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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Concept. Go to the latest version.

Supplementary Angles

Two angles that add to 180 degrees.

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Supplementary Angles

What if you were given two supplementary angles? How would you determine their angle measures? After completing this Concept, you'll be able to use the definition of supplementary angles to solve problems like this one.

Guidance

Two angles are supplementary when they add up to 180\begin{align*}180^\circ\end{align*}. Supplementary angles do not have to be congruent or touching.

Example A

The two angles below are supplementary. If mMNO=78\begin{align*}m \angle MNO = 78^\circ\end{align*} what is mPQR\begin{align*}m \angle PQR\end{align*}?

Set up an equation.

78+mPQR=180mPQR=102

Example B

What are the measures of two congruent, supplementary angles?

Supplementary angles add up to 180\begin{align*}180^\circ\end{align*}. Congruent angles have the same measure. Divide 180\begin{align*}180^\circ\end{align*} by 2, to find the measure of each angle.

180÷2=90

So, two congruent, supplementary angles are right angles, or 90\begin{align*}90^\circ\end{align*}.

Example C

Name one pair of supplementary angles in the diagram below.

One example is INM\begin{align*} \angle INM\end{align*} and MNL\begin{align*} \angle MNL\end{align*}.

Watch this video for help with the Examples above.

Vocabulary

Two angles are supplementary when they add up to 180\begin{align*}180^\circ\end{align*}.

Guided Practice

Find the measure of an angle that is supplementary to ABC\begin{align*}\angle ABC\end{align*} if mABC\begin{align*}m \angle ABC\end{align*} is

1. 45\begin{align*}45^\circ\end{align*}

2. 118\begin{align*}118^\circ\end{align*}

3. 32\begin{align*}32^\circ\end{align*}

4. x\begin{align*}x^\circ\end{align*}

1. 135\begin{align*}135^\circ\end{align*}

2. 62\begin{align*}62^\circ\end{align*}

3. 148\begin{align*}148^\circ\end{align*}

4. 180x\begin{align*}180-x^\circ\end{align*}

Practice

Find the measure of an angle that is supplementary to ABC\begin{align*}\angle ABC\end{align*} if mABC\begin{align*}m\angle ABC\end{align*} is:

1. 112\begin{align*}112^\circ\end{align*}
2. 15\begin{align*}15^\circ\end{align*}
3. 97\begin{align*}97^\circ\end{align*}
4. 81\begin{align*}81^\circ\end{align*}
5. 57\begin{align*}57^\circ\end{align*}
6. (xy)\begin{align*}(x-y)^\circ\end{align*}
7. (x+y)\begin{align*}(x+y)^\circ\end{align*}

Use the diagram below for exercises 8-9. Note that NK¯¯¯¯¯¯IL\begin{align*}\overline{NK} \perp \overleftrightarrow{IL}\end{align*}.

1. Name another pair of supplementary angles.
1. If mINJ=63\begin{align*}m\angle INJ = 63^\circ\end{align*}, find mJNL\begin{align*}m\angle JNL\end{align*}.

For exercises 10-13, determine if the statement is true or false.

1. Supplementary angles add up to 180\begin{align*}180^\circ\end{align*}.
2. Two angles on a straight line are supplementary angles.
3. To be supplementary, two angles must be touching.
4. It's possible for two angles in a triangle to be supplementary.

For 14-15, find the value of x\begin{align*}x\end{align*}.

Vocabulary Language: English

Supplementary angles

Supplementary angles

Supplementary angles are two angles whose sum is 180 degrees.


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# Thread: Find the point on the line 3x + 2y -3 = 0, which is closest to the point (1, -3)

1. ## Find the point on the line 3x + 2y -3 = 0, which is closest to the point (1, -3)

Find the point on the line 3x + 2y -3 = 0, which is closest to the point (1, -3).

( , ) ?

2. Originally Posted by B-lap
Find the point on the line 3x + 2y -3 = 0, which is closest to the point (1, -3).
Where does $2x-3y-11=0$ intersect the given line?

3. Originally Posted by B-lap
Find the point on the line 3x + 2y -3 = 0, which is closest to the point (1, -3).

( , ) ?
Call this point (a,b). Solve line equation for y. Now use the distance formula between the point (1,-3) and (a,b).

Two hints: 1) (a,b) = (a,f(a)) where f(a) is the y-value for x=a
2) The distance formula has a nasty square-root. If you minimize D^2, you also minimize D. So you can drop the square root.

Differentiate and find critical point.

format: plain


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# fix propel/self command

## Syntax

fix ID group-ID propel/self mode magnitude keyword values ...

• ID, group-ID are documented in fix command

• propel/self = style name of this fix command

• mode = velocity or quat

• magnitude = magnitude of the active force

• one or more keyword/value pairs may be appended to args

• keyword = types

types values = one or more atom types

## Examples

fix active_group all propel/self velocity 1.0
fix constant_velocity all viscous 1.0

fix active_group all propel/self quat 1.0

fix active all propel/self quat 1.0 types 1 2 4

## Description

Adds a force of a constant magnitude to each atom in the group. The nature in which the force is added depends on the mode.

For mode = velocity, the active force acts along the velocity vector of each atom. This can be interpreted as a velocity-dependent friction, such as proposed by (Erdmann).

For mode = quat the force is applied along the axis obtained by rotating the x-axis along the atom’s quaternion. In other words, the force is along the x-axis in the atom’s body frame. This mode requires all atoms in the group to have a quaternion, so atom_style should either be ellipsoid or body. In combination with Langevin thermostat for translation and rotation in the overdamped regime, the quaternion mode corresponds to the active Brownian particle model introduced by (Henkes), (Bialke) and (Fily).

By default, this fix is applied to all atoms in the group. You can override this behavior by specifying the atom types the fix should work on through the types keyword.

Restart, fix_modify, output, run start/stop, minimize info:


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Giáo trình

College Physics

Science and Technology

Gyroscopic Effects: Vector Aspects of Angular Momentum

Tác giả: OpenStaxCollege

Angular momentum is a vector and, therefore, has direction as well as magnitude. Torque affects both the direction and the magnitude of angular momentum. What is the direction of the angular momentum of a rotating object like the disk in [link]? The figure shows the right-hand rule used to find the direction of both angular momentum and angular velocity. Both $\mathbf{\text{L}}$ and $\mathbf{\text{ω}}$ are vectors—each has direction and magnitude. Both can be represented by arrows. The right-hand rule defines both to be perpendicular to the plane of rotation in the direction shown. Because angular momentum is related to angular velocity by $\mathbf{\text{L}}=I\mathbf{\text{ω}}$, the direction of $\mathbf{\text{L}}$ is the same as the direction of $\mathbf{\text{ω}}$. Notice in the figure that both point along the axis of rotation.

Now, recall that torque changes angular momentum as expressed by

$\text{net}\phantom{\rule{0.25em}{0ex}}\mathbf{\text{τ}}=\frac{\text{Δ}\mathbf{\text{L}}}{\text{Δ}t}.$

This equation means that the direction of $\text{Δ}\mathbf{\text{L}}$ is the same as the direction of the torque $\mathbf{\text{τ}}$ that creates it. This result is illustrated in [link], which shows the direction of torque and the angular momentum it creates.

Let us now consider a bicycle wheel with a couple of handles attached to it, as shown in [link]. (This device is popular in demonstrations among physicists, because it does unexpected things.) With the wheel rotating as shown, its angular momentum is to the woman's left. Suppose the person holding the wheel tries to rotate it as in the figure. Her natural expectation is that the wheel will rotate in the direction she pushes it—but what happens is quite different. The forces exerted create a torque that is horizontal toward the person, as shown in [link](a). This torque creates a change in angular momentum $\mathbf{\text{L}}$ in the same direction, perpendicular to the original angular momentum $\mathbf{\text{L}}$, thus changing the direction of $\mathbf{\text{L}}$ but not the magnitude of $\mathbf{\text{L}}$. [link] shows how $\text{Δ}\mathbf{\text{L}}$ and $\mathbf{\text{L}}$ add, giving a new angular momentum with direction that is inclined more toward the person than before. The axis of the wheel has thus moved perpendicular to the forces exerted on it, instead of in the expected direction.

This same logic explains the behavior of gyroscopes. [link] shows the two forces acting on a spinning gyroscope. The torque produced is perpendicular to the angular momentum, thus the direction of the torque is changed, but not its magnitude. The gyroscope precesses around a vertical axis, since the torque is always horizontal and perpendicular to $\mathbf{\text{L}}$. If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque ($\mathbf{\text{L}}=\text{Δ}\mathbf{\text{L}}$), and it rotates around a horizontal axis, falling over just as we would expect.

Earth itself acts like a gigantic gyroscope. Its angular momentum is along its axis and points at Polaris, the North Star. But Earth is slowly precessing (once in about 26,000 years) due to the torque of the Sun and the Moon on its nonspherical shape.

Section Summary

• Torque is perpendicular to the plane formed by $r$ and $\mathbf{\text{F}}$ and is the direction your right thumb would point if you curled the fingers of your right hand in the direction of $\mathbf{\text{F}}$. The direction of the torque is thus the same as that of the angular momentum it produces.
• The gyroscope precesses around a vertical axis, since the torque is always horizontal and perpendicular to $\mathbf{\text{L}}$. If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque ($\mathbf{\text{L}}=\text{Δ}\mathbf{\text{L}}$), and it rotates about a horizontal axis, falling over just as we would expect.
• Earth itself acts like a gigantic gyroscope. Its angular momentum is along its axis and points at Polaris, the North Star.

Conceptual Questions

While driving his motorcycle at highway speed, a physics student notices that pulling back lightly on the right handlebar tips the cycle to the left and produces a left turn. Explain why this happens.

Gyroscopes used in guidance systems to indicate directions in space must have an angular momentum that does not change in direction. Yet they are often subjected to large forces and accelerations. How can the direction of their angular momentum be constant when they are accelerated?

Problem Exercises

Integrated Concepts

The axis of Earth makes a 23.5° angle with a direction perpendicular to the plane of Earth’s orbit. As shown in [link], this axis precesses, making one complete rotation in 25,780 y.

(a) Calculate the change in angular momentum in half this time.

(b) What is the average torque producing this change in angular momentum?

(c) If this torque were created by a single force (it is not) acting at the most effective point on the equator, what would its magnitude be?

(a) $5\text{.}\text{64}×{\text{10}}^{\text{33}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot {\text{m}}^{2}\text{/s}$

(b) $1\text{.}\text{39}×{\text{10}}^{\text{22}}\phantom{\rule{0.25em}{0ex}}\text{N}\cdot \text{m}$

(c) $2\text{.}\text{17}×{\text{10}}^{\text{15}}\phantom{\rule{0.25em}{0ex}}\text{N}$


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# The spaces $\ell^p, \; 1 \leq p < + \infty$ are separable. On the other side, $\ell^\infty$ is not.

Exercise :

Show that the spaces $$\ell^p, \; 1 \leq p < + \infty$$ are separable.

Attempt :

In order to show that $$\ell^p$$ is separable for $$\ell^p, \; 1 \leq p < + \infty$$, we need to work over a set $$D$$ proving that it is countable and dense over $$\ell^p$$, while showing that for $$x \in \ell^P$$ and $$y \in D$$, it is $$\|x-y\|<\varepsilon.$$

Since $$x \in \ell^p$$, where $$x=(x_n)_{n}$$, this obviously means that : $$\sum_{n=1}^\infty |x_n|^p <\infty \implies \left( \sum_{n=1}^\infty |x_n|^p \right)^{1/p} < \infty \implies \left( \sum_{n=1}^\infty |x_n|^p \right)^{1/p}<\varepsilon$$ $$\Leftrightarrow$$ $$\sum_{n=1}^\infty |x_n|^p < \varepsilon^p, \quad \text{for some } \varepsilon >0$$

But that would mean that $$\exists k \in \mathbb N$$, such that :

$$\sum_{n=k+1}^\infty |x_n|^p < \left(\frac{\varepsilon}{2}\right)^p$$

Now, if I let $$y=(y_n)_n$$ be the sequence

$$y_n = \begin{cases} x_n& n \leq k \\ 0& n> k \end{cases}$$

then, one can easily see that :

$$\left\| x-y\right\|_p = \left( \sum_{n=k+1}^\infty |x_n|^p \right)^{1/p}<\frac{\varepsilon}{2}$$

Now, let $$D_n$$ be the set :

$$D_n = \left\{\sum_{i=1}^n q_ie_i : q_i \in \mathbb Q \right\}$$

This set is countable as $$D_n$$ can be correlated to $$\mathbb Q^n$$. Now, the union $$\bigcup_{n=1}^\infty D_n$$ is also countable as a union of countable sets. Now, it is :

$$\|x-z\|_p \leq \|x-y\|_p + \|y-z\|_p < \varepsilon/2 + \varepsilon/2 \equiv \varepsilon$$

This means that $$D = \bigcup_{n=1}^\infty D_n$$ is countable and dense over $$\ell^p$$ which means that $$\ell^p$$ is separable.

Question : How does one show that $$\ell^\infty$$ is not a separable space ?

• Take $D$ to be all the finite sequences (i.e. they get only $0$ eventually) with rational values and argue almost as you did. Feb 9 '19 at 20:12
• For the second question. Consider all the $\{0,1\}$ sequences in $l^\infty$. This is a non-countable set of elements with distance $1$ from one another. [I believe you can answer your own question now, if you still struggle I can try to write a more detailed answer] Feb 9 '19 at 20:14
• @Yanko Kindly thanks for your inputs ! I just updated my answer based on your hint and some intuition. Is it correct ? Thanks for the hint over the final part, it's straighforward based on that intuition ! Feb 9 '19 at 20:17
• It's good. but you have some index problems with the definition of $D_n$ (you want $\sum_{i=1}^n$ instead of $\sum_{n=1}^k$.) Feb 9 '19 at 20:18
• @Yanko Thanks, I messed up. Appreciate all your help ! Feb 9 '19 at 20:19

As Yanko suggested in the comments, you can do the following:

In order to show that $$\ell_\infty$$ is not separable it is enough to find $$A \subset \ell_\infty$$ that is not countable s.t. for each $$x\ne y \in A$$ , $$||x-y|| =\delta \gt 0$$. since then we can surround those points of $$A$$ with $$\dfrac{1}{2}\delta$$-Balls and then there couldn't be a countable set that is dense since it should've element in each such ball (and distinct ones in each ball).

So take $$A = \{x=(x_n) : x_n \in \{0,1\} \}\subset \ell_\infty$$. $$A$$ is uncountable.

Show that $$||x-y||_{\infty} = 1$$ iff $$x\ne y$$ and you are done .

• That was Yanko's comment and hint from all along so I guess that he at least deserves some credits. Thanks for your input though. Feb 9 '19 at 20:30
• Im sorry , i really didn't read the comments before i answered(i had this exact question in my problem set a while ago so i just answered) i will edit it. @Rebellos
– user335501
Feb 9 '19 at 20:32
• Np, just mentioned ! Feb 9 '19 at 20:34


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# Differentiate the following functions:

Question:

Differentiate the following functions:

(i) $6 \times 5+4 \times 3-3 \times 2+2 \times-7$

(ii) $5 x^{-3 / 2}+\frac{4}{\sqrt{x}}+\sqrt{x}-\frac{7}{x}$

(iii) $a \times 3+b \times 2+c x+d$, where $a, b, c, d$ are constants

Solution:

(i) $6 x^{5}+4 x^{3}-3 x^{2}+2 x-7$

Formula:-

$\frac{d}{d x} x^{n}=n x^{n-1}$

Differentiating with respect to $\mathrm{x}$,

$\frac{d}{d x}\left(6 x^{5}+4 x^{3}-3 x^{2}+2 x-7\right)=30 x^{5-1}+12 x^{3-1}-6 x^{2-1}+2 x^{1-1}+0$

$=30 x^{4}+12 x^{2}-6 x^{1}+2 x$

(ii) $5 x^{-3 / 2}+\frac{4}{\sqrt{x}}+\sqrt{x}-\frac{7}{x}$

Formula:-

$\frac{d}{d x} x^{n}=n x^{n-1}$

Differentiating with respect to $\mathrm{x}$,

$\frac{d}{d x}\left(5 x^{-3 / 2}+\frac{4}{\sqrt{x}}+\sqrt{x}-\frac{7}{x}\right)$

$=5 \times-\frac{3}{2} x^{-\frac{3}{2}-1}+4 \times-\frac{1}{2} x^{-\frac{1}{2}-1}+\frac{1}{2} x^{\frac{1}{2}-1}-7 \times-1 x^{-1-1}$

$=-\frac{15}{2} x^{-\frac{5}{2}}-2 x^{-\frac{3}{2}}+\frac{1}{2} x^{\frac{-1}{2}}+7 x^{-2}$

(iii) $a x^{3}+b x^{2}+c x+d$, where $a, b, c, d$ are constants

Formula:-

$\frac{d}{d x} x^{n}=n x^{n-1}$

Differentiating with respect to $\mathrm{x}$,

$\frac{d}{d x}\left(a x^{3}+b x^{2}+c x+d\right)=3 a x^{3-1}+2 b x^{2-1}+c x^{1-1}+d \times 0$

$=3 a x^{2}+2 b x+c$


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# The Crystal Structure of Benzyloxycarbonyl-${(\alpha-aminoisobutyryl)}_2$-L-Alanyl Methyl Ester

Prasad, Venkataram BV and Shamala, N and Nagaraj, R and Balaram, P (1980) The Crystal Structure of Benzyloxycarbonyl-${(\alpha-aminoisobutyryl)}_2$-L-Alanyl Methyl Ester. In: Acta Crystallographica Section B: Structural Crystallography and Crystal Chemistry, 36 (1). pp. 107-110.

 Preview
PDF
benzyl.pdf

Crystals of the title compound, $C_{20}H_{29}N_3O_6$, are monoclinic, space group ${P2}_1$, with a = 8.839 (3), b = 10.818 (3), c = 11.414 (2) A, \beta = 95.69 (2) deg, Z = 2; final R = 0.053. The molecular conformation is defined by the following angles (\phi, \psi): Aib-1 58.1, 36.8; Aib-2 68.3, 18.6; Ala-3 (\phi) = 136.2 deg. The molecule adopts a type III' \beta-turn conformation stabilized by an intramolecular hydrogen bond between the CO of the benzyloxycarbonyl group and the NH of the alanyl residue. The hydrogen-bond parameters are N . . . O 2.904 A and \angle NH . . . O 156.9 deg.


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##### This cache has been archived.

OReviewer: As there's been no response to my earlier post, I'm archiving this cache to prevent it from blocking other cache placements. If you wish to repair/replace the cache sometime in the future, just contact me (by email), including the GC Code or a link to the page, and assuming it meets the guidelines, I'll be happy to unarchive it.

More
<

## Buster's Spot

A cache by Brookdale Boomers Message this owner
Hidden : 10/07/2007
Difficulty:
Terrain:

Size:  (small)

Join now to view geocache location details. It's free!

#### Watch

How Geocaching Works

Please note Use of geocaching.com services is subject to the terms and conditions in our disclaimer.

### Geocache Description:

This is an easy cache in a public park that allows geocaching.

Brookdale Park is the third largest park in the Essex County system. It was designed by the Olmsted Brothers who also designed Central Park in New York City. Construction began in 1928 but was interrupted by the Great Depression and the major work on the park was not completed until 1937, using labor from the WPA and ERA.

Enjoy the park and good luck.

The cache is in a plastic mayonnaise jar which has had the paper lid liner replaced with a rubber seal. It contains:
Small Log Book
Old Town Koosie
Auto Fuses
Retractable USB Cord
Parkway Tokens
$2 bill for www.wheresgeorge.com (no need to leave$, spend it an put it in circulation after you log it on WG)
Toy Globe
Toy Narwhal Whale

Additional Hints (Decrypt)

Ybbx bss gur cngu vafvqr gur gerr :)

Decryption Key

A|B|C|D|E|F|G|H|I|J|K|L|M
-------------------------
N|O|P|Q|R|S|T|U|V|W|X|Y|Z

(letter above equals below, and vice versa)

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text:
# Center aligning a shrunk frame in beamer

I have a beamer related question. I am using [shrink=50] command to shrink a frame. I want the shrunk frame to be center aligned but by default it is aligned left of the slide. How do I do this?

\documentclass{beamer}
\usetheme{Warsaw}
\begin{document}
\begin{frame}[shrink=50]
\textbf{Title of the slide}
\vskip10pt
Hi
\end{frame}
\end{document}

-
I assume you've read Till Tantau's advice: "Never ever use the evil option shrink." – Gonzalo Medina Dec 2 '11 at 2:22
You can just wrap the contents in a center environment. – Werner Dec 2 '11 at 2:26
:D that worked. My question sounds so stupid now. I tried all sorts of things except this. – John Smith Dec 2 '11 at 2:32

As I mentioned in a comment, Till Tantau advices: "Never ever use the evil option shrink."

That being said, you can the \centering command or the center environment to get the desired horizontal centering:

\documentclass{beamer}
\usetheme{Warsaw}
\begin{document}
\begin{frame}[shrink=50]
\centering
\textbf{Title of the slide}
\vskip10pt
Hi\par
\end{frame}
\end{document}

-


text:
Find the number of specific permutations

Find the number of all permutations of the set $\left\{ 1,2,...,2n \right\}$ such that does not have compact subsequence: $\langle i, \ i+1\rangle$ or $\langle i+1, \ i\rangle$ for all $1\le i\le 2n-1$.

I'm not sure if I translated above text properly, so please correct me if needed. For example, for $n=3$ permutation: $1,6,4,2,5,3$ is ok, but $4,6,2,1,3,5$ is not.

I've tried inclusion-exclusion principle, but it didn't help. Probably because I was always weak at this principle. How can I solve this problem?

-
To arrive at a good permutation of $\{1,2,...,n\}$, you can start with a good permutation of $\{1,2,...,n-1\}$ and insert $n$ in any of the $n-2$ positions not adjacent to $n-1$, or you can start with a permutation of $\{1,2,...,n-1\}$ with a single adjacent pair and insert $n$ between its members, breaking them up. – mjqxxxx Aug 30 '12 at 19:37
possible duplicate of OEIS A000255 recursion. Not quite, as that one didn't do both reversals. – Ross Millikan Aug 30 '12 at 20:42
OEIS A002464 gives formulae, generating functions, references etc. – Henry Aug 30 '12 at 20:45
@mjqxxxx, I understand your point, but I have problems with the second part. Single adjacent pair can be $k,k+1$ (which is easy) or $k+1,k$, which I don't know how to examine. So far I got: $a_n=(n-2)a_{n-1}+(n-2)(a_{n-2}+?)$, where $?$ should be the number of all permutations that contain substring $k+1,n,k$ but I don't know how to count them. – ray Aug 31 '12 at 10:12


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# LaTeXiT and ghostscript 9.27

update As of ghostscript version 9.50 (possibly earlier), this is no longer issue, so you can unpin and brew upgrade to your heart’s content.

A couple weeks ago, I started my day with a usual brew upgrade, then got to work on an upcoming paper deadline… only to find that LaTeXiT broke, and everything would be half-rendered:

It didn’t take long to find the culprit: ghostscript was a likely candidate, so I checked what brew had upgraded, and as expected Ghostscript had been upgraded to 9.27. However, it took about an hour to figure out how to actually roll back to 9.26, as a lot of previous methods to do this in homebrew have been deprecated.

I finally managed to fix this by directly adjusting the ghostscript formula. I finally fixed it, and figured this might be useful for others. You can roll things back with:

1. go to the homebrew-core folder: cd /usr/local/Homebrew/Library/Taps/homebrew/homebrew-core
2. find the commit for ghostscript 9.26: git log master -- Formula/ghostscript.rb
3. check it out: git checkout 6ec0c1a03ad789b62
4. uninstall ghostscript brew uninstall ghostscript
5. reinstall it making sure not to touch anything else:HOMEBREW_NO_AUTO_UPDATE=1 brew install ghostscript
6. pin ghostscript so that it won’t be upgraded anymore: brew pin ghostscript

You can now continue your life!

In the meantime I’ve pinged Pierre Chachatelier, the maintainer of LaTeXiT, and he suggested also an alternative: install a patched version of ghostscript. I’m lazy, so I like letting hombrew do everything (also meaning, I didn’t test this), but if you need ghostscript 9.27 for whatever reason, there’s this.

# JSPS fellowship, final interview

In a previous post I documented the paper application for the JSPS scholarship fellowship. A number of applicants are accepted outright after this step, however a number are required to give a final interview in the form of a 4-minute powerpoint presentation. I was in this second set, and here is my advice for this final part.

## TL;DR

• Read the JSPS ‘About us’ page. This will give you a good idea of what they are looking for.

• Impact and Originality are two key messages that should be emphasized.

• Independent Thought. Which parts of the project are your own ideas, from your own initiative (as opposed to your professor’s)? This is another key point that should be clear.

• Add Japanese Subtitles. Or alternatively make the slides completely in Japanese. You’re allowed to present in English, but having Japanese on the slides will make it easier for the audience to find the thread again if they lose track for a moment.

• Get feedback from a wide audience. The panel will be a mixed crowd (from your general field), so you need to make sure your presentation is both understandable and appealing to a wide audience. Skype with old buddies from back home, ask other PhDs at your uni, hell, go knock on the door of some professors and ask them too. Not all of them will indulge you, but the ones who do will probably give you some really good feedback.

## The story

The TL;DR pretty much covers it and most of the reasoning is explained in this post, so I won’t go into too many details. Of course being able to give a solid and compelling presentation is key, but that isn’t specific to the JSPS interview. So I will focus here on the Q&A section. Most of these questions are rather standard, so you should prepare and practice them. With some practice, you can often answer questions in a way that emphasizes things that you think make you a strong candidate but aren’t directly asked for (i.e. I wanted to point out that I have had a lot of international experience, which I think is one of the things that sets me apart from the average Japanese student).

## Q&A

Q1: Why do you want to study in Japan?
This is a very standard question, and a perfect excuse to talk about something more important. I quickly mentioned that I like Japanese/Asian culture, then explained the main reason was specifically to join the Ishiguro lab. I had met prof. Ishiguro a year earlier during a summer-school, we got along very well and I really liked both the research topics his lab had as well as the approach they take. Then I took the opportunity to explain that his lab was very much focused on hardware development, whereas my background is more heavy on optimization and optimal control, so this provided a lot of synergy: I would be bringing new know-how into the lab, and at the same time there was a lot of new stuff for me to learn. Lo and behold, the very next question was…

Q2: What is your contribution, and what is from the lab? Actually, this was phrased as “Well, actually you already mentioned this in the previous question, but …”. Having known that this was important to them, I managed to preemptively cover a good part of this question already! This meant that now I could go a little deeper into the details (I had some back-up slides to help illustrate) without losing the audience (with only 6 minutes Q&A, I didn’t want to spend too much time on a single question. Being able to answer concisely is important). They seemed quite impressed and very satisfied.

Q3: What do you expect to be most challenging? This is actually a very standard question that I hadn’t thought of and prepared for. Fortunately, I had been working on my topic for a while already and was able to give a good answer, but you should definitely be ready for this one.

Q4: What are your plans after you PhD? “Do you already have plans to stay with your current lab?”. Reading between the lines… will you stay in Japan? This is a question I was expecting and I was worried might count negatively, since I doubt I will stay in Japan. At the same time, I wanted to be completely honest. I answered that I plan to stay in academia, which means I will have to be flexible and go somewhere that has an interesting project and also funding, so I don’t know what I will do next. I then added (which perhaps wasn’t necessary) that this meant probably I would leave Japan, also because I think it is important to move around in order to continue learning.
Since JSPS is of course interested in fostering the Japanese scientific community, this was a risky statement… but I got the fellowship, so I think they appreciate the honesty and also know that even if you leave the country, it is still likely that you will indirectly help by being a good contact outside of Japan. Regardless, my advice is in any case to be honest and keep your integrity.

Q5: Energy is a big challenge for field robotics, how does your topic relate to this?
This was the last question! A somewhat technical question, but not in the specific details. Also relatively easy to answer, since my research is quite closely related to this problem (or rather, this problem is very related to my research :P).

I have heard from friends that they had more specific technical questions. In general I had the impression that the questions were prepared beforehand: expect that they have read your application well and know exactly what they want to double-check before recommending you. This also means that if you’ve changed your research plan from your original application (indeed, about half a year passed between the paper submission and the interview, so this is possible if not likely), you should be ready to explain what/why you changed the plan.

# JSPS fellowship, paper application

I’m a grad student at this lab in Japan, and after many trials and tribulations I managed to get the JSPS DC 1 scholarship fellowship. This fellowship is rather competitive, and I felt that due to lack of English information as well as many cultural differences, the application is somewhat challenging and confusing for foreigners. This post is meant to give some suggestions and hopefully help other would-be applicants.

## TL;DR

• Read the JSPS ‘About us’ page. This will give you a good idea of what they are looking for.

• Start Early! Making a strong application will take several iterations. The good thing is, after this you should also already have a good road-map for your PhD.

• Get help for Japanese. I wouldn’t write the entire application in Japanese unless you can actually write it yourself. But I do think it helps a lot to have a short mini-summary in Japanese at the beginning of each section.

• Keep it Simple! The panel of reviewers should be from your general field (unless you apply to the interdisciplinary section), but not necessarily from your specialization. Remember, make it simple and clear, but don’t dumb it down!

• Get feedback from different people. This is the best way to find out what parts of your explanations are too field-specific, which parts are unclear and which parts are actually interesting to a wide audience.

• What is your Story? What is the story that links your master’s thesis/project to your PhD research? In Japan it is generally assumed that they are strongly related, so think carefully about it.

• List of Publications is really important. Not so much advice as a general warning.

## The Story

### Some general information

Let’s start with some ‘admin’ stuff. There are 2 tracks, DC1 and DC2, with the same application procedure and deadlines. The difference is that DC2 is for people who already in their first year of PhD studies and therefore only covers 2 years (in Japan the standard PhD is 3 years), whereas you apply for the DC1 before starting. You do have to apply through your university(the university will then submit your application to JSPS), as well as know which lab you will be in and what your project will be: you will need to describe in great detail what your research will be. This description can be in English, however the rest of the application must be done in Japanese. This means it will be very difficult if not practically impossible to apply without actually being in your lab in Japan.
Start Early! The application deadline is usually sometime in April, but you need to pre-apply through your univisty a couple months earlier, and in particular you will need plenty of time to prepare the application. I spent close to 2 months focused mostly on this; I started sneaking in sessions of programming optimization exercises as a way to let off steam, it was that bad!
This fellowship is really good for Japan: not only is the financial support good, you’ll get some funding for your research and I’ve gotten the impression that getting it actually boosts the rep both for the recipient but also for the lab and the school. Presumably because of this, Tohoku U has special sessions where you bring your first drafts to be checked by a “mock” panel of various professors to get feedback and help improve student’s chances. So check if there is similar support at your uni. Note that chances are the announcement of this support will be in Japanese(I wouldn’t have known about this if not for the Japanese PhD in my lab who also applied), and the support itself might well be in Japanese too. Use it anyway. Your reviewers will/should be a mixed panel of people from different fields (should be your general field, but not your specific field), and they will be Japanese. One of the hardest parts for me was to understand and adapt what is actually impressive (and what isn’t) in Japan, and getting real honest feedback is critical and will probably surprise you.

### Research Description

The biggest part of the application will be a description of your research. This is structured as follows:

1. Research Description so far (i.e. master thesis): 1.5 pages
2. Research Topic (i.e. your PhD)
• Background, challenge, approach in relation to part 1.: 0.5 page
• Purpose and Contents of the research: 1 page
• Originality and Impact: 0.5 page
• Time plan: 0.5 page
• Legal issues (if for example you conduct experiments on animals, or take surveys or whatnot…): 0.5 page

It is important at this point to understand that Japanese academic careers work very differently from the west: just as people usually join and be loyal to the same company for life (Shūshin-Koyō or ‘lifetime-employment’), students are generally not encouraged to move between labs but often join a lab very early (here at Tohoku, usually at the end of the bachelor degree) and stay with that lab throughout their undergrad, grad and post-grad studies.
This is so commonplace that it is assumed that you will do your PhD not only in the same lab you did your masters-project on, but on the same project. This is why the description of the research topic (part 2) is in relation to part 1 (the Japanese explanation actually asks how your master project led you to the new topic, etc.). This is generally not the case for foreigners (at least it wasn’t for me), as I finished my studies in Switzerland before coming to Japan. Luckily (and this is actually the reason I decided to come this lab in Japan specifically), my previous project was closely related to my new topic, so I could weave a nice story. In anycase, this is the take-home message: when explaining your master project make sure to highlight and emphasize the parts that help explain what your PhD research will be about. After all it’s 1.5 pages at your disposal to convince the audience about funding your PhD.

A very important point is the language. You are allowed to write this section in English, however I strongly advise you to get someone to help translate at least parts into Japanese. During one of the Tohoku U feedback sessions (see above), I was told “You should use more figures and less text. A reviewer who sees this much English might not bother reading it thoroughly.”. This is a hard truth; you should not assume all your reviewers are completely comfortable with English, and if it is difficult to understand your ideas they will be less convincing. In my case, in addition to ensuring the explanations were simple and well-illustrated (which is a good thing regardless of the language/circumstance), a lab-mate graciously helped me translate the entire “Personal Motivation” section, and for the technical explanations my professor translated a short “mini-abstract” paragraph for each section. I am very convinced this greatly helped my chances.

Finally, when you explain your project you are supposed to emphasize the originality in two ways: both the originality of the research with respect to the field itself, and also which parts are really your own ideas and not the brainchild of your professor/lab. JSPS is actively trying to foster independent thinkers, and as the standard in Japan is for students to continue working on the same project as they transition from undergrad to grad studies, this is something they actively check for and it should be clearly explained.

If you are collaborating with other labs (especially internationally), you probably want to mention that. This is all very much conjecture, but I have the impression that although a large part of Japanese society is still very isolated in their world-view and aren’t very interested in or aware of what happens off the island, I feel that the upper echelons have identified the importance of opening up and made it a priority. This might be a bonus point. It will probably depend a bit on who actually reads it.

### List of Publications

This part was a bit of a shocker for me. In Europe (at least in Switzerland), it’s rather unusual for a master student to have any publications, and quite impressive if you have just one. This is because the education system is completely different, and you aren’t really a “graduate” until you finish your masters (bachelors degree is 3 years, and everybody does a masters).
In Japan (and from what I understand, it is similar in the US) however, the situation is different: it is quite common to enter the industry after a bachelors, which lasts 4 years.
So when I went to the first feedback session, having listed my 1 conference proceeding + 1 conference paper, I was feeling pretty confident and even cocky. Only to have a prof frown at the list and ask “Can’t you fill out all this white-space a little more?”.
You see, what I had not considered is that once a student joins a lab in Japan, he will publish regularly in domestic conferences, which have relaxed or even no review. It’s a different system, with a focus on giving young students practice and experience, rather than original research. Indeed from what I’ve seen, it is about equally unusual for a graduating Japanese student to have a paper at an international conference. This notwithstanding, most Japanese students will have a very impressive-looking list of perhaps a dozen publications and achievements, and this is one of the most important criteria for selection. The good news is that international publications are given more points, however it’s still difficult to make up for the quantity that your competition will have.
So if you have the luxury of spending a semester or two at the lab before starting your PhD, you can make it clear to your prof that you plan on staying there, and participate in some of these domestic conferences. From what I’ve seen, it is common practice to send specific students who will continue as PhDs to more conferences, specifically for this reason.

### Personal Motivation and Achievements

This section is more standard, and separated in 2 parts. The first is a free-form regular motivation letter, and in the second you are encouraged to list actual accomplishments and things you did that lend weight to what you claimed in the first part. As an example, I wrote that I believe active communication (‘science communication’) of research to people from other disciplines as well as the general public is essential. To back it up, I explained I’ve taught robot-programming workshops for elementary-grade children, and also got a poster award at a biology conference (my background is in engineering).
If you can, I advise to get a friend to translate this part into Japanese. Since it doesn’t need to be so technically precise, I think it makes more sense that this part is in Japanese (I didn’t feel comfortable not writing the technical part myself, hence in English).


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# Intuition for the prime number theorem

(surprisingly, it appears that this question has not been asked before)

Let $\pi(n)$ denote the number of primes $\leq n$. The prime number theorem states that

$$\pi(n) \sim \frac{n}{\log n} \ \text{as} \ n \to +\infty$$

After painstakingly reading through Erdos's elementary proof of this theorem, I think I understand the mechanics of it from a formal perspective. However, I still don't seem to understand intuitively why this theorem is true. I would like some intuitive insight as to why this theorem holds.

I understand that for a result as deep as this one, even the intuition is going to contain some nitty-gritty details. It's probably not the sort of thing that you could explain to a child, for example. Nevertheless, I will ask this question regardless. There has to be some convincing argument for this theorem beyond the technical details of the proofs.

• If you move the $n$ to the other side, you get a statement about the density of the primes in the naturals. – Simply Beautiful Art Feb 15 '17 at 2:09
• Fair enough. It's just that this seems to be the more common way of presenting it. – MathematicsStudent1122 Feb 15 '17 at 2:13
• – Winther Feb 15 '17 at 2:33
• Wasn't asked before but will be asked plenty of times after. – Robert Soupe Jul 18 '18 at 3:36

Not a full explanation, but it is too long for a comment.

Consider the Sieve of Eratosthenes.

Start with the first $n$ numbers. Remove (one less than) $\frac 12$ of them as multiples of $2$, Of the remainder, remove $\frac 13$ of them as multiples of $3$. Of the remainder, remove $\frac 15$ as multiples of $5$, etc. You should be left with about $$n\prod_{p\le n, \text{ prime}}1 - \frac 1p$$

values as primes below $n$. Now, when multiplied out

$$\prod_{p\le n, \text{ prime}}1 - \frac 1p = 1 - \sum_{k \in S_n} \frac 1k$$ Where $S_n$ is the set of all square-free integers $> 1$ whose prime factors are $\le n$.

It remains to estimate that $1 - \sum_{k \in S_n} \frac 1k\approx \frac 1{\log n}$.

Edit:

Since it was already chosen as the solution and Winther has kindly provided Merten's 3rd theorem which says just what was needed, I could just let this go. But Merten's theorem strikes me as hardly more intuitively obvious than the prime number theorem itself, so I've been thinking on heuristic concepts to explain it.

Now for $|x| < 1$, $\frac1{1-x} = 1 + x + x^2 + ...$ Therefore $$\frac 1{\prod\limits_{p\le n}1 - \frac 1p} = \prod\limits_{p\le n}\left(1 + \frac 1p + \frac 1{p^2} + ...\right)$$

Multiplying the right side out, we get $\sum_{k \in R_n} \frac 1k$, where $R_n$ is the set of all integers whose prime factors are all $\le n$. Of these, it should be expected (I'm being heuristic here, so I can get away with that weasel-wording) that the sum for $k > n$ will be significantly smaller than for $k \le n$. Thus it seems reasonable that $$\sum_{k \in R_n} \frac 1k \sim \sum_{k \in R_n, k\le n} \frac 1k = \sum_{k=1}^n \frac 1k \sim \log n$$

• :D you capture everything I think of in a nut-shell. +1 – Simply Beautiful Art Feb 15 '17 at 12:01
• By Mertens' 3rd theorem the asymptotic behavior of $\prod_{p\leq n} \left(1 - \frac{1}{p}\right)$ is $\sim \frac{e^{-\gamma}}{\log(n)} \approx \frac{0.56}{\log(n)}$. – Winther Feb 15 '17 at 12:27

The intuition behind the Prime Number Theorem is indeed simple enough that you can explain it to a child.

It’s very intuitive that, other things being equal, as you lower the price of a product, the more of the product will be sold. Because of the phenomenon of discount pricing for wholesale purchases, you can lower your retail price, and still make a profit, by making larger wholesale purchases. The mechanism of discount pricing can be implemented, qualitatively, by means of the (natural) logarithm. That is, the total cost for x items is log(x). (Technically, log(1 + x), but we’re assuming that x is large enough that we can approximate log(1 + x) by log(x).) The unit cost is then log(x)/x. If you grant that, qualitatively, the number of sales is the reciprocal of the unit cost, then you have a real-world analog of, or at least a mnemonic for, the Prime Number Theorem, by taking the number of retail sales (assuming the entire stock is sold) to be pi(x). Then pi(x) times the unit cost is approximately 1. Then pi(x) times log(x)/x is approximately 1. Then pi(x) times log(x) is approximately x. Then x divided by (pi(x) times log(x)) is approximately 1. Then (x divided by log(x)) divided by pi(x) is approximately 1, which is the Prime Number Theorem. To reiterate: the real-world analog / mnemonic is that the number of retail sales is approximately equal to the reciprocal of the unit price, which is something that the first ten-year-old you meet coming down the sidewalk would understand.


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## Approximation argument in geometric flows

I’m studying by myself Mean Curvature Flow and I’m reading the paper “Interior estimates for hypersurfaces moving by mean curvature” by Klaus Ecker and Gerhard Huisken and I’m stuck in the following theorem: Theorem 5.1: M0≡F0(Rn) be a locally Lipschitz continuous entire graph over Rn, then the initial value problem (1) has a smooth solution … Read more

## Double commutant of compact operators

So my question is straightforward. Let $\mathfrak{X}$ be a (complex, if necessary) Banach space and $K\colon\mathfrak{X}\to\mathfrak{X}$ a nonzero compact operator. Denote by $\mathcal{C}(K)$ the commutant of $K$—i.e. the algebra of operators that commute with $K$, and $\mathcal{C}^2(K):=\mathcal{C}(\mathcal{C}(K))$. For every $T\in \mathcal{C}^2(K)$ that is not a multiple of the identity, is it the case that $\mathcal{C}^2(T)$ … Read more

## Weak compactness of order intervals in L1L^1

Let (Ω,μ) be a measure space, say σ-finite for the sake of simplicity, and let L1:=L1(Ω,μ) denote the real-valued L1-space over (Ω,μ). For all f,h∈L1 we call the set [f,h]:={g∈L1:f≤g≤h} the order interval between f and h. Order intervals have the following property: Proposition 1. Every order interval in L1 is weakly compact. However, the … Read more

## Can the degree of kk-nilpotence of a simple simply connected compact Lie group be in (0,1)(0,1)?

Let G be a simple (i.e. every proper normal subgroup is discrete) simply connected compact Lie group. Define the degree of k-nilpotence of G to be the Haar measure of the set {(x1,…,xk+1):[x1,…,xk+1]=1}. ([x,y]=x−1y−1xy and [x1,…,xk+1]:=[[x1,…,xk],xk+1].) The following question is raised in Martino, Tointon, Valiunas, and Ventura – Probabilistic nilpotence in infinite groups: Does a … Read more

## When is a Nemytskii map between Sobolev spaces compact?

Let f:R→R be a smooth function with bounded derivative. Define the Nemytskii map F:H1(Ω)→H1(Ω) by F(u)(x):=f(u(x)). Here Ω is a bounded smooth domain. There exists work where we can deduce continuity and differentiability of F under some assumptions on f. What conditions on f will ensure that F satisfies the following compactness criterion: if un⇀u … Read more

## How to describe the compact real forms of the exceptional Lie groups as matrix groups?

I know that $G_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe in his answer to the post A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$ a nice description of the complex $E_6$ … Read more

## Pullback of Morse form satisfies Palais Smale

Let $(\alpha,g)$ be a Morse-Smale pair on a closed smooth manifold $M$, i.e. $\alpha$ is a Morse form and $g$ a Riemannian metric on $M$ such that stable and unstable manifolds of the gradient vector field $X$ intersect transversally. Let $\pi \colon \tilde{M} \to M$ be the associated cover to $\ker [\alpha]$. Denote by \$\tilde{f} … Read more

## Is this property of continuous maps equivalent to some more familiar condition?

Let f:X→Y be a continuous map. Suppose that, for each collection of open sets {Vi}i∈I⊂X, ⋃U⊂Y open, f−1(U)⊂⋃i∈IViU=⋃i∈I⋃U⊂Y open, f−1(U)⊂ViU. I wonder whether this property is equivalent to being proper for locally compact hausdorff spaces. Below I have a proof of one direction. Write f∗(V)=⋃f−1(U)⊂VU. (1) says that f∗ preserves joins, so that it has a right adjoint f!. … Read more

## Reeb stability counterexample: foliation in Sn−2×S1×S1S^{n-2}\times S^1\times S^1 with non-diffeomorphic leaves

Reeb’s global stability theorem requires the foliation to be of codimension 1. As a counterexample, in “Geometric theory of foliations”, Camacho and Lins Neto present the following. Consider the manifold Sn−2×S1×S1 with coordinates (x1,…,xn−1,φ,θ) such that ∑n−1i=1x2i=1 , and two differential one-forms η1=dθ and η2=((1−sin(θ))2+x21)dφ+sin(θ)dx1. Frobenius’ integrability condition is satisfied as can be easily checked, so there’s … Read more

## Compactness of symmetric power of a compact space

Suppose I have a compact metric space (X,d) and let X=XK be the product space. Consider the equivalence relation ∼ on X given as: for α,β∈X, α∼β iff there exists a permutation τ of {1,…,K} such that αi=βτ(i) for all i. Consider the quotient space X/∼ and define a metric ρ on this quotient space … Read more


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# 26.5 Telescopes  (Page 2/4)

 Page 2 / 4

The image in most telescopes is inverted, which is unimportant for observing the stars but a real problem for other applications, such as telescopes on ships or telescopic gun sights. If an upright image is needed, Galileo’s arrangement in [link] (a) can be used. But a more common arrangement is to use a third convex lens as an eyepiece, increasing the distance between the first two and inverting the image once again as seen in [link] .

A telescope can also be made with a concave mirror as its first element or objective, since a concave mirror acts like a convex lens as seen in [link] . Flat mirrors are often employed in optical instruments to make them more compact or to send light to cameras and other sensing devices. There are many advantages to using mirrors rather than lenses for telescope objectives. Mirrors can be constructed much larger than lenses and can, thus, gather large amounts of light, as needed to view distant galaxies, for example. Large and relatively flat mirrors have very long focal lengths, so that great angular magnification is possible.

Telescopes, like microscopes, can utilize a range of frequencies from the electromagnetic spectrum. [link] (a) shows the Australia Telescope Compact Array, which uses six 22-m antennas for mapping the southern skies using radio waves. [link] (b) shows the focusing of x rays on the Chandra X-ray Observatory—a satellite orbiting earth since 1999 and looking at high temperature events as exploding stars, quasars, and black holes. X rays, with much more energy and shorter wavelengths than RF and light, are mainly absorbed and not reflected when incident perpendicular to the medium. But they can be reflected when incident at small glancing angles, much like a rock will skip on a lake if thrown at a small angle. The mirrors for the Chandra consist of a long barrelled pathway and 4 pairs of mirrors to focus the rays at a point 10 meters away from the entrance. The mirrors are extremely smooth and consist of a glass ceramic base with a thin coating of metal (iridium). Four pairs of precision manufactured mirrors are exquisitely shaped and aligned so that x rays ricochet off the mirrors like bullets off a wall, focusing on a spot.

A current exciting development is a collaborative effort involving 17 countries to construct a Square Kilometre Array (SKA) of telescopes capable of covering from 80 MHz to 2 GHz. The initial stage of the project is the construction of the Australian Square Kilometre Array Pathfinder in Western Australia (see [link] ). The project will use cutting-edge technologies such as adaptive optics    in which the lens or mirror is constructed from lots of carefully aligned tiny lenses and mirrors that can be manipulated using computers. A range of rapidly changing distortions can be minimized by deforming or tilting the tiny lenses and mirrors. The use of adaptive optics in vision correction is a current area of research.

## Section summary

• Simple telescopes can be made with two lenses. They are used for viewing objects at large distances and utilize the entire range of the electromagnetic spectrum.
• The angular magnification M for a telescope is given by
$M=\frac{\theta \prime }{\theta }=-\frac{{f}_{\text{o}}}{{f}_{\text{e}}}\text{,}$
where $\theta$ is the angle subtended by an object viewed by the unaided eye, $\theta \prime$ is the angle subtended by a magnified image, and ${f}_{\text{o}}$ and ${f}_{\text{e}}$ are the focal lengths of the objective and the eyepiece.

## Conceptual questions

If you want your microscope or telescope to project a real image onto a screen, how would you change the placement of the eyepiece relative to the objective?

## Problem exercises

Unless otherwise stated, the lens-to-retina distance is 2.00 cm.

What is the angular magnification of a telescope that has a 100 cm focal length objective and a 2.50 cm focal length eyepiece?

$-\text{40}\text{.}0$

Find the distance between the objective and eyepiece lenses in the telescope in the above problem needed to produce a final image very far from the observer, where vision is most relaxed. Note that a telescope is normally used to view very distant objects.

A large reflecting telescope has an objective mirror with a $\text{10}\text{.}0 m$ radius of curvature. What angular magnification does it produce when a $3\text{.}\text{00 m}$ focal length eyepiece is used?

$-1\text{.}\text{67}$

A small telescope has a concave mirror with a 2.00 m radius of curvature for its objective. Its eyepiece is a 4.00 cm focal length lens. (a) What is the telescope’s angular magnification? (b) What angle is subtended by a 25,000 km diameter sunspot? (c) What is the angle of its telescopic image?

A $7.5×$ binocular produces an angular magnification of $-7\text{.}\text{50}$ , acting like a telescope. (Mirrors are used to make the image upright.) If the binoculars have objective lenses with a 75.0 cm focal length, what is the focal length of the eyepiece lenses?

$+\text{10.0 cm}$

Consider a telescope of the type used by Galileo, having a convex objective and a concave eyepiece as illustrated in [link] (a). Construct a problem in which you calculate the location and size of the image produced. Among the things to be considered are the focal lengths of the lenses and their relative placements as well as the size and location of the object. Verify that the angular magnification is greater than one. That is, the angle subtended at the eye by the image is greater than the angle subtended by the object.

What is the difference between a principle and a law
the law is universally proved. The principal depends on certain conditions.
Dr
what does the speedometer of a car measure ?
Car speedometer measures the rate of change of distance per unit time.
Moses
describe how a Michelson interferometer can be used to measure the index of refraction of a gas (including air)
using the law of reflection explain how powder takes the shine off a person's nose. what is the name of the optical effect?
WILLIAM
is higher resolution of microscope using red or blue light?.explain
WILLIAM
can sound wave in air be polarized?
Unlike transverse waves such as electromagnetic waves, longitudinal waves such as sound waves cannot be polarized. ... Since sound waves vibrate along their direction of propagation, they cannot be polarized
Astronomy
A proton moves at 7.50×107m/s perpendicular to a magnetic field. The field causes the proton to travel in a circular path of radius 0.800 m. What is the field strength?
derived dimenionsal formula
what is the difference between mass and weight
assume that a boy was born when his father was eighteen years.if the boy is thirteen years old now, how is his father in
Isru
what is airflow
derivative of first differential equation
why static friction is greater than Kinetic friction
draw magnetic field pattern for two wire carrying current in the same direction
An American traveler in New Zealand carries a transformer to convert New Zealand’s standard 240 V to 120 V so that she can use some small appliances on her trip.
What is the ratio of turns in the primary and secondary coils of her transformer?
nkombo
what is energy
Yusuf
How electric lines and equipotential surface are mutually perpendicular?
The potential difference between any two points on the surface is zero that implies È.Ŕ=0, Where R is the distance between two different points &E= Electric field intensity. From which we have cos þ =0, where þ is the angle between the directions of field and distance line, as E andR are zero. Thus
sorry..E and R are non zero...


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# Tag Info

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The short answer: the really minimum knowledge of math to understand the first half of the plan of GCT, once you've seen a little of groups, rings, and fields, is basically laid out in Chapter 3 of my thesis (shameless self plug). That chapter is, however, incomplete, in that I don't get to the representation theory part of things. The representation theory ...

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The usual answer to "what could cause you to divide by a log?" is a combination of two things: a model of computation in which constant time arithmetic operations on word-sized integers are allowed, but in which you want to be conservative about how long the words are, so you assume $O(\log n)$ bits per word (because any fewer than that and you couldn't ...

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I did a survey on Twitter about this a while back, results here. A few of my favorites: Parametric Polymorphism through Runtime Sealing, or, Theorems for Low, Low Prices! by Jacob Matthews and Amal Ahmed, ESOP 2008 DOI:10.1007/978-3-540-78739-6_2 F-ing Modules by Andreas Rossberg, Claudio Russo, and Derek Dreyer, TLDI 2010 Cons should not cons its arguments ...

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The Rubik's Cube is a very natural (and to me, unexpected) example. An $n\times n\times n$ cube requires $\Theta(n^2/\log n)$ steps to solve. (Note that this is theta notation, so that's a tight upper and lower bound). This is shown in this paper [1]. It may be worth mentioning that the complexity of solving specific instances of the Rubik's cube is open,...

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The Planar Separator Theorem states that in any planar $n$-vertex graph $G$ there exists a set of $O(\sqrt{n})$ vertices whose removal leaves the graph disconnected into at least two roughly balanced components. Moreover, such a set can be found in linear time. This (tight) result, proved by Lipton and Tarjan (improving on a previous result by Ungar) is a ...

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I used to like quirky titles when I started out in computer science but got bored eventually. Some authors manage to write titles that are clever, memorable and relevant but most attempts at funny titles results in unnecessarily long, uninformative and kludgy phrases that I find difficult to remember and look up. There are papers like Pnueli's The Temporal ...

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The Knot Equivalence Problem. Given two knots drawn in the plane, are they topologically the same? This problem is known to be decidable, and there do not seem to be any computational complexity obstructions to its being in P. The best upper bound currently known on its time complexity seems to be a tower of $2$s of height $c^n$, where $c = 10^{10^{6}}$, ...

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The recent paper of László Babai showing that Graph Isomorphism is in Quasi-P is already a classic. Here is a more accessible exposition of the result published in the ICM 2018 proceedings.

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This is really a false belief in math, but comes up often in TCS contexts: If random variables $X$ and $Y$ are independent, then conditioned on $Z$ they remain independent. (false even if $Z$ is independent of both $X$ and $Y$.)

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Karp-Lipton Theorem states that $\mathsf{NP}$ cannot have polynomial-size boolean circuits unless the Polynomial hierarchy collapses to its second level. Two implications of this theorem for complexity theory: $\mathsf{NP}$ probably has no polynomial-size boolean circuits; proving lower bounds on circuit sizes is therefore a possible approach for ...

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In a recent preprint, Harvey and Van Der Hoeven show how to compute Integer multiplication in time $O(n \log n)$ on a multi-tape Turing machine, culminating some 60 years of research (Karatsuba, Toom–Cook, Schönhage–Strassen, Fürer, Harvey–Van Der Hoeven–Lecerf). The paper has not yet been peer-reviewed, but prior work of the authors on this problem makes it ...

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Random Self-Reducibility of the Permanent. Lipton showed that if there exist an algorithm that correctly computes the permanent of $1-1/(3n)$ fraction of all $\mathbb{F}^{n\times n}$, where $\mathbb{F}$ is a finite field of size at least $3n$, then this algorithm can be used as a black box to compute the permanent of any matrix with high probability. The ...

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For a general audience you have to stick to things that they can see. As soon as you start theorizing they'll start up their mobile phones. Here are some ideas which could be worked out to complete examples: There is a surface which has only one side. A curve may fill an entire square. There are constant width curves other than a circle. It is possible to ...

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Extending the example pointed out by Emil Jerabek in the comments, $\mathsf{EXPSPACE}$-complete problems arise naturally all over algebraic geometry. This started (I think) with the Ideal Membership Problem (Mayr–Meyer and Mayr) and hence the computation of Gröbner bases. This was then extended to the computation of syzygies (Bayer and Stillman). ...

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Here's a version of the minimum circuit size problem (MCSP): given the $2^n$ bit truth table of a Boolean function, does it have a circuit of size at most $2^{n/2}$? Known to be not in $AC0$. Contained in $NP$. Generally believed to be $NP$-hard, but this is open. I believe it's not even known to be $AC0[2]$-hard. Indeed, recent work with Cody Murray (to ...

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The complexity of computing a bit (specified in binary) of an irrational algebraic number (such as $\sqrt{2}$) has the best known upper bound of $\mathsf{P^{{{PP}^{PP}}^{PP}}}$ via a reduction to the problem $\mathsf{BitSLP}$ which known to have this upper bound [ABD14]. On the other hand we do not even know if this problem is harder than computing the ...

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There is no Zoo-style reference yet, but a recent automata-theoretic survey of Giovanni Pighizzini has been useful to me, especially the slides from his talk. Giovanni Pighizzini, Investigations on Automata and Languages over a Unary Alphabet, CIAA 2014. doi:10.1007/978-3-319-08846-4_3

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It's the base rather than the exponent, but there's an $O(\varphi^k n^2)$ FPT time bound in "An Efficient Fixed Parameter Tractable Algorithm for 1-Sided Crossing Minimization", Vida Dujmovic, Sue Whitesides, Algorithmica 40:15–31, 2004. Also, it's a lower bound rather than an upper bound, but: "An $n^{1.618}$ lower bound on the time to simulate one queue ...

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I know of many theoretical computer scientists whose first inspiration came from reading Godel, Escher, Bach Its becoming a bit dated at this point, but is still an excellent read.

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Another natural topological problem, similar in spirit to Peter Shor's answer, is embeddability of 2-dimensional abstract simplicial complexes in $\mathbb{R}^3$. In general it's natural to ask when can we effectively/efficiently decide that an abstract $k$-dimensional simplicial complex can be embedded in $\mathbb{R}^d$. For $k=1$ and $d=2$ this is the graph ...

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Importance is in the eyes of the beholder. However, I would say that the Feder–Vardi CSP dichotomy conjecture, proved independently by A. Bulatov and D. Zhuk, is a seminal result.

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Are NP-completeness in the sense of Cook and NP-completeness in the sense of Karp different concepts, assuming P $\neq$ NP?

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For a given graph $G$ and an integer $k\ge 1$, the $k$-th power of $G$, denoted by $G^k$, has the same vertex set such that two distinct vertices are adjacent in $G^k$ if their distance in $G$ is at most $k$. The $k$-th power of split graph problem asks if a given graph is the $k$-th power of a split graph. For $k=1$, the problem solvable in linear time....

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Non-Uniform ACC Circuit Lower Bounds by Ryan Williams: https://people.csail.mit.edu/rrw/acc-lbs.pdf and Classical Verification of Quantum Computations by Urmila Mahadev: http://ieee-focs.org/FOCS-2018-Papers/pdfs/59f259.pdf seem like good candidates

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Some graph classes allow polynomial-time algorithms for problems that are NP-hard for the class of all graphs. For instance, for perfect graphs, one can find a largest independent set in polynomial time (thanks to vzn in a comment for jogging my memory). Via a product construction, this also allows a unified explanation for several apparently quite ...

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Lattice-basis reduction (LLL algorithm). This the basis for efficient integer polynomial factorization and some efficient cryptanalytic algorithms like breaking of linear-congruential generators and low-degree RSA. In some sense you can view the Euclidean algorithm as a special case.

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Many problems that are PSPACE-complete become EXPSPACE-complete when the input is given "succinctly", i.e., via some encoding that lets you describe inputs that would normally be of exponential size. Here is an example on finite automata (equivalently, on directed graphs with labeled edges): deciding whether two automata accept the same language (have the ...

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$SL = L$. $RL$ stands for randomized logspace and $RL=L$ is a smaller version of the problem $RP=P$. A major stepping stone was the proof of Reingold in '04 ("Undirected S-T Connectivity in Logspace") that $SL = L$, where $S$ stands for "symmetric" and $SL$ is an intermediate class between $RL$ and $L$. The idea is that you can think of a randomized ...

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(from my comment above) The Fortnow and Melkebeek time/space lower-bound for SAT solvability ($n^{\phi - \epsilon}$ time and $n^{o(1)}$ space) contained the golden ratio in the exponent; but it has been improved later by Ryan Williams.

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Algebraic geometry Noether's Normalization Lemma (NNL) for explicit varieties is currently only known to be in $\mathsf{EXPSPACE}$ (like general NNL), but is conjectured to be in $\mathsf{P}$ (and is in $\mathsf{P}$ assuming that PIT can be black-box derandomized). Update 4/18/18: It was recently shown that for the variety $\overline{\mathsf{VP}}$ it is in \$...

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Category Archives: Teaching

Teaching

Mentoring…

Ever since I graduated I’ve always wanted to learn new things and for that reason I’ve pretty much always had at least three different types books on the go at any given time.  Typically I have a fiction book to read when my brain is really tired, a technical book linked to engineering, maths, or programming, and some sort of self-improvement book.   I want to be the best I can be, to reach my full potential and not be limited by my lack of understanding.

Over the past few years I’ve been spending quite a bit of time thinking about how mentoring (and what I believe to be the lack of) is affecting the environment for modern graduates in industry.  As margins become tighter and tighter in industry, the experienced engineers have less and less wriggle room to spend mentoring graduates.  Instead the experienced engineers merely push the less-experienced engineers to a quick solution so that deadlines are met and fee structures are not blown – resulting in a shallow learning experience.

I’ve always tried to make time to help less experienced engineers understand new concepts, even if I think they should know them already from University.  This has meant frequently giving up my dinner hour and time after work to help someone understand a new concept.  I’ve not always been successful and that’s a limitation of my skill and something that I’m getting better at with time as I improve as a mentor, but as time goes by I wonder if my approach is mentoring or coaching?

Most of the time I spend on a 1 to 1 basis with students takes the form of conversations and specific questioning to change the way they think about things.  To change or reinforce their perceptions about how something can be improved in the future.  This is the commonest form of mentoring that I’ve taken over the years, but through what I’ve read of Starr (2010) this approach isn’t mentoring at all, it’s more like coaching.  It’s essentially a series of conversations to help change a future outcome.  One of the favourite activities I teach is the supervision of dissertations, this is the module I can see the biggest leap in a student’s abilities, particularly with regards critical thinking and it presents the biggest opportunity that I get to mentor and build a connection with the students.  It’s the one area that probably sucks a disproportionate amount of time out of my week but it’s definitely where my mojo (Goldsmith, 2010, p17) lies when I’m teaching.

It’s this connection with the students and their topics I think that I enjoy the most, especially as I watch them grow and I still get emails and calls from some of the students that I supervised a few years ago simply so they can let me know what they’re up to.  Of course quite a few students only get in touch when they’re chasing a reference or contacting me because they want something from me but when someone drops you a note to say hi or let you know what they’re up to now without wanting anything from you it’s personally very rewarding and it’s this type of contact that I treasure as it’s then that you know you’ve made a real impact on someone’s perspective on life.

Dissertations are also one of the key activities that is keeping me in academia, as it’s giving me an opportunity to continually learn and grow on a personal level although the longer I stay in academia the lower my career progression opportunities and the lower my earning potential become…. Perhaps Robert Greene (2012) is right, when looking back on my life nobody will remember the wonderful report I sacrificed weekends and evenings to complete, but perhaps they’ll remember the time I gave them to help improve their critical thinking skills and understanding of structural behaviour and I know my kids will hopefully appreciate the time we have together now I’m no longer being continually shipped out all over the place to design buildings.

After all I’ve changed quite a few skylines around the world when I had a proper job, maybe there’s more satisfaction to be had creating brilliant engineers to create even more radical designs.  Or perhaps there’s a compromise to be had by spending some of my time teaching and another chunk of my time working with brilliant engineers and architects to design really radical and life-changing designs… I think this could be where my future lies in all honesty, it’s about time I seized the steering wheel again as these things don’t happen by themselves.

References.

Goldsmith, M. (2010). Mojo. London: Profile Books.

Greene, R. (2012). Mastery. London: Profile Books.

Starr, J. (2010). The Coaching Manual (Third ed.). London: Pearson Business.

Inspiration…

Whilst I’ve been in academia for about 2 and a half years now, I’ve only just seen off my second batch of first years and attended my first graduation (after much cajoling by a few final years).  Having a significant birthday approaching I’ve been thinking a great deal about what am I going to do for the next few years until retirement, I enjoy the teaching a lot and working on dissertations gives me tremendous pleasure, but I miss designing buildings is the dilemma that I’ve been having.  But there’s one aspect I’d not really considered when I moved to academia, and that’s that I genuinely love learning and essentially I’m getting paid to learn… not as well paid as I would be as a consultant, but I get a lot of time to be very self-indulgent and learn new skills.

I try hard to not be the kind of lecturer that I had to endure when I was doing my degree and typically put anywhere between 5 and 12 hours of preparation behind every hour of lecture that I deliver.  I’m not saying that my lecturers weren’t prepared, they certainly knew their stuff, but they were really dry and tedious in their delivery and at times it felt a little like they were padding.  I had hoped to at least inspire a few of the engineers of the future and hopefully I’ve done that by combining my industry background with my enthusiasm, but what I hadn’t counted on was that my students would inspire me and teach me all sorts of things about the world and myself.

Having been fortunate to work with engineers, architects, and clients who have really been brave and pushed the boundaries of my abilities with the result that we’ve created some truly fantastic buildings together, I foolishly thought that all of this experience meant that as the lecturer the ability to inspire was my sole domain, but I’m frequently humbled and inspired by my students.  I’ll comfortably pull 17 hour days when I’ve a bee in my bonnet and can sustain this pace for several months at a go, but I’ve students who put my work ethic and self-discipline to shame.

The beauty of teaching classes far in excess of a hundred students (many of whom are international) is that they come with a multitude and wealth of experience and perspectives.  Each of them are complex individuals with differing opinions and backgrounds and in all honesty I’d never really encountered this level of diversity in industry.  Each week I’m humbled and gain new perspectives from the students, some of whom have faced more adversity than I ever could have imagined and yet still manage to achieve impressive results and retain their ability to be great humans.  Something that perhaps I’ve lost track of at times during my career, but that hopefully I’m regaining as the days tick by.

When I joined academia, I had a clear exit strategy… 3 years and then exit back to industry.  It’s not that clear cut any longer though, I was prepared to be the lecturer… but I don’t think I was prepared to become the student and I’m discovering that I’ve still so much to learn that I may just have to hang around a little while longer than I had planned.

Critical thinking…

One of the biggest challenges that I face as a lecturer is helping the students to develop their critical thinking skills, especially with regards problem solving as this is one of the key skills an engineer should possess.

It’s a fundamental skill for a practicing engineer to not only problem solve, but to critically review the problem and reconstruct it so that only the important parts are focused upon.  Essentially honing in on the nub of the problem.

I’m a fan of using stories in lectures and I used to work with an Italian engineer who unfortunately died last year and he had a plethora of stories and anecdotes that he paraded out whenever the occasion arose.  One of his numerous stories that sticks in my mind and that I think I’ll start using in my lectures about critical thinking involves three construction professionals on a train ride to Scotland for the first time.

There is an Architect, a Project Manager, and an Engineer on a train together and as they cross the border into Scotland for the first time on a train racing through the countryside they spot a brown cow in a field.

The Architect, being the first to spot the cow, declares that clearly all Scottish cows are brown, the Project Manager who feels that he clearly has the organisational skills to rationalise this observation and thinks that the architect hasn’t been critical enough, decides to correct the Architect and notes that in reality what the Architect has actually observed is that in Scotland there is a field which has brown cows.

The Engineer meanwhile has been listening intently to this discussion and decides to correct them both by sharing his observation, that in fact there is in Scotland, a field, that contains a cow, one side of which is brown.

Teaching

Golden ratio…

A quick link through to my first ever screen case written for a Maths MOOC that’s hopefully going live soon… this was much more hard work than I gave it credit for when I agreed to write it.

The slides are available from below as a slideshare link.

Fibonacci…

I was recently approached to see if I could tinker about and create a short video associated with the Fibonacci series and more specifically the golden ratio for a maths MOOC that’s about to launch.  As per usual my inability to say no came into play and then I suddenly realised that I’ve never created a teaching video before and really what do I know about the Fibonacci series apart from fluffing about with a few Mathematica notebooks, and so blind panic set in… and with blind panic comes a quick trip to XKCD to check out the latest strips for a bit of inspiration…

Now the golden ratio (φ) doesn’t sound like the sort of everyday mathematics that you would use will pushing a trolley around TESCO’s but the key thing you have to appreciate is that it’s a very sneaky number.  Things are that are proportioned using the golden ratio are naturally and inexplicably appealing to the eye, we find the ratio a thing of beauty even though most people don’t even know what it is.  It turns out that it’s used in everything from composing pleasing photographs (using the rule of thirds as an approximation) through to designing the Parthenon in Greece which is proportioned using this ratio.  It’s cool stuff, and to be frank if Mother Nature sees fit to use the principle to establish geometries in plants and other animals… who am I to argue?!

One of the things I like about XKCD are the graphs that they create, they have a certain charm about them and I love the style… and as sketchy as it looks, I’ve always struggled to replicate these diagrams using Visio or Omnigraffle.

One of my other digital haunts is the Stack Exchange forum, which as a (very) amateur coder I constantly find myself sitting reading through in a sense of amazement at some of the nifty tricks that people pull to make some quite elegant and funky code.  One of the sub-forums I like there is the Mathematica one, which recently asked the question if it was possible to create XKCD graphs using Mathematica, a copy of the thread is here.  I’ve replicated the code below, as much to save it for myself should the thread ever get closed.

The author of this code has created some sample images using this code and I think they’re brilliant.

But coming back to the reason I needed to create some diagrams, the fibonacci sequence and the golden ratio… starting with the golden ratio this is defined by the equation below.

$\varphi = \frac{{1 + \sqrt 5 }}{2}$

$\varphi = 1.61803…$

The reason that this number is expressed as a fraction is that it is an irrational number, which means it’s similar to π and the decimal places keep on repeating forever with no repetition and so it can’t accurately be approximated a decimal.  So how would you approximate this number? Well to start with you need to understand the Fibonacci sequence, which sounds quite grand and complex, but is simple once you understand the trick to determine the sequence… essentially you add together the previous two numbers in the sequence and this gives you the next number.  So to start with 1+0=1;  1+1=2; 2+1=3; 3+2=5; etc etc.

${\text{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…}}$

But how does this get you to the golden ratio?  Well, if you divide one number, by the number before it you get a series of numbers that after a few iterations start to get close to the golden ratio (φ).

$1,2,\frac{3}{2},\frac{5}{3},\frac{8}{5},\frac{{13}}{8},\frac{{21}}{{13}},\frac{{34}}{{21}},\frac{{55}}{{34}},\frac{{89}}{{55}},\frac{{144}}{{89}}…$

Which shows that the number starts to approximate towards the golden ratio with a reasonable level of accuracy after about the fifth iteration or so (well reasonable to a simple engineering brain anyway).. but let’s not be blinded by numbers, let’s draw an XKCD style graph using our new mathematica code…

The last ImageSize term is important to get a reasonable resolution on the image when you export it from Mathematica for embedding into other documents, but the code above gives the graph below… which I think is quite a good attempt at a XKCD style diagram.

Now that we can determine the Fibonacci sequence and the golden mean easily in Mathematica, we could start to see how these relate to the large sunflower seed heads that mother nature creates time after time…

Just a single line of code will help us generate some nice pretty Golden Ratio inspired spirals in Mathematica which look strangely familiar.

So it turns out that I did know enough about the Fibonacci sequence and the Golden Ratio to be almost dangerous… the trick’s going to be can I record a short YouTube video for a maths MOOC as I promised to a half decent standard?  We’ll have to wait and see… but now I know I can muck about with formatting diagrams and the code that goes along with the Golden Ratio in Mathematica I can start to mess around with tessalating spheres to create geodesic domes…

And here’s a Mathematica graphic after following the procedure set out in this blog post.

But I don’t want to go off another tangent just yet with Bucky Fuller… but he was inspired by nature and the cleanness of the geometry that the golden ratio presented.

Teaching

Code…

I’m spending a lot of my time tinkering in Mathematica lately, with two objectives really.

1.) I want to write some code to calculate the tedious geometrical aspects for me for my research.

2.) I think that Mathematica has some real potential for students who are trying to learn structures, particularly through the CDF files.

I’ve learned various programming languages over the years most of which I’ve taught myself, admittedly not to the level that perhaps a computer programmer would, but I’ve armed myself with just enough knowledge to be dangerous and get the task done that I want to achieve.  I’ve created countless spreadsheets that can do all sorts of elaborate calculations and also to use as validation calculations for more complex analysis models that I’ve created.

This is a skill that I take for granted for engineers, particularly of my age, and in most engineers that are a couple of years older or so I tend to find that they’ve spent some time abusing Fortran code in some fashion.  Fortran isn’t a language that I’ve ever learned, but it’s probably one of those languages that a lot of engineers have dabbled with at some point.

One trend I’ve noticed in a lot of younger engineers is a reluctance to create computer code and indeed even in creating what I would consider simple spreadsheets to make a calculation tool that can run several scenarios for them and validate an approach to determine the boundaries of their calculations.  Increasingly there seems to be a preponderancy for young engineers to select their FEM weapon of choice and throw triangles at it until the model begs for mercy, or to download an app that some other bright spark has written to do at least part of the process of what they want to happen.  None of this promotes a deeper understanding of how the process works though unfortunately.

Perhaps the days of writing your own code are long gone for young engineers? But part of me thinks that they are missing out on a real opportunity to structure their understanding of the steps needed to do certain tasks as they assemble their code.  Writing code to perform a task, whether on a spreadsheet or in a more formal programming language, is a great way to develop understanding and to explore the intricacies of design codes and I feel that younger engineers are missing out on this experience by only using off the shelf packages.

One encouraging aspect is that with the launch of the Raspberry Pi I can see that if this continues to gather momentum that younger generations will again an ability to tinker and break programming code and gain an appreciation of the advantages that breaking down engineering problems into a series of procedural steps can bring.

At least for the next year or so I’m still responsible for sculpting young engineers minds and I’m determined to develop a fun way of getting them to extend their understanding of how to write code or spreadsheets, I just need to work out a way of doing this within the confines of the resources I do or don’t have available.

What do you think? Should engineers know how to programme or at least be able to create simple spreadsheets?

Kriss Kross…

I’ve been really busy with various things associated with my new house lately and not really had much time to tinker and meddle with Mathematica or my PhD.  One of the key elements that I need to investigate is to calculate the internal area of a polygon where cables cross…  This will vary, depending on how many internal cables that I have and so I need a quick way of finding out the internal area for lots of different permutations.

This is a simple polygon problem which can be easily calculated once you know the cartesian co-ordinates for each of the key points using the following equation which is presented for the general condition.  It’s not a hard calculation, as really it’s just triangles, but it is a tedious calculation, which means it’s perfect for Mathematica or Excel.

$Area = \left| {\frac{{\left( {{x_1}{y_2} – {y_1}{x_2}} \right) + \left( {{x_2}{y_3} – {y_2}{x_3}} \right) \ldots + \left( {{x_n}{y_1} – {y_n}{x_1}} \right)}}{2}} \right|$

However, the tricky part comes from determining where the cables actually cross over as that is where the cartesian co-ordinates will come from that are needed to determine the areas.  The calculation of the co-ordinates can be done a few different ways, but I’ve decided to solve the problem through using some vector based geometry, consider the simple condition below which is just for two lines that cross, how do we find the point where the lines intersect?

You could describe the two lines in the general term of y=mx+c, but this isn’t what I want to do within Mathematica as I want to automate this for a wide range of values and lines and so iterative geometry using vectors is likely to be far more beneficial.  Representing these two lines in vector format using cartesian co-ordinates gives:

$\begin{array}{l} Lin{e_1} = \left( {1 – \lambda } \right){P_1} + \lambda {P_2}\\ Lin{e_2} = \left( {1 – \mu } \right){P_3} + \mu {P_4} \end{array}$

Now we know the equations in vector format it becomes possible to solve both of these equations to determine λ and μ which when back substituted into either equation will give us the co-ordinate for the intersection point p5 in (x,y) format.

This isn’t that difficult to do by hand, but requires a bit of vector manipulation to expand the brackets and given my low attention span I would inevitably get it wrong after a few run throughs, but in Mathematica the problem gets much more simple, so for the example below we’ll assume that the four co-ordinates are at 4m away from the origin (0,0) that is point p1.

{p1, p2, p3, p4} := {{0, 0}, {4, 4}, {0, 4}, {4, 0}};

Solve[{(1 – λ) p1 + λ*p2 == (1 – μ) p3 + μ*p4}, {λ, μ}, Reals];

p5 = Flatten[(1 – λ) p1 + λ*p2 /. %]

Executing this codes gives the intersection as being at point (2,2) in a fraction of a second, which is where you would expect it to be.  Now to explain the three lines of code and what they do, I’m sure a more experienced user could make this code much neater but doing it this way makes sense to my tiny brain.

The first line simply sets the variables p1-p4 with their corresponding cartesian co-ordinate position, giving the point two in the top right corner 4m up and 4m across, p3 is along the bottom 4m away, p4 is straight up in the top left corner 4m above point p1, as shown in the figure below.

The second line then solves the two vector equations of the lines for λ and μ and makes sure that they are real numbers, rather than imaginary.  The real part probably isn’t necessary, but it’s just a force of habit that I put this in when using the solver as a just incase.

The final line substitutes the λ value back into the equation and then stores this as variable p5 so that I can use this in up and coming calculations, this was the fiddly part for me and makes use of the strange /. % input that allows the equation to effectively go back and substitute the solutions from the Solve command into the equation specified and recalculate and because my equation has the λ within it Mathematica knows this is what I want substituting in even without me being explicit.   I spent ages trying to work out this final stage, but now I’ve cracked it as a principle I can start to roll it out on the other sheets that I’ve been working on and this was the reason for writing this post incase it too helps someone else.  Now I can directly write the solution outputs from Solve, FindRoot, and NSolve and then carry on using these within my calculations rather than having to manually type them in.

This process lets me know that the intersection point is at (2,2) and now I know that the process works I can start to extrapolate this algebraically for the general conditions and start to calculate the areas of my polygons.

All of the figures in this post have been written using OmniGraffle, which I’m slowly getting to grips with, I can’t make it dance as well as I can Visio, but I think it will help create some nice looking diagrams for my thesis once I get more proficient with it.

Mathematica…

One of the reasons that I started this blog was so that I could mess about with embedding some Mathematica files to help with testing out some ideas.  For this to make sense it’s easiest if I embed a few simple examples in this blog post.  Now if you want to interact with these examples, I’m afraid you’re going to have to download the Wolfram CDF player, which is completely free and works on PC’s and Mac’s alike.  Imagine it as a sort of PDF viewer but it lets you interact with the files as opposed to a PDF which is typically just a static and lifeless document.

Consider the following equation:

$Sin\left( {2x} \right)$

Most text books would draw the graph for this over whichever range they deemed to be suitable and then students would try and learn from these dull and boring diagrams.

[WolframCDF source=”https://dl.dropboxusercontent.com/u/22612196/Wordpress_Test5.cdf” width=”653″ height=”405″ altimage=”” altimagewidth=”https://dl.dropboxusercontent.com/u/22612196/Wordpress_Test5.png” altimageheight=””]

Now this is how I was taught maths and in fairness, it’s pretty dull and it’s difficult to gain any form of intuition as to how it might behave if the 2 became a 3 for example, this is where Mathematica’s CDF files come in handy because it has some nice tricks for letting you explore maths in an interactive fashion… let’s consider the following equation, from the previous graph most people wouldn’t really know how it would affect the graph.

$Sin\left( {a.{\rm{ }}x} \right)$

But if we crank this through Mathematica we can create a really nice interactive widget that can be shared with anyone for free!  As you change the slider, the graph updates in real time, and if you want to know what number you’re changing ‘a’ to be then simply click the little + sign next to the slider itself to expand the input values beneath it.  In fact if you think that messsing with sliders is far too much like hard work, then simply click the little play button in the top right and the widget will work the sliders for you… sit back and watch the pattern.

[WolframCDF source=”http://dl.dropbox.com/u/22612196/Wordpress_Test4.cdf” CDFwidth=”752″ CDFheight=”574″]

If you’re not familiar with Mathematica, you may be concerned that this sort of widget is really difficult to create, but actually I’m still on Chapter 3 on the text that I’m working through and the code is incredibly simple to create this kind of interactive learning tool and I’ve replicated it below to show how few lines of text can create this level of interaction.

Manipulate[ Plot[ Sin[a x], {x, -10, 10}], {a, 1, 5}]

Essentially this code starts with “I want a slider widget”, “Plot me a graph of Sin(a.x) over a range of values for x from -10 to 10”, then “make the slider vary a from 1 to 5”.

Now this seems ok, but the Manipulate command is actually incredibly powerful and with a little more twiddling, high quality interactive 3D plots can be created, so let’s consider the following expression.

$f{\rm{ }}Sin\left( x \right) + g{\rm{ }}Sin\left( y \right)$

This expression has four variables: f,g,x, and y.  Of course, I bet you’re dying to know what the graph looks like for this function so you can boost your maths skills…

[WolframCDF source=”http://dl.dropbox.com/u/22612196/Wordpress_Test3.cdf” CDFwidth=”752″ CDFheight=”758″]

This is where the CDF player starts to flex its muscles a little, not only can you mess around with the sliders to change the values of f and g… but you can click and rotate the 3D graph itself to get a better view of how you think it’s working.  For me this level of interaction is a real opportunity for playing with the maths to help build up a level of intuition and feeling of how the maths will behave.  And once again the code to get it to work is fairly straight forward even for a novice such as myself.

Manipulate[Plot3D[(f ) Sin[ x] + (g)  Sin [y], {x, 1, 10}, {y, 1, 10}], {f, -10, 10}, {g, -10, 10}]

Now here’s the rub, a full Mathematica licence is the best part of £1,000 for a lecturer to use, in these hard times that’s a lot of money.  But because I carry ‘dual’ status as I’m studying 2 degrees as well as working full time as a lecturer I was able to pick up a student licence for roughly £80.  Normally the cost for a student licence is a shade over £100 but it is possible to reduce the normal student price by 15% by using the discount code PD1637 at the Wolfram store checkout and I still retain the full functionality of sharing my CDF files via export.

I hope this helps someone, if you’ve any feedback on this post or would like to ask any questions, please get in touch or leave a comment below.


text:
### Challenges in cyber security - Ransomware Phenomenon

##### Abstract

Ransomware has become one of the major threats nowadays due to its huge impact and increased rate of infections around the world. CryptoWall 3, was responsible for damages of over 325 millions of dollars, since its discovery in 2015. Recently, another family of ransomware appeared in the cyber space which is called WannaCry over 230.000 computers around the world, in over 150 countries were infected. Ransomware usually uses the RSA algorithm to protect the encryption key and AES for encrypting the files. If these algorithms are correctly implemented then it is impossible to recover the encrypted information. Some attacks, nonetheless, work against the implementation of RSA. These attacks are not against the basic algorithm, but against the protocol. In the following sections we present the fully analysis on three representative ransomware: Spora, DMA Locker and WannaCry.

Available format(s)
Publication info
Preprint. MINOR revision.
Contact author(s)
emil simion @ upb ro
History
Short URL
https://ia.cr/2018/125

CC BY

BibTeX

@misc{cryptoeprint:2018/125,
author = {Pasca Vlad-Raul and Simion Emil},
title = {Challenges in cyber security - Ransomware Phenomenon},
howpublished = {Cryptology ePrint Archive, Paper 2018/125},
year = {2018},
note = {\url{https://eprint.iacr.org/2018/125}},
url = {https://eprint.iacr.org/2018/125}
}

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