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# Countability of local maxima on continuous real-valued functions

I am working through a bank of previous exams and couldn't figure a problem out to my satisfaction.

Let $f(x) : \mathbb{R} \to \mathbb{R}\,$ be a continuous function.

1. Show that $f$ can have at most countably many strict local maxima.
2. Assume that $f$ is not monotone on any interval. Then show that the local maxima of $f$ are dense in $\mathbb{R}$.
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– Martin Sleziak Jun 26 '15 at 7:22

1. For each $\delta>0$, the set of all $x\in\mathbb{R}$ such that $f(y)<f(x)$ for all $y$ with $0<|x-y|<\delta$ is countable. This can be seen by noting that the set contains at most one element of the interval $[k\frac{\delta}{2},(k+1)\frac{\delta}{2}]$ for each integer $k$, and these intervals cover $\mathbb{R}$. The set of strict local maxima is a countable union of such sets, for example taking $\delta=\frac{1}{n}$ as $n$ ranges over the positive integers.

2. Suppose that $f$ is a continuous function that is not monotone on any interval. We want to show that $f$ has a local maximum in every interval. The argument is the same if the interval is $(0,2)$, so to slightly reduce notation let's work there. Note that not being monotone in any interval means that each interval contains pairs $x_1<x_2$ and $y_1<y_2$ such that $f(x_1)<f(x_2)$ and $f(y_1)>f(y_2)$. So $(0,1)$ contains a pair $x_1<x_2$ such that $f(x_1)<f(x_2)$, and $(1,2)$ contains a pair $y_1<y_2$ such that $f(y_1)>f(y_2)$. Because $f$ is continuous, there is a point $x_0\in [x_1,y_2]$ where $f$ has its maximum value. Because $f(x_2)>f(x_1)$ and $f(y_1)>f(y_2)$, $x_0$ is not an endpoint of $[x_1,y_2]$. Therefore $f$ has a local maximum at $x_0$.

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How do you use the continuity requirement in showing 1? – davidk01 Mar 19 '11 at 4:11
Continuity isn't needed for 1. – Jonas Meyer Mar 19 '11 at 4:13
Then I don't see how what you've said is true. Why can't I just take $f(y)$ and move it down creating a hole in the function where all the values around it are above the new value where I moved $f(y)$. That would make the entire interval $(y-\delta,y+\delta)$ except for $y$ satisfy your stated condition and $(y-\delta,y+\delta)$ is uncountable. – davidk01 Mar 19 '11 at 4:17
@davidk01: I am saying that for each $\delta>0$, the set $\{x:\text{ for all }$y$, |x-y|<\delta\Rightarrow f(y)<f(x)\}$ is countable. Changing the value at a single point in the domain does not affect this. These are not the points $x$ where $f(x)>f(y)$ for some fixed $y$ close to $x$, but for all $y$ close to $x$. Hence the statement that there can be only one such point in an interval of length less than $\delta$. – Jonas Meyer Mar 19 '11 at 4:25
@Jonas Meyer: That wasn't clear in your original statement. I'll look this over. – davidk01 Mar 19 '11 at 4:27

For #1, prove that you can find disjoint intervals around each local maximum. Then pick a rational number in each interval.

Edit: This answer is wrong but the comments below are instructive, so I'll leave it here.

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That is not true, by 2. – Jonas Meyer Mar 19 '11 at 3:05
@Jonas Meyer: Why not? Since it is continuous it must be bounded on any small enough interval and in that interval it must obtain a maximum somewhere. Now we can chop out this interval and move on. As long as we keep chopping out non-trivial intervals we should be able to cover the real line with the chopped intervals. – davidk01 Mar 19 '11 at 3:19
@davidk01: In a situation where the strict local maxima are dense, which happens for functions that are monotone on no interval, if you take an interval around a point where $f$ has a local maximum, the interval will contain other points where $f$ has a local maximum, and therefore it will intersect all intervals containing those points. – Jonas Meyer Mar 19 '11 at 3:42
@Jonas Meyer: I still don't see how the density of the local maxima makes a difference? Being bounded and continuous the function must obtain an absolute maximum at some point of whatever bounded, non-trivial interval I'm looking at. So I can always find a single absolute maximum on any non-trivial, bounded interval and since such intervals cover the real line there can only be countably many absolute maximums. I should probably look up a definition of strict local maximum. – davidk01 Mar 19 '11 at 3:57
@davidk01: Continuous functions take on their maxima on closed bounded intervals, but that isn't relevant for the first question. As for how density makes a difference: Being able to cover a set with disjoint intervals each of which contains only one element of the set is a lot stronger than saying that the set is not dense. If a set is dense in $\mathbb{R}$, then every interval contains infinitely many elements of the set. – Jonas Meyer Mar 19 '11 at 4:08

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# regular language proof

Question:

Prove that a finite language is a regular language.

How would I go about solving this? I tried my own approach (below) but didn't get far because I don't understand how I am supposed to approach this proof.

My Approach:

Consider a finite automata M

($$\delta$$* denotes $$\delta$$ hat(^) which means all states w travels to until empty)

$$M=(Q,\Sigma,\delta,q_0,F)$$

if w is a string and w $$\in$$L* (any finite language)

=> $$\delta$$*($$q_0$$, w)$$\in$$F (then i make a logic statement that says for all w this is true?)

and if w$$\in\epsilon$$

=> $$\delta$$*($$q_0$$, $$\epsilon$$ )$$\in$$F

Let $$n$$ be the maximum length of the words in the given finite language $$L$$, and let $$m$$ be the size of the (occuring letters from the) alphabet.
Build an automata with $$\sum_{k\le n}m^k$$ many states, one for each possible word, and mark exactly those states as accepting which correspond to a word in $$L$$.


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# \paragraph{} sectioning with eledmac/eledpar

Maybe just a short question: In a layout with parallel pages using eledpar and eledmac I try to use the \paragraph sectioning command (and KOMA-Script), which causes the layout going off perfect sync.

Is there a equivalent to the \paragraph{}-command for eledmac/eledpar, or is it better to build something like: \noindent\textbf{1.}\hspace{0.7em} text… manually?

Thanks in advance for any help.

Here's an MWE. The first two paragraphs go out of sync:

\documentclass[11pt,a4paper,BCOR=12mm,DIV=14]{scrbook}
\usepackage[utf8]{inputenc}
\usepackage[ngerman, polutonikogreek, latin]{babel}
\usepackage[T1]{fontenc}
\usepackage{eledmac}
\usepackage{eledpar}

\begin{document}
\begin{pages}
\begin{Leftside}
\selectlanguage{latin}
\beginnumbering
\pstart
\eledchapter*{Titel}
\pend\pstart
\eledsection*{Section}
\pend\pstart
\paragraph{bla} phantomtext phantomtextphantomtext phantomtext phantomtext phantomtext
\pend\pstart\itshape
\paragraph{2.} phantomtext2 phantomtext2 phantomtext2 phantomtext2 phantomtext2 phantomtext2
\pend\pstart\itshape
\paragraph{3.} phantomtext3 phantomtext3 phantomtext3 phantomtext3 phantomtext3 phantomtext3 phantomtext3.
\pend
\endnumbering
\end{Leftside}

%BEGINN RECHTE SEITE
\begin{Rightside}
\selectlanguage{ngerman}
\beginnumberingR
\pstart
\eledchapter*{Titel}
\pend\pstart
\eledsection*{Section}
\pend\pstart
\paragraph{bla} phantomtext phantomtextphantomtext phantomtext phantomtext phantomtext phantomtext
\pend\pstart\itshape
\paragraph{2. bla} phantomtext2 phantomtext2 phantomtext2 phantomtext2 phantomtext2 phantomtext2
\pend\pstart\itshape
\paragraph{3.} phantomtext3 phantomtext3 phantomtext3 phantomtext3 phantomtext3.
\pend
\endnumberingR
\end{Rightside}
\end{pages}
\Pages
\end{document}

• What's going in the \paragraph or \textbf? Are you counting something, or might it be a combination or text and numbers? – jon Jul 4 '15 at 22:20
• Please edit your question and add there a little complete document illustrating the problem mentioned. – Gonzalo Medina Jul 5 '15 at 1:30
• I'm not counting anything but would like to have a graphical sectioning, with text going on in the same line. An MWE is coming. – mxordn Jul 5 '15 at 8:03
• They line up for me: in Leftside, there is a blank line in the output between paragraphs "bla." and "2." such that all three paragraphs start at the same place with respect to the corresponding paragraph on the "Rightside" page. – jon Jul 5 '15 at 15:41
• with book, it's working. with scrbook, there is problem of sync (even without your \eledsection, which are not need for your MWE). I think your problem is with scrbook. Plese open an github issue, even if I can't assure to look for it quickly. – Maïeul Jul 5 '15 at 17:33


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# Omar Khayyam

### Quick Info

Born
18 May 1048
Nishapur, Persia (now Iran)
Died
4 December 1131
Nishapur, Persia (now Iran)

Summary
Omar Khayyam was an Islamic scholar who was a poet as well as a mathematician. He compiled astronomical tables and contributed to calendar reform and discovered a geometrical method of solving cubic equations by intersecting a parabola with a circle.

### Biography

Omar Khayyam's full name was Ghiyath al-Din Abu'l-Fath Umar ibn Ibrahim Al-Nisaburi al-Khayyami. A literal translation of the name al-Khayyami (or al-Khayyam) means 'tent maker' and this may have been the trade of Ibrahim his father. Khayyam played on the meaning of his own name when he wrote:-
Khayyam, who stitched the tents of science,
Has fallen in grief's furnace and been suddenly burned,
The shears of Fate have cut the tent ropes of his life,
And the broker of Hope has sold him for nothing!
The political events of the 11th Century played a major role in the course of Khayyam's life. The Seljuq Turks were tribes that invaded southwestern Asia in the 11th Century and eventually founded an empire that included Mesopotamia, Syria, Palestine, and most of Iran. The Seljuq occupied the grazing grounds of Khorasan and then, between 1038 and 1040, they conquered all of north-eastern Iran. The Seljuq ruler Toghrïl Beg proclaimed himself sultan at Nishapur in 1038 and entered Baghdad in 1055. It was in this difficult unstable military empire, which also had religious problems as it attempted to establish an orthodox Muslim state, that Khayyam grew up.

Khayyam studied philosophy at Naishapur and one of his fellow students wrote that he was:-
... endowed with sharpness of wit and the highest natural powers ...
However, this was not an empire in which those of learning, even those as learned as Khayyam, found life easy unless they had the support of a ruler at one of the many courts. Even such patronage would not provide too much stability since local politics and the fortunes of the local military regime decided who at any one time held power. Khayyam himself described the difficulties for men of learning during this period in the introduction to his Treatise on Demonstration of Problems of Algebra (see for example [1]):-
I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles in the vagaries of time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him.
However Khayyam was an outstanding mathematician and astronomer and, despite the difficulties which he described in this quote, he did write several works including Problems of Arithmetic, a book on music and one on algebra before he was 25 years old. In 1070 he moved to Samarkand in Uzbekistan which is one of the oldest cities of Central Asia. There Khayyam was supported by Abu Tahir, a prominent jurist of Samarkand, and this allowed him to write his most famous algebra work, Treatise on Demonstration of Problems of Algebra from which we gave the quote above. We shall describe the mathematical contents of this work later in this biography.

Toghril Beg, the founder of the Seljuq dynasty, had made Esfahan the capital of his domains and his grandson Malik-Shah was the ruler of that city from 1073. An invitation was sent to Khayyam from Malik-Shah and from his vizier Nizam al-Mulk asking Khayyam to go to Esfahan to set up an Observatory there. Other leading astronomers were also brought to the Observatory in Esfahan and for 18 years Khayyam led the scientists and produced work of outstanding quality. It was a period of peace during which the political situation allowed Khayyam the opportunity to devote himself entirely to his scholarly work.

During this time Khayyam led work on compiling astronomical tables and he also contributed to calendar reform in 1079. Cowell quotes The Calcutta Review No 59:-
When the Malik Shah determined to reform the calendar, Omar was one of the eight learned men employed to do it, the result was the Jalali era (so called from Jalal-ud-din, one of the king's names) - 'a computation of time,' says Gibbon, 'which surpasses the Julian, and approaches the accuracy of the Gregorian style.'
Khayyam measured the length of the year as 365.24219858156 days. Two comments on this result. Firstly it shows an incredible confidence to attempt to give the result to this degree of accuracy. We know now that the length of the year is changing in the sixth decimal place over a person's lifetime. Secondly it is outstandingly accurate. For comparison the length of the year at the end of the 19th century was 365.242196 days, while today it is 365.242190 days.

In 1092 political events ended Khayyam's period of peaceful existence. Malik-Shah died in November of that year, a month after his vizier Nizam al-Mulk had been murdered on the road from Esfahan to Baghdad by the terrorist movement called the Assassins. Malik-Shah's second wife took over as ruler for two years but she had argued with Nizam al-Mulk so now those whom he had supported found that support withdrawn. Funding to run the Observatory ceased and Khayyam's calendar reform was put on hold. Khayyam also came under attack from the orthodox Muslims who felt that Khayyam's questioning mind did not conform to the faith. He wrote in his poem the Rubaiyat :-
Indeed, the Idols I have loved so long
Have done my Credit in Men's Eye much Wrong:
Have drowned my Honour in a shallow cup,
And sold my reputation for a Song.
Despite being out of favour on all sides, Khayyam remained at the Court and tried to regain favour. He wrote a work in which he described former rulers in Iran as men of great honour who had supported public works, science and scholarship.

Malik-Shah's third son Sanjar, who was governor of Khorasan, became the overall ruler of the Seljuq empire in 1118. Sometime after this Khayyam left Esfahan and travelled to Merv (now Mary, Turkmenistan) which Sanjar had made the capital of the Seljuq empire. Sanjar created a great centre of Islamic learning in Merv where Khayyam wrote further works on mathematics.

The paper [18] by Khayyam is an early work on algebra written before his famous algebra text. In it he considers the problem:-
Find a point on a quadrant of a circle in such manner that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal.
Khayyam shows that this problem is equivalent to solving a second problem:-
Find a right triangle having the property that the hypotenuse equals the sum of one leg plus the altitude on the hypotenuse.
This problem in turn led Khayyam to solve the cubic equation $x^{3} + 200x = 20x^{2} + 2000$ and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by ruler and compass methods, a result which would not be proved for another 750 years. Khayyam also wrote that he hoped to give a full description of the solution of cubic equations in a later work [18]:-
If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared.
Indeed Khayyam did produce such a work, the Treatise on Demonstration of Problems of Algebra which contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. In fact Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations. Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and al-Khazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of al-Khwarizmi). However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations. Khayyam wrote (see for example [9] or [10]):-
In the science of algebra one encounters problems dependent on certain types of extremely difficult preliminary theorems, whose solution was unsuccessful for most of those who attempted it. As for the Ancients, no work from them dealing with the subject has come down to us; perhaps after having looked for solutions and having examined them, they were unable to fathom their difficulties; or perhaps their investigations did not require such an examination; or finally, their works on this subject, if they existed, have not been translated into our language.
Another achievement in the algebra text is Khayyam's realisation that a cubic equation can have more than one solution. He demonstrated the existence of equations having two solutions, but unfortunately he does not appear to have found that a cubic can have three solutions. He did hope that "arithmetic solutions" might be found one day when he wrote (see for example [1]):-
Perhaps someone else who comes after us may find it out in the case, when there are not only the first three classes of known powers, namely the number, the thing and the square.
The "someone else who comes after us" were in fact del Ferro, Tartaglia and Ferrari in the 16th century. Also in his algebra book, Khayyam refers to another work of his which is now lost. In the lost work Khayyam discusses the Pascal triangle but he was not the first to do so since al-Karaji discussed the Pascal triangle before this date. In fact we can be fairly sure that Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients. This follows from the following passage in his algebra book (see for example [1], [9] or [10]):-
The Indians possess methods for finding the sides of squares and cubes based on such knowledge of the squares of nine figures, that is the square of 1, 2, 3, etc. and also the products formed by multiplying them by each other, i.e. the products of 2, 3 etc. I have composed a work to demonstrate the accuracy of these methods, and have proved that they do lead to the sought aim. I have moreover increased the species, that is I have shown how to find the sides of the square-square, quatro-cube, cubo-cube, etc. to any length, which has not been made before now. the proofs I gave on this occasion are only arithmetic proofs based on the arithmetical parts of Euclid's "Elements".
In Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution to non-euclidean geometry, although this was not his intention. In trying to prove the parallels postulate he accidentally proved properties of figures in non-euclidean geometries. Khayyam also gave important results on ratios in this book, extending Euclid's work to include the multiplication of ratios. The importance of Khayyam's contribution is that he examined both Euclid's definition of equality of ratios (which was that first proposed by Eudoxus) and the definition of equality of ratios as proposed by earlier Islamic mathematicians such as al-Mahani which was based on continued fractions. Khayyam proved that the two definitions are equivalent. He also posed the question of whether a ratio can be regarded as a number but leaves the question unanswered.

Outside the world of mathematics, Khayyam is best known as a result of Edward Fitzgerald's popular translation in 1859 of nearly 600 short four line poems the Rubaiyat. Khayyam's fame as a poet has caused some to forget his scientific achievements which were much more substantial. Versions of the forms and verses used in the Rubaiyat existed in Persian literature before Khayyam, and only about 120 of the verses can be attributed to him with certainty. Of all the verses, the best known is the following:-
The Moving Finger writes, and, having writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a Line,
Nor all thy Tears wash out a Word of it.

### References (show)

1. B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Omar-Khayyam
3. J L Coolidge, The mathematics of the great amateurs (Oxford, 1949).
4. J N Crossley, The emergence of number (Singapore, 1980).
5. D S Kasir, The Algebra of Omar Khayyam, trans. from Arabic (1972).
6. C H Mossaheb, Hakim Omare Khayyam as an Algebraist (Tehran, 1960).
7. R Rashed and A Djebbar (eds), L'Oeuvre algébrique d'al-Khayyam (Arabic), Sources and Studies in the History of Arabic Mathematics 3 (Aleppo, 1981).
8. B A Rozenfel'd and A P Yushkevich, Omar Khayyam (Russian), Akademija Nauk SSSR Izdat. 'Nauka' (Moscow, 1965).
9. R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).
10. R Rashed, Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984).
11. S G Tirtha, The Nectar of Grace, Omar Khayyam's Life and Works (Allahbad, 1941).
12. A R Amir-Moéz, Khayyam, al-Biruni, Gauss, Archimedes, and quartic equations, Texas J. Sci. 46 (3) (1994), 241-257.
13. R C Archibald, Notes on Omar Khayyam (1050-1122) and recent discoveries, Pi Mu Epsilon J. 1 (1953), 350-358.
14. A V Dorofeeva, Omar Khayyam (1048-1131) (Russian), Mat. v Shkole (2) (1989), i, 145-147.
15. A E-A Hatipov, Omar Khayyam and Newton's binomial (Russian), Trudy Samarkand. Gos. Univ. (N.S.) 181 (1970), 84-88.
16. A E-A Hatipov, A trigonometric treatise of Omar Khayyam (?) (Russian), Trudy Samarkand. Gos. Univ. (N.S.) 181 (1970), 83-84.
17. A E-A Hatipov, The first book of Omar Khayyam's treatise on geometry (Russian), Trudy Samarkand. Gos. Univ. (N.S.) Vyp. 107 (1960), 9-16.
18. O Khayyam, A paper of Omar Khayyam, Scripta Math. 26 (1963), 323-337.
19. O Khayyam, The mathematical treatises of Omar Khayyam (Russian), Istor.-Mat. Issled. 6 (1953), 9-112.
20. K M Mamedov and O Khayyam, Newton's binomial formula was first published by Omar Khayyam (Azerbaijani), Izv. Akad. Nauk Azerbaidzan. SSR Ser. Fiz.-Tehn. Mat. Nauk (3) (1972), 3-8.
21. V A Ogannisjan, Omar Khayyam (Russian), Armjan. Gos. Ped. Inst. Sb. Nauv cn. Trud. Ser. Fiz.-Mat. Vyp. 3 (1966), 89-98.
22. B A Rozenfel'd and A P Yushkevich, Notes to the mathematical treatises of Omar Khayyam (Russian), Istor.-Mat. Issled. 6 (1953), 113-172.
23. D Struik, Omar Khayyam, Mathematics Teacher 4 (1958), 280-285.
24. B Vahabzadeh, Al-Khayyam's conception of ratio and proportionality, Arabic Sci. Philos. 7 (2) (1997), 159, 161, 247-263.
25. H J J Winter and W Arafat, The algebra of Omar Khayyam, J. Roy. Asiatic Soc. Bengal. Sci. 16 (1950), 27-77.
26. P D Yardley, Graphical solution of the cubic equation developed from the work of Omar Khayyam, Bull. Inst. Math. Appl. 26 (5-6) (1990), 122-125.
27. A P Yushkevich, Omar Khayyam and his 'Algebra' (Russian), Akad. Nauk SSSR. Trudy Inst. Istorii Estestvoznaniya 2 (1948), 499-534.


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# Why is SpaceX not also working on a smaller version of Starship?

SpaceX is planning to retire Falcon 9, which would leave it with only Starship. While a Starship launch is expected to be cheaper than a Falcon 9 one, a downscaled Starship launch would be cheaper still.

Moreover, a downscaled Starship would be easier to develop and iterate upon, paving the way for the full scale Starship. It could be as similar as possible to the full scale Starship (but of course allometrically scaled). So it might even lower the total development cost and risk.

It could conceivably use seven times less Falcon engines, ending up with five on the first stage, and one on the second. That smaller Starship could be used for almost all the satellite launches, as it would still be a respectable size.

• They just trimmed the top off of it? twitter.com/NASASpaceflight/status/1197265917589303296 – ta.speot.is Nov 21 '19 at 2:49
• "a downscaled Starship launch would be cheaper still" why would you think so? Things don't scale effortlessly, up or down. Even the mere fact of having two launchers instead of one introduces extra expenses. And that's completely ignoring R&D, which is a significant part of the costs of dealing with rockets in general. – Luaan Nov 21 '19 at 12:11
• The original design a few years ago was based around a 12m diameter. Musk has also said an 18m design might come at some point. Maybe the current 9m design is the smaller version... – James Thorpe Nov 22 '19 at 9:18
• The Falcon 9 uses two types of Merlin engines: sea-level optimized, and vacuum-optimized variants. The Merlin is a LOX / RP1 gas-generator cycle engine with thrust around 900 kn. The Starship and Superheavy booster are designed to use two variants of the new new Raptor engine. The Raptor is a LOX / Liquid Methane full-flow staged combustion engine and has a thrust of around 2400 kn. – Dragongeek Nov 24 '19 at 13:48

There are several compelling engineering and design reasons why a bigger spaceship makes sense and several reasons why making a mini-starship does not make sense for SpaceX specifically (and their vision).

First and foremost, Elon Musk has made it clear that his goal for the company and the future isn't to provide cheap satellite launch capabilities, it's to put people on Mars. Building an inferior version of the Starship isn't on the "critical path" to putting people on Mars for Elon. SpaceX is very much Elon Musk's company and it follows his vision.

That said, here are a couple other reasons why SpaceX specifically might not want to make a Mini-Starship (although a Mini-Starship might make sense for some manufacturers):

• Bigger rockets can be more efficient, fuel-consumption wise, as dry mass can be a smaller portion of the overall rocket mass. For example, if your flight computer and avionics system masses 100 kg, it would have a large effect on a rocket which can only lift 1000kg to LEO. If your rocket is able to put up 100,000kg at a time, the 100kg flight computer suddenly becomes much less of the overall dry mass and thus your rocket is more efficient.

• Building big vs building small. One of the things that is "revolutionary" in Starship's current design/construction progress is that it's being welded together in a field, outdoors. SpaceX believes that, when your rocket is big, you can get away with looser tolerances, which in turn equates to money saved. For example, the Sea Dragon proposal was designed to be assembled in a shipyard and had an enormous lift capability for a very low cost. Starship is similar. If a Mini-Starship were constructed, it would probably require tighter tolerances and smaller, more exact parts. Really, this boils down to my first point again. If you're building a small rocket, every weld, bolt, and wire has a more significant impact on a lighter, smaller rocket's efficiency.

• SpaceX believes that a design similar to the two Starships currently being prototyped will work. If they instead scrapped what they have developed so far and built a Mini-Starship, they'd have to essentially start from zero again. The design for a rocket is so complex, that you can't just resize it and have a functional design afterwards. SpaceX learned this lesson the hard way when developing Falcon Heavy. Initially, Elon had thought that it would just be "strapping three boosters together" but in the end, SpaceX had to develop the center core almost entirely from scratch.

• SpaceX has a limited amount of employees and money. If they decided to work on a Starship and Mini-Starship concurrently, the pace on both projects would be cut in half if not more. Elon has stated that development speed is critically important to him and diluting his engineers with extra projects means that Starship takes longer to build.

• The smallsat launch market is heating up. Several companies, notably Rocket Labs, are focused on sending small satellites to orbit. SpaceX is betting that there will be no shortage of demand for mass sent to space and then it won't matter how big the rocket is that goes up, as they'll be able to fill every rocket with payloads. Even then, the 2 million launch cost is so low that even if Starship flies mostly empty, it would still be making a profit. Addendum: In the comments several concerns have been raised: • Building a Mini-Starship would be cheaper than a full sized Starship • On this, I don't disagree. If you were starting from scratch, building a 1/8 scale starship would probably be cheaper than a full sized one. • SpaceX has already sunk a lot of money and development time into ITS/BFR/Starship over the years. Simply shelving these efforts to work on an inferior rocket would be seen as a waste of money. • While it would be cheaper, I don't think it would be significantly cheaper. The big costs are the Raptor engines as they're highly complex and are a revolutionary design. Still, all Starship prototypes we've seen so far (Starhopper and the late MK1) haven't been equipped with the full compliment of engines. Starhopper only had one and the MK1 was only briefly test-fit with three engines. • Starship and the Superheavy booster are two different craft. Starship is designed to be equipped with three Sea-level raptor engines and three vacuum-optimized engines for redundancy. On a Mini-Starship, only a single Raptor engine would fit which wouldn't be efficient and wouldn't provide the redundancy that multiple engines allow. An entirely new Merlin-sized Methalox engine would need to be developed which just isn't going to happen when SpaceX has the working Raptor in production. • A Mini-Starship could be iterated more rapidly • Maybe, but neither the Mini-Starship or Starship are small desk-top or garage-shop construction projects. While working with smaller components would be easier, cranes and heavy equipment would still be required for both. • The labor that's connected to the actual size of the rocket doesn't take up much of the overall assembly time. Yes, welding together a bigger rocket takes longer than a small one, but installing avionics, sensors, engines, wires, and writing software wouldn't suddenly become quicker if the rocket were physically smaller. • Mini-Starship could provide cheap launches while Starship is developed • Starship (or Mini-Starship) are not SSTO vehicles. They both require the additional development of the Superheavy booster. Also, there is no reason why SpaceX needs to stop operating the Falcon 9 series. They have plenty of rockets with plenty of reuse cycles left and despite the non-reusable second stage, Falcon 9 can easily undercut the price of any other launch provider. Until the Vulcan or Ariane 6 rocket become operational, SpaceX won't even have to lower their prices (and profit margins) for Falcon 9 to be the cheapest provider. • Beliefs that upscaling the Mini-Starship to a full sized one would be easy or lots could be shared during development • I'm not quite sure where this belief is coming from. In the engineering world, you can't just scale something up and expect it to work. For example, take the Cessna 172, which is the most built aircraft of all time. Why don't larger planes just look like the Cessna but bigger? Because, simply scaling up a design, no matter how fantastic just doesn't always make sense. • The parts of development that could be shared don't justify building an entirely new rocket. The engines are already built and building fuel tanks and the actual structure isn't simple per se but it's rather straightforward compared to developing a FFSC engine. Aerodynamics and in-flight control would also be explored in a Mini-Starship and maybe heat-shield tech, and would be transferable, but again, SpaceX clearly doesn't believe this would be cost-effective • SpaceX could reap profits in the smallsat market with Mini-Starship • Just because the smallsat market is heating up doesn't mean that bigger satellites are going anywhere. Especially with Starlink, SpaceX isn't hurting for cash or launch contracts. Dealing with smallsat launches where each satellite only pays like 50k isn't where SpaceX sees profits. • "When your rocket is big, you can get away with looser tolerances, " this is yet to be proven. The sloppy-looking welds on the vehicle in Boca Chica scare me. – Organic Marble Nov 20 '19 at 18:38 • @OrganicMarble Fixed – Dragongeek Nov 20 '19 at 18:40 • @OrganicMarble Especially on the windward side, during the hypersonic flight regime. How will those bumps and welds affect the hypersonic flow? Terrifying to think about. And how do you model that in a wind tunnel? – geoffc Nov 20 '19 at 18:53 • @OrganicMarble Prescient comment. It blew up a few hours after you made that comment. – ceejayoz Nov 21 '19 at 2:18 • "Blew up" -- they were conducting an over pressure test and reached the point it couldn't take the over pressure. It looks dramatic, but they knew going in it was a possibility. – Rob Crawford Nov 21 '19 at 22:01 There was some talk of modifying Falcon 9's second stage to give it full reusability in November of 2018. The idea was that a reusable second stage would be used to test out Starship technologies. This effort was scrapped 10 days later in favor of accelerating development on the current stainless steel design of starship. The superheavy/starship architecture is designed to facilitate exploration and exploitation of space beyond simple Low Earth Orbit, which necessitates a larger booster and spacecraft. Since development costs tend to outpace materials, manufacturing, and fuel costs; it makes more sense to build a single large vehicle that can do many jobs than two vehicles for separate missions. • This answer assumes that developing a smaller Starship in parallel would significantly increase the total development costs, which is unclear to me. As I wrote in my question, developing a smaller version could even decrease some of the total costs. Falcon 9 is a very different rocket, so it being scrapped doesn't seem like strong evidence. – Stephane Bersier Nov 23 '19 at 23:55 Similar to how a small plane can't really carry any people around the world, but a larger plane can, it turns out there are challenges to making really small rockets. These are magnified when you take in to account full reusability. It turns out that Starship is about the smallest spacecraft that makes sense for a fully reusable spacecraft. A few things to consider. The pressure of the tanks has to remain about the same. The strength of a tank is pretty much dependent on the thickness of the tank, thus you get a lot more stored fuel with a larger rocket for the weight. Rocket engines have a maximum thrust to weight ratio when they are larger. Starship is intended to support missions to orbit even when some rockets fail. The size pretty much again comes to Starship sized. Heat shielding is a bit trickier, but I believe there is a similar minimum mass that is required to truly be effective. Essentially it has to absorb the energy of the spacecraft and dissipate it out. With a larger spacecraft, the density per area ratio is lower, allowing for more effective slowing down, and also less heat shielding required. Lastly, there was a similar thought process with Falcon 1 vs Falcon 9. It turns out that many of the costs of a launch are fixed, the flight analysis, coupled loads analysis, etc all have to be done regardless if you have a single small satellite or a huge one. Concentrating on a larger load allows those costs to be minimized. Bottom line is, it is far more efficient for a spacecraft of Starship's goals to focus purely on a spacecraft roughly the size of Starship. Robert Zubrin even mentioned this at a recent Mars Society meeting. • In particular, the surface area of fuel tanks/heat shields goes up as the square of the ship scale, but the volume goes up as the cube, which is what provides the favorable economies of larger ships. Of course, this doesn't continue indefinitely, as you become limited by materials strength. Thus, there is an optimal size. – Lawnmower Man Nov 21 '19 at 3:40 • Cube-square law does not apply to pressurized tanks. The structural mass of a tank is proportional to pressure and volume. See Pressure Vessel#Scaling – Rainer P. Nov 21 '19 at 8:49 • @RainerP. Right, but that doesn't apply very much to rockets. Unless you're using pressure-fed engines, the internal pressure in the tanks is relatively low, and the resulting loads are much less important than the axial compression due to thrust and dynamic pressure or aerodynamic bending loads. Also, with liquid propellants in the tanks, only hydrostatic pressure matters for the tanks, and that only scales with length. – TooTea Nov 21 '19 at 10:06 • Most of the arguments of this answer are flawed. – Stephane Bersier Nov 23 '19 at 23:37 • For paragraph one: in the atmosphere, smaller planes have a smaller range because of the square–cube law. This doesn't apply in space. While rockets launched from Earth do have to contend with the atmosphere initially, the energy lost to drag is already very minor for a Falcon 9. – Stephane Bersier Nov 23 '19 at 23:38 The simple answer, as already stated by others, is that small rockets do not align with Musk's goals of putting people on Mars. Additionally, Spacex already have a small (OK, medium sized) launcher in the form of Falcon 9. They claim that Starship + Falcon Superheavy will be able to undercut it. One possiblity I wouldn't disregard (if Spacex chose do go down that route in future) is to use the upper stage of Starship without a booster to launch payloads to suborbital speeds. A payload would then need its own kicker stage to get into orbit. Spacex have already revised their Earth-to-Earth passenger scheme to one without a booster so using the upper stage of Starship without a booster is not an entirely new idea. The kicker stage could be a simple solid rocket, or a reusable stage - whatever is decided in the future. I would note that when Falcon 9 is retired, Spacex will have several hundred surplus Merlin engines which could be given one last use in an expendable kicker stage. Spacex has gone for the upper size end of the market, with very few competitors, all of them expendable: SLS (projected to be ready soon) and Long March 9 and Yenisei (projected to be ready late 2020's.) They have avoided the crowded lower size end of the market. If one of the current players in that market becomes big enough to compete with Spacex, they may regret it in 20 years. But there is no sign of that happening yet. I wouldn't discount spacex developing a smaller launch system later - but it would be done in an opportunistic way on the back of the Starship program. Spacex does have one Spinoff project not directly aligned to the goal of getting to Mars, and in typical Musk style, it is unique: the Starlink Satellite constellation project. Spinoffs like this are obviously a necessary way to raise funds. • I feel like your last paragraph contradicts your first one... – Stephane Bersier Nov 23 '19 at 23:28 • @StephaneBersier How so? Musk's main goal with Spacex is to build a large rocket to take people to Mars. For that he needs both money and a way to gain experience with his rocket. One way to make money is with the starlink constellation, which is something nobody else is doing (or at least, not on anything like the same scale.) Another way would be to build a small rocket, which is something a lot of others are competing to do, therefore SpaceX have avoided this. – Level River St Nov 25 '19 at 2:22 • SpaceX is actually already planning to compete in the smallsat market with their Smallsat Program. Given the expected specs, a mini-Starship would have no trouble competing in the smallsat market while still maintaining large profit margins, serving the same role as the Falcon 9 in their current Smallsat Program plans. – Stephane Bersier Nov 25 '19 at 18:45 • @StephaneBersier Spacex's Smallsat program is a rideshare program, not a small rocket program. It's a great option if you don't care too much what orbit you're going into or when you launch, since all satellites on the launch go into virtually the same orbit on the same launch date. Small rocket operators like Rocketlab will give you a bespoke orbit and launch when you want. That's the advantage they have over rideshare, and in theory they can charge more for clients who need it. The trouble is there's a lot of other small rocket operators either up and running or coming soon. – Level River St Nov 25 '19 at 23:54 • Exactly. I didn't say it's a small rocket program. My arguments still hold: SpaceX could make money for its ultimate goal with a mini-Starship, just as it's doing with Starlink. If SpaceX can make money with its current Smallsat program using a Falcon 9, then a fortiori it could do so with a mini-Starship. – Stephane Bersier Nov 27 '19 at 0:14 Right now Starship and it Superheavy booster are both open design. Current plan is for 35 engines on 1st stage, 6 on 2st stage, 150ton to LEO ascend payload, dry mass 85t, return to Earth payload 50t. All of this could change during rocket development, because of technological challenges, but also money which will have SpaceX available during development process. They already change Starship design many times ( number of fins, TPS types, design of legs ) and they could keep changing it constantly after first test flights. At the end they can finish with much smaller, medium version of Starship with less engines, less payload to LEO, smaller gross mass. For example 20,30t to LEO Starship/Superheavy with just total of 15 Raptors, but maybe 20,30t to LEO payload, but still with 30-40 Raptors total. It depends, on what would be optimal for lowest possible cost per kilogram to LEO, GTO, which will be main goal. Figure 2 million per flight is just < aspiring goal > and shouldn't be taken as fact. For example ESA Arianespace want with expendable Ariane 6 ( payload 20t to LEO, 10t to GTO ) achieve same cost per kilogram as F9R, which is about 5000 per kilogram to LEO ( more than twice to GTO ) and with Ariane next program, which could use potentially reusable LOX/Methane engine Prometheus, improve Ariane 6 cost per kilogram further by factor of two. But that of course is not yet given.

Same as is not given that Starship/Superheavy cost per kilogram to LEO or GTO will be better than F9R, since second stage reuse is much harder than 1st stage ( both technically and economically ), and no rocket with massive size of Starship/Superheavy ( it will have see level trust 3-5 times of Saturn 5 ) was ever build, let alone try to be reused.

• The Starship design is indeed not final. However, it is to be at least a super heavy-lift launch vehicle, which excludes the possibility of it being as small as the mini-Starship described in the question. – Stephane Bersier Nov 30 '19 at 0:44


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An item is discounted by 15% from its usual selling price. What is the usual selling price?

(1) The dollar value of the discount is $45. (2) The discounted price is$255.
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An item is discounted by 15% from its usual selling price. What is the usual selling price?

(1) The dollar value of the discount is $45. (2) The discounted price is$255.

Statement 1: Discount is $$15$$ , if usual selling price is $$100$$. $$45$$ discount corresponds to usual selling price of $$= \frac{100*45}{15}$$. Sufficient

Statement 2: Discounted price is $$100-15=85$$ , if usual selling price is $$100$$. $$255$$ discounted price corresponds to usual selling price of $$= \frac{100*255}{85}$$. Sufficient

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## Thomas' Calculus 13th Edition

$x=1-t; y=1+t; z=-1$
We are given that the equation of plane is: $1+y+z=1 ; 1+y=2$ Since, $r=r_0+tv$ and $v=\lt -1,1,0 \gt$ Now, let us consider $x=1$ and $y=1$ and $z=0-y=0-1=-1$ Thus, $r_0=\lt 1,1,-1 \gt$ The parametric equations are $x=1-t; y=1+t; z=-1+(0)t=-1$ or, $x=1-t; y=1+t; z=-1$


text:
Giant Planets, Tiny Stars: Producing Short-period Planets around White Dwarfs with the Eccentric Kozai–Lidov Mechanism
Abstract The recent discoveries of WD J091405.30+191412.25 (WD J0914 hereafter), a white dwarf (WD) likely accreting material from an ice-giant planet, and WD 1856+534 b (WD 1856 b hereafter), a Jupiter-sized planet transiting a WD, are the first direct evidence of giant planets orbiting WDs. However, for both systems, the observations indicate that the planets’ current orbital distances would have put them inside the stellar envelope during the red-giant phase, implying that the planets must have migrated to their current orbits after their host stars became WDs. Furthermore, WD J0914 is a very hot WD with a short cooling time that indicates a fast migration mechanism. Here, we demonstrate that the Eccentric Kozai–Lidov Mechanism, combined with stellar evolution and tidal effects, can naturally produce the observed orbital configurations, assuming that the WDs have distant stellar companions. Indeed, WD 1856 is part of a stellar triple system, being a distant companion to a stellar binary. We provide constraints for the orbital and physical characteristics for the potential stellar companion of WD J0914 and determine the initial orbital parameters of the WD 1856 system.
Authors:
; ;
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Publication Date:
NSF-PAR ID:
10328723
Journal Name:
The Astrophysical Journal
Volume:
922
Issue:
1
Page Range or eLocation-ID:
4
ISSN:
0004-637X


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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are reading an older version of this FlexBook® textbook: CK-12 Middle School Math - Grade 8 Go to the latest version.

# 12.2: Adding and Subtracting Polynomials

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Pyramid Building

As the students rounded the corner on fifth street, they spotted a peculiar looking building. It was in the shape of a pyramid.

“Here we go again,” said Michael as he read the next problem to his friends.

“A pyramid-shaped building has rectangular floors that get increasingly smaller as you go higher up in the building. If the $87^{th}$ floor has a length of $6x+16$ and a width of 28, and each floor’s length and width decrease by 4 as you ascend, find the total area of the $87^{th}, 88^{th},$ and $89^{th}$ floor.”

This is the next dilemma.

To work on this problem, you will need to understand area as well as adding and subtracting polynomials. Pay close attention to the rules of this lesson and you will be able to figure out this problem by the end of the lesson.

What You Will Learn

In this lesson, you will learn the following skills.

• Add polynomials vertically.
• Add polynomials horizontally.
• Subtract polynomials vertically.
• Subtract polynomials horizontally.
• Solve real – world problems by writing, simplifying, and evaluating polynomial expressions regarding rectangle areas.

Teaching Time

I. Add Polynomials Vertically

Now that you understand how to identify polynomials, we can work on adding and subtracting them. Let’s start with adding. To begin, let’s look at how we can add whole numbers with many digits.

Example

Add the numbers 5026 and 3210.

You might add it like this, right?

$5026 \\\underline{+3210} \\{8236}$

If you think about it, you might notice that the same addition could be thought of in this way.

$& \qquad \qquad \qquad \text{thousands} \quad \qquad \text{hundreds} \quad \qquad \text{tens} \quad \qquad \text{ones} \\& \quad 5026 \quad \rightarrow \qquad \ \ 5000 \qquad \qquad \qquad \qquad \qquad \ \ 20 \qquad \qquad 6\\& \underline{+ \ 3210 \quad \rightarrow \quad \ + \ 3000 \qquad \qquad \quad \ \ 200 \qquad \ \ \quad10 \qquad \qquad \qquad}\\& \quad 8236 \quad \leftarrow \qquad \ \ 8000 \quad \ + \qquad \quad 200 \quad + \quad \ 30 \quad \ + \quad \ 6$

Here we have shown 5026 to be $5000 + 20 + 6$. The number 3210 is $3000 + 200 + 10$.

Each of the similar places has been lined up vertically (one on top of the other) so that 3000 is beneath 5000 in the thousands place and 10 is beneath 20 in the tens place. Also, 200 is by itself because the first number had no digits in the hundreds place. Likewise, 6 is by itself because the second number had no digits in the ones place. Although this is not a practical way of writing a simple addition problem, it does demonstrate the technique we can use to add polynomials.

You learned in the previous lesson about like terms—terms are alike when they have exactly the same variable(s) to exactly the same power(s). When terms are alike, we can combine them by adding their coefficients.

Example: $5x^3+9x^3=14x^3$

We also learned that polynomials are one or more terms in an expression.

Example of polynomials: $x^2+ 5 \qquad 3x-8+4x^5 \qquad -7a^2+9b-4b^3+6$

Polynomials can be added in the same manner as we added 5026 and 3210.

Example

Add the polynomials $(7x^2+9x-5)$ and $(6x^2+3x+10)$.

$& (7x^2+9x-5) \quad \rightarrow \quad 7x^2 \quad + \quad 9x \quad + \quad -5 \\& \qquad \quad + \\ & \underline{(6x^2+3x+10) \ \ \rightarrow \quad 6x^2 \quad + \quad 3x \quad + \quad 10 \;} \\& 13x^2+12x+5 \quad \leftarrow \ \ 13x^2 \ \ + \quad 12x \ \ + \quad \ 5$

Each of the like terms was aligned vertically, one on top of the other. Notice that the negative sign on -5 was kept with the number 5. Be careful when you add the integers.

Example

Add the polynomials $(-2x^3+9x^2-3)$ and $(8x^2+5x-14)$.

$& \quad (-2x^3+9x^2-3) \qquad \ \rightarrow \quad -2x^3 \quad + \quad 9x^2 \quad \ + \quad \quad \quad + \ \ -3\\& \underline{\ +(8x^2+5x-14) \qquad \quad \rightarrow \ \qquad \qquad \qquad 8x^2 \ \ \ \ + \quad 5x \quad + \ -14 \ \ } \\& -2x^3+17x^2+5x-17 \ \leftarrow \quad -2x^3 \quad + \quad 17x^2 \ \ + \quad 5x \quad + \ \ -17$

Each of the terms was again aligned vertically. Notice this time that the space underneath $-2x^3$ is empty. That is because there was no like term in the second polynomial that could be combined with $-2x^3$. There was a like term of $9x^2$. The $8x^2$ is the like term that could be combined with $9x^2$ so it was placed beneath. Their sum was $17x^2$. There was no like term of $5x$, either. But -3 and -14, the constants, were like terms so were aligned together and combined. The result was $-2x^3+17x^2+5x-17$. Unneeded addition signs are removed.

Be sure that you have the definitions for polynomials and like terms down in your notebooks.

II. Add Polynomials Horizontally

A second method for adding polynomials is horizontally—in a single line. Just as you might add $6 + 19 = 25$ without placing them one on top of the other, so can a polynomial be added to another.

Example

$& =(7x^2+3x-11)+(3x^2-9x+5)\\&=7x^2+3x-11+3x^2-9x+5 \\&=10x^2-6x-6$

Step 1: rewrite without parenthesis

Step 2: combine like terms

In Step 1, the polynomial can be rewritten without parentheses because the parentheses serve only to show the separation of the polynomials. Notice, we do not align them vertically by like terms as before. However, we had to take care to recognize and combine like terms correctly.

This method can be a little trickier. If you find yourself getting confused, then go back and add the polynomials horizontally.

III. Subtract Polynomials Vertically

Just like we could add polynomials, we can subtract them too. We can perform this operation both vertically and horizontally. Let’s start with vertically.

When we subtract polynomials, we can use a similar procedure as with addition—we can subtract vertically. However, remember that subtraction is the same as “adding the opposite.” In other words, $5 - 8$ is the same as $5 + (-8)$. We can add the opposite of 8 instead of subtracting 8. We will use the same idea with polynomials.

Remember that subtracting is the same as adding the opposite. Write this down in your notebook.

Example

$(9x^2+4x-7)-(2x^2+6x-4)$

Set up the problem vertically by aligning the like terms.

$& \quad (9x^2+4x-7) \quad \rightarrow \ \ \quad 9x^2 \quad + \ \ \quad 4x \quad + \quad -7 \quad \text{Lines up like terms.} \\& \underline{-(2x^2+6x-4) \quad \rightarrow \quad -2x^2 \quad + \quad -6x \quad + \ \quad 4 \quad \ \text{Add the opposite.} \quad \ \ } \\& \quad 7x^2-2x-3 \quad \ \ \leftarrow \quad \ \ 7x^2 \quad + \ \quad -2x \ \ + \quad -3 \quad \text{Combine lilke terms.}$

When you add the opposite, the sign changes on each of the terms in the subtracted polynomial. Inside the parentheses, the coefficient of $2x^2$ is positive. But when you add the opposite, the sign changes to negative, or $-2x^2$. We also changed the sign on the $6x$ to $-6x$ and the -4 to 4.

Example

$(-7x^3+3x^2-x+4)-(-6x^2+9)$

$& \quad (-7x^3+3x^2-x+4) \quad \rightarrow \ \ \quad -7x^3 \quad + \ \ \quad 3x^2 \quad + \quad -x \quad + \quad \quad 4 \quad \ \text{Lines up like terms.} \\& \underline{\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; -(-6x^2+9) \quad \rightarrow \quad \quad \quad \quad \quad \quad \quad \ \ 6x^2 \quad + \ \quad \quad \quad \ + \quad -9 \ \ \ \text{Add the opposite.}\; \; \; \; \; \;} \\& \quad -7x^3+9x^2-x-5 \quad \ \leftarrow \quad \quad -7x^3 \quad + \quad \ 9x^2 \quad + \quad -x \quad + \quad -5 \quad \ \text{Combine lilke terms.}$

Now we can look at subtracting polynomials horizontally.

IV. Subtract Polynomials Horizontally

When we added polynomials, we used two methods—adding vertically and adding horizontally. You just learned to subtract polynomials vertically. As you have guessed, we can also subtract polynomials horizontally. First we will review the distributive property.

The distributive property: For all real numbers $a, b,$ and $c, \ a(b + c) = ab + ac$.

Examples:

$5(3x+7)&=15x+35 \\4(2y-7)&=8y-28 \\-2(9x+3)&=-18x-6 \\-3 (-2y-4)&=6y+12$

Remember to be careful with negative signs when using distributive property.

Now, let us remember that coefficients are the numerical factors of variables. The coefficient of $3x$ is 3. The coefficient of $9x^2$ is 9. When we see the term $-x$, the coefficient is -1. Although you could write $-1x$, we normally do not because the 1 is considered unnecessary. How does this relate to the distributive property? The negative sign could be in front of the parentheses, like this: $-(3x - 2)$. This is similar to $-x$ where the coefficient is the unwritten -1. Just like you could write $-1x$, you could also write $-1(3x - 2)$. The distributive property is now more apparent in that each term will now be multiplied by -1.

Example

$& -(7x+5)=-1(7x+5)=-7x-5 \\& -(x^2-3x+14)=-1(x^2-3x+14)=-x^2+3x-14 \\& \qquad \qquad \quad \uparrow$

Here you can insert the -1 and then multiply. As with adding the opposite, the sign changes on each of the terms in the polynomial.

We can now use this method to subtract polynomials horizontally. First we’ll distribute the negative sign to each of the terms in the subtracted polynomial and then we will combine like terms just as we did when we added polynomials.

Example

$& (5x+3)-(2x-8)\\&=(5x+3)-1(2x-8) \\&=5x+3-2x+8 \\&=3x+11$

It may seem that way, but if you go step by step and remember that subtracting is adding the opposite, then you will be able to subtract polynomials vertically and horizontally.

V. Solve Real – World Problems by Writing, Simplifying and Evaluating Polynomial Expressions Regarding Rectangle Areas

As seen in previous lessons, polynomials can be applied to geometrical shapes. Keep in mind some basic geometrical ideas.

Area refers to the space inside an object or space. It is measured in square units. The area of land is generally measured with units such as acres, square miles, or square kilometers. The area formula of a rectangle is $A = lw$ or area equals length times width. Shapes that are not rectangular are often broken up into rectangular parts. Their total area is calculated by adding the areas of the separate parts.

Example

A concrete walkway surrounds a rectangular swimming pool. In order to know how much weather treatment to buy, the owner must know how many square units of concrete he has. The walkway is 5 feet wide on all sides. The swimming pool has a length of $7x$ and a width of 14 feet. How many square units concrete does he have?

In order to find the area of the concrete, we must find the area of the large rectangle and then subtract the area of the swimming pool.

The length of the large rectangle measures $7x + 5 + 5$. Its width measures $14 + 5 + 5$.

So its area will be $(7x + 5 + 5) \cdot (14 + 5 + 5)$

The area of the swimming pool will be its length times its width or $7x \cdot 14$.

Area of the swimming pool is $98x$.

To find the area of the concrete, subtract the area of the swimming pool from the total area:

$&(168x + 240)-98x \\&=70x + 240$

This is the answer.

Now let’s go back to the problem from the introduction.

## Real-Life Example Completed

The Pyramid Building

Here is the problem from the introduction. Reread it and then figure out the areas of the other floors of the building.

As the students rounded the corner on fifth street, they spotted a peculiar looking building. It was in the shape of a pyramid.

“Here we go again,” said Michael as he read the next problem to his friends.

“A pyramid-shaped building has rectangular floors that get increasingly smaller as you go higher up in the building. If the $87^{th}$ floor has a length of $6x + 16$ and a width of 28, and each floor’s length and width decrease by 4 as you ascend, find the total area of the $87^{th}, 88^{th},$ and $89^{th}$ floor.”

This is the next dilemma.

Remember there are three parts to your answer.

Solution to Real – Life Example

Here is the work to figure out the area of the other three floors and the total area too.

$87^{th} \ \text{floor} \quad \text{Area} &= 28(6x + 16) = 168x + 448 \\88^{th} \ \text{floor} \quad \text{Area} &= (28 - 4)(6x + 16 - 4) = 24(6x + 12) = 144x + 288\\89^{th} \ \text{floor} \quad \text{Area} &= (28 - 4 - 4)(6x + 16 - 4 - 4) = 20(6x + 8) = 120x + 160 \\\text{Total area} &= (168x + 448) + (144x + 288) + (120x + 160) \\&=168x + 448 + 144x + 288 + 120x + 160 \\&=432x + 896$

## Vocabulary

Here are the words that are found in this lesson.

Polynomial
one or more terms in an expression, often referred to specifically in situations where there are more than three terms.
Like Terms
terms that have the same variable and power.
Area
the space inside an object or area. It is measured in square units.

## Time to Practice

Directions: Add the following polynomials vertically. Be sure to align like terms.

1. $(4x^2+7x-2)+(3x-17)$
2. $(-4x^4-x^3+8)+(-2x^3+5x+6)$
3. $(10x^3-4x^2-2x+5)+(-x^2+9x-5)$
4. $(6x^2+5x+9)+(4x^2+3x+6)$
5. $(9x^2-3x+4)+(6x^2-9x+2)$
6. $(3y^2+4x-9)+(-5y^2-6x+10)$

Directions: Add the following polynomials horizontally.

1. $(-3x-8)+(15x+5)$
2. $(x^4+7x^3-2x+7)+(-8x^3+9x^2-4)$
3. $(4x^2y-3x^2y^2+7xy)+(9x^2y^2-5xy+3x^2)$
4. $(5xy-3x+19)+ (4xy-9x-22)$

Subtract the following polynomials vertically.

1. $(x^2+5x)-(3x-14)$
2. $(7y^3+4y^2-3y-1)-(y^3+6y^2-4)$

Subtract the following polynomials horizontally.

1. $(m^2+17m-11)-(3m^2+8m+12)$
2. $(z^2+3z)-(3z^2+7z+16)-(4z-13)$
3. Find the sum of the area of the two rectangles.
4. Divide this shape into rectangles. Then find its total area.
5. A wooden rectangular patio has a length of 27 and a width of $6x - 2$. Cut into the patio are three equal rectangular planters of length $4x + 1$ and width 3. What is the total area of the wooden patio without the planters? Use a sketch to help you.
6. An ancient pyramid has a walkway about half-way up. The pyramid has four sides. On each side of the pyramid, the rectangular walkway is 10 feet wide and $40x + 8$ long. There is a second walkway closer to the top which is half as wide and half as long. What is the area of the long walkway?
7. What is the area of the short walkway?
8. What is the total area of the two walkways?

At Grade

Jan 14, 2013

## Last Modified:

Oct 29, 2015
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text:
# Marginal Revenue with definite integral

• December 7th 2009, 07:33 PM
mellowdano
Marginal Revenue with definite integral
Hello,

I was given a graph of a marginal revenue function which i have attached. (If for some reason it does not come up, these are main points on the graph (0,100), (500,0) and (1000,-100).)
The first thing i needed to do is find the equation of the marginal revenue function

What i have done so far:
found slope which is -0.2
y intercept is 100
Therefore marginal revenue function is R'(x)=--0.2x + 100

However my prof tells me she wants me to find it as a definite integral with limit 0 to 1000.

Would I start the integral as
∫(100 - 0.2x)dx [0,1000]
and finish with
= 100x -0.1x^2 + C [0,1000]
• December 7th 2009, 08:34 PM
VonNemo19
Quote:

Originally Posted by mellowdano
Hello,

I was given a graph of a marginal revenue function which i have attached. (If for some reason it does not come up, these are main points on the graph (0,100), (500,0) and (1000,-100).)
The first thing i needed to do is find the equation of the marginal revenue function

What i have done so far:
found slope which is -0.2
y intercept is 100
Therefore marginal revenue function is R'(x)=--0.2x + 100

However my prof tells me she wants me to find it as a definite integral with limit 0 to 1000.

Would I start the integral as
∫(100 - 0.2x)dx [0,1000]
and finish with
= 100x -0.1x^2 + C [0,1000]

$\int_0^{1000}(-0.2x+100)dx=(-\frac{1}{10}x^2+100x)|_0^{1000}$

Now evaluate...

This being a definte integral, there will be no constant C.
• December 8th 2009, 10:41 AM
mellowdano
when i evaluates the answer is 0.
= ∫1000 (100 – 0.2x) dx
0
= (100x – 0.1x2)|1000
0
= [100(1000) – 0.1(1000)2] – [100(0) – 0.1(0)2]
=[100000 – 100000] – [0 – 0]
= 0-0
= 0

Would I conclude from this that the 2 x intercepts are therefore x=0 and x=1000 without graphing it?


text:
# Photometric Redshifts in the Hawaii-Hubble Deep Field-North (H-HDF-N)

 Title Photometric Redshifts in the Hawaii-Hubble Deep Field-North (H-HDF-N) Publication Type Journal Article Year of Publication 2014 Authors Yang, G, Xue, Y Q, Luo, B, Brandt, W N, Alexander, D M, Bauer, F E, Cui, W, Kong, X, Lehmer, B D, Wang, J-X, Wu, X-B, Yuan, F, Yuan, Y-F, Zhou, H Y Journal \apjs Volume 215 Pagination 27 Keywords catalogs, cosmology: observations, galaxies: distances and redshifts, galaxies: evolution, galaxies: formation, surveys Abstract {We derive photometric redshifts (z $_{phot}$) for sources in the entire (\~{}0.4 deg$^{2}$) Hawaii-Hubble Deep Field-North (H-HDF-N) field with the EAzY code, based on point-spread-function-matched photometry of 15 broad bands from the ultraviolet (U band) to mid-infrared (IRAC 4.5 {$μ$}m). Our catalog consists of a total of 131,678 sources. We evaluate the z $_{phot}$ quality by comparing z $_{phot}$ with spectroscopic redshifts (z $_{spec}$) when available, and find a value of normalized median absolute deviation {$\sigma$}$_{NMAD}$ = 0.029 and an outlier fraction of 5.5% (outliers are defined as sources having |z$_{phot}$ - z$_{spec}$ |/(1 + z$_{spec}$ ) {\gt} 0.15) for non-X-ray sources. More specifically, we obtain {$\sigma$}$_{NMAD}$ = 0.024 with 2.7% outliers for sources brighter than R = 23 mag, {$\sigma$}$_{NMAD}$ = 0.035 with 7.4% outliers for sources fainter than R = 23 mag, {$\sigma$}$_{NMAD}$ = 0.026 with 3.9% outliers for sources having z łt} 1, and {$\sigma$}$_{NMAD}$ = 0.034 with 9.0% outliers for sources having z {\gt} 1. Our z $_{phot}$ quality shows an overall improvement over an earlier z $_{phot}$ work that focused only on the central H-HDF-N area. We also classify each object as a star or galaxy through template spectral energy distribution fitting and complementary morphological parameterization, resulting in 4959 stars and 126,719 galaxies. Furthermore, we match our catalog with the 2 Ms Chandra Deep Field-North main X-ray catalog. For the 462 matched non-stellar X-ray sources (281 having z $_{spec}$), we improve their z $_{phot}$ quality by adding three additional active galactic nucleus templates, achieving {$\sigma$}$_{NMAD}$ = 0.035 and an outlier fraction of 12.5%. We make our catalog publicly available presenting both photometry and z $_{phot}$, and provide guidance on how to make use of our catalog. } DOI 10.1088/0067-0049/215/2/27


text:
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# Wall pressure spectra on a DU96-W-180 profile from low to pre-stall angles of attack

Suryadi, Alexandre and Herr, Michaela (2015) Wall pressure spectra on a DU96-W-180 profile from low to pre-stall angles of attack. 21st AIAA/CEAS Aeroacoustics Conference, 22 - 26 June 2015, Dallas, Texas.

 PDF 6MB

## Abstract

In an effort to characterize noise induced by separated turbulent boundary layers, surface pressure fluctuations on a DU96-W-180 airfoil were measured using miniature pressure sensors. Because of limitation in amplifier channels and available sensors, a rearrangeable configuration of sensors was applied. Chordwise distributions of the surface pressure were obtained for aerodynamic angles of attack of $-0.8^\circ\leq\alpha\leq10.3^\circ$ and at three Reynolds numbers $(0.8, 1.0, and 1.2) \times 10^6$. The boundary layer profile at 1%c behind the trailing edge was measured using constant temperature anemometry. The boundary layer thickness compares well with that simulated using XFOIL for $alpha\leq7.8^\circ�$. Within the limits of the simulation, other relevant boundary layer properties from XFOIL were used to calculate the surface pressure spectrum predicted from published empirical models for zero and non-zero pressure gradient turbulent boundary layers. Finally, a modified Blake-TNO semi-empirical model was used to predict the surface pressure spectrum near the trailing edge for separated flow. The modification is introduced to the so called `moving axis spectrum' and the chord-normal correlation length scale. It is found that in the low frequency range, the modified semi-empirical model �fits well with the measured surface pressure spectrum of a separated turbulent boundary layer.

Item URL in elib:https://elib.dlr.de/97164/
Document Type:Conference or Workshop Item (Speech)
Title:Wall pressure spectra on a DU96-W-180 profile from low to pre-stall angles of attack
Authors:
AuthorsInstitution or Email of AuthorsAuthors ORCID iD
Herr, Michaelamichaela.herr (at) dlr.deUNSPECIFIED
Date:23 June 2015
Refereed publication:Yes
Open Access:Yes
Gold Open Access:No
In SCOPUS:No
In ISI Web of Science:No
Status:Published
Keywords:measurements, wall pressure spectra, wind energy, separated flow, flow modeling
Event Title:21st AIAA/CEAS Aeroacoustics Conference
Event Location:Dallas, Texas
Event Type:international Conference
Event Dates:22 - 26 June 2015
Organizer:AIAA
HGF - Research field:Energy
HGF - Program:Renewable Energies
HGF - Program Themes:other
DLR - Research area:Energy
DLR - Program:E SF - Solar research
DLR - Research theme (Project):E - Wind Energy (old), L - Simulation and Validation
Location: Braunschweig
Institutes and Institutions:Institute of Aerodynamics and Flow Technology > Technical Acoustics
Deposited By: Suryadi, Alexandre
Deposited On:20 Jul 2015 16:09


text:
# Nodes and Weights of Gauss-Hermite

## Calculates the nodes and weights of the Gauss-Hermite integration.

 $\normal(1)\ {\large\int_{\small -\infty}^{\hspace{25}\small \infty}}e^{-x^2}f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\[10](2)\ g(x)=e^{-x^2}f(x)\\\hspace{20}{\large\int_{\small -\infty}^{\hspace{25}\small \infty}}g(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}e^{x_i^2}g(x_i)\\[20]\hspace{5}nodes\hspace{35} x_i:\hspace{10} H_n(x_i)=0\\\hspace{5}weights\hspace{20} w_i={\large\frac{2^{n+1}{\large n!\sqrt{\pi}}}{[H_{n+1}(x_i)]^2}}\\$
 nodes half(x>=0) all order n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 n=2,3,4,..,100
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 $\normal\ i=1,2,...,n\\(1)\ x_{10}=\sqrt{2n+1}-\frac{1.85575}{(2n+1)^{1/6}}\\\hspace{25}x_{20}=x_1-{\large\frac{1.14n^{0.426}}{x_1}},\\\hspace{25}x_{30}=1.86x_{2}-0.86x_{1}\\\hspace{25}x_{40}=1.91x_{3}-0.91x_{2},\\\hspace{25}x_{i0}=1.86x_{i-1}-0.86x_{i-2}\\\hspace{30}..........\\[15](2)\ solve\ x_i\hspace{20}H_n(x_i)=0\\ \hspace{20}Halley's\ method\hspace{20}x\leftarrow x-{\large\frac{2yy^'}{2[y^']^2-yy^{''}}}\\ \hspace{20}y=H_n(x)\\ \hspace{20}y^'=H^'_n(x)=2nH_{n-1}(x)\\ \hspace{20}y^{''}=H^{''}_n(x)=2xH^'_n(x)-2nH_n(x)\\[10](3)\ w_i={\large\frac{2^{n+1}{\large n!\sqrt{\pi}}}{[H_{n+1}(x_i)]^2}}\\$

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text: 2019-08-05 07:33:01 -0600 asked a question Why GF(256) matrix only have zero and one number? like this sage: matrix(GF(2**8), [1, 2, 3, 4]) [1 0 1 0]  I just want do some matrix calculate mod 256, can any help me, please?


text:
$K = \frac{k_{forward}}{k_{reverse}}$

Suraj Doshi 2G
Posts: 100
Joined: Fri Aug 02, 2019 12:15 am

Can someone explain the difference between adsorption and absorption as it relates to catalysis?

Asha Agarwal 1E
Posts: 51
Joined: Fri Aug 30, 2019 12:16 am

Absorption is when something takes in something else like a sponge absorbing water. Adsorption on the other hand is when a reactant sits on the surface of a catalyst.

Matt F
Posts: 100
Joined: Sat Aug 17, 2019 12:17 am

^To add onto this, adsorption occurs with a heterogeneous catalyst where the catalyst is in a different phase as the reactants (usually solids), causing them to sit on the surface

805307623
Posts: 99
Joined: Fri Aug 09, 2019 12:17 am


text:
# Mapdef (decl)

Mapdefs are declarations that contain a list of general map properties. These declarations are required if you wish for your map to be displayed during the setup of a multiplayer game.

## Syntax

mapDef [path] {
"name" [name]
"devname" [name]
"singleplayer" [boolean]
"Deathmatch" [boolean]
"Team DM" [boolean]
"Last Man" [boolean]
"Tourney" [boolean]
"size0" [size]
"size1" [size]
"size2" [size]
"size3" [size]
}

MapDef declarations are started with a header which consists of the keyword mapDef followed by the name of the map. They are then followed by the body of the declaration which is enclosed in curly brackets.

The name and devname values you enter here will determine how the map is named in the server window. The key values with the boolean field accept 0, 1 to determine which game modes are available for play. The four size keys are a performance optimization and you should set them with the " com_updateLoadSize 1" CVar after loading the map. You need to do this for all four quality settings .

Note: It’s normal if size0 and size1 use the same value.

Alternatively you can create a batch file to do this for you. A batch file is an ASCII text file with the .BAT extension thus you can use notepad to create one. Here is an example of a batch file named “mapsize.bat”.

doom3 +com_updateLoadSize 1 +map <mappath>\<mapname>.map +quit

The batch file itself can be anywhere if you set the right paths but in the example above it’s supposed to be put in the Doom 3 folder. This will update the corresponding MapDef entry with the correct sizes. If the map is multiplayer only then do:

doom3 +com_updateLoadSize 1 +si_map <mappath>\<mapname>.map +si_pure 0 +spawnserver +quit


text:
# How do I add a remote Git repository to an Ubuntu Server?

I have created a Git repository on my Desktop machine (Windows 7) with:

git init
git commit -m "added my files"

Now I have installed a new Ubuntu Server 10.10 on a machine on my LAN and installed OpenSSH. My home directory is /home/jonas and I created a directory ~/code/ to contain my projects. I can log in to the Ubuntu Server from Windows 7 with Putty.

I installed Git on the server with sudo apt-get install git

Now I want to add my Git repository on my Desktop to the Server. I tried to follow the instructions from Pragmatic Version Control Using Git.

From my Desktop I run these commands:

git remote add origin jonas@192.168.1.10/home/jonas/code/myproject.git
git push origin master

But I got this error message:

fatal: 'jonas@192.168.1.180/home/jonas/code/myproject.git' does not appear to be
a git repository
fatal: The remote end hung up unexpectedly

What is the problem? How do I create the remote repository?

As PerfectlyNormal suggested, I added a : in the address. Now it worked better, and I had to type my password to the server, but then I got a similar error message:

fatal: '/home/jonas/code/myproject.git' does not appear to be a git repository
fatal: The remote end hung up unexpectedly

Do I have to initialize a Git repository on the server before I can git push to it?

-

Did you setup the repository on the remote server? You need to run

mkdir -p /home/jonas/code/myproject.git
cd /home/jonas/code/myproject.git
git init --bare

on the server in order to set it up. I recommend taking a look at how to setup a git server in the free ProGit book.

-
Thanks, this seem to work. But how can I checkout the project on the server? If I do git status then I get a message that I must be on a "work tree" –  Jonas Feb 27 '11 at 22:28
You can't. A remote repository must be a bare repository, which cannot be used for working on. See this answer for more information. If you want to be able to work on the repo on the server as well, you need to checkout the repository in another directory on the server as well and use it just as if you were on a different computer (i.e. you must pull and push as usual). –  Andrew Marshall Feb 27 '11 at 22:34
@Andrew: Ah, that was informative. Thanks. –  Jonas Feb 27 '11 at 22:35
@andrew-marshall Is that true? Couldn't he use a non-bare repo on the server and then push to working-dir/.git? Clearly, this is not a typical setup but isn't it possible? –  Tilman Vogel Feb 28 '11 at 0:05
It's possible but will lead to unexpected results if you push to a branch on a remote repository that is checked-out on the remote machine. In fact, I don't think recent versions of git (1.7+ perhaps?) will even allow you to do so. –  Andrew Marshall Feb 28 '11 at 0:10
git remote add origin jonas@192.168.1.10/home/jonas/code/myproject.git

When using SSH, remote repository addresses can be expressed in two ways. One using absolute paths and one using relative paths from the users home directory. You've mixed them up.

The corrected command would be one of the following.

git remote add origin jonas@192.168.1.10:code/myproject.git

-

First thing I notice is that you're missing a ':'. Should be git remote add origin jonas@192.168.1.10:/home/jonas/code/myproject.git

-

you need a colon:

git remote add origin jonas@192.168.1.10/home/jonas/code/myproject.git

should be:

git remote add origin jonas@192.168.1.10:/home/jonas/code/myproject.git

-

I normally create a bare repository locally then scp that repository to the server when I'm setting up a remote repository.

example

cd c:\gits

git clone --bare c:\path\to\local\repository\some_project

which creates some_project.git

then


text:
# Weekly Papers on Quantum Foundations (27)

Reply to the comment on “Quantum priniple of relativity”. (arXiv:2206.15247v1 [quant-ph])

We refute criticisms by Del Santo and Horvat towards our paper “Quantum principle of relativity”: most of their counterarguments can be dismissed, and the rest provides further evidence to our claims.

Stochastic Bohmian and Scaled Trajectories. (arXiv:2206.15260v1 [quant-ph])

In this review we deal with open (dissipative and stochastic) quantum systems within the Bohmian mechanics framework which has the advantage to provide a clear picture of quantum phenomena in terms of trajectories, originally in configuration space. The gradual decoherence process is studied from linear and nonlinear Schr\”odinger equations through Bohmian trajectories as well as by using the so-called quantum-classical transition differential equation through scaled trajectories. This transition is governed by a continuous parameter, the transition parameter, covering these two extreme open dynamical regimes. Thus, two sources of decoherence of different nature are going to be considered. Several examples will be presented and discussed in order to illustrate the corresponding theory behind each case, namely: the so-called Brownian-Bohmian motion leading to quantum diffusion coefficients, dissipative diffraction in time, dissipative tunnelling for a parabolic barrier under the presence of an electric field and stochastic early arrivals for the same type of barrier. In order to simplify the notations and physical discussion, the theoretical developments will be carried out in one dimension throughout all this wok. One of the main goals is to analyze the gradual decoherence process existing in these open dynamical regimes in terms of trajectories, leading to a more intuitive way of understanding the underlying physics in order to gain new insights.

Quantum mechanics? It’s all fun and games until someone loses an $i$. (arXiv:2206.15343v1 [quant-ph])

QBism regards quantum mechanics as an addition to probability theory. The addition provides an extra normative rule for decision-making agents concerned with gambling across experimental contexts, somewhat in analogy to the double-slit experiment. This establishes the meaning of the Born Rule from a QBist perspective. Moreover it suggests that the best way to formulate the Born Rule for foundational discussions is with respect to an informationally complete reference device. Recent work [DeBrota, Fuchs, and Stacey, Phys. Rev. Res. 2, 013074 (2020)] has demonstrated that reference devices employing symmetric informationally complete POVMs (or SICs) achieve a minimal quantumness: They witness the irreducible difference between classical and quantum. In this paper, we attempt to answer the analogous question for real-vector-space quantum theory. While standard quantum mechanics seems to allow SICs to exist in all finite dimensions, in the case of quantum theory over the real numbers it is known that SICs do not exist in most dimensions. We therefore attempt to identify the optimal reference device in the first real dimension without a SIC (i.e., $d=4$) in hopes of better understanding the essential role of complex numbers in quantum mechanics. In contrast to their complex counterparts, the expressions that result in a QBist understanding of real-vector-space quantum theory are surprisingly complex.

Thermalization of Gauge Theories from their Entanglement Spectrum. (arXiv:2107.11416v2 [quant-ph] UPDATED)

Using dual theories embedded into a larger unphysical Hilbert space along entanglement cuts, we study the Entanglement Structure of $\mathbf{Z}_2$ lattice gauge theory in $(2+1)$ spacetime dimensions. We demonstrate Li and Haldane’s conjecture, and show consistency of the Entanglement Hamiltonian with the Bisognano-Wichmann theorem. Studying non-equilibrium dynamics after a quench, we provide an extensive description of thermalization in $\mathbf{Z}_2$ gauge theory which proceeds in a characteristic sequence: Maximization of the Schmidt rank and spreading of level repulsion at early times, self-similar evolution with scaling coefficients $\alpha = 0.8 \pm 0.2$ and $\beta = 0.0 \pm 0.1$ at intermediate times, and finally thermal saturation of the von Neumann entropy.

Platonic Bell inequalities for all dimensions. (arXiv:2112.03887v2 [quant-ph] UPDATED)

In this paper we study the Platonic Bell inequalities for all possible dimensions. There are five Platonic solids in three dimensions, but there are also solids with Platonic properties (also known as regular polyhedra) in four and higher dimensions. The concept of Platonic Bell inequalities in the three-dimensional Euclidean space was introduced by Tavakoli and Gisin [Quantum 4, 293 (2020)]. For any three-dimensional Platonic solid, an arrangement of projective measurements is associated where the measurement directions point toward the vertices of the solids. For the higher dimensional regular polyhedra, we use the correspondence of the vertices to the measurements in the abstract Tsirelson space. We give a remarkably simple formula for the quantum violation of all the Platonic Bell inequalities, which we prove to attain the maximum possible quantum violation of the Bell inequalities, i.e. the Tsirelson bound. To construct Bell inequalities with a large number of settings, it is crucial to compute the local bound efficiently. In general, the computation time required to compute the local bound grows exponentially with the number of measurement settings. We find a method to compute the local bound exactly for any bipartite two-outcome Bell inequality, where the dependence becomes polynomial whose degree is the rank of the Bell matrix. To show that this algorithm can be used in practice, we compute the local bound of a 300-setting Platonic Bell inequality based on the halved dodecaplex. In addition, we use a diagonal modification of the original Platonic Bell matrix to increase the ratio of quantum to local bound. In this way, we obtain a four-dimensional 60-setting Platonic Bell inequality based on the halved tetraplex for which the quantum violation exceeds the $\sqrt 2$ ratio.

A comment on Bell’s Theorem Logical Consistency. (arXiv:2202.09639v2 [quant-ph] UPDATED)

In their recent paper, Lambare and Franco correctly claim that Bell deterministic model and inequalities may be derived using only local causality, perfect correlations and measurement independence, without talking about joint probabilities. However, measurement independence, as we explain, should not be called no-conspiracy or freedom of choice. Measurement independence should be called noncontextuality, because it allows implementing random variables, describing incompatible random experiments, on a unique probability space, on which they are jointly distributed. Using a precise terminology proposed by Dzhafarov and Kujala in the Contextuality by Default approach , such implementation defines a probabilistic coupling, which we explain in this paper. The frequentists proof of inequalities fails, if this probabilistic coupling and joint probabilities do not exist. We construct also a probabilistic coupling for their counterexample to prove, that there is no contradiction with Fine Theorem. Nobody questions Bell Theorem logical consistency and nobody claims that Fine disproved Bell Theorem. Various metaphysical assumptions, such as local realism, classicality or counterfactual definiteness may motivate a choice of a probabilistic model. However, once a model is chosen, its meaning and its implications may only be discussed rigorously in a probabilistic framework. Bell inequalities are violated in various Bell Tests; for us it proves that hidden variables depend on settings confirming contextual character of quantum observables and an active role played by measuring instruments. Bell was a realist, thus he thought that he had to choose between nonlocality and super-determinism. From two bad choices he chose nonlocality. Today he would probably choose contextuality

Law of Total Probability in Quantum Theory and Its Application in Wigner’s Friend Scenario. (arXiv:2204.12285v2 [quant-ph] UPDATED)

It is well-known that the law of total probability does not hold in general in quantum theory. However, the recent arguments on some of the fundamental assumptions in quantum theory based on the extended Wigner’s Friend scenario show a need to clarify how the law of total probability should be formulated in quantum theory and under what conditions it still holds. In this work, the definition of conditional probability in quantum theory is extended to POVM measurements. Rule to assign two-time conditional probability is proposed for incompatible POVM operators, which leads to a more general and precise formulation of the law of total probability. Sufficient conditions under which the law of total probability holds are identified. Applying the theory developed here to analyze several quantum no-go theorems related to the extended Wigner’s friend scenario reveals logical loopholes in these no-go theorems. The loopholes exist as a consequence of taking for granted the validity of the law of total probability without verifying the sufficient conditions. Consequently, the contradictions in these no-go theorems only reconfirm the invalidity of the law of total probability in quantum theory, rather than invalidating the physical statements that the no-go theorems attempt to refute.

Interaction between macroscopic quantum systems and gravity. (arXiv:2206.07574v3 [gr-qc] UPDATED)

We review experiments and theoretical models about the possible mutual interplay between the gravitational field and materials in the superconducting state or other macroscopic quantum states. More generally, we focus on the possibility for quantum macrosystems in a coherent state to produce local alterations of the gravitational field in which they are immersed. This fully interdisciplinary research field has witnessed a conspicuous progress in the last decades, with hundreds of published papers, and yet several questions are still completely open.

A comment on Bell’s Theorem Logical Consistency. (arXiv:2202.09639v2 [quant-ph] UPDATED)

Authors: Marian Kupczynski

In their recent paper, Lambare and Franco correctly claim that Bell deterministic model and inequalities may be derived using only local causality, perfect correlations and measurement independence, without talking about joint probabilities. However, measurement independence, as we explain, should not be called no-conspiracy or freedom of choice. Measurement independence should be called noncontextuality, because it allows implementing random variables, describing incompatible random experiments, on a unique probability space, on which they are jointly distributed. Using a precise terminology proposed by Dzhafarov and Kujala in the Contextuality by Default approach , such implementation defines a probabilistic coupling, which we explain in this paper. The frequentists proof of inequalities fails, if this probabilistic coupling and joint probabilities do not exist. We construct also a probabilistic coupling for their counterexample to prove, that there is no contradiction with Fine Theorem. Nobody questions Bell Theorem logical consistency and nobody claims that Fine disproved Bell Theorem. Various metaphysical assumptions, such as local realism, classicality or counterfactual definiteness may motivate a choice of a probabilistic model. However, once a model is chosen, its meaning and its implications may only be discussed rigorously in a probabilistic framework. Bell inequalities are violated in various Bell Tests; for us it proves that hidden variables depend on settings confirming contextual character of quantum observables and an active role played by measuring instruments. Bell was a realist, thus he thought that he had to choose between nonlocality and super-determinism. From two bad choices he chose nonlocality. Today he would probably choose contextuality

Space and time transformations with a minimal length. (arXiv:2206.15422v1 [gr-qc])

Authors: Pasquale Bosso

Phenomenological studies of quantum gravity have proposed a modification of the commutator between position and momentum in quantum mechanics so to introduce a minimal uncertainty in position in quantum mechanics. Such a minimal uncertainty and the consequent minimal measurable length have important consequences on the dynamics of quantum systems. In the present work, we show that such consequences go beyond dynamics, reaching the definition of quantities such as energy, momentum, and the same Hamiltonian. Furthermore, since the Hamiltonian, defined as the generator of time evolution, results to be bounded, a minimal length implies a minimal time.

The intrinsic pathology of self-interacting vector fields. (arXiv:2205.07784v3 [gr-qc] UPDATED)

Authors: Andrew CoatesFethi M. Ramazanoğlu

We show that self-interacting vector field theories exhibit unphysical behaviour even when they are not coupled to any external field. This means any theory featuring such vectors is in danger of being unphysical, an alarming prospect for many proposals in cosmology, gravity, high energy physics and beyond. The problem arises when vector fields with healthy configurations naturally reach a point where time evolution is mathematically ill-defined. We develop tools to easily identify this issue, and provide a simple and unifying framework to investigate it.

Newton’s Third Rule and the Experimental Argument for Universal Gravity

Domski, Mary (2022) Newton’s Third Rule and the Experimental Argument for Universal Gravity. Routledge Focus on Philosophy . Routledge, New York. ISBN 978-1-032-02036-5

Hilbert-Style Axiomatic Completion: On von Neumann and Hidden Variables in Quantum Mechanics

Mitsch, Chris (2022) Hilbert-Style Axiomatic Completion: On von Neumann and Hidden Variables in Quantum Mechanics. [Preprint]

Complete Physical Characterization of Quantum Nondemolition Measurements via Tomography

Author(s): L. Pereira, J. J. García-Ripoll, and T. Ramos

We introduce a self-consistent tomography for arbitrary quantum nondemolition (QND) detectors. Based on this, we build a complete physical characterization of the detector, including the measurement processes and a quantification of the fidelity, ideality, and backaction of the measurement. This fra…

[Phys. Rev. Lett. 129, 010402] Published Wed Jun 29, 2022

Humeanism and the Measurement Problem

Gao, Shan (2022) Humeanism and the Measurement Problem. [Preprint]

Consciousness and Complexity: Neurobiological Naturalism and Integrated Information Theory

Ellia, Francesco and Chis-Ciure, Robert (2022) Consciousness and Complexity: Neurobiological Naturalism and Integrated Information Theory. [Preprint]


text:
Dataplot Vol 1 Vol 2

# FRECHET PLOT

Name:
FRECHET PLOT
Type:
Graphics Command
Purpose:
Generates a Frechet plot.
Description:
The Frechet plot can be used to determine whether the 2-parameter Frechet distribution is an appropriate distributional model for a set of data.

The Frechet plot is formed by

1. Sort the data. Call these points yi.

2. The x-axis coordinates are $$\ln(y_{i})$$.

3. The y-axis coordinates are $$-\ln(-\ln(p_{i}))$$ where

$$p_{i} = \frac{i - 0.3}{n + 0.4}$$

If the data come from a 2-parameter Frechet distribution, then the points on this plot should be approximately linear.

If you fit a line to the points on this plot, the intercept can be used as an estimate of the scale parameter and the slope can be used as an estimate of the shape parameter. However, the plot is generally used to determine if a 2-parameter Frechet is appropriate. If the plot is approximately linear, then parameter estimates would usually be determined by maximum likelihood using the command

FRECHET MAXIMUM LIKELIHOOD Y

Maximum likelihood estimates would generally be preferred over the Frechet plot estimates since its statistical properties are better understood.

Currently, this plot is supported for the maximum version of the Frechet distribution. It is also only supported for the case with no censoring. It is also not supported for grouped data.

The characteristics of these components are controlled through the LINE and CHARACTER commands. This is demonstrated in the program examples below.

If you have groups in the data, you can specify a "highlight" option to draw the points in the different groups with different attributes. For example, this can be used to draw outliers in a different color. This is demonstrated in the second program example below.

Syntax 1:
FRECHET PLOT <y>             <SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 2:
HIGHLIGHT FRECHET PLOT <y> <x>
<SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable;
<x> is a group-id variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
FRECHET PLOT Y1
HIGHLIGHT FRECHET PLOT Y1 X
Note:
The following internal parameters are saved after a FRECHET PLOT. These parameters can be used like any user created parameter by the analyst.

 SHAPE - the slope of the fitted line as an estimate of the shape parameter SCAL - the intercept of the fitted line as an estimate of the scale parameter BPT1 - the 0.1% point of the best fit distribution BPT5 - the 0.5% point of the best fit distribution BP1 - the 1% point of the best fit distribution BP5 - the 5% point of the best fit distribution BP10 - the 10% point of the best fit distribution BP20 - the 20% point of the best fit distribution BP50 - the 50% point of the best fit distribution BP80 - the 80% point of the best fit distribution BP90 - the 90% point of the best fit distribution BP95 - the 95% point of the best fit distribution BP99 - the 99% point of the best fit distribution BP995 - the 99.5% point of the best fit distribution BP999 - the 99.9% point of the best fit distribution

The percent point estimates are computed by using the estimates of scale (SCAL) and shape (SHAPE) obtained from the plot and plugging these values into the Frechet percent point function.

Note:
The Frechet plot is similar in concept to a Frechet probability plot. However, a few distinctions should be noted.

1. The Frechet plot is based on a 2-parameter Frechet distribution. A Frechet probability plot is based on a 3-parameter Frechet distribution.

2. The Frechet plot linearization is specific to the Frechet distribution. The probability plot linearization can be used for any distribution. For the probability plot method, the ppcc plot is used to estimate the shape parameter and then the probability plot is used to estimate the location and scale parameters and to assess the goodness of fit.
Default:
None
Synonyms:
SUBSET is a synonym for HIGHLIGHT
Related Commands:
 LINES = Sets the type for plot lines. CHARACTER = Sets the type for plot characters. PROBABILITY PLOT = Generates a probability plot. PPCC PLOT = = Generates a ppcc plot. NORMAL PLOT = Generates a normal plot. WEIBULL PLOT = Generates a Weibull plot. MAXIMUM LIKELIHOOD = Estimate distribution parameters using maximum likelihood.
Reference:
Karl Bury (1999), "Statistical Distributions in Engineering", Cambridge University Press, chapter 16.
Applications:
Distributional Modeling
Implementation Date:
2013/10
Program 1:

. Step 1:  Read the data
.
skip 25
skip 0
.
. Step 2:  Set plot control features
.
label case asis
title case asis
title offset 2
character X blank
line blank solid
title Frechet Plot for FRECHET.DAT
y1label -LN(-LN(P(i)))
x1label LN(y(i))
x2label P(i) = (i - 0.3)/(n + 0.4)
.
. Step 3:  Generate the plot
.
frechet plot y

Program 2:

. Demonstrate the HIGHLIGHT option
.
. Step 1:  Read the data
.
skip 25
skip 0
let n = size y
let x = 0 for i = 1 1 n
let x = 1 subset y > 22
let x = 1 subset y < 14
.
. Step 2:  Set plot control features
.
label case asis
title case asis
title offset 2
character circle circle blank
character hw 1 0.75 all
character fill on on
character color black red
line blank blank solid
title Frechet Plot for FRECHET.DAT
y1label -LN(-LN(P(i)))
x1label LN(y(i))
x2label P(i) = (i - 0.3)/(n + 0.4)
.
. Step 3:  Generate the plot
.
highlight frechet plot y x

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Date created: 01/31/2015
Last updated: 01/31/2015

Please email comments on this WWW page to alan.heckert.gov.