits_{k=0}^n a_k b_{n-k}$) emerge?

Let me clarify: People have been working with some notion of polynomials for centuries, but mostly without a rigorous concept of what a polynomial is. From what I understand, "equations" rather than polynomials were the focus, and demarcating the precise extent of what an "equation" is was secondary to actually solving equations of given, fairly specific types. When the need for a rigorous definition became clear, it seems that the go-to solution was to define a polynomial as a polynomial **function** (e.g., §1 of [Roland Weitzenböck, _Invariantentheorie_ , 1923](https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV0733)). This worked well for polynomials over $\mathbb R$ or $\mathbb C$, but not so much for polynomials over finite fields (which may be why E. H. Moore had to define $\mathbb{F}_{p^n}$ as a quotient ring of a polynomial ring over $\mathbb Z$ rather than of a polynomial ring over $\mathbb{F}_p$ [in his 1893 talk](https://projecteuclid.org/download/pdf_1/euclid.bams/1183407788)).

What I call the "modern definition" of a polynomial is one of the following two definitions:

  * either as an infinite sequence $\left(a_0, a_1, a_2, \ldots\right)$ of coefficients (with all but finitely many $n$ satisfying $a_n = 0$), with addition defined entrywise and multiplication defined by the $\left(ab\right)_n = \sum\limits_{k=0}^n a_k b_{n-k}$ formula (and the indeterminate defined as the sequence $\left(0,1,0,0,0,\ldots\right)$ rather than being some mythic "symbol" or "indeterminate value" or "variable"),

  * or as the monoid ring of the additive monoid $\left\\{0,1,2,\ldots\right\\}$.

The second of these two definitions appears in §15 of B. L. van der Waerden, _Moderne Algebra I_ , 2te Auflage 1937. (I can't check the 1st edition, as I don't have it.) This gives a lower bound on the age of the modern definition. As for an upper bound, I am perplexed. Nicolas Bourbaki's _Elements of the History of Mathematics_ grazes the question on its page 82:

> In all this research, the fields that arise consist of "concrete" elements in the sense of classical mathematics - (complex) numbers or functions of complex variables. 32 But already Kronecker, in 1882, takes full account of the fact (obscurely sensed by GaUBB and Galois) that "indeterminates" only play the role in his theory of the basis elements of an algebra, and not that of variables in the sense of Analysis ([186 a], v. II, pp. 339-340); and, in 1887, he develops this idea, linked to a vast programme that aims at nothing less than recasting all mathematics by rejecting all that cannot be reduced to algebraic operations on the integers. It is on this occasion that, taking up an idea of Cauchy ([56 a], (1), v. X, pp. 312 and 351) who had defined the field $\mathbb C$ of complex numbers as the field of remainders $\mathbb R\left[X\right]/\left(X^2+1\right)$, Kronecker shows how the theory of algebraic numbers is completely independent of the "fundamental theorem of algebra" and even of the theory of real numbers, all fields of algebraic numbers (of finite degree) being isomorphic to a field of remainders $\mathbb Q\left(X\right)/\left(f\right)$ ($f$ an irreducible polynomial over $\mathbb Q$)([186 a], v. 1111, pp. 211-240).

Following the references here, I see some signs of the modern definition slowly dawning in the 19th Century: [Note: "Polynomials" are often called "forms" before ca. the 1930s, particularly when they are homogeneous.]

  * I don't know what exactly Cauchy had done; the [56 a] reference goes to his Oeuvres, which I don't have.

  * The first Kronecker reference goes to [his _Grundzüge einer arithmetischen theorie der algebraischen Grössen_ (1882)](https://archive.org/details/grundzgeeinerar01krongoog/page/n129), where he talks of "veränderliche oder unbestimmte Grössen" but never constructs any.

  * [J. Molk, _Sur une notion qui comprend celle de la divisibilité et sur la théorie générale de l'élimination_ , Acta Math. **6** (1885), pp. 1--165](https://projecteuclid.org/euclid.acta/1485888021), spends its section 2 (and maybe the whole Chapitre I?) arguing about the difference between variables and indeterminates. Alas, the argument appears to veer into philosophical terrain rather quickly, which puts it beyond my French comprehension skills; thus I have no idea what conclusion he actually comes to.

  * [H. Weber, _Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie_ , Mathematische Annalen **43** (1893), pp. 521--549](https://eudml.org/doc/157689) tries to define polynomials in its §3 (after defining the notions of groups and fields in ways that are equivalent to their modern definitions). The definition pays obvious tribute to the idea that polynomials should not be functions; in particular it explicitly says that two polynomials are understood to be equal if and only if their respective coefficients are equal (rather than their values being equal). However, a vague notion of "expressions of the form $\Phi\left(x,y,z,\ldots\right) = \sum a x^r y^s z^t \cdots$" underlies this definition, and (e.g.) the multiplication is never explicitly defined (nor its associativity proven). (To this day, some algebra textbooks geared towards non-math majors work on this level.)

So my question is: Who completed this definition? Weber (later)? Kronecker? Dedekind? Noether? van der Waerden? And while at that:

> **Question 2.** Who defined formal power series?
>
> **Question 3.** Who defined polynomials in noncommutative indeterminates?

These are litmus tests for Question 1; a rigorous definition of polynomials needs only a few trivial modifications to define formal power series, and a definition of noncommutative polynomials is not far away either.

The concept of a polynomial is, in a way, similar to the number $0$: It's an idea that, on its own, appears pedantic and meatless. But once it is established, it becomes a building block of a whole discipline that no one would want to be missing. (19th Century algebraists like Sylvester tend to use the unique factorization property of a polynomial ring without ever explaining what it is that they are factoring; from a modern perspective, this is a gaping hole in their proofs.) My hope is that, as this specific matter is much younger than the number $0$, we may have better clues to its authors.

Not in the modern sense, but that can be said about most of algebra until the late 18th century. Hudde further treated polynomials purely algebraically in the 17th. Dieudonne, before reinterpreting Euler's manipulations in terms of formal power series, states the following:"These series appear in print in Hilbert's Grundlagen der Geometrie, apparently for the first time".

Interesting! This is pretty much the modern definition. I take it he didn't prove anything, though? (The question is nontrivial, because he had something very much resembling induction in his work, so it's not like the toolkit was entirely missing.)

What you call the "modern definition" was the original one given by al-Karaji and al-Samawal back in the 12th century. The latter wrote powers and the corresponding coefficients as columns of a table, and gave a multiplication rule equivalent to the convolution formula (al-Karaji's  descriptions were verbal and more obscure). He even allowed negative powers (hence Laurent polynomials), and gave an algorithm for the "synthetic division". See e.g. Katz, History of mathematics, 9.3.3


Unintentionally proven false theorem with type-in-type outside logic and foundations?

We are all familiar with Russell's paradox, and [it is known](https://proofassistants.stackexchange.com/a/1220) that Per Martin-Löf proved that type-in-type is normalizing and consistent (which is false), by accidentally using an assumption in his meta-theory that is essentially type-in-type. But all these arise in the areas of logic and foundations. Has anyone proven some statement Q by accidentally using some type-in-type assumption, without being aware that such an assumption is unsound, and such that Q can be shown to be surely false (say in a typical predicative type theory)?

The linked example would be considered to be under foundations. Here is another example that is not quite under logic or foundations but anyone who proves it will certainly be immediately aware that the type-in-type assumption is unsound: Let W be the type of all rooted (downward-directed) trees with no infinite (downward) path. Let T be a rooted tree whose root has exactly the members of W as children. Now T is in W...

So is there any example where the person who proved the result was still unaware (at least for a reasonable period of time) that the type-in-type assumption was unsound?

And negation is not even needed, nor is any notion at all about 'categories'. To see why, if type∈type, then let S be the 'type' of all types x such that ∀t∈{ 0 : x∈x } ( ⊥ ), and it is trivial to obtain ⊥ by constructive proof: Given any t∈{ 0 : S∈S }, we have S∈S and so ∀t∈{ 0 : S∈S } ( ⊥ ), hence ⊥. Therefore S∈S, and so ⊥. By the way, if we require a type's defining predicate to quantify over only predicative types, then we can have type∈type in a predicative system!

@AndrejBauer: Yes, but that has nothing to do with category theory; it is just plain foundational concerns, because it just comes down to saying that sets are closed under function types but classes are not (if you only have two levels). In turn, this is ultimately like Frege's Axiom V; because it is just another result of 'wishful thinking' that certain concepts have reifications.  If people want to reason about any 'large' things without giving solid justification why those things can even be reified in the first place, then it's just like asking for Russel's paradox all over again.

Still running on vague memories, I think one of the pertinent observations was that “$f : A \to B$ is epi” in a large category involves universal quantification over a class. Speaking type-theoretically, the type of epis in a category $C$ living in a universe $U$ might live outside $U$. So for instance, if we take the subcategory of epis in $C$, we've bumped up the universes. But ultimately, we'd have to find the specific objections to see in what way they're problematic.

@AndrejBauer: I vaguely recall reading some examples of that sort, yes, but to me those are due to wishful thinking way beyond higher-order arithmetic. Ultimately, what is the difference between those mistakes and Frege's Axiom V? The point is that a lot of these explorations were based on dubious axioms that arose out of logic and foundations, and were not things that mathematicians genuinely believed to be true in a platonic sense.

I vaguely remember Benabou, or perhaps Dobus, telling some cautionary tales about faulty reasoning in category theory, where mistakes can be made easily enough if one does not pay attention to class-set distinction. But I don't know the reference.

@AndrejBauer: Let me put it another way; if an example was actually utilized to obtain a theorem in the language of higher-order arithmetic that is believed to be incorrect, then I would agree that that obtained theorem is an actual example of the type I am looking for. If not, then I'm unlikely to buy that it is outside of logic and foundations.

Mike's example looks like it's "just" about type theory, but it actually has a much wider applicability.

I don't count Mike's example by the way, as it is literally about type theories, which is squarely in logic and foundations. After all, I did link to that thread in my question!

@AndrejBauer: It need not be in type theory. Naive set theory also has a type-in-type assumption, even if it is not called "type". I'm not intending to restrict answers to only type theories of a particular narrow tradition.

@MevenLennon-Bertrand: But did that result in any false theorem that was believed to be a good theorem?

Or in the glory era of early Unimath, when they decided to use type-in-type to avoid issues with universes? Which looks like the time to which Mike Shulman's example belongs.

Just about anyone who learns type theory nowadays knows about type-in-type. So if there is an example of what you're asking for, it would have to have taken place in the narrow period between Martin-Löf's type theory with type-in-type, and the realization that it's inconsistent.

@AndrejBauer is this the one: proofassistants.stackexchange.com/questions/1219/… ? I don't know if this fits the question, though: I would say it is outside of logic and foundation, but Mike Shulman says he does not know whether it holds or not, so it probably cannot "be shown to be surely false" with our current knowledge…

I seem to remember some discussion about accidental universe inconsistencies that were masked at first. Maybe Mike Shulman communicated them, I'll try to remember.


Rules for mutual inductive/coinductive types

Some proof assistants, like Agda and maybe Coq, allow families of mutually defined types, or nested definitions of types, in which some are inductive and others are coinductive. I have no idea what the type-theoretic introduction/elimination rules should be for such a family, and as shown in [this question](https://proofassistants.stackexchange.com/q/961/98) their behavior can be kind of unexpected. The implementation, I guess, allows general "matches" and "comatches", with a separate "termination/productivity checker"; but this is not a semantic explanation suitable for, say, finding category-theoretic models.

Has anyone written down an inference-rule presentation of a type theory including mutual or nested inductive/coinductive definitions?

I'm aware of Hening Basold's thesis. And Danielsson and Altenkirch's Mixing Induction and Coinduction has a long list of related work, some of which may have rules (e.g. the work on sized types).

I mean the 'partiality monad', which you can model as a HIT/QIT via the free ω-complete partial order on a type. As a mixed definition you try to define the same thing similarly to the coinductive definition of the extended natural numbers. Conceivably it might be better behaved than quotienting a coinductive type after the fact (which I think requires countable choice to be a monad).

I might add that even if there's a more rigorous presentation of what Agda/Coq does, it'd probably also be nice to have something better. A while back it was observed that trying to define ω-completion as a mixed higher-inductive/coinductive definition gets you something contractible. But now I wonder if it's just because you can't write down the right thing in Agda (i.e. you really want uses of the higher constructor to be well-founded, but they aren't in Agda).


What's the difference between proliferation and diffusion when talking about changes in tumor density?

Cell proliferation and cell diffusion seem to be important quantities to estimate when trying to understand or measure tumor growth, but I don't really understand a) the difference between them or b) their relationship to tumor growth.

Cell proliferation refers to cellular subdivision. This doesn't necessarily mean tumor growth because something can subdivide but still have the same net mass, right?

Cell diffusion on the other hand, refers to changes in cell density. Supposing cell proliferation is constant and we have positive diffusion, that would mean that cells are 'spreading out' (ie the mass is less dense but bigger). Is this correct? Would positive diffusion be equivalent to tumor growth?

This question came up when reading [this paper](https://www.ncbi.nlm.nih.gov/pubmed/27164582). It talks about estimating diffusion and proliferation parameters in order to solve the reaction-diffusion equation which is, in turn, used to model tumor growth.

A tumor by definition is an abnormal proliferation of cells but I don't think diffusion has anything to do with cells actually diffusing. Rather, the carrying capacity of a tumor depends on such parameters as nutrients or oxygen, which must diffuse further inwards until they cannot reach cells. Hence, the core of large solid tumors tends to be necrotic. It's also my understanding that there is a natural pressure that forces older cells inwards to the core of the tumor, as the outside layers continue to divide and grow outwards, increasing the diameter of the tumor.

I would also like to see a reference to these terms if you have one.


ある一定のダウンタイムがある並列システムを、特定の確率以上で常に必要並列数を上回らせるために必要な並列数は?

例えば web サービスは一般的にロードバランサーと並列数`N`で表されるワーカープロセスによって構築される場合が多いです。

今、あるワーカーが故障した時、そのワーカーが復帰するために必要な時間を`t`とします。ワーカーは、単位時間あたり`λ`の確率で故障し、その故障確率はそれまでの稼動時間に依らない、とします。
今、このサービスが問題なく稼動してワークロードをさばくためには、最低`m`個のワーカーがダウンしていない状態で稼動している必要があるとします。

このとき、確率論的に、ある一定の期間 `T` が与えられたとき、その期間の間のすべてのタイミングにおいて、上記の `m` 個のワーカーが稼動している確率 `p` が定義できると思い、これはこれまで出てきた `N`, `m`, `t`, `λ`, `T` についての関数になっていると思うのですが、その算出方法が分からずにいます。

# 質問

上記のような、ある並列システムが最低`m`の並列度を指定期間において達成している確率を求めたいのでが、これはどのようにすると求められますか?

これが分かることで、逆に算出された確率を信頼度とすることで、求められる信頼度に対して必要な `N` を求めたいと思い、質問しています。

> "N, m, t, λ, T についての関数になっていると思うのですが"  関数形を解析的に書けないように思います。 パラメータの値を具体的に与えたうえでシミュレーションして確率を求める方針をとることはできます。

課題か何かでしょうか。実務的にはほとんど意味がない計算にしかならないと思います(構成要素の網羅度が足りなさすぎる、ある程‌度網羅したら複雑すぎて計算は困難)


Plessey computer at Stanford?

While updating the wiki article on the 4004, I came across an interesting claim by Ted Hoff, who stated the idea of using a general-purpose CPU in a calculator came to him due to a minicomputer that had been donated to Stanford by the UK company Plessey.

I'm at a loss as to what this might be... I know Plessey had a range of realtime machines used for telephone exchanges and the Linesman radar system (rather unsuccessfully) but these were mainframe sized and I cannot find a hint of a mini.

I suppose this could be any old mini that Plessey bought and donated, but it seems less likely that Hoff would refer to it that way, if AT&T donated a PDP-11 to Stanford someone talking about it later would likely refer to it as "a dec mini".

On the face of it, the dates don't work, but 1. the information in Wikipeda is not sourced 2. a prototype might have been given to a university before it was ready for sale - perhaps hardware would have been available before suitable software was written.

@mikado - had not heard of this - a little late for the timeline but an interesting device... and apparently actually used in the 90s!

en.wikipedia.org/wiki/Plessey_System_250?wprov=sfla1 sounds like an interesting and unusual system

Whatever it was, it needs to have been in Stanford in 1969, to fit the 4004 narrative.

forum.stanford.edu/wiki/index.php/…. has no indication of a Plessey as one of the early computers at Stanford. Doesn't rule it out though...

@another-dave Right. At least that's the only one I know. In fact, I've seen one not long ago at the VCL collection in Neubiberg.

Per Wikipedia, Plessey was in the PDP-11-compatible business.  But that might be a little late for 4004 work.


Looking for late 90s shareware open world racing game

I'm trying to track down a shareware racing game released somewhere between 1999 and 2004 - or at least, my parents found it bundled in a computer magazine during that time.

Here's what I remember about it:

  * It was a Win9x game, 3D Flat shaded with hard edges and basically no textures, no actual transparency and rather sparse geometry; It wasn't due to low settings, it was just stylized like that;
  * It was open world;
  * The scope of the game was basically collecting (pinkish?) waypoints randomly placed around the map within a time limit;
  * It was probably made by an European dev;
  * the "buy the game" screen showed a few other maps, including a "rainbows in the sky" fantasy setting.

It seems that most websites curating shareware games have just disappeared from the face of the earth, so I haven't found any reference about it online.

Does my description ring any bells to anyone?

Are you really sure that it is a Win9x game? Low Poly graphics without texturing could also mean, that it was a DOS game.

@thkala you were absolutely alone against time, and the car in the demo was just one extremely low poly sedan. The physics and handling weren’t too arcades, but very far from current day realism level.

Do you remember anything about the cars? Were they street cars? F1-type? Off-road vehicles? Were you alone against time, or were there NPCs?

I wish I could give some more details but there's very little to add. I mean, I thought it was popular enough given it was deemed worthy of being bundled on a magazine. I'll check it out on discord.

Since I see there's one close vote, I won't wait to mention that, if you don't get an answer here, try /r/TipOfMyJoystick or the #identify-a-game channel on the MobyGames Discord.

An italian magazine by Future Media called "Giochi per il mio computer" which was the local edition of "PC Gamer". I've already tried browsing archive.org for some clues but couldn't find anything.

Can you remember which magazine it might have been ?

Nope, it was clearly a small team or one-man made game. Much more lo-fi than the Interstate games.