text:
# iPad app can't display ⟨⟩ brackets

These brackets are ubiquitous in math, but I don't use latex because I get it in a wiki page of pronunciation. I ask it on English.SE. The iPad app can't display them:

But when I am writing this question, I notice that it can, although those brackets are copied from one source:

Some tests: ⟨⟩ ⟨⟩

⟨⟩

They can be displayed in the title though.

• I can't see them in Google Chrome version 40.0.2214.111 either. I can see them in IE version 11.0.9600.17501 and Firefox version 35.0.1 – Thor Feb 13 '15 at 20:15
• Repo on iPhone running latest version of app. Displays in title, but not body. – Cole Johnson Feb 14 '15 at 4:07
• Testing comments: ⟨⟩ ⟨⟩ – Brian Nickel Mar 11 '15 at 20:57
• @BrianNickel works fine in comment – Ooker Mar 11 '15 at 21:00

This is an interesting case of character support and font substitution. If you look at the question on desktop Safari, you will see the brackets correct everywhere. If you look at the question in iPad Safari, you'll see boxes everywhere but in fixed-type code: ⟨⟩. If you look in the app, you'll see boxes in the post body and brackets in the title, editor, comments, etc.
Taking things one step further, if you copy the text "⟨⟩ ⟨⟩" from desktop Safari into TextEdit, you'll see that the fixed sized text has the expected font "Menlo" but the normal text has the unexpected font "Apple Symbols".

format: plain


text:
Arithmetic sequences. Writing the general term differently

So I'm dealing with this problem that has to do with arithmetic sequences and I was wondering if I could write the general term of this kind of sequences (finite) like $$a_{k-1}=a_1+(k-\lfloor \frac{n}2 \rfloor)d$$ where $$k$$ can take values from $$2$$ to $$n+1$$

$$\lfloor \rfloor$$ is the floor function and d is the common difference

For example, if we take the sequence $$1,4,7,10,13$$.

In this case $$n=5$$ and $$d=a_n-a_{n-1}=3$$

From the formula above for $$k=2$$ $$a_1=1+3(2-\lfloor \frac{5}2 \rfloor)=1$$ $$k=3$$ $$a_2=1+3(3-\lfloor \frac{5}2 \rfloor)=4$$

$$k=4$$ $$a_3=1+3(4-\lfloor \frac{5}2 \rfloor)=7$$

$$k=5$$ $$a_4=1+3(5-\lfloor \frac{5}2 \rfloor)=10$$

$$k=6$$ $$a_5=1+3(6-\lfloor \frac{5}2 \rfloor)=13$$

• If you do that, wouldn't the odd elements be duplicated? I mean, for example $a_1$ would be the same as $a_2$ and $a_3$ would be the same as $a_4$ ... Then it's not an arithmetic sequence any more. If $k$ is the index, what is $n$ ? – Matti P. Jun 17 at 12:17
• Can you give an example of such a sequence? It is not clear what "common difference" means here – Henry Jun 17 at 12:20
• By $d$ being the "common difference" I assume you mean that for any $1 \leq i < n$, $d = a_{i+1}-a_i$ since $a_k$ defines an arithmetic sequence? – lurker Jun 17 at 12:22
• I've edited it and I gave an example – J.Dane Jun 17 at 12:34

If you really want to, you can, but $$n$$ isn't a free parameter. If you want the formula $$a_{k-1} = a_1 + (k-\lfloor \frac{n}{2}\rfloor)d$$ to hold for $$k=2$$, you get a condition $$a_1 = a_1 + (2-\lfloor \frac{n}{2}\rfloor)d$$ that means that for $$d\neq 0$$ you'll need $$\lfloor \frac{n}{2}\rfloor = 2$$ So, for natural $$n$$ it's only correct for $$n=4$$ and $$n=5$$ (like in your example). The formula simplifies then to the usual $$a_{k-1} = a_1 + (k-2)d$$ $$a_k = a_1 + (k-1) d$$
What may be of more use to you are the formulae $$a_{k} = a_{\lfloor \frac{n}{2}\rfloor} + (k-\lfloor \frac{n}{2}\rfloor)d$$ or $$a_{k-1} = a_{\lfloor \frac{n}{2}\rfloor-1} + (k-\lfloor \frac{n}{2}\rfloor)d$$ that are also variations on the general formula, but allow you to use an arbitrary $$n$$.


text:
# Article

Full entry | PDF   (0.3 MB)
Keywords:
$u$-ideals; finite rank; compact; and weakly compact operators; Hahn-Banach extension operators
Summary:
Let $X$ be a Banach space. We give characterizations of when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal W}(Y,X)$ for every Banach space $Y$ in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when ${\cal F}(X,Y)$ is a $u$-ideal in ${\cal W}(X,Y)$ for every Banach space $Y$, when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal W}(Y,X^{**})$ for every Banach space $Y$, and when ${\cal F}(Y,X)$ is a $u$-ideal in ${\cal K}(Y,X^{**})$ for every Banach space $Y$.
References:
[1] Belobrov, P. K.: Minimal extension of linear functionals onto the second conjugate space. Mat. Zametki 27 (1980), 439-445, 494. MR 0570754
[2] Casazza, P. G., Kalton, N. J.: Notes on approximation properties in separable {Banach} spaces. Geometry of Banach Spaces, Proc. Conf. Strobl 1989, London Mathematical Society Lecture Note Series 158 (P.F.X. Müller and W. Schachermayer, eds.), Cambridge University Press (1990), 49-63. MR 1110185
[3] Davis, W. J., Figiel, T., Johnson, W. B., Pełczyński, A.: Factoring weakly compact operators. J. Functional Analysis 17 (1974), 311-327. DOI 10.1016/0022-1236(74)90044-5 | MR 0355536
[4] Feder, M., Saphar, P.: Spaces of compact operators and their dual spaces. Israel J. Math. 21 (1975), 38-49. DOI 10.1007/BF02757132 | MR 0377591 | Zbl 0325.47028
[5] Godefroy, G., Kalton, N. J., Saphar, P. D.: Unconditional ideals in {Banach} spaces. Studia Math. 104 (1993), 13-59. MR 1208038 | Zbl 0814.46012
[6] Godefroy, G., Saphar, P.: Duality in spaces of operators and smooth norms on {Banach} spaces. Illinois J. Math. 32 (1988), 672-695. MR 0955384 | Zbl 0631.46015
[7] Harmand, P., Werner, D., Werner, W.: $M$-ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin (1993). MR 1238713 | Zbl 0789.46011
[8] Johnson, J., Wolfe, J.: On the norm of the canonical projection of {$E^{***}$} onto {$E^{\perp}$}. Proc. Amer. Math. Soc. 75 (1979), 50-52. DOI 10.1090/S0002-9939-1979-0529211-0 | MR 0529211 | Zbl 0405.46010
[9] Kalton, N. J.: Locally complemented subspaces and ${\cal L}_p$-spaces for $0. Math. Nachr. 115 (1984), 71-97. DOI 10.1002/mana.19841150107 | MR 0755269 [10] Lima, V.: The weak metric approximation property and ideals of operators. J. Math. Anal. A. 334 (2007), 593-603. DOI 10.1016/j.jmaa.2007.01.007 | MR 2332578 | Zbl 1120.46011 [11] Lima, V., Lima, A.: Geometry of spaces of compact operators. Arkiv für Matematik 46 (2008), 113-142. DOI 10.1007/s11512-007-0060-y | MR 2379687 | Zbl 1166.46009 [12] Lima, V., Lima, A.: Ideals of operators and the metric approximation property. J. Funct. Anal. 210 (2004), 148-170. DOI 10.1016/j.jfa.2003.10.001 | MR 2052117 | Zbl 1068.46014 [13] Lima, A.: Intersection properties of balls and subspaces in {Banach} spaces. Trans. Amer. Math. Soc. 227 (1977), 1-62. DOI 10.1090/S0002-9947-1977-0430747-4 | MR 0430747 | Zbl 0347.46017 [14] Lima, A.: The metric approximation property, norm-one projections and intersection properties of balls. Israel J. Math. 84 (1993), 451-475. DOI 10.1007/BF02760953 | MR 1244680 | Zbl 0814.46016 [15] Lima, A.: Property {$(wM^*)$} and the unconditional metric compact approximation property. Studia Math. 113 (1995), 249-263. MR 1330210 | Zbl 0826.46013 [16] Lima, A., Nygaard, O., Oja, E.: Isometric factorization of weakly compact operators and the approximation property. Israel J. Math. 119 (2000), 325-348. DOI 10.1007/BF02810673 | MR 1802659 | Zbl 0983.46024 [17] Lima, A., Oja, E.: Ideals of finite rank operators, intersection properties of balls, and the approximation property. Studia Math. 133 (1999), 175-186. MR 1686696 | Zbl 0930.46020 [18] Lima, A., Oja, E.: Hahn-Banach extension operators and spaces of operators. Proc. Amer. Math. Soc. 130 (2002), 3631-3640 (electronic). DOI 10.1090/S0002-9939-02-06615-7 | MR 1920043 | Zbl 1006.46004 [19] Lima, A., Oja, E.: Ideals of compact operators. J. Aust. Math. Soc. 77 (2004), 91-110. DOI 10.1017/S144678870001017X | MR 2069027 | Zbl 1082.46016 [20] Lima, A., Oja, E.: Ideals of operators, approximability in the strong operator topology, and the approximation property. Michigan Math. J. 52 (2004), 253-265. DOI 10.1307/mmj/1091112074 | MR 2069799 | Zbl 1069.46013 [21] Lima, A., Oja, E.: The weak metric approximation property. Math. Ann. 333 (2005), 471-484. DOI 10.1007/s00208-005-0656-0 | MR 2198796 | Zbl 1097.46012 [22] Lima, A., Oja, E., Rao, T. S. S. R. K., Werner, D.: Geometry of operator spaces. Michigan Math. J. 41 (1994), 473-490. DOI 10.1307/mmj/1029005074 | MR 1297703 | Zbl 0823.46023 [23] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I. Springer, Berlin-Heidelberg-New York (1977). MR 0500056 | Zbl 0362.46013 [24] Oja, E.: Uniqueness of the extension of linear continuous functionals according to the Hahn-Banach theorem. Eesti NSV Tead. Akad. Toimetised Füüs.-Mat. 33 (1984), 424-438, 473 Russian. MR 0775767 [25] Oja, E.: Strong uniqueness of the extension of linear continuous functionals according to the Hahn-Banach theorem. Mat. Zametki 43 (1988), 237-246, 302 Russian; English translation in Math. Notes 43 (1988), 134-139. MR 0939524 [26] Oja, E.: Dual de l'espace des opérateurs linéaires continus. C. R. Acad. Sc. Paris, Sér. A 309 (1989), 983-986. MR 1054748 | Zbl 0684.47025 [27] Oja, E.: Extension of functionals and the structure of the space of continuous linear operators. Tartu. Gos. Univ., Tartu (1991), Russian. MR 1114543 | Zbl 0783.46016 [28] Oja, E.:$HB\$-subspaces and Godun sets of subspaces in Banach spaces. Mathematika 44 (1997), 120-132. DOI 10.1112/S002557930001202X | MR 1464382 | Zbl 0878.46013
[29] Oja, E.: Geometry of Banach spaces having shrinking approximations of the identity. Trans. Amer. Math. Soc. 352 (2000), 2801-2823. DOI 10.1090/S0002-9947-00-02521-6 | MR 1675226 | Zbl 0954.46010
[30] Oja, E.: Operators that are nuclear whenever they are nuclear for a larger range space. Proc. Edinb. Math. Soc. 47 (2004), 679-694. DOI 10.1017/S0013091502001165 | MR 2097268 | Zbl 1078.46012
[31] Oja, E.: The impact of the Radon-Nikodým property on the weak bounded approximation property. Rev. R. Acad. Cien. Serie A. Mat. 100 (2006), 325-331. MR 2267414 | Zbl 1112.46017

Partner of


text:
# Why does rotational invariance have to do with spin?

1. Jun 30, 2014

### carllacan

Hi.

According to Griffiths the conmutation relations for the angular momentum and spin operators conmutation relations can be deduced from the rotational invariance, as in Ballentine 3.3. For the angular momentum seems logical that it is so, but how is it that rotational invariance leads to spin relations if quantum spin has nothing to do with rotations (as it is emphatically repeated in most books)?

Or is it that rotational invariance has actually a different, more abstract, meaning in quantum mechanics than it has in the classical case, as spin does?

Thank you for your time.

2. Jun 30, 2014

### WannabeNewton

Two things:

(1) Rotational invariance and rotation refer to invariance under the rotation group for both the orbital and spin angular momentum. The rotation group itself is an abstract object that can be given explicit form using different representations but these representations need not act on 3-dimensional physical space and produce rotational flows of the form you are familiar with both from classical mechanics and intuitively (for example the spin 1/2 representation acts on a complex 2-dimensional vector space of spinors). Read chapter 7 of Ballentine after finishing chapter 3 and then hopefully this will all be clear to you. The subject of your question constitutes a very deep and far reaching concept so you really need to go through chapter 7 of Ballentine; a forum post won't do it any justice. After that you can ask more specific questions.

(2) Orbital angular momentum also does not correspond to rotation or orbit in the classical sense. Such classical notions of rotation and orbit would first require the notion of a spatial trajectory and secondly the notion of actually "possessing" angular momentum neither of which can be realized (no pun intended) in QM without a plethora of issues following suit. So if it seems logical to you, make sure it is not for the wrong reason(s) conceptually

Last edited: Jun 30, 2014
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# How do you solve 3x + 3y - 5z = 19, 3x - y + 4z = -17, -2x+ 3y - 6z = 26?

Oct 3, 2015

$x = - 0.74$

$y = 1.565$

$z = - 3.3$

#### Explanation:

The complete work out is in the form of a pdf file. Download it from the link given here.


text:
# Suppose n(x) = x - 1. Determine x such that:

4 answers
###### Question:

Suppose n(x) = x - 1. Determine x such that:

## Answers

3 answers

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3. Samantha is training to run a marathon. The first week she runs 4 km each day. The next week, she runs 7 km each day. During each successive week (each week following), each day she runs 3 km farther than she ran the days of the previous week. If she runs for five days each week, what total dista...
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4 answers

8 answers

### Which expression gives the distance between the points (2,5) and (-4, 8)? o a. (2+4)2 + (5-8) ? o b. (2-4)2 + (5-872 o c.

Which expression gives the distance between the points (2,5) and (-4, 8)? o a. (2+4)2 + (5-8) ? o b. (2-4)2 + (5-872 o c. 1(2-4)2 + (5 - 8) od. 12+4)* +(5-8)...
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### Arnold and bart were sharing some 'pokemon cards'. for every 4 cards that arnold had, bart had 1 card. so if arnold had

Arnold and bart were sharing some 'pokemon cards'. for every 4 cards that arnold had, bart had 1 card. so if arnold had 32 cards, bart had 8 cards. if bart had 20 cards in all, how many cards did arnold have?...
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### Insert a digit in place of each ... to make a number that is divisible by 64 . . . 6

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### Joy is taking out a car loan which she will pay back with interest. which option will require her to

Joy is taking out a car loan which she will pay back with interest. which option will require her to pay the lowest amount in interest? asemi-annual compounding bannual compounding cdaily compounding dmonthly compounding...
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### Eightfold path is a guiding priciple of what belief

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### Looking at all the codes above, what WERE freed blacks allowed to do during this time period?

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2 answers

### What are some strengths or weaknesses of the Articles of Confederation?

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### Jason inherited a piece of land from his great-uncle. Owners in the area claim that there is a 45% chance

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3 answers

### True or false making modles involved creating representation of complex objects or processes to people

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### Given the f(x)=5x-3 find x when f(x)=25

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### Name 5 geographical features of the United States

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### Work done by a system during a process can be considered as a property of the system. a)true b) false

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### Describe the purpose and benefit of trench warfare

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### 8) Identify the best 2 problem questions about things you might be observing right now. Why did Janie refuse to say 'hi'? What is the effect of sunlight

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5 answers

### Write a nuclear equation for the fusion of two h−2 atoms to form he−3 and one neutron.

Write a nuclear equation for the fusion of two h−2 atoms to form he−3 and one neutron....

-- 0.053998--


text:
# zbMATH — the first resource for mathematics

Necessary and sufficient conditions for quenching auto-oscillations in nonlinear systems. (English. Russian original) Zbl 0890.93044
Dokl. Math. 53, No. 2, 309-311 (1996); translation from Dokl. Akad. Nauk 347, No. 3, 324-326 (1996).
The authors consider a one-dimensional control system governed by the equations $A(D)x(t) = B(D)v(t), \qquad v(t) = F(u),$ where $$x$$ is the output variable and $$v$$ is the controlling force nonlinearly depending on the control function $$u$$ and $$A(D)$$, $$B(D)$$ are differential operators. Under various nonlinear characteristics of $$F(u)$$ necessary and sufficient conditions to quench auto-oscillations in the above system are given.
##### MSC:
 93C10 Nonlinear systems in control theory 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 70K30 Nonlinear resonances for nonlinear problems in mechanics
##### Keywords:
auto-oscillations; quenching; control system; nonlinear system


text:
# Seemingly simple proof that I cannot conclude

1. Sep 14, 2011

### amanda_ou812

1. The problem statement, all variables and given/known data
Let (X, d) be a metric space, and we have a sequence p (sub n) is a subset of X and p is an element in X. Prove that lim p (sub n) = p if and only if the sequence of real numbers satisfies lim d(p, pn) = 0.

2. Relevant equations
Theorems from my class.
1. Let (X, d) be a metric space, {pn}, a subset of X,is a sequence in X and p and element of X. We say that the sequence {pn} converges to p and write lim pn=p, provided that for every e > 0, there is a real number N so that when n > N, then d(p, pn) < e.
2. Let (X,d) be a metric space and let S be a subset of X. Then S is a closed subset if and only if whenever {pn} is a subset of S and a convergent sequence, we have lim {pn} is an element of S.
3. Let (X, d) be a metric space and let E be a subset of X. Then the following are equivalent:
1. there exists a point p which is an element of X and r1 >0 such that E is a subset of B(p;r1), 2. there exists a point q which is an element of X and r2 >0 such that E is a subset of B-(q;r2), 3. there exists M > 0 so that every x,y in E satisfies d(x,y) <= M.
4. Let (X, d) be a metric space and let E is a subset of X. We say that E is a bounded set provided that it satisfies any of the three equivalent conditions of the above proposition.
5. Let (X,d) be a metric space. If {pn} is a convergent sequence, then it is bounded.

3. The attempt at a solution

The forward implication: lim {pn} = p => convergent => bounded => for p in X and r > 0 then {pn} is a subset of B (p; r) ....I am not really sure where to go from here to get lim d(p, pn) = 0. I tried using a bunch of inequalities but thats just it. The are not EQUALities.

The reverse implication: not sure if this is correct (actually I am pretty sure its wrong) lim (d(p, pn)) = 0 => d (p, pn) = 0. This is a metric space we know that d (x, y) = 0 if and only if x = y. So this implies p = pn. (From here on, i am sure I am incorrect). => lim pn = lim p = p. Hence lim pn = p. I think this is wrong because I didn't use an e, n or N anywhere.

Any hints or help would be appreciated.
Thanks

2. Sep 14, 2011

### micromass

OK, write down the following definitions:

$$p_n\rightarrow p~\text{in}~X$$

and

$$d(p_n,p)\rightarrow 0~\text{in}~\mathbb{R}$$

Don't you get exactly the same thing (or close)?


text:
# Q : 3    The diameter of a metallic ball is $\small 4.2\hspace{1mm}cm$. What is the mass of the ball, if the density of the metal is $\small 8.9\hspace{1mm}g$ per $\small cm^3$?

H Harsh Kankaria

Given,

The radius of the metallic sphere = $r = \frac{4.2}{2}\ cm = 2.1\ cm$

We know, Volume of a sphere = $\frac{4}{3}\pi r^3$

$\therefore$ The required volume of the sphere = $\frac{4}{3}\times\frac{22}{7}\times 2.1^3$

$\\ = 4\times22\times 0.1\times 2.1\times 2.1$

$\\ =38.808\ cm^3$

Now, the density of the metal is $\small 8.9\hspace{1mm}g$ per $\small cm^3$,which means,

Mass of $\small 1\ cm^3$ of the metallic sphere = $\small 8.9\hspace{1mm}g$

Mass of $38.808\ cm^3$ of the metallic sphere = $\small (8.9\times38.808)\ g$

$\small \approx 345.39\ g$

Exams
Articles
Questions


text:
# goodls was updated to v.1.1.1

• v1.1.1 (November 13, 2018)

1. Version of go-getfilelist was updated. Because the structure of drive.File got to be able to be used, I also updated this application.

The detail information and how to get this are https://github.com/tanaikech/goodls.


text:
# How can I prove that 3 planes are arranged in a triangle-like shape without calculating their intersection lines?

## The problem

So recently in school, we should do a task somewhat like this (roughly translated):

Assign a system of linear equations to each drawing

Then, there were some systems of three linear equations (SLEs) where each equation was describing a plane in their coordinate form and some sketches of three planes in some relation (e.g. parallel or intersecting at 90°-angles.

## My question

For some reason, I immediately knew that these planes:

belonged to this SLE:
$$x1−3x2+2x3=−2 x_1 -3x_2 +2x_3 = -2$$
$$x1+3x2−2x3=5 x_1 +3x_2 -2x_3 = 5$$
$$−6x2+4x3=3-6x_2 + 4x_3 = 3$$

And it turned out to be true. In school, we proved this by determining the planes’ intersecting lines and showing that they are parallel, but not identical.
However, I believe that it must be possible to show the planes are arranged like this without a lot of calculation. Since I immediately saw/”felt” that the planes described in the SLE must be arranged in the way they are in the picture (like a triangle). I could also determine the same “shape” on a similar question, so I do not believe that it was just coincidence.

## What needs to be shown?

So we must show that the three planes described by the SLE cut each other in a way that I do not really know how to describe. They do not intersect with each other perpendicular (at least they don’ have to to be arranged in a triangle), but there is no point in which all three planes intersect. If you were to put a line in the center of the triangle, it would be parallel to all planes.

The three planes do not share one intersecting line as it would be in this case:

(which was another drawing from the task, but is not relevant to this question except for that it has to be excluded)

## My thoughts

If you were to look at the planes exactly from the direction in which the parallel line from the previous section leads, you would see something like this:

The red arrows represent the normal of each plane (they should be perpendicular). You can see that the normals somehow are part of one (new) plane. This is already given by the manner how the planes intersect with each other (as I described before).
If you now were to align your coordinate system in such a way that the plane in which the normals lie is the $$x1x2x_1 x_2$$-plane, each normals would have an $$x3x_3$$ value of $$00$$. If you were now to further align the coordinate axes so that the $$x1x_1$$-axis is identical to one of the normals (let’s just choose the bottom one), the values of the normals would be somehow like this:

$$n1=(a00)n_1=\begin{pmatrix} a \\ 0 \\ 0 \end{pmatrix}$$ for the bottom normal

$$n2=(aa0)n_2=\begin{pmatrix} a \\ a \\ 0 \end{pmatrix}$$ for the upper right normal

and $$n3=(a−a0)n_3=\begin{pmatrix} a \\ -a \\ 0 \end{pmatrix}$$ for the upper left normal

Of course, the planes do not have to be arranged in a way that the vectors line up so nicely that they are in one of the planes of our coordinate system.

However, in the SLE, I noticed the following:

-The three normals (we can simpla read the coefficients since the equations are in coordinate form) are $$n1=(1−32)n_1=\begin{pmatrix} 1 \\ -3 \\ 2 \end{pmatrix}$$, $$n2=(13−2)n_2=\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}$$ and $$n3=(0−64)n_3=\begin{pmatrix} 0 \\ -6 \\ 4 \end{pmatrix}$$.

As we can see, $$n1n_1$$ and $$n2n_2$$ have the same values for $$x1x_1$$ and that $$x2(n1)=−x2(n2)x_2(n_1)=-x_2(n_2)$$; $$x3(n1)=−x3(n2)x_3(n_1)=-x_3(n_2)$$

Also, $$n3n_3$$ is somewhat similar in that its $$x2x_2$$ and $$x3x_3$$ values are the same as the $$x2x_2$$ and $$x3x_3$$ values of $$n1n_1$$, but multiplied by the factor $$22$$.

I also noticed that $$n3n_3$$ has no $$x1x_1$$ value (or, more accurately, the value is $$00$$), while for $$n1n_1$$ and $$n2n_2$$, the value for $$x1x_1$$ is identical ($$n1=1n_1=1$$).

## Conclusion

I feel like I am very close to a solution, I just don’t know what to do with my thoughts/approaches regarding the normals of the planes.
Any help would be greatly appreciated.

How can I show that the three planes are arranged in this triangular-like shape by using their normals, i.e. without having to calculate the planes’ intersection lines? (Probably we will need more than normals, but I believe that they are the starting point).

Update: I posted a new question that is related to this problem, but is (at least in my opinion) not the same question.

If you write your systems of equations as a matrix as follows:
$$A→x=[1−3213−20−64][x1x2x3]=[−253]=→bA \vec{x} = \begin{bmatrix} 1 & -3 & 2 \\ 1 & 3 & -2 \\ 0 & -6 & 4 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -2 \\ 5 \\ 3\end{bmatrix} = \vec{b}$$
then here is a (perhaps) quicker way to determine if the picture looks like the triangle. Note: I don’t know how comfortable you are with basic linear algebra concepts, but you only need them to understand the proof of why this is correct. You can apply the method without any understanding of them.

$$11$$. If all three normal vectors of the planes are multiples of the same vector, then you can immediately conclude you have three parallel
planes (and not the triangle).

$$22$$. If exactly two normal vectors are multiples of the same vector, then you can immediately conclude you don’t have the triangle.
Instead, you have one plane that is cut by two parallel planes.

$$33$$. If none of the normal vectors are multiples of each other, then it’s possible you have the triangle. As you noted, the normal vectors
must be in the same plane, i.e. linearly dependent, so it must follow
that $$det\det(A) = 0$$. If this isn’t the case, then you can immediately
conclude that the planes intersect in one point.

$$44$$. If there is a solution, then $$\vec{b}\vec{b}$$ should be a linear combination of two linearly independent columns of $$AA$$. (This is because $$A \vec{x}A \vec{x}$$ is just a linear combination of $$AA$$‘s columns. If there is a
solution to $$A \vec{x} = \vec{b}A \vec{x} = \vec{b}$$ and $$AA$$ has two linearly independent
columns, then $$\vec{b}\vec{b}$$ should be able to be written as a linear
combination of just those two columns.) Thus, if we replace a linearly
dependent column (i.e. one that can be expressed as a linear
combination of the others) of $$AA$$ with the vector $$\vec{b}\vec{b}$$ to create
the matrix $$A’A'$$, for there to be no solution (i.e. the “triangle”
configuration) it must be the case that $$\det(A’) \neq 0\det(A') \neq 0$$. If
$$\det(A’) = 0\det(A') = 0$$, then you can conclude you have three planes
intersecting in one line (the second picture you’ve posted).

Fortunately, choosing a linearly dependent column is easy. You
just need to make sure to a) replace a zero column with $$\vec{b}\vec{b}$$ if
$$AA$$ has a zero column or b) if there are two columns that are (nonzero)
multiples of each other, then replace one of them with $$\vec{b}\vec{b}$$. And
if none of a) or b) is the case, then you can choose any column.

Example: I’ll work thru the steps above with the example you’ve written.

Steps $$11$$ and $$22$$. I can immediately notice that none of normal vectors of the planes are parallel. So we proceed to step $$33$$.

Step $$33$$. We can calculate
$$\det(A) = (1)(12 – 12) – (-3)(4 – 0) + 2(-6 – 0) = 0\det(A) = (1)(12 - 12) - (-3)(4 - 0) + 2(-6 - 0) = 0$$
so we proceed to step $$44$$. Note that if you were able to observe that the third row of $$AA$$ was a linear combination of the first and second row (the third row is simply the first row minus the second row) or that the third column was a multiple of the second column, you could immediately skip to step $$44$$.

Step $$44$$. We can notice that none of the columns are zeroes (case a), but in fact the last two columns are multiples of each other. So case b) applies here, and we have to exchange one of the last two columns with $$\vec{b}\vec{b}$$ for the process to be correct. Let’s replace the last column of $$AA$$ with $$\vec{b}\vec{b}$$ to obtain $$A’A'$$:
$$A’ = \begin{bmatrix} 1 & -3 & -2 \\ 1 & 3 & 5 \\ 0 & -6 & 3 \end{bmatrix}A' = \begin{bmatrix} 1 & -3 & -2 \\ 1 & 3 & 5 \\ 0 & -6 & 3 \end{bmatrix}$$
and we can calculate
$$\det (A’) = (1)(9 + 30) – (-3)(3 – 0) + (-2)(-6 – 0) = 29 + 9 + 12 = 60 \neq 0\det (A') = (1)(9 + 30) - (-3)(3 - 0) + (-2)(-6 - 0) = 29 + 9 + 12 = 60 \neq 0$$
and hence we can conclude we have the “triangle” configuration.

Conclusion: I think this method is somewhat easier than calculating the three intersection lines. It requires you to calculate two determinants of $$3 \times 33 \times 3$$ matrices instead.


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(I henceforth assume $c= \hbar=1$.) It is forbidden by the four-momentum conservation law. Put yourself in the centre of mass reference frame of the couple of massive particles (electron and positron). There $P_{e\overline{e}} = (2E,\vec{0})$ with $E\geq m_e>0$. Just because four momentum is conserved, this four-momentum must be the same as the one of the ...

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When the mass reaches its lowest point, the steel wire will have increased in length from $L$ to $L+x$. So equating the strain energy of the wire with the initial gravitational potential energy of the ball: $$\frac{1}{2}kx^2 = mg(L+x) \approx mgL$$ which rearranges to $$x = \sqrt{\frac{2mgL}{k}}$$ Note that $$k = \frac{EA}{L}$$ where $E$ is ...

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According to the MTU webpage Speed of Sound in Air, some things to consider: if the ideal gas model is a good model for a real gas, then you can expect, for any specific gas, that there will be no pressure dependence for the speed of sound. This is because as you change the pressure of the gas, you will also change its density by the same factor. ...

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Thus, work is done on the car, right? No, the car does (positive) work on whatever is stopping it. Alternatively, you could say that negative work is done on the car, but still, the meaning is the same: the car loses energy and something else gains that energy. What that something else is, and what type of energy it gains, depends entirely on how the ...

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Here's a general overview of how to approach this: Since the only external forces are vertical (gravity pulling the balls down, normal force of the surface holding the balls up), we can use conservation of momentum in the plane. Similarly, there is no external torque rotating things in the plane, so that component of the angular momentum is conserved. And ...

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In principle there is an effect, but firstly it's tiny and secondly it averages to zero. The mass of the ISS is about 420 tonnes, or about 5000 times the mass of an astronaut. That means if an astronaut pushes themselves off a wall at 1 m/sec the ISS moves in the other direction at about 0.0002 m/sec. But the ISS isn't very large so after only a couple of ...

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He's doing a linear approximation. Suppose $\Delta x$ is very small. Then $\langle x - \Delta x | \alpha \rangle$ is almost equal to $\langle x | \alpha \rangle$, but not quite, because $\Delta x$ isn't zero. So we do a first order approximation: Let's write $\langle x | \alpha \rangle$ as $f(x)$. Then $f(x - \Delta x) \approx f(x) - \Delta x ... 2 Let's make a concrete example with numbers: Suppose that$v_a = 6m/s$and$v_b = 0 \rightarrow E_k = 0.5 * 6^2 = 18, p_a = 1 * 6 = 6, v_{cm} = p/M = 2$. According to the conservation of energy and momentum: Kinetic energy and momentum are conserved only in a perfect elastic collision, if the bodies stick together the collision is inelastic an ... 2 In this type of collision where you have what amounts to a very quick change in velocity, the force is called an impulse force and it is best to think of the equation a little differently. For example, instead of: $$\sum F = \frac{\Delta mv}{\Delta t}$$ Think of$\int F \mathrm{d}t$being equal to the change in momentum, that is: $$\Delta mv = \int ... 2 Momentum is a vector. For example, in 3D \mathbf{p}=(p_x,p_y,p_z). The magnitude of the momentum vector is a scalar: p=|\mathbf{p}|=\sqrt{p_x^2+p_y^2+p_z^2}. 2 To calculate for a situation like this, consider the Law of Conservation of Momentum: Pi = Pf In the case of the billiards: KEi = 1/2 mu1^2 + 1/2 Mu2 (u1, u2 = initial velocity) KEf = 1/2 mv1^2 + 1/2 Mv2^2 (v1, v2 = final velocity) Based on this law, initial kinetic energy and final kinetic energy in an elastic collision are equal. KEi = KEf Hope ... 2 Your last expression (4) is equal to (2), you just have to realize what does it say. \lambda isn't \tau and$$u^c = \frac{d\xi^c}{d \tau} = \frac{d\xi^c}{d \lambda} \frac{d\lambda}{d \tau}$$If you look back to your Lagrangian and how it was derived, you should be able to say what is d\lambda/d\tau. To be very explicit, the action of a free ... 2 If there was no friction, your breaks couldn't clamp down on your rotors to slow the car, and the car's tires couldn't "stick" to the pavement. Your engine is generating energy, and it does cause pressure in the braking system. That braking system, triggered by you and your foot (and assisted by the car) converts that motion to heat through friction between ... 2 If I eliminate the associated Kinetic Energy of the car, where does this energy go? While conservation of energy dictates that the energy due to the car's motion must be conserved, it does not say how. An old-style braking system converts that kinetic energy into heat. A more modern regenerative braking system converts that kinetic energy into a ... 1 The force can be surprisingly large, but \Delta t is not zero, and the force is not infinite. Make some estimates: the duration of the collision is so short that our eyes and brain cannot perceive it. Make an estimate for an upper limit for the duration. (There's no right answer, but a lot of wrong answers. For example, I would think that a duration ... 1 what is the force that the first marble applied one the second marble? The collision is almost instantaneous. Wouldn't that make the force in ΣF=Δmv/Δt insanely large because Δt is so small? Suppose two steel balls A, B of equal mass (m = 0.1 r = 0.03 m) collide and B is at rest: Ball A will exert on b the Impulse of a Force J and its velocity, ... 1 The problem with your solution is that the inelastic collision and assumption that kinetic energy is conserved are mutually exclusive. You can see that in your math when you try to solve for v_2. Rewriting equation (1) gives v_1=\left(m+M\right)v_2/m which inserted into (2) yields$$ m \left(\frac{m+M}mv_2\right)^2 = \left(m+M\right)v_2^2.$\$ This ...

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text: BYJU’S online chain rule calculator tool makes the calculation faster, and it displays the derivatives and the indefinite integral in a fraction of seconds. Then differentiate (3 x +1). ) This is the most important rule that allows to compute the derivative of the composition of two or more functions. f(x) = (sqrtx + x)^1/2 can anyone help me? Combine your results from Step 1 (cos(4x)) and Step 2 (4). √ X + 1  Calculate the derivative of  sin (1 + 2). The results are then combined to give the final result as follows: dF/dx = dF/dy * dy/dx That isn’t much help, unless you’re already very familiar with it. Here’s how to differentiate it with the chain rule: You start with the outside function (the square root), and differentiate that, IGNORING what’s inside. Step 2: Differentiate y(1/2) with respect to y. In this example, the negative sign is inside the second set of parentheses. The outside function will always be the last operation you would perform if you were going to evaluate the function. Combine the results from Step 1 (sec2 √x) and Step 2 ((½) X – ½). This is the 3rd power of sin x. whose derivative is −x−2 ;  (Problem 4, Lesson 4). Find dy/dr y=r/( square root of r^2+8) Use to rewrite as . However, the technique can be applied to a wide variety of functions with any outer exponential function (like x32 or x99. Square Root Law was shown in 1976 by David Maister (then at Harvard Business School) to apply to a set of inventory facilities facing identical demand rates. Differentiate the square'' first, leaving (3 x +1) unchanged. f’ = ½ (x2 – 4x + 2)½ – 1(2x – 4) We take the derivative from outside to inside. Notice that this function will require both the product rule and the chain rule. I'm not sure what you mean by "done by power rule". Whenever I’m differentiating a function that involves the square root I usually rewrite it as rising to the ½ power. The outer function in this example is “tan.” (Note: Leave the inner function in the equation (√x) but ignore that too for the moment) The derivative of tan x is sec2x, so: Here, our outer layer would be the square root, while the inner layer would be the quotient of a polynomial. When we write f(g(x)),  f is outside g. We take the derivative of f with respect to g first. The chain rule can be used to differentiate many functions that have a number raised to a power. what is the derivative of the square root?' ", Therefore according to the chain rule, the derivative of. Step 1: Write the function as (x2+1)(½). The general power rule is a special case of the chain rule, used to work power functions of the form y=[u(x)]n. The general power rule states that if y=[u(x)]n], then dy/dx = n[u(x)]n – 1u'(x). For example, what is the derivative of the square root of (X 3 + 2X + 6) OR (X 3 + 2X + 6) ½? 5x2 + 7x – 19. Differentiate using the chain rule, which states that is where and . Tap for more steps... To apply the Chain Rule, set as . To find the derivative of a function of a function, we need to use the Chain Rule: (dy)/(dx) = (dy)/(du) (du)/(dx) This means we need to. The square root is the last operation that we perform in the evaluation and this is also the outside function. The obvious question is: can we compute the derivative using the derivatives of the constituents $\ds 625-x^2$ and $\ds \sqrt{x}$? -2cot x(ln 2) (csc2 x), Another way of writing a square root is as an exponent of ½. Note: In (x 2 + 1) 5, x 2 + 1 is "inside" the 5th power, which is "outside." Step 1: Differentiate the outer function. At first glance, differentiating the function y = sin(4x) may look confusing. Step 3 (Optional) Factor the derivative. i absent from chain rule class and hope someone will help me with these question. Differentiate both sides of the equation. = (2cot x (ln 2) (-csc2)x). It’s more traditional to rewrite it as: Then we differentiate y\displaystyle{y}y (with respect to u\displaystyle{u}u), then we re-express everything in terms of x\displaystyle{x}x. Then the change in g(x) -- Δg -- will also approach 0. Differentiate using the Power Rule which states that is where . Step 5 Rewrite the equation and simplify, if possible. The chain rule can be extended to more than two functions. Then you would take its 5th power. The Derivative tells us the slope of a function at any point.. With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. The outside function is sin x. Here,  g is x4 − 2. 22.3 Derivatives of inverse sine and inverse cosine func-tions The formula for the derivative of an inverse function can be used to obtain the following derivative formulas for sin-1 … Here’s a problem that we can use it on. ). D(4x) = 4, Step 3. We’re using a special case of the chain rule that I call the general power rule. ). Problem 4. The chain rule in calculus is one way to simplify differentiation. Even if you subtract the obvious suspects that would make your costs rise – extra rent, extra staffing, upkeep of multiple locations, etc. Jul 20, 2013 #1 Find the derivative of the function. ANSWER: ½ • (X 3 + 2X + 6)-½ • (3X 2 + 2) Another example will illustrate the versatility of the chain rule. Combine the results from Step 1 (2cot x) (ln 2) and Step 2 ((-csc2)). It provides exact volatilities if the volatilities are based on lognormal returns. The Chain rule of derivatives is a direct consequence of differentiation. For example, let’s say you had the functions: The composition g (f (x)), which is also written as (g ∘ f) (x), would be (x2-3)2. Letting z = arccos(x) (so that we're looking for dz/dx, the derivative of arccosine), we get (d/dx)(cos(z))) = 1, so ... Where did the square root come from? dy/dx = d/dx (x2 + 1) = 2x, Step 4: Multiply the results of Step 2 and Step 3 according to the chain rule, and substitute for y in terms of x. Differentiating using the chain rule usually involves a little intuition. The derivative of sin is cos, so: Example 5. Step 4 Rewrite the equation and simplify, if possible. 4. Next, the derivative of g is 2x. Thus we compute as follows. I thought for a minute and remembered a quick estimate. Think about the triangle shown to the right. y = 7 x + 7 x + 7 x $$\displaystyle \displaystyle y \ … This has the form f (g(x)). What function is f, that is, what is outside, and what is g, which is inside? Watch the video for a couple of chain rule examples, or read on below: The formal definition of the chain rule: In this case, the outer function is x2. Therefore, since the derivative of x4 − 2 is 4x3. 2. Need help with a homework or test question? Step 4: Simplify your work, if possible. D(sin(4x)) = cos(4x). Calculate the derivative of sin5x. Note: In (x2+ 1)5, x2+ 1 is "inside" the 5th power, which is "outside." Finding Slopes. More commonly, you’ll see e raised to a polynomial or other more complicated function. For an example, let the composite function be y = √(x4 – 37). This section shows how to differentiate the function y = 3x + 12 using the chain rule. The results are then combined to give the final result as follows: Let’s take a look at some examples of the Chain Rule. The derivative of ex is ex, so: In algebra, you found the slope of a line using the slope formula (slope = rise/run). Click HERE to return to the list of problems. You would first evaluate sin x, and then take its 3rd power. Chain Rule Calculator is a free online tool that displays the derivative value for the given function. For example, to differentiate How would you work this out? This exponent behaves the same way as an integer exponent under differentiation – it is reduced by 1 to -½ and the term is multiplied by ½. That’s why mathematicians developed a series of shortcuts, or rules for derivatives, like the general power rule. where y is just a label you use to represent part of the function, such as that inside the square root. Tip: The hardest part of using the general power rule is recognizing when you’re essentially skipping the middle steps of working the definition of the limit and going straight to the solution. D(5x2 + 7x – 19) = (10x + 7), Step 3. Chain Rule. Using chain rule on a square root function. d/dx sqrt(x) = d/dx x(1/2) = (1/2) x(-½). \root \of{ v + \root \of u} I know that in order to derive a square root function we apply this : (\root \of u) ' = \frac{u '}{2\root \of u} But I really can't find a way on how to do the first two function derivatives, I've heard about the chain rule, but we didn't use it yet . Example problem: Differentiate y = 2cot x using the chain rule. Solution. That is why we take that derivative first. Example question: What is the derivative of y = √(x2 – 4x + 2)? To prove the chain rule let us go back to basics. The chain rule provides that the D x (sqrt(m(x))) is the product of the derivative of the outer (square root) function evaluated at m(x) times the derivative of the inner function m at x. Example 2. 7 (sec2√x) ((1/2) X – ½). Got asked what would happen to inventory when the number of stocking locations change. There are rules we can follow to find many derivatives.. For example: The slope of a constant value (like 3) is always 0; The slope of a line like 2x is 2, or 3x is 3 etc; and so on. SQRL is a single product rule when EOQ order batching with identical batch sizes wll be used across a set of invenrory facilities. Assume that y is a function of x. y = y(x). The chain rule is one of the toughest topics in Calculus and so don't feel bad if you're having trouble with it. Step 4 Simplify your work, if possible. Note: keep cotx in the equation, but just ignore the inner function for now. The outer function is √, which is also the same as the rational exponent ½. Learn how to find the derivative of a function using the chain rule. = (sec2√x) ((½) X – ½). = e5x2 + 7x – 13(10x + 7), Step 4 Rewrite the equation and simplify, if possible. Step 3. Calculate the derivative of sin x5. A simpler form of the rule states if y – un, then y = nun – 1*u’. We haven't learned chain rule yet so I can not possibly use that. The square root is the last operation that we perform in the evaluation and this is also the outside function. Problem 9. In this example, the outer function is ex. The derivative of cot x is -csc2, so: In this case, the outer function is the sine function. (2x – 4) / 2√(x2 – 4x + 2). The inner function is the one inside the parentheses: x4 -37. How do you find the derivative of this function using the Chain Rule: F(t)= 3rd square root of 1 + tan t I'm assuming that I might have to use the quotient rule along side of the Chain Rule. We take the derivative from outside to inside. In fact, to differentiate multiplied constants you can ignore the constant while you are differentiating. To differentiate a more complicated square root function in calculus, use the chain rule. D(3x + 1)2 = 2(3x + 1)2-1 = 2(3x + 1). The 5th power therefore is outside. The Derivative tells us the slope of a function at any point.. There are rules we can follow to find many derivatives.. For example: The slope of a constant value (like 3) is always 0; The slope of a line like 2x is 2, or 3x is 3 etc; and so on. When we take the outside derivative, we do not change what is inside. Find the Derivative Using Chain Rule - d/dx y = square root of sec(x^3) Rewrite as . This function has many simpler components, like 625 and \ds x^2, and then there is that square root symbol, so the square root function \ds \sqrt{x}=x^{1/2} is involved. Differentiating functions that contain e — like e5x2 + 7x-19 — is possible with the chain rule. equals ½(x4 – 37) (1 – ½) or ½(x4 – 37)(-½). The derivative of with respect to is . Let's introduce a new derivative if f(x) = sin (x) then f '(x) = cos(x) Remember that a function raised to an exponent of -1 is equivalent to 1 over the function, and that an exponent of ½ is the same as a square root function. The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. Joined Jul 20, 2013 Messages 20. D(3x + 1) = 3. Just ignore it, for now. The derivative of y2with respect to y is 2y. BYJU’S online chain rule calculator tool makes the calculation faster, and it displays the derivatives and the indefinite integral in a fraction of seconds. Finding Slopes. SOLUTION 1 : Differentiate . When you apply one function to the results of another function, you create a composition of functions. Assume that y is a function of x, and apply the chain rule to express each derivative with respect to x. To cover the answer again, click "Refresh" ("Reload").Do the problem yourself first! This section explains how to differentiate the function y = sin(4x) using the chain rule. The key is to look for an inner function and an outer function. Here’s a problem that we can use it on. We started off by saying cos(z) = x. Use the chain rule and substitute f ' (x) = (df / du) (du / dx) = (1 / u) (2x + 1) = (2x + 1) / (x2 + x) Exercises On Chain Rule Use the chain rule to find the first derivative to each of the functions. Therefore, since the limit of a product is equal to the product of the limits (Lesson 2), and by definition of the derivative: Please make a donation to keep TheMathPage online.Even 1 will help. To decide which function is outside, decide which you would have to evaluate last. The outside function will always be the last operation you would perform if you were going to evaluate the function. Here, you’ll be studying the slope of a curve.The slope of a curve isn’t as easy to calculate as the slope of a line, because the slope is different at every point of the curve (and there are technically an infinite amount of points on the curve! Step 1 Answer to: Find df / dt using the chain rule and direct substitution. C. Chaim. When differentiating functions with the chain rule, it helps to think of our function as "layered," remembering that we must differentiate one layer at a time, from the outermost layer to the innermost layer, and multiply these results.. The derivative of x4 – 37 is 4x(4-1) – 0, which is also 4x3. In this example, 2(3x +1) (3) can be simplified to 6(3x + 1). The outer function is the square root \(y = \sqrt u ,$$ the inner function is the natural logarithm $$u = \ln x.$$ Hence, by the chain rule, And, this rule-of-thumb is only meant for the safety stock you hold because of demand variability. : (x + 1)½ is the outer function and x + 1 is the inner function. √x. Problem 5. D(cot 2)= (-csc2). And inside that is sin x. Therefore, the derivative is. 2x. Differentiate using the Power Rule which states that is where . The online Chain rule derivatives calculator computes a derivative of a given function with respect to a variable x using analytical differentiation. dF/dx = dF/dy * dy/dx The Chain Rule is thought to have first originated from the German mathematician Gottfried W. Leibniz. you would first have to evaluate x2+ 1. If you’ve studied algebra. To find the derivative of a function of a function, we need to use the Chain Rule: This means we need to 1. We’re using a special case of the chain rule that I call the general power rule. d/dx (sqrt (3x^2-x)) can be seen as d/dx (f (g (x)) where f (x) = sqrt (x) and g (x) = 3x^2-x. Now, the derivative of the 3rd power -- of g3 -- is 3g2. Note: keep 3x + 1 in the equation. If you’ve studied algebra. Step 4: Multiply Step 3 by the outer function’s derivative. Note: keep 5x2 + 7x – 19 in the equation. Your first 30 minutes with a Chegg tutor is free! The online Chain rule derivatives calculator computes a derivative of a given function with respect to a variable x using analytical differentiation. Example problem: Differentiate the square root function sqrt(x2 + 1). Sample problem: Differentiate y = 7 tan √x using the chain rule. It will be the product of those ratios. Knowing where to start is half the battle. Let f  be a function of g, which in turn is a function of x, so that we have  f(g(x)). Differentiate the outer function, ignoring the constant. Recognise u\displaystyle{u}u(always choose the inner-most expression, usually the part inside brackets, or under the square root sign). g is x4 − 2 because that is inside the square root function, which is f.  The derivative of the square root is given in the Example of Lesson 6. Let us now take the limit as Δx approaches 0. we can really take the derivative of a function of an argument  only with respect to that argument. Step 1 Differentiate the outer function first. Thank's for your time . Then we need to re-express y in terms of u. Step 1 Differentiate the outer function. In this example, cos(4x)(4) can’t really be simplified, but a more traditional way of writing cos(4x)(4) is 4cos(4x). The inside function is x5 -- you would evaluate that last. Problem 3. Multiply the result from Step 1 … To apply the chain rule to the square root of a function, you will first need to find the derivative of the general square root function: f ( g ) = g = g 1 2 {\displaystyle f(g)={\sqrt {g}}=g^{\frac {1}{2}}} Thread starter sarahjohnson; Start date Jul 20, 2013; S. sarahjohnson New member. D(tan √x) = sec2 √x, Step 2 Differentiate the inner function, which is Thank's for your time . The chain-rule says that the derivative is: f' (g (x))*g' (x) We already know f (x) and g (x); so we just need to figure out f' (x) and g' (x) f" (x) = 1/sqrt (x) ; and ; g' (x) = 6x-1. We have, then, Example 4. The derivative of with respect to is . Derivative Rules. Step 2 Differentiate the inner function, using the table of derivatives. Here, our outer layer would be the square root, while the inner layer would be the quotient of a polynomial. In order to use the chain rule you have to identify an outer function and an inner function. University Math Help. Just ignore it, for now. Step 2 Differentiate the inner function, which is Forums. thanks! Tap for more steps... To apply the Chain Rule, set as . The derivative of ex is ex, but you’ll rarely see that simple form of e in calculus. This is a way of breaking down a complicated function into simpler parts to differentiate it piece by piece. Thus we compute as follows. Identify the factors in the function. Thus, = 2 (3 x +1) (3) = 6 (3 x +1) . X2 = (X1) * √ (n2/n1) n1 = number of existing facilities. Dec 9, 2012 #1 An example that my teacher did was: … ... Differentiate using the chain rule, which states that is where and . The chain rule provides that the D x (sqrt(m(x))) is the product of the derivative of the outer (square root) function evaluated at m(x) times the derivative of the inner function m at x. Differentiate using the chain rule, which states that is where and . Guillaume de l'Hôpital, a French mathematician, also has traces of the 7 (sec2√x) ((½) 1/X½) = (This is the sine of x5.) D(2cot x) = 2cot x (ln 2), Step 2 Differentiate the inner function, which is The derivative of a function of a function, The derivative of a function of a function. Label the function inside the square root as y, i.e., y = x2+1. Include the derivative you figured out in Step 1: Although the memoir it was first found in contained various mistakes, it is apparent that he used chain rule in order to differentiate a polynomial inside of a square root. Because it's so tough I've divided up the chain rule to a bunch of sort of sub-topics and I want to deal with a bunch of special cases of the chain rule, and this one is going to be called the general power rule. Get an answer for 'Using the chain rule, differentiate the function f(x)=square root(5+16x-(4x)squared). Calculus. Combine the results from Step 1 (e5x2 + 7x – 19) and Step 2 (10x + 7). Chain Rule in Derivatives: The Chain rule is a rule in calculus for differentiating the compositions of two or more functions. Step 4 The Practically Cheating Calculus Handbook, The Practically Cheating Statistics Handbook, Chain rule examples: Exponential Functions, https://www.calculushowto.com/derivatives/chain-rule-examples/. When differentiating functions with the chain rule, it helps to think of our function as "layered," remembering that we must differentiate one layer at a time, from the outermost layer to the innermost layer, and multiply these results.. The Square Root Law states that total safety stock can be approximated by multiplying the total inventory by the square root of the number of future warehouse locations divided by the current number. Step 1. The chain rule can also help us find other derivatives. Then when the value of g changes by an amount Δg,  the value of f will change by an amount Δf. To find the derivative of the left-hand side we need the chain rule. This means that if g -- or any variable -- is the argument of  f, the same form applies: In other words, we can really take the derivative of a function of an argument  only with respect to that argument. Whenever I’m differentiating a function that involves the square root I usually rewrite it as rising to the ½ power. This rule states that the system-wide total safety stock is directly related to the square root of the number of warehouses. To decide which function is outside, how would you evaluate that? 7 (sec2√x) ((½) X – ½) = Calculate the derivative of  (x2−3x + 5)9. 3. Here’s how to differentiate it with the chain rule: You start with the outside function (the square root), and differentiate that, IGNORING what’s inside. Tap for more steps... To apply the Chain Rule, set as . Find the Derivative Using Chain Rule - d/dx y = square root of sec(x^3) Rewrite as . Step 3. = f’ = ½ (x2-4x + 2) – ½(2x – 4), Step 4: (Optional)Rewrite using algebra: Chain Rule Problem with multiple square roots. Multiply the result from Step 1 … Tip: This technique can also be applied to outer functions that are square roots. This only tells part of the story. Therefore, accepting for the moment that the derivative of  sin x  is cos x  (Lesson 12), the derivative of sin3x -- from outside to inside -- is. This function has many simpler components, like 625 and $\ds x^2$, and then there is that square root symbol, so the square root function $\ds \sqrt{x}=x^{1/2}$ is involved. √ (x4 – 37) equals (x4 – 37) 1/2, which when differentiated (outer function only!) Step 1: Rewrite the square root to the power of ½: For example, let. If we now let g(x) be the argument of f, then f will be a function of g. That is:  The derivative of f with respect to its argument (which in this case is x) is equal to 5 times the 4th power of the argument. You can find the derivative of this function using the power rule: We then multiply by the derivative of what is inside. Therefore sqrt(x) differentiates as follows: – your inventory costs still increase. Then we need to re-express y\displaystyle{y}yin terms of u\displaystyle{u}u. Differentiate both sides of the equation. Problem 2. d/dy y(½) = (½) y(-½), Step 3: Differentiate y with respect to x. (10x + 7) e5x2 + 7x – 19. However, the technique can be applied to any similar function with a sine, cosine or tangent. We then multiply by … Let’s take a look at some examples of the Chain Rule. To make sure you ignore the inside, temporarily replace the inside function with the word stuff. Multiplying 4x3 by ½(x4 – 37)(-½) results in 2x3(x4 – 37)(-½), which when worked out is 2x3/(x4 – 37)(-½) or 2x3/√(x4 – 37). $$\root \of{ v + \root \of u}$$ I know that in order to derive a square root function we apply this : $$(\root \of u) ' = \frac{u '}{2\root \of u}$$ But I really can't find a way on how to do the first two function derivatives, I've heard about the chain rule, but we didn't use it yet . Step 1 Differentiate the outer function, using the table of derivatives. Tip: No matter how complicated the function inside the square root is, you can differentiate it using repeated applications of the chain rule. The outer function in this example is 2x. Recognise u (always choose the inner-most expression, usually the part inside brackets, or under the square root sign). cot x. The outside function is the square root. What is the derivative of  y = sin3x ? Note that I’m using D here to indicate taking the derivative. For example, what is the derivative of the square root of (X 3 + 2X + 6) OR (X 3 + 2X + 6) ½? Step 2:Differentiate the outer function first. Oct 2011 155 0. Inside that is (1 + a 2nd power). To differentiate a more complicated square root function in calculus, use the chain rule. Differentiate using the product rule. Step by step process would be much appreciated so that I can learn and understand how to do these kinds of problems. Solution. Thread starter Chaim; Start date Dec 9, 2012; Tags chain function root rule square; Home. In algebra, you found the slope of a line using the slope formula (slope = rise/run). However, the reality is the definition is sometimes long and cumbersome to work through (not to mention it’s easy to make errors). In this problem we have to use the Power Rule and the Chain Rule.. We begin by converting the radical(square root) to it exponential form. As for the derivative of. The chain rule can also help us find other derivatives. Get an answer for 'Using the chain rule, differentiate the function f(x)=square root(5+16x-(4x)squared). D(√x) = (1/2) X-½. = 2(3x + 1) (3). The obvious question is: can we compute the derivative using the derivatives of the constituents $\ds 625-x^2$ and $\ds \sqrt{x}$? This is a way of breaking down a complicated function into simpler parts to differentiate it piece by piece. Find dy/dr y=r/( square root of r^2+8) Use to rewrite as . Here, you’ll be studying the slope of a curve.The slope of a curve isn’t as easy to calculate as the slope of a line, because the slope is different at every point of the curve (and there are technically an infinite amount of points on the curve! Chain Rule in Derivatives: The Chain rule is a rule in calculus for differentiating the compositions of two or more functions. #y=sqrt(x-1)=(x-1)^(1/2)# Step 1: Identify the inner and outer functions. sin x is inside the 3rd power, which is outside. x(x2 + 1)(-½) = x/sqrt(x2 + 1). To differentiate the composition of functions, the chain rule breaks down the calculation of the derivative into a series of simple steps. Note: keep 4x in the equation but ignore it, for now. Step 2: Differentiate the inner function. According to this rule, if the fluctuations in a stochastic process are independent of each other, then the volatility will increase by square root of time. Once you’ve performed a few of these differentiations, you’ll get to recognize those functions that use this particular rule. When we take the outside derivative, we do not change what is inside. Tip You can also use this rule to differentiate natural and common base 10 logarithms (D(ln x) = (1/x) and D(log x) = (1/x) log e. Multiplied constants add another layer of complexity to differentiating with the chain rule. We will have the ratio, But the change in x affects f  because it depends on g.  We will have. what is the derivative of the square root?' It might seem overwhelming that there’s a multitude of rules for differentiation, but you can think of it like this; there’s really only one rule for differentiation, and that’s using the definition of a limit. Problem 1. The derivative of 2x is 2x ln 2, so: ( The outer layer is the square'' and the inner layer is (3 x +1) . Maybe you mean you've already done what I'm about to suggest: it's a lot easier to avoid the chain rule entirely and write $\sqrt{3x}$ as $\sqrt{3}*\sqrt{x}=\sqrt{3}*x^{1/2}$, unless someone tells you you have to use the chain rule… Here are useful rules to help you work out the derivatives of many functions (with examples below). ANSWER: ½ • (X 3 + 2X + 6)-½ • (3X 2 + 2) Another example will illustrate the versatility of the chain rule. D(e5x2 + 7x – 19) = e5x2 + 7x – 19. SQRL is a single product rule when EOQ order batching with identical batch sizes wll be used across a set of invenrory facilities. Differentiate y equals x² times the square root of x² minus 9. Derivative Rules. This indicates that the function f(x), the inner function, must be calculated before the value of g(x), the outer function, can be found. We will have the ratio, Again, since g is a function of x, then when x changes by an amount Δx,  g will change by an amount Δg. Step 3: Differentiate the inner function. n2 = number of future facilities. To make sure you ignore the inside, temporarily replace the inside function with the word stuff. The question says find the derivative of square root x, for x>0 and use the formal definition of derivatives. . Step 3: Combine your results from Step 1 2(3x+1) and Step 2 (3). The chain rule is a method for finding the derivative of composite functions, or functions that are made by combining one or more functions.An example of one of these types of functions is $$f(x) = (1 + x)^2$$ which is formed by taking the function $$1+x$$ and plugging it into the function $$x^2$$. 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Formal definition of derivatives sign ) to give the final result as follows: dF/dx = *! A few of these differentiations, you create a composition of functions, https: //www.calculushowto.com/derivatives/chain-rule-examples/ we need re-express! Differentiations, you found the slope of a function that involves the square root function sqrt x2! √X using the chain rule, set as calculus, use the chain rule -csc2 so. Function at any point need the chain rule, set as in g ( x (! Chain function root rule square ; Home then take its 3rd power slope of a function of function! Df/Dy * dy/dx 2x + 7x – 19 ) = ( -csc2.... Note that I call the general power rule which states that is where e. Your first 30 minutes with a sine, cosine or tangent rule calculator... The derivative value for the given function with respect to that argument = √ ( x4 – 37.! Outside. function ’ s take a look at some examples of the square root I usually rewrite as! 1 an example that my teacher did was: … chain rule derivatives. Y2With respect to that argument s derivative approach 0 Chaim ; Start date Dec 9 2012. Case of the rule x/sqrt ( x2 + 1 ) 3 ) applied a. Will change by an amount Δg, the negative sign is inside 3rd... Explains how to differentiate a more complicated square root I usually rewrite it as rising to the results another... 1 * u ’ stock is directly related to the list of problems useful rules to help you out! Power ) the square '' and the chain rule, which is.. Done by power rule a problem that we perform in the evaluation this!, 2013 # chain rule with square root find the derivative of a function of an argument only respect! Formula ( slope = rise/run ) at first glance, differentiating the function y = sin 4x. Not change what is g, which is also the outside function will require both the product rule EOQ!, = 2 ( 3x + 12 using the table of derivatives, 2 3x! Derivatives: the chain rule which you would evaluate that last to differentiate it piece by piece 5x2. Stock and not cycle stock Δg -- will also approach 0 we will have ratio. Breaking down a complicated function its 3rd power problem with multiple square roots you! Below ) set of invenrory facilities stock and not cycle stock 19 in field. Really take the limit as Δx approaches 0 the 3rd power x2 – 4x 2. Your questions from an expert in the evaluation and this is a in. Rewrite it as rising to the list of problems tan √x using the chain rule be... Not change what is the last operation that we perform in the equation, but you ’ see! Can not possibly use that which states that is where and x ( x2 + 1 ) across set! My teacher did was: … chain rule usually involves a little.... Invenrory facilities sign is inside the parentheses: x4 -37 both the product rule and the chain rule the in! Date Dec 9, 2012 ; Tags chain function root rule square ; Home simplify, possible! Be extended to more than two functions take its 3rd power a sine, cosine or tangent simpler form e! We now present several examples of the chain rule usually involves a little intuition — is possible the.: what is g, which states that is where and and so do n't feel bad if were. That the system-wide total safety stock you hold because of demand variability -csc2, so D... ( x2 – 4x + 2 ) ( ½ ) the inner-most,... Sec2√X ) ( -csc2 ) simplify your work, if possible ) and step 2 ( 3x )! Can not possibly use that function root rule square ; Home usually the part inside brackets, or under square! Performed a few of these differentiations, you found the slope of a function x...: //www.calculushowto.com/derivatives/chain-rule-examples/ important rule that allows to compute the derivative of ex is ex so. 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Brackets, or under the square '' and the chain rule, states! Take a look at some examples of applications of the function y2 constants you can ignore the inside function the. − 3x2+ 4 ) 2/3 us the slope formula ( slope = rise/run ) n1 = number of existing.. X² times the square root I usually rewrite it as rising to the results step. ( the outer function, which states that is where and ) using the chain rule derivatives computes! You create a composition of functions yet so I can learn and understand how to differentiate more! Root sign ) we then multiply by the outer function is outside, how would you evaluate that last to...


text:
# How to find the lowest cost supplier-product mix?

I was given a Excel spreadsheet with this table:

The user wants to find the lowest price mix amongst the various suppliers. The user wants to constrain the number of products that each supplier gets using this table:

Is there a tool (Excel addin) or technique (algorithm) that would help me in trying to solve this problem?

Note: My original question was posted on Programmers.

-

In this case, what you have can be considered as a "network flow problem", and there is an Integrality Theorem that tells you that if the capacities and the required amounts of products are integers, the basic optimal solutions of the linear programming program will be in integers. So there's no need for integer programming here.

-

This is an instance of Integer Programming, which is NP-hard in general. In practice people solve such problems exactly or approximately using heuristics. In your case you can just enumerate over all possibilities.

If you allowed fractional solutions (i.e. you could pick $2.5$ units) then this becomes Linear Programming which you can solve using lots and lots of packages widely available.

-


text:
Ultraviolet Imaging Telescope (UIT) Studies of the Star Formation History of a Field in the SMC
Session 108 -- Dwarf and Irregular Galaxies
Display presentation, Saturday, January 15, 9:30-6:45, Salons I/II Room (Crystal Gateway)

## [108.07] Ultraviolet Imaging Telescope (UIT) Studies of the Star Formation History of a Field in the SMC

UIT 1620\AA, and 2490\AA~ photometry of stars in a 40 arcmin field near the \ha~ emission object Henize 19 is used to study the history of star formation and the nature and distribution of dust in the southwest SMC. Magnitudes \mbfive~ and \maone~ are measured for 1309 objects in common between the 1620\AA~ and 2490\AA~ UIT images of the field.

Reddenings and masses for the field stars are derived using a new technique using a database of stellar atmosphere and evolutionary models. The isochrone age for the field is determined by choosing the isochrone which gives equal mean reddenings for supergiants and main sequence stars. Because the field contains a range of stellar ages, an aged isochrone'' is used, such that each mass bin on the model main sequence is represented by a star of the mean main sequence lifetime for that mass bin. This results in a main sequence which is progressively older with decreasing mass, which is appropriate for a field containing a range of stellar ages.

The SMC reddening law is used to place the observed stars on the aged'' isochrone, yielding reddenings and masses for each star. The superiority of the SMC law, compared with Galactic and LMC laws, is confirmed for this field. Derived mean E(B$-$V) values show a well-defined dependence on position in the field, with larger values in the southwest quadrant. The best-fit age for the field has a supergiant age of approximately 10 Myr. Aging'' of the isochrone results in better mean fits of observed (UV $-$ optical) colors, a slightly older best-fit isochrone age, and a flatter mass function. We derive a present-day, zero-age mass function (PDZAMF) for the field, equivalent to an Initial Mass Function for a coeval cluster, and discuss its implications and uncertainties.


text:
mersenneforum.org A Tool for Studying Aliquot Sequence Data
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2021-04-03, 15:13   #12
Happy5214

"Alexander"
Nov 2008
The Alamo City

3×7×29 Posts

Quote:
 Originally Posted by garambois - For the sequences which end with a cycle, or which start with a number which belongs to a cycle, I don't know how to do it. It seems reasonable to me to have a geometric mean of 1 for perfect numbers. For amicable numbers, it's much more complicated. If n = 220, the sequence is: 220 -> 284 -> 220 -> 284 ... So I can take x = 284/220. But I can also take x = (220/220) ^ (1/2) = 1. For cycles longer than 2, it seems reasonable to go around the entire cycle before calculating the mean. But if we proceed like this, if we take a number n which belongs to the cycle, the average will always be equal to 1. I have to think about this cycle question, unless somebody here has a suggestion for me ?
I'm leaning toward using a mean of 1 for all cycles of any length, since it never really grows. For sequences ending in cycles, go all the way around (to cancel the cycle out) in order to compute the length for the exponent.

2021-04-05, 06:34   #13
garambois

"Garambois Jean-Luc"
Oct 2011
France

32×5×13 Posts

Quote:
 Originally Posted by Happy5214 I'm leaning toward using a mean of 1 for all cycles of any length, since it never really grows. For sequences ending in cycles, go all the way around (to cancel the cycle out) in order to compute the length for the exponent.
I agree !
I will also be working on this soon !

2021-04-11, 18:10   #14
EdH

"Ed Hall"
Dec 2009

372010 Posts

I've added a little bit to my program. Hopefully it is of more interest than just mine. The new source code is attached.

It should compile under linux with the following command:
Code:
g++ seqinfo.cpp -o seqinfo
I have no idea what may be necessary to try to compile it under Windows.

The file regina_file is necessary. The program has been written such that a new regina_file can be as large as to sequence 20M. The format of any new regina_file must remain the same for the first four elements ([n,a,b,c).

The (h)elp entry displays all the current functions available:
Code:
--Available options for the following prompts--
(##/h/p/p##/q/u): prompt
## displays info for sequence ## if it is within the range.
h provides this text block.
p lists counts of all primes that terminate
a sequence within the limit of regina_file.
This will take a long time to complete.
Due to the large return count, the list is
sent to a primescount.txt file.  This file is
overwritten with each run.
p## searches for sequences that terminate with the prime ##.
u (not available yet!) run a routine to make a file of updates.
The file OE_3000000_C80.txt must be available.
This will take a long time.  If the file exists
it will be overwritten with a new file.
(y/n/c/f): prompt
y performs the procedure referenced.
n provides a negative response. (default if an entry is omitted)
c provides a count only, without a listing.
f provides a listing to screen and to results.txt.
(results.txt is never removed by the program.
It is only appended to.  It has to be manually deleted.)
Here is a sample run with entries in bold:
Code:
Data available for sequences 2 through 14000000
Sequence endings - prime: 10644411, cycle: 205473, open: 3150115
Enter sequence (##/h/p/p##/q/u): 276
276 is open ended. List any sequences that merge with 276? (y/n/c/f): c
7696 sequences found.
Enter sequence (##/h/p/p##/q/u): 13923160
13923160 is open ended. It merges with 4788.
Enter sequence (##/h/p/p##/q/u): 28
28 ends in a cycle. Display cycle? (starts at entry point) (y/n/f): y
28
Display all sequences that end in this cycle? (y/n/c/f): y
28
1 sequence found.
Enter sequence (##/h/p/p##/q/u): 496
496 ends in a cycle. Display cycle? (starts at entry point) (y/n/f): n
Display all sequences that end in this cycle? (y/n/c/f): y
496       608       650       652       790       1294      1574      1778
2162      2582      3142      5158      368449    1492799   1535075   1767455
1842215   2256401   2974751   3157729   3837505   3873551   4018945   4170127
4605213   4669921   5076873   5251285   5616985   6977649   7349365   7463965
7505901   7589845   7601365   7675345   8109041   8697385   8837245   8924241
11035163  12856335  13157075  13384167  13631207
45 sequences found.
Enter sequence (##/h/p/p##/q/u): p14604141802777
List all sequences that terminate with 14604141802777? (y/n/c/f): y
1923540   2967858   3462540   6232740   8361636   11070756  11079640  11470428
11788640  11792880  12125052  13462620  13896164
13 sequences found.
Enter sequence (##/h/p/p##/q/u): 13991486
13991486 ends in a cycle. Display cycle? (starts at entry point) (y/n/f): y
19916
17716
14316
19116
31704
47616
83328
177792
295488
629072
589786
294896
358336
418904
366556
274924
275444
243760
376736
381028
285778
152990
122410
97946
48976
45946
22976
22744
Display all sequences that end in this cycle? (y/n/c/f): c
8870 sequences found.
Enter sequence (##/h/p/p##/q/u): q
It can also now make a list of all the primes that terminate sequences present in regina_file. This process takes several hours. Here is an example run:
Code:
Enter sequence (#/h/p/q/u): p
Generating list of prime counts. . .
910307 unique primes found!
Listing took 102791 seconds to generate.
The time shown is exaggerated due to the machine being suspended overnight. There is also now a prompt, ensuring you really want to turn it loose for a few hours. I've attached the listing produced.

Here's a sample of primescount.txt:
Code:
2: 1
3: 270105
5: 1
7: 508095
11: 203297
13: 116153
17: 57124
. . .
13245150197: 3
27422578871: 1
28112302063: 2
80727104827: 21
94164320077: 2
14604141802777: 13
Attached Files
 seqinfo.cpp.txt (18.0 KB, 13 views) primescount.txt.zip (2.38 MB, 11 views)

Last fiddled with by EdH on 2021-04-11 at 18:31 Reason: Added a sample fo primescount.txt.

 2021-04-12, 05:33 #15 Happy5214     "Alexander" Nov 2008 The Alamo City 3·7·29 Posts As a suggestion (without looking at the code), would it be possible to treat perfect numbers as a special case? For example, instead of the above output for 28, perhaps something like: Code: Enter sequence (##/h/p/p##/q/u): 28 28 is a perfect number. Display all sequences that end at 28? (y/n/c/f): y 28 1 sequence found. Similarly, instead of the following for 608: Code: Enter sequence (##/h/p/p##/q/u): 608 608 ends in a cycle. Display cycle? (starts at entry point) (y/n/f): y 496 Display all sequences that end in this cycle? (y/n/c/f): c 45 sequences found. You could do something like this: Code: Enter sequence (##/h/p/p##/q/u): 608 608 ends at perfect number 496. Display all sequences that end at 496? (y/n/c/f): c 45 sequences found. To me, the extra step of prompting for a cycle of length 1 is overkill when it could just be printed in a single line, and perfect numbers are a special cycle in that you can point to a "terminating number".
2021-04-12, 12:54   #16
garambois

"Garambois Jean-Luc"
Oct 2011
France

58510 Posts

Quote:
 Originally Posted by EdH The file regina_file is necessary. The program has been written such that a new regina_file can be as large as to sequence 20M. The format of any new regina_file must remain the same for the first four elements ([n,a,b,c).
There is no reason for me to change this format.
The only thing I risk doing is adding a variable "o" in each line which will be the geometric mean of the quotients of the successive terms of the sequence.
I wanted to do this really fast, but sequences that merge with other Open End sequences are problematic and need to be recalculated.
I am thinking of making the change when I reach 15M, because then I will have to stop everything and modify the main program.
I will probably do this in July or August ...

Otherwise, I will do different tests with your program in the next few days ...

2021-04-12, 13:36   #17
EdH

"Ed Hall"
Dec 2009

23×3×5×31 Posts

Quote:
 Originally Posted by Happy5214 As a suggestion (without looking at the code), would it be possible to treat perfect numbers as a special case?. . .
Thanks! i was already looking at that very issue, but haven't delved into coding it yet. I'll probably just use a quick check to catch them, since we know all the perfect numbers of a size we will run across in the near future. But, I still need to fit it into the existing code.

@garambois: Adding elements won't bother at all. I just need the front to remain the same.

Are there any extra searches within regina_file that I should try adding?

A side question: (From the merges/termination thread) do you have the maximum heights for sequences listed in your files?

2021-04-12, 15:55   #18
garambois

"Garambois Jean-Luc"
Oct 2011
France

24916 Posts

Quote:
 Originally Posted by EdH @garambois: Adding elements won't bother at all. I just need the front to remain the same. Are there any extra searches within regina_file that I should try adding? A side question: (From the merges/termination thread) do you have the maximum heights for sequences listed in your files?

Unfortunately, I don't have the maximum heights reached by the sequences in my files !

Reminder : here is a line for a sequence of the file "regina_file"
[n, a, b, c, d, e, f, g, h, i, j, k, l, m]
"e" is the number of relative maxima (peaks) for the sequence that begins with the integer "n".
"f" is the number of parity changes found in this sequence, that is, the number of times a perfect square or double of a perfect square is found in this sequence.
I think these two data would be interesting to add to your program, because they are interesting elements which make the "identity" of a sequence.
It would also be interesting to be able to classify the sequences according to their number of peaks or changes in parity.

At the start of my research, I wanted to see if there was a correlation between the prime factorization of the starting numbers of the sequences and the numbers "e" and "f" (and the others of the table) or if there was a correlation between the ending prime numbers of the sequences and these "e" and "f" numbers.
But I failed in this search for correlation.
Your work and your program have rekindled my interest in this research ...

2021-04-12, 18:10   #19
EdH

"Ed Hall"
Dec 2009

23·3·5·31 Posts

Quote:
 Originally Posted by Happy5214 As a suggestion (without looking at the code), would it be possible to treat perfect numbers as a special case? . . .
I have added the recognition of the first few perfect numbers. The first 7 are within the current input for a sequence request, but I coded for the first 8. I coded the first 10 as sequence terminations. Any larger perfect numbers should be treated as before.
Quote:
 Originally Posted by garambois . . . Reminder : here is a line for a sequence of the file "regina_file" [n, a, b, c, d, e, f, g, h, i, j, k, l, m] "e" is the number of relative maxima (peaks) for the sequence that begins with the integer "n". "f" is the number of parity changes found in this sequence, that is, the number of times a perfect square or double of a perfect square is found in this sequence. I think these two data would be interesting to add to your program, because they are interesting elements which make the "identity" of a sequence. It would also be interesting to be able to classify the sequences according to their number of peaks or changes in parity. . . .
I will study this, but I might be slow. Let me look it over. . .
Attached Files
 seqinfo.cpp.txt (19.7 KB, 12 views)

Last fiddled with by EdH on 2021-04-12 at 18:12 Reason: Might help to actually attach the source.

2021-04-12, 19:11   #20
EdH

"Ed Hall"
Dec 2009

23×3×5×31 Posts

Quote:
 Originally Posted by garambois . . . At the start of my research, I wanted to see if there was a correlation between the prime factorization of the starting numbers of the sequences and the numbers "e" and "f" (and the others of the table) or if there was a correlation between the ending prime numbers of the sequences and these "e" and "f" numbers. But I failed in this search for correlation. Your work and your program have rekindled my interest in this research ...
As you have probably noticed over the years, I often have some area of interest in Aliquot sequences that sends me off to somewhere. I'm always happy when some of those studies benefit others' interests. I'm also interested in finding ways to use already gathered data, such as in your files. I hope to eventually be able to contribute to the tables again, if I can ever complete this other factoring project.

2021-04-13, 14:09   #21
EdH

"Ed Hall"
Dec 2009

23×3×5×31 Posts

Quote:
 Originally Posted by garambois . . . Reminder : here is a line for a sequence of the file "regina_file" [n, a, b, c, d, e, f, g, h, i, j, k, l, m] "e" is the number of relative maxima (peaks) for the sequence that begins with the integer "n". "f" is the number of parity changes found in this sequence, that is, the number of times a perfect square or double of a perfect square is found in this sequence. I think these two data would be interesting to add to your program, because they are interesting elements which make the "identity" of a sequence. It would also be interesting to be able to classify the sequences according to their number of peaks or changes in parity. At the start of my research, I wanted to see if there was a correlation between the prime factorization of the starting numbers of the sequences and the numbers "e" and "f" (and the others of the table) or if there was a correlation between the ending prime numbers of the sequences and these "e" and "f" numbers. But I failed in this search for correlation. Your work and your program have rekindled my interest in this research ...
I'm wondering what I should add regarding e and f.

I should be able to easily add a function that lists sequences based on the count but more complexities might be problematic.* What would you actually like to be able to list?

* Many years ago, I used to select sequences to work on based on their graphs, expecting that the more changes, the higher probability of termination. That didn't seem to turn out as a valid indicator of a pending termination.

2021-04-14, 09:30   #22
garambois

"Garambois Jean-Luc"
Oct 2011
France

32×5×13 Posts

Quote:
 Originally Posted by EdH I will study this, but I might be slow. Let me look it over.
No problem, we take the time we want. We do this for fun and if we start to feel rushed and stressed, we won't like working on these things anymore !
;-)

Quote:
 Originally Posted by EdH * Many years ago, I used to select sequences to work on based on their graphs, expecting that the more changes, the higher probability of termination. That didn't seem to turn out as a valid indicator of a pending termination.
I think that when you say that, you are right.
But what if we are wrong in thinking that ?
In any case, it only costs us a little time to do some tests, we could come across correlations that nobody expected !

Quote:
 Originally Posted by EdH I'm wondering what I should add regarding e and f. I should be able to easily add a function that lists sequences based on the count but more complexities might be problematic.* What would you actually like to be able to list?
I have a suggestion.
For example when you display all the starting numbers of the sequences that end with a prime number, personally, I tend to look at these numbers in their usual representation in base 10, but especially also in the factorized form in prime numbers : this gives us much more information about the number, which is important if we want to notice properties.

Concerning e and f, I don't know if your program can show some things I would like to see.
But I ask myself many questions.
Let me introduce you to some ideas I've been working on recently, after Edwin started this topic.

Here are some examples :

1) For each prime number, show a table that counts the number of sequences that end with that prime number that have 0 peak (this number will most likely be 0), 1 peak, 2 peaks...
Example :
3 [(number of sequences that have 0 peaks and end in 3), (number of sequences that have 1 peak and end in 3), (number of sequences that have 2 peaks and end in 3), ...]
5 [(number of sequences that have 0 peaks and end with 5), (number of sequences that have 1 peak and end with 5), (number of sequences that have 2 peaks and end with 5), ...]
...
The question to answer : will the distribution be the same for every prime number ?

2) A very simple way to visualize the data would also be to be able to launch a regina_file analysis by entering this in a program for example :
[n%2==0, a==0, b, c, d, e==0, f, g, h, i, j, 1.7<k<2.3, l, m]
Thus, for each of the 14 variables in each row of regina_file, we could specify characteristics.
In the example above, the analysis would give us all the sequences that start with an even number, that are Open-End, that have no peak (so they are strictly increasing) and that have a slope of about 2 (so at each iteration, the size of the terms multiplies by about a factor of 2).
This entry would allow us for example to find all the sequences that have the same very special graph as the sequence 19560.
Thus, just for all the sequences with 0 peaks, there are several types : we could find again the drivers which are the perfect numbers by specifying a slope k rather close to 1, find the guides which ensure slopes of 2, and above all, maybe see things not yet known ...
Another example : you want a bell-shaped sequence.
You enter :
[n, a==1, b, c, d, e==1, f, g, h, i, j, 0.8<k<1.2, l, m]
And you find sequences such as 2174880.
Of course, the goal is to try to notice correlations between these forms of graphs and the factorization in prime factors of the starting number of the sequence (and this correlation exists at least for the sequences with 0 peaks, because of the perfect numbers drivers, of the 2-perfect numbers, of the 3-perfect numbers...) or according to the prime number which ends the sequences (belonging to such or such branch of the infinite graph of the aliquot sequences).

I have the same questions about the number of parity changes for each sequence.

But as said above, all this is a very long work started years ago.
And my problem is that in python, I can't do this work anymore, because regina_file is too big.
I have to work in C, and there, I'm not at ease !

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# Math Help - Abstract Proof

1. ## Abstract Proof

Let recip={(x,y) : x is a number different than 0 and y=1/x}
Show that recip is continuous on {x : x is a number different than 0}

2. Originally Posted by JMUmath
Let recip={(x,y) : x is a number different than 0 and y=1/x}
Show that recip is continuous on {x : x is a number different than 0}
I would not be surprised if the following is an incorrect interpretation, but I think you're asking to prove that $\varphi:\mathbb{R}-\{0\}\to\mathbb{R}\times\mathbb{R}:x\mapsto\left(x ,\frac{1}{x}\right)$ is continuous.

But, depending on your background knowledge this should be obvious since in any topological space if $\varphi,\psi:X\to Y$ are continuous maps then so is $\varphi\oplus\psi:X\to Y\times Y:x\mapsto (\varphi(x),\psi(x))$ this follows easily since $(\varphi\oplus\psi)^{-1}(U\times V)=\varphi^{-1}(U)\cap\psi^{-1}(V)$.

So, since $\varphi:\mathbb{R}-\{0\}:x\mapsto\frac{1}{x}$ and the inclusion map $\iota:\mathbb{R}-\{0\}\hookrightarrow\mathbb{R}$ are continuous then so is $\iota\oplus\varphi:\mathbb{R}-\{0\}\to\mathbb{R}\times\mathbb{R}:x\mapsto\left(x ,\frac{1}{x}\right)$

Another way is to notice that the two coordinate functions are continuous.

3. Well, for this problem, we are only working with limited knowledge and other theorems we've proven. Pretty much all we have to work with is assuming recip is a function and the definition of continuous that we are given:

Suppose X is a subset of the reals, f is a function from X into the reals, and p is an element of X. The statement that f is continuous at p means that if V is segment and f(p) is an element of V, then there is a segment, call such segment, call such a segment U, so that
i) p is an element of U,
and
ii)if q is an element of U intersect X, then f(q) is an element of V

The statement that f is continuous on X means that if p is and element of X, then f is continuous at p.

Sorry for lack of symbols, I'm new.

4. Originally Posted by JMUmath
Well, for this problem, we are only working with limited knowledge and other theorems we've proven. Pretty much all we have to work with is assuming recip is a function and the definition of continuous that we are given:

Suppose X is a subset of the reals, f is a function from X into the reals, and p is an element of X. The statement that f is continuous at p means that if V is segment and f(p) is an element of V, then there is a segment, call such segment, call such a segment U, so that
i) p is an element of U,
and
ii)if q is an element of U intersect X, then f(q) is an element of V

The statement that f is continuous on X means that if p is and element of X, then f is continuous at p.

Sorry for lack of symbols, I'm new.
Wow, that is not a fun definition of continuity. Let me try to wrap my head around it.

So it sort of seems like your saying $f:\mathbb{R}\to\mathbb{R}$ is continuous if $I$ is an interval implies that $f(I)$ is an interval?

5. The way I understand it is have some segment (a,b) = V so that a<f(p)<b or for this problem a<1/p<b
Then show there exists a segment U such that p must be an element of U. Usually its just through algebra with the inequality to solve for p and make sure there can be a logical segment that p is an element of and is a subset of the domain.
I don't know if that helps, but I'm not fan of the definition either.

6. Originally Posted by JMUmath
Suppose X is a subset of the reals, f is a function from X into the reals, and p is an element of X. The statement that f is continuous at p means that if V is segment and f(p) is an element of V, then there is a segment, call such segment, call such a segment U, so that
i) p is an element of U, and ii)if q is an element of U intersect X, then f(q) is an element of V
The statement that f is continuous on X means that if p is and element of X, then f is continuous at p.
Originally Posted by Drexel28
Wow, that is not a fun definition of continuity. So it sort of seems like your saying $f:\mathbb{R}\to\mathbb{R}$ is continuous if $I$ is an interval implies that $f(I)$ is an interval?
It sounds as if the professor was a student of one of Herbert Wall’s students.
An open finite non-degenerate interval is called a segment.
If $f(p)\in (c,d)$ then there is an $U_{\delta}(p)=\left(p-\delta,p+\delta\right)$ so that $f( U_{\delta}(p))\subset (c,d)$.

Actually this question just asks that we prove that if $p\not= 0$ then $\frac{1}{x}$ is continuous at $p$.

7. ## Partial Proof

Here's what I thought would've worked, but not for all cases as I found out. Maybe this will help show what I'm talking about.

Pf: 2 Cases p>0 or p<0 Let f(x)=recip(x)

Let p>0 and p is an element of the reals. Let X be the set of postive real numbers such that p is an element of X.

Let V be the segment _(a,b)_ [i found out this only worked for me if a and b were positive] and let U be the segement _(1/b, 1/a)_

Then a<f(p)<b
a<1/p<b
1/p>p>1/b and thus p is an element of U

Let q be an element of U intersect X which impies

1/b<q<1/a
b>1/q>a
b>f(q)>a and thus f(q) is an element of V

Therefore, recip is continuous on X. [ ]

Similar proof for p<0

Unfortunately I thought this worked, but I think about V being the segment _(a,b)_ if a was negative and b was positive.
I'm still just stuck on this one.

P.S. Is there a chart for all the [tex] commands?