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# X and Y components of velocity and acceleration vectors

## Homework Statement

(Translating from a Polish high school textbook, so if anything is unclear please let me know).

An object moves on a trajectory described by the parabola ##y=\frac{1}{2\lambda}x^2## such that the ##x## component of its velocity is constant and equal to ##v_0##. The ##\lambda## parameter, which is in units of length, is constant. At time ##t=0## the object was located at ##(x_0, y_0) = (0,0)##.

Show how the following depend on time:

a) The ##x## and ##y## coordinates of the velocity vector.
b) The ##x## and ##y## coordinates of the tangential acceleration vector.

None

## The Attempt at a Solution

For part (a), I took advantage of the fact that ##v_0## in the ##x## direction is constant and rewrote the parabola equation as:

$$y=\frac{1}{2\lambda}(tv_0)^2$$

Differentiating, I get that the change in ##y## with respect to time = ##\frac{v_0^2t}{\lambda}##

So in answer to (a), the ##x## and ##y## coordinates of the velocity vector are

$$(v_0, \frac{v_0^2t}{\lambda})$$

Unfortunately, I'm at a loss as to how to tackle (b). I started by differentiating velocity with respect to time, which yields ##\frac{v_0^2}{\lambda}##, but don't know how to proceed from here.

Related Introductory Physics Homework Help News on Phys.org
The velocity vector is the vector whose components are the derivates of the respective position components. The position vector can be parameterized as ##(x,\frac {1}{2\lambda}x^2)##

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Thanks for the reply, but I am not sure how I can use that information to find the ##x## and ##y## coordinates of the tangential acceleration vector. Can you elaborate?

Just as the velocity vector is defined as the rate of change of the position vector in an infinitesimal amount of time, and it is tangencial to it (geometrical motivation of the derivate), the acceleration vector is the change of the velocity vector in an infinitesimal amount of time, hence it's also tangential.
In the position vector ##S = (x, y)##, we can write it as ##S = (x, \frac {1}{2\lambda}x^2)## the first component in the ##x## direction, the second in the ##y## direction (we know that the second component is this because it was given that ##y = \frac {1}{2\lambda}x^2##, if this is confusing to you, you might should take a look into parametrization)

all I'm saying is that to find the acceleration vector thought the position vector, you must derivate it twice
##a = (\frac {d^2}{dx^2} x, \frac {d^2}{dx^2} \frac {1}{2\lambda}x^2)##

Thanks for the explanation. I'm aware that the acceleration vector is the derivative of the velocity with respect to time, but I feel like the question has a bit of subtlety to it that's confusing me. Basically, simply taking the derivative of the position vector doesn't work here because that would get me the change of Y with respect to X, but that's not what the question is asking for - it's asking for the change of Y (and X) coordinates of velocity with respect to time. The same applies to the acceleration vector.

Thanks for the explanation. I'm aware that the acceleration vector is the derivative of the velocity with respect to time, but I feel like the question has a bit of subtlety to it that's confusing me. Basically, simply taking the derivative of the position vector doesn't work here because that would get me the change of Y with respect to X, but that's not what the question is asking for - it's asking for the change of Y (and X) coordinates of velocity with respect to time. The same applies to the acceleration vector.
lol I'm sorry, I completely misunderstood the problem. It's a little bit more complex, the first thought that came to my mind though was writing ##v_{0}## as ##\frac {dx}{dt}##
Then find ##y## applying the chain rule (we don't know how ##y## changes with respect to time, but we know how ##y## changes with respect to ##x## and how ##x##changes with respect to time. ##\frac {dy}{dt} = \frac {dy}{dx} \frac {dx}{dt}##

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Thank you, this gets me the velocity coordinates with respect to time (see the original posting - I already managed to find that). Do you have suggestions for how to move forward from the velocity vector to get the change of the x and y coordinates of the acceleration vector with time?

Thank you, this gets me the velocity coordinates with respect to time (see the original posting - I already managed to find that). Do you have suggestions for how to move forward from the velocity vector to get the change of the x and y coordinates of the acceleration vector with time?
try doing the same thing, after finding the vector ##V = (\frac{dx}{dt}, \frac{dy}{dx} \frac{dx}{dt}) = (v_{0}, \frac{x}{\lambda} v_{0}) ##, ##a = \frac{d}{dt}\vec{V}##
##a =(\frac{dv_{0}}{dt}, \frac{d}{dt}\frac{v_{0}x}{\lambda}\frac{dx}{dy}) ##
I hope this is correct now

marksyncm
ehild
Homework Helper

## The Attempt at a Solution

For part (a), I took advantage of the fact that ##v_0## in the ##x## direction is constant and rewrote the parabola equation as:

$$y=\frac{1}{2\lambda}(tv_0)^2$$

Differentiating, I get that the change in ##y## with respect to time = ##\frac{v_0^2t}{\lambda}##

So in answer to (a), the ##x## and ##y## coordinates of the velocity vector are

$$(v_0, \frac{v_0^2t}{\lambda})$$

Unfortunately, I'm at a loss as to how to tackle (b). I started by differentiating velocity with respect to time, which yields ##\frac{v_0^2}{\lambda}##, but don't know how to proceed from here.
##\frac{v_0^2}{\lambda}## is the y component of the acceleration vector. What is the x component?
You need the tangential acceleration. The tangent is parallel to the velocity vector. So you need the component of the acceleration parallel to the velocity vector. How do you get the projection of a vector to a given direction?

marksyncm
What is the x component?
Isn't the ##x## component equal to zero because the object is not accelerating in the ##x## direction?

haruspex
Homework Helper
Gold Member
2020 Award
Isn't the ##x## component equal to zero because the object is not accelerating in the ##x## direction?
Yes.

marksyncm
If the ##x## component is equal to zero, doesn't this mean that the acceleration vector is pointing straight up (parallel to the ##y## axis), so that it can never be parallel to the velocity vector?

haruspex
Homework Helper
Gold Member
2020 Award
If the ##x## component is equal to zero, doesn't this mean that the acceleration vector is pointing straight up (parallel to the ##y## axis), so that it can never be parallel to the velocity vector?
It will not as a whole be parallel to the velocity vector, but unless it is orthogonal to it there will be a component parallel to the velocity vector.

marksyncm
ehild
Homework Helper
If the ##x## component is equal to zero, doesn't this mean that the acceleration vector is pointing straight up (parallel to the ##y## axis), so that it can never be parallel to the velocity vector?
The red vector is the acceleration, the blue is the tangent vector to the curve, parallel with the velocity. The tangential acceleration was the question, that is the component of acceleration in the tangential direction. Is it really zero? How do you determine it?

#### Attachments

• 747 bytes Views: 242
marksyncm
Thank you for the image, it's very helpful.

Is the tangential acceleration equal to the acceleration vector times the cosine of the angle between the acceleration vector and the velocity vector? Is that angle ##\alpha## equal to ##\frac{\pi}{4} - \arctan(y/x)##, where ##y## and ##x## are the coordinates of the velocity vector? (EDIT: on second though, I don't think this approach would work in all cases).

Alternatively, could we maybe obtain the angle using the vector product? So $$\frac{\vec{v} \cdot \vec{a}}{|\vec{v}| |\vec{a}|} = \cos(\alpha)$$

Or is there something much more obvious that I'm missing?

Yes, the scalar product.

Thank you all, I have learned a lot from this thread and from the questions that were asked.

ehild
Homework Helper
Instead of the angle, you need the projection of the acceleration vector onto the tangent unit vector which is ##\frac{\vec{v} \cdot \vec{a}}{|\vec{v}| } ## That is the magnitude of the tangential acceleration.
The other approach is: the tangential acceleration along a curve is the time derivative of the speed. Make it a vector by multiplying with the tangential unit vector .

marksyncm
Thanks, I will take a look at this answer soon (as I'm in the middle of creating another thread).

Instead of the angle, you need the projection of the acceleration vector onto the tangent unit vector which is
Yes, thank you.

The other approach is: the tangential acceleration along a curve is the time derivative of the speed. Make it a vector by multiplying with the tangential unit vector.
I'm not sure I follow. I understand that taking the derivative of the velocity gives us the equation for acceleration, but I do not understand what is meant by "multiplying with the tangential unit vector| to turn it into a vector. For example, differentiating velocity I get ## \frac{dv}{dt} = \frac{v_0^2}{\lambda}##. What happens now?

I still have a follow-up question.

We were able to determine the magnitude of the tangential acceleration, but I just realized that the question asks for the ##x## and ##y## coordinates of that vector, not its magnitude. I'm not sure how to get the coordinates from here given that what we have is a scalar value?

ehild
Homework Helper
I still have a follow-up question.

We were able to determine the magnitude of the tangential acceleration, but I just realized that the question asks for the ##x## and ##y## coordinates of that vector, not its magnitude. I'm not sure how to get the coordinates from here given that what we have is a scalar value?
The tangential acceleration is a vector. It has magnitude and direction parallel with the tangent of the parabola.

ehild
Homework Helper
I still have a follow-up question.

We were able to determine the magnitude of the tangential acceleration, but I just realized that the question asks for the ##x## and ##y## coordinates of that vector, not its magnitude. I'm not sure how to get the coordinates from here given that what we have is a scalar value?
What did you get for the magnitude of the tangential acceleration? It is not ## \frac{v_0^2}{\lambda}##

What did you get for the magnitude of the tangential acceleration? It is not ## \frac{v_0^2}{\lambda}##
I got this for the magnitude.

$$\frac{v_{0}^3t}{\lambda^2 \sqrt{1+\frac{t^2}{\lambda^2}}}$$

Does it make sense?


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# How to center part,section,… (LyX)

How Can I center a part or a section in LyX?
I saw this question - Center part title in Lyx - but I want to know to do it every time without write any part, section,... \begin{center}...\end{center}

It is possible?
Thank you!

• Did you try using \sectionfont{\centering} and \subsectionfont{\centering} to center the section and subsection headings? – leandriis Apr 14 '18 at 16:51
• @leandriis - it's doesn't recognize it.. :-( – heblyx Apr 14 '18 at 21:15

In LyX itself, I don't know, but you can delve into the underlying code and write in the preamble:

\usepackage{titlesec}
\titleformat*{\part}{\centering}
\titleformat*{\section}{\centering}

For more complex changes you'll have to use the \titleformat command (without a star), which has 4 mandatory and 1 optional arguments.

Edit: Here is an ‘advanced’ code:

\titleformat{\part}[display]{\filcenter\huge\bfseries}{\partname \thepart}{4ex} {}
\titleformat{\section}[block]{\filcenter\LARGE\bfseries} {\thesection.}{0.5em}{}

• I think the only LyX-specific instruction would be to put the above commands in Document > Settings > LaTeX Preamble – scottkosty Apr 14 '18 at 18:27
• It's give an error. it says: Package titlesec Error: Not allowed in `easy' settings. What can I do? – heblyx Apr 14 '18 at 21:34
• Could you post a full compilable code producing this error? I tested this solution with a minimal code before posting, so probably there's something else which is not compatible with this code. – Bernard Apr 14 '18 at 21:38
• @Bernard - Sure! this is the error: anotepad.com/notes/j2tg7h – heblyx Apr 15 '18 at 8:43
• And here is the full code of the document - anotepad.com/notes/47bsmc – heblyx Apr 15 '18 at 8:45

format: plain


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# Logic Seminar - Vadim Kulikov - Borel Reducibility in Generalised Descriptive Set Theory"

I will review some recent results in the Borel reducibility on uncountable cardinals of the Helsinki logic group. Borel reducibility on the generalised Baire space \kappa^\kappa for uncountable \kappa is defined analogously to that for \kappa=\omega. One of the corollaries of this work is that under some mild cardinality assumptions on kappa, if T1 is classifiable and T2 is unstable or superstable with OTOP, then the ISOM(T1) is continuously reducible ISOM(T2) and ISOM(T2) is not Borel reducible to ISOM(T1). Another result, on which we will focus more in the talk is that under certain cardinality assumptions on kappa the inclusion modulo the non-stationary ideal is a Sigma^1_1-complete quasiorder which has the corollary that if V=L, then the embeddability of dense linear orders of cardinality kappa is Sigma^1_1-complete.

## Date:

Wed, 24/01/2018 - 11:00 to 13:00

Ross 63


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# Hyperspace

In 2016 Oxford University Press started a Landmark Science series with cheap reprints of classic books, the 'must-reads', about topics that have shaped current science. Here I review two of the more mathematical/physical ones from the first release if seven books in April 2016: this book and The Emperor's New Mind (R. Penrose).

Like Penrose, Kaku gives in this book also an overview of how physics and our understanding of the laws that govern nature and the cosmos. Where Penrose's main objective was to understand the human mind and brain and uses a lot of technical material, Kaku's story is much more accessible and sticks more to the quest of a unifying theory that captures all the laws of physics. As physicists of the 20th century constructed more complex theories that picked up more of the laws that describe phenomena, considered of different nature before, they needed to extend the dimension of the space in which they had to be described. This need for higher dimensions provides the thread that runs through the book and that also was the inspiration for the title.

Kaku divides his book in four parts. In the first part he prepares the reader for the roller coaster he is about to enter. The four forces that need to be unified are Maxwell electromagnetic force, strong and weak nuclear force, and gravitational force. We are introduced to the queer higher dimensional world and a world that is not flat. We get a sneak preview of where this may lead us: black holes, wormholes, and time travel. He more or less assumes we are familiar with the classical theory of Newtonian mechanics and presents a first unification by introducing the Riemannian tensor for the Kaluza-Klein theory which consists of the 4-dimensional Einstein block, bordered by a row and column to include Maxwell's theory.

In part two, we get the more recent history in search for the unification. It starts with the observations that have led to the development of quantum theory that, when followed consequently, has strange and unexplainable consequences. The Yang-Mills field theory replaced the classical Maxwell theory. Here we also meet with Einstein's reluctance towards quantum theory's 'spooky action at a distance'. Till then the theory was carved in marble he said, quantum theory was made from wood, and he wanted to build a theory that was just marble, no wood. This metaphore of opposing viewpoints is taken along by Kaku as his book develops. Einstein's relativity theory where gravity is explained by space-time curvature is of a different nature that does not mix well with quantum field theory. That is why the Standard Model could explain and unify the electromagnetic and the strong and weak nuclear forces but leaves out gravitation which is too weak a force to fit. Gradually more wood is turned into marble, i.e. using geometric arguments, by bordering the Riemann tensor with the Yang-Mills field. The many subatomic quarks and leptons were rather messy and symmetry was lost but that problem was solved by supergravity (and supersymmetry), lifting the tensor to an even higher N-dimensional space, so that more symmetry could be assumed.

Enter string theory, exit supergravity in the effort to include gravity. This is where Kaku himself has contributed. Here the general N is naturally' forced to be 10 for mathematical reasons. According to Kaku, this is something that could be explained by studying modular functions in Ramanujan's lost notebook. Recall that Kaku wrote his book in 1994, and the theory was still young. Since experimental verification is not possible with the energy we have available here on Earth, we have to look outside to cosmic phenomena and the origin of the universe. Kaku even takes the time to go through several historical proofs for the existence of God as a creator of it all.

The two remaining parts will certainly be appreciated by lovers of Science Fiction. Kaku explores in part 3 what the theory so far would make possible in principle. So we learn the possibilities created by black holes, wormholes and time travel. The possible interpretation of quantum theory dealing with many universes. The last part speculates on the what if' aspect of the future. What if our civilization ever reaches a type II level where it becomes possible to control the amounts of energy of our sun, or even type III where it becomes possible to work with the energy of an entire galaxy. Do such civilizations exist and if so, why didn't they contact us? Perhaps they never reach it because of disastrous events happened as they did on our planet according to geological observations. And what will eventually happen with the whole universe?

Although the book is about physics, the battlefield is on the mathematical structure that is used. Topology is what should bind the Riemannian geometry of Einstein and the Lie groups of quantum physics. Mathematics and physics were in close harmony for many centuries. They started to develop in different directions late 19th and early 20th century. It turned out that at the dawn of the 21st century they need each other more than ever, which raises the philosophical question of all times: why this `unreasonable effectiveness of mathematics in the natural sciences' as E. Wigner formulated it. If you are looking for the mathematics, that you will not find in this book, but it will convince you, if that is still needed, that mathematics is the key to understand nature.

Of course, theoretical physics and mathematics did not stop evolving after this book was written, gravitational waves were observed in 2016, the Higgs boson was identified in 2012, the tau neutrino in 2000, etc. Neither did Kaku stop form 'spreading the news'. He is present in the media and wrote several other popular books since, just a couple of examples: Parallel Worlds: A Journey through Creation, Higher Dimensions, and the Future of the Cosmos (2004), Physics of the Impossible: A Scientific Exploration into the World of Phasers, Force Fields, Teleportation, and Time Travel (2008), and a recent one that relates to the book of Penrose mentioned in the beginning: The Future of the Mind: The Scientific Quest to Understand, Enhance, and Empower the Mind (2014). However, the current Hyperspace book was his biggest success so far. So one should be aware that as Kaku brings the reader of this book to the level of superstrings, this is in the context of the euphoria of the early 1990s. Nevertheless, it is still mostly entertaining to read this book about the evolution of theoretical physics up to that point. It is the best documentation for the layman of this history that I know of. He does avoid the technical stuff that most of those readers will probably skip anyway, and yet it gives so much details that takes you along on A Scientific Odyssey Trough Parallel Universes, Time Warps, and The Tenth Dimension which is rightfully the subtitle of the book.

Reviewer:
Book details

This is a reprint of the original book of 1994. Kaku introduces the reader to theoretical physics from Einstein's relativity theory and quantum physics to string theory. He takes the reader along on an entertaining story, avoiding in gracious slaloms all the technical material. In the later part he amazes with possibilities of wormholes and time machines and civilizations that are far beyond our lifespan but that have inspired many an SF author.

Author:  Publisher:
Published:
2016
ISBN:
9780198785033 (pbk)
Price:
£9.99 (pbk)
Pages:
384
Categorisation


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# 7.16: Vinyl Halides and Aryl Halides

## Nucleophilic substitution in the aryl halides

We'll look in some detail at the structure of chlorobenzene. Bromobenzene and iodobenzene are just the same. The simplest way to draw the structure of chlorobenzene is:

To understand chlorobenzene properly, you need to dig a bit deeper than this. There is an interaction between the delocalised electrons in the benzene ring and one of the lone pairs on the chlorine atom. This overlaps with the delocalised ring electron system . . .

to giving a structure rather like this:

This delocalisation is by no means complete, but it does have a significant effect on the properties of both the carbon-chlorine bond and the polarity of the molecule. The delocalisation introduces some extra bonding between the carbon and the chlorine, making the bond stronger. This has a major effect on the reactions of compounds like chlorobenzene.

There is also some movement of electrons away from the chlorine towards the ring. Chlorine is quite electronegative and usually draws electrons in the carbon-chlorine bond towards itself. In this case, this is offset to some extent by the movement of electrons back towards the ring in the delocalisation. The molecule is less polar than you would otherwise have expected.

Simple aryl halides like chlorobenzene are very resistant to nucleophilic substitution. It is possible to replace the chlorine by -OH, but only under very severe industrial conditions - for example at 200°C and 200 atmospheres. In the lab, these reactions do not happen. There are two reasons for this - depending on which of the above mechanisms you are talking about.

1. The extra strength of the carbon-halogen bond in aryl halides: The carbon-chlorine bond in chlorobenzene is stronger than you might expect. There is an interaction between one of the lone pairs on the chlorine atom and the delocalized ring electrons, and this strengthens the bond. Both of the mechanisms above involve breaking the carbon-halogen bond at some stage. The more difficult it is to break, the slower the reaction will be.
2. Repulsion by the ring electrons: This will only apply if the hydroxide ion attacked the chlorobenzene by a mechanism like the first one described above. In that mechanism, the hydroxide ion attacks the slightly positive carbon that the halogen atom is attached to.If the halogen atom is attached to a benzene ring, the incoming hydroxide ion is going to be faced with the delocalized ring electrons above and below that carbon atom. The negative hydroxide ion will simply be repelled.

## Contributors

Jim Clark (Chemguide.co.uk)

This page titled 7.16: Vinyl Halides and Aryl Halides is shared under a not declared license and was authored, remixed, and/or curated by Layne Morsch.


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## "Weighted" sum of divisor function

Hi, I am trying to solve a problem about the divisor function. Let us call d(n) the classical divisor function, i.e. d(n) is the number of divisors of the integer n. It is well known that the sum of d(n) over all positive integers from n=1 to x, when x tends to infinite, is asymptotic to x Ln(x) + (2 gamma-1) x + O(sqrt(x)).
I would like to calculate a similar asymptotic expression for the sum of d(n)/n, again calculated from n=1 to x and for x that tends to infinite. I have made some calculations and believe that this formula may be 1/2 (Ln(x))^2 + 2 gamma Ln (x) +O(1), where the last term of the expression, that is to say the intercept, approximately tends to 0.96. Since I am not sure of this formula, could someone help me? If the formula is correct, I would particularly be interested to know what that intercept exactly is.
As an additional question, I would be very interested in obtaining similar asymptotic formulas for the same sum of d(n)/n calculated over all odd integers from 1 to x, and calculated over all even integers from 1 to x.
Many thanks.


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# The Mixed Strategy Algorithm

This lesson shows the algorithm we use to solve for mixed strategy Nash equilibrium in simple 2×2 games.

Takeaway Points

1. If there is a mixed strategy Nash equilibrium, it usually is not immediately obvious. (Do not let matching pennies lull you into believing this is easy!)
2. However, there is a straightforward algorithm that lets you calculate mixed strategy Nash equilibria. We will employ it frequently.
3. The algorithm involves setting the payoffs for a player’s two pure strategies equal to each other and solving for the mixed strategy of the other player that makes this equation true.
4. The mixed strategy algorithm does not involve any fancy mathematics. All one needs is basic knowledge of algebra. However, the process requires that the analyst be careful in putting the right values in the right places.
5. The trickiest part of the logic of mixed strategy Nash equilibrium is that we must use one player’s payoffs to solve for the player’s strategy. This is because the point of my mixed strategy is to make you indifferent, and vice versa.

Back to Game Theory 101


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# How did washingtons policy on matters at home differ from his foreign policy

###### Question:

How did washingtons policy on matters at home differ from his foreign policy

### Plz solve. plz solve by this Friday thxx

plz solve. plz solve by this Friday thxx ...

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### What is the answer to this ? ????????

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### What is part A and part B

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### Write a brief summary of the story “The flight” in your summary answer the question who what when where how

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### How did Spain help the American colonies when they first entered the war? A.They trained the American soldiers B.in secret C.They lent the American colonies millions of dollars D.by opening the Mississippi river to them               i got it its B.if anyone wants the points the answer is B.

How did Spain help the American colonies when they first entered the war? A.They trained the American soldiers B.in secret C.They lent the American colonies millions of dollars D.by opening the Mississippi river to them               i got it its B.if anyone wants the points the a...

### Maddie has a retangular garden in her backyard. The perimeter of the garden is eighty-two feet. The width is five shorter than the length. What are the length and width of the garden?

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### A closed, rigid tank is filled with a gas modeled as an ideal gas, initially at 27°C and gauge pressure of 300 kPa. The gas is heated, and the gauge pressure at the final state is 367 kPa. Determine the final temperature, in °C. The local atmospheric pressure is 1 atm.

A closed, rigid tank is filled with a gas modeled as an ideal gas, initially at 27°C and gauge pressure of 300 kPa. The gas is heated, and the gauge pressure at the final state is 367 kPa. Determine the final temperature, in °C. The local atmospheric pressure is 1 atm....

### Which was not a result of the growth of the railroads in 18,000 A : Industries grew because they could ship more Goods cheaply B : cities grew into industrial centers C : raw materials could be shipped to factories year round D : the price of goods increased

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### BRAINLIEST ANSWER PLEASE! There are benefits and challenges when you live in a diverse society. Explain at least two benefits and two challenges (4 Points). Then tell us why do the benefits, or why don't the benefits, outweigh the challenges (2 points). (6 points total)

BRAINLIEST ANSWER PLEASE! There are benefits and challenges when you live in a diverse society. Explain at least two benefits and two challenges (4 Points). Then tell us why do the benefits, or why don't the benefits, outweigh the challenges (2 points). (6 points total)...

### Akamai Technologies is a dominant player in the content delivery network (CDN) market. Akamai is not very diversified (i.e., is dependent on the CDN market). If rival CDN providers such as Limelight Networks and Level 3 Communications lower their basic CDN service prices, which of the following would Akamai likely do? a. Do nothing since it is the market leader b. Exit the industry c. Lower its prices d. Raise its prices

Akamai Technologies is a dominant player in the content delivery network (CDN) market. Akamai is not very diversified (i.e., is dependent on the CDN market). If rival CDN providers such as Limelight Networks and Level 3 Communications lower their basic CDN service prices, which of the following woul...


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# kg to MeV Conversion

For kg to MeV conversion, use $E=mc^{2}$ to convert 1 kg to Joules and then use the fact that 1 MeV is the energy required to move a charge $e$ across a potential difference of $10^{6}$ volts. Thus 1 MeV is $1.6 \times 10^{-19} \times 10^{6}$ Joules. Therefore
$E({\rm MeV}) = \frac{m ({\rm kg}) \ c^{2}}{10^{6} \ e}$
$E({\rm MeV}) = 5.6175 \times 10^{29} \ m ({\rm kg})$


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# zbMATH — the first resource for mathematics

Conformal geometry and quasiregular mappings. (English) Zbl 0646.30025
Lecture Notes in Mathematics. 1319. Berlin etc.: Springer-Verlag. xix, 209 p. (1988).
Apart from Yu. G. Reshetnyak’s books, available only in Russian, this is the first book on quasiregular mappings in $$R^ n.$$ These mappings form a generalization of plane analytic functions to higher dimensional Euclidean spaces: A continuous mapping f of a domain G into $$R^ n$$ is K-quasiregular (QR) if f is ACL$$^ n$$ and $$| f'(x)|^ n\leq KJ(x,f)$$ a.e. in G.
The book stresses geometric properties of QR-maps. After a good survey on the current state of affairs Chapter I deals with conformal geometry: Möbius transformations, hyperbolic and quasihyperbolic geometry. There are several useful formulas hard to find in the literature. Quasihyperbolic geometry has also been dealt in F. W. Gehring’s lecture notes [F. W. Gehring, Characteristic properties of quasidisks (1982; Zbl 0495.30018)]. Chapter II is devoted to the modulus and capacity. Here the author has mostly collected the known properties of the n-modulus, the main source of information being J. Väisälä’s book on n-dimensional quasiconformal $$(=$$ homeomorphic QR-) mappings [J. Väisälä: Lectures on n-dimensional quasiconformal mappings (1971; Zbl 0221.30031)]. However, the moduli of Teichmüller and Grötzsch condensers get a careful analysis. At the end of the chapter two conformal invariants, the modulus metric and its dual, are introduced. The author has used these invariants in several occasions to study quasiconformal and QR-mappings, especially their distortion properties.
QR-mappings are introduced in Chapter III. Their topological behavior and the change of the moduli of condensers under QR-maps are stated without proof and the main emphasis is put to the change of the aforementioned conformal invariants. Using these the author develops a rather precise distortion theory: several Hölder-continuity type results are obtained. The last chapter is devoted to the boundary behavior: the author mostly considers quasiconformal mappings since very little is known on the boundary properties of QR-maps. Lindelöf type problems get most attention: When does a quasiconformal mapping of the unit ball of $$R^ n$$ into $$R^ n$$ have an angular limit at a boundary point? The final, short section is devoted to Dirichlet finite mappings f, i.e. $$\int_{B^ n}| f'(x)|^ n dm(x)<\infty.$$
QR-mappings of B n do not satisfy this condition, but since this is true locally for bounded QR-mappings, it can be used to study uniform continuity of these mappings in the hyperbolic metric.
The book contains plenty of examples and exercises. The bibliography list is extensive. Those interested in the semilocal distortion theory of QR and quasiconformal mappings will find the book most valuable.
Reviewer: O.Martio

##### MSC:
 30C62 Quasiconformal mappings in the complex plane 30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
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# If the light velocity is a vector quantity, why vector addition cannot be applied to it?

If the light velocity is a vector quantity, why vector addition cannot be applied to it?

Or the light velocity is not a vector quantity?

-

Update 2: It might also be useful add the relation to usual Galilean physics. This will probably only make sense after reading Update 1.

Recall that one can parametrize boosts in (1+1) by $[\cosh(\eta), \sinh(\eta)]$. As it so happens this equals to $[\gamma, \gamma {v \over c}]$. The classical physics correspond to the asymptotic case $c \to \infty$ (i.e. no limit on the speed of light). So this reduces to $\gamma \to 1$, ${v \over c} \to 0$ and that means $\eta \to 0$. So in Galilean case all boosts degenerate to trivial transformation and this is why time and space separate and addition of velocities starts to work.

First, there exists a concept of rapidity. This is a natural variable for parametrization of boosts. First consider a circle. Why circle? Because it has to do with rotations and rotations are very similar to boosts.

Circle is an object given by an equation $x^2 + y^2 = 1$. You can decide that you'll parametrize (part of) it by coordinates $[x, \sqrt{1 - x^2}]$. Now what if you want to rotate the circle by some angle? Can you figure out how will the parametrization change? Maybe you can but I assure you this is not pretty. But there is a nicer way. Let's try parametrizing the circle by an angle $\phi$ so that it would become set of points $[\cos(\phi), \sin(\phi)]$ and now the rotation by angle $\psi$ corresponds to parametrization $[\cos(\phi + \psi), \sin(\phi + \psi)]$. So our parameter is additive! You won't find any better parametrization of the circle than this.

Okay, so we understand circles a little better now. But as already said, rotation in two space dimensions is almost the same thing as boost in (1+1) space-time dimensions. In the same way that rotations (around origin) preserve circles, boosts preserve hyperbolas. So instead of working with $\sin$ and $\cos$ you'll work with hyperbolic functions $\sinh$ and $\cosh$ and instead of $\phi$ you'll obtain rapidity $\eta$.

Now, this only works this nicely in (1+1)-dimensions. In (3+1) you'll have many more interesting effects (similarly to like rotations are strange beasts in 3 dimensions as opposed in 2). But it's still true that like the general 3-dimensional rotations are nicely parametrized by an axis and rotation angle, the boosts are nicely parametrized by the direction of boost and rapidity. So if you perform two rotations about the same axis it's the same as rotation by a sum of the angles and if you perform two boosts in one direction it's the same as doing a boost with added rapidities.

I'll assume you're talking about Galilean principle of relativity whereby the velocities transform by pure addition. This concept breaks down when the speeds one is dealing with are too large. Speed of light is an extreme case of such speed. Then one has to use Special Relativity and instead consider four-vectors transforming by Lorentz transformation. Now, this transformations preserve the Minkowski length of four-vectors (in the same way that rotations preserve length of usual vectors).

The point is that velocity of light corresponds to zero Minkowski length and so light moves at the speed of light in every inertial frame. This is the famous Einstein's postulate.

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Also check out this wikipedia article which contains some actual computations. – Marek Nov 27 '10 at 12:58
You can also mention that there is a kind of adapted addition law. – Raskolnikov Nov 27 '10 at 13:52
Yeah, I was about to submit my own comment on that before I saw yours, @Raskolnikov. Vector addition does apply perfectly well to velocities (including that of light) if you modify the definition of "addition" to the law explained in the Wikipedia article. – David Z Nov 27 '10 at 20:59
@Raskolnikov, @David: yeah, so you want to just add up four-vectors and pretend that it is just "modified" addition of usual vectors; do I get that right? I always hated this approach and I never understood SR until Lorentz transformations and four-vectors where presented to me as what they really where (instead of just some "modified" laws). – Marek Nov 28 '10 at 5:40

The speed of light is a vector quantity and vector summation works perfectly well for it (at least in Special Relativity). You just cannot change the frame of reference.

For example if you have one object moving at c in one direction and another object moving at 1/2c in the opposite direction, then the middle between the two will move at c/2 in the same direction as the first. This is from point of view of a stationary observer.

The distance between the two objects grows as 3/2c. This is in the stationary reference frame of course, the objects themselves will see each other moving at speed of light.

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# Toronto Math Forum

## MAT244--2019F => MAT244--Test & Quizzes => Quiz-3 => Topic started by: Siyan Chen on October 11, 2019, 02:22:45 PM

Title: TUT0301 Quiz3
Post by: Siyan Chen on October 11, 2019, 02:22:45 PM
quiz 2 (tut 0301)
$$y’’ + 4y = 0,\ y_{1}(t) = \cos(2t),\ y_{2}=\sin(2t)$$
The given solution of differential equation is: $y_{1}(t) = \cos(2t)$
So, $y_{1}’(t) = -2 \sin(2t)$
$y_{1}’’(t) = -4 \cos(2t)$

Then, plug into the given differential equation: $y’’ + 4y = 0$,
i.e. $-4 \cos(2t)\ + 4(\cos(2t)) = 0$ => 0=0
So, $y_{1}(t) = \cos(2t)$ is a solution of this equation

Similarly, we have $y_{2}(t)=\sin(2t)$
So, $y_{2}’(t) = 2 \cos(2t)$
$y_{2}’’(t) = -4 \sin(2t)$

Then, plug into the given differential equation: $y’’ + 4y = 0$,
i.e. $-4 \sin(2t)\ + 4(\sin(2t)) = 0$ => 0=0
So, $y_{2}(t)=\sin(2t)$ is also a solution of this equation

To check whether $y_{1}$ and $y_{2}$ constitute a fundamental set of solutions, we will find the Wronskian $W(y_{1}(t), y_{2})(t)\ = \begin{vmatrix} y_{1}(t) & y_{2}(t)\\ y_{1}’(t) & y_{2}’(t)\\ \end{vmatrix}$
= $\begin{vmatrix} \cos(2t) & \sin(2t)\\ -2\sin(2t) & 2\cos(2t)\\ \end{vmatrix}$
= $2\cos^2(2t)+2\sin^2(2t)$
= $2(\cos^2(2t)+\sin^2(2t))$
= $2 (1)$
= $2$
Since, $W(y_{1}(t), y_{2})(t)\neq0$, then we say that $y_{1}$ and $y_{2}$ constitute a fundamental set of solutions.


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# Chapter 4 Managing Portfolios

In this chapter we show how to explore and analyze data using the dataset created in Chapter @ref(#s_2Data):

At first we will learn how to full-sample optimize portfolios, then (in the next chapters) we will do the same thing in a rolling analysis and also perform some backtesting. The major workhorse of this chapter is the portfolioAnalytics-package developed by Peterson and Carl (2018).

portfolioAnalytics comes with an excellent introductory vignette vignette("portfolio_vignette") and includes more documents, detailing on the use of ROI-solvers vignette("ROI_vignette"), how to create custom moment functions vignette("custom_moments_objectives") and how to introduce CVaR-budgets vignette("risk_budget_optimization").

## 4.1 Introduction

SHORT INTRODUCTION TO PORTFOLIOMANAGEMENT

We start by first creating a portfolio object, before we…

## 4.2 Tools for Portfolio Management

### 4.2.1 The Portfolio Object

The portfolio object is a so-called S3-object, which means, that it has a certain class (portfolio) describing its properties, behavior and relation to other objects. Usually such an objects comes with a variety of methods. To create such an object, we reuse the stock data set that we have created in Chapter @ref(#s_2Data):

load("stocks.RData")
glimpse(stocks.final)
## Observations: 2,160
## Variables: 10
## $symbol <chr> "AAPL", "AAPL", "AAPL", "AAPL", "AAPL", "AAPL", "AAPL... ##$ date     <S3: yearmon> Jan 2000, Feb 2000, Mrz 2000, Apr 2000, Mai 2...
## $return <dbl> -0.07314358, 0.10481940, 0.18484208, -0.08651642, -0.... ##$ adjusted <dbl> 2.489997, 2.750997, 3.259497, 2.977497, 2.015998, 2.5...
## $volume <dbl> 175420000, 92240400, 101158400, 62395200, 108376800, ... ##$ sp500    <dbl> -0.041753145, -0.020108083, 0.096719828, -0.030795756...
## $Mkt.RF <dbl> -0.0474, 0.0245, 0.0520, -0.0640, -0.0442, 0.0464, -0... ##$ SMB      <dbl> 0.0505, 0.2214, -0.1728, -0.0771, -0.0501, 0.1403, -0...
## $HML <dbl> -0.0045, -0.1057, 0.0794, 0.0856, 0.0243, -0.1010, 0.... ##$ RF       <dbl> 0.0041, 0.0043, 0.0047, 0.0046, 0.0050, 0.0040, 0.004...
stocks.final %>% slice(1:2)
## # A tibble: 2 x 10
##   symbol date   return adjusted volume   sp500  Mkt.RF    SMB     HML
##   <chr>  <S3:>   <dbl>    <dbl>  <dbl>   <dbl>   <dbl>  <dbl>   <dbl>
## 1 AAPL   Jan ~ -0.0731     2.49 1.75e8 -0.0418 -0.0474 0.0505 -0.0045
## 2 AAPL   Feb ~  0.105      2.75 9.22e7 -0.0201  0.0245 0.221  -0.106
## # ... with 1 more variable: RF <dbl>

For the portfolioAnalytics-package we need our data in xts-format (see @ref(#sss_112xts)) and therefore first spread the dataset returns in columns of stocks and the convert to xts using tk_xts() from the timetk-package.

returns <- stocks.final %>%
select(symbol,date,return) %>%
tk_xts(silent = TRUE)

Now its time to initialize the portfolio.spec() object passing along the names of our assets. Afterwards we print the object (most S3 obejcts come with a printing methods that nicely displays some nice information).

pspec <- portfolio.spec(assets = stocks.selection$symbol, category_labels = stocks.selection$sector)
print(pspec)
## **************************************************
## PortfolioAnalytics Portfolio Specification
## **************************************************
##
## Call:
## portfolio.spec(assets = stocks.selection$symbol, category_labels = stocks.selection$sector)
##
## Number of assets: 10
## Asset Names
##  [1] "AAPL" "MSFT" "AMZN" "CSCO" "NVDA" "ORCL" "AMGN" "ADBE" "QCOM" "GILD"
##
## Category Labels
## Information Technology : AAPL MSFT CSCO NVDA ORCL ADBE QCOM
## Consumer Discretionary : AMZN
## Health Care : AMGN GILD
str(pspec)
## List of 6
##  $assets : Named num [1:10] 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 ## ..- attr(*, "names")= chr [1:10] "AAPL" "MSFT" "AMZN" "CSCO" ... ##$ category_labels:List of 3
##   ..$Information Technology: int [1:7] 1 2 4 5 6 8 9 ## ..$ Consumer Discretionary: int 3
##   ..$Health Care : int [1:2] 7 10 ##$ weight_seq     : NULL
##  $constraints : list() ##$ objectives     : list()
##  $call : language portfolio.spec(assets = stocks.selection$symbol, category_labels = stocks.selection$sector) ## - attr(*, "class")= chr [1:2] "portfolio.spec" "portfolio" Checking the structure of the object str() we find that it contains several elements: assets which contains the asset names and initial weights that are equally distributed unless otherwise specified (e.g. portfolio.spec(assets=c(0.6,0.4))), category_labels to categorize assets by sector (or geography etc.), weight_seq (sequence of weights for later use by random_portfolios), constraints that we will set soon, objectives and the call that initialised the object. Before we go and optimize any portfolio we will show how to set contraints. ### 4.2.2 Constraints Constraints define restrictions and boundary conditions on the weights of a portfolio. Constraints are defined by add.constraint specifying certain types and arguments for each type, as well as whether the constraint should be enabled or not (enabled=TRUE is the default). #### 4.2.2.1 Sum of Weights Constraint Here we define how much of the available budget can/must be invested by specifying the maximum/minimum sum of portfolio weights. Usually we want to invest our entire budget and therefore set type="full_investment" which sets the sum of weights to 1. Alternatively we can set the type="weight_sum" to have mimimum/maximum weight_sum equal to 1. pspec <- add.constraint(portfolio=pspec, type="full_investment") # print(pspec) # pspec <- add.constraint(portfolio=pspec,type="weight_sum", min_sum=1, max_sum=1) Another common constraint is to have the portfolio dollar-neutral type="dollar_neutral" (or equivalent formulations specified below) # pspec <- add.constraint(portfolio=pspec, # type="dollar_neutral") # print(pspec) # pspec <- add.constraint(portfolio=pspec, type="active") # pspec <- add.constraint(portfolio=pspec, type="weight_sum", min_sum=0, max_sum=0) #### 4.2.2.2 Box Constraint Box constraints specify upper and lower bounds on the asset weights. If we pass min and max as scalars then the same max and min weights are set per asset. If we pass vectors (that should be of the same length as the number of assets) we can specify position limits on individual stocks pspec <- add.constraint(portfolio=pspec, type="box", min=0, max=0.4) # print(pspec) # add.constraint(portfolio=pspec, # type="box", # min=c(0.05, 0, rep(0.05,8)), # max=c(0.4, 0.3, rep(0.4,8))) Another special type of box constraints are long-only constraints, where we only allow positive weights per asset. These are set automatically, if no min and max are set or when we use type="long_only" # pspec <- add.constraint(portfolio=pspec, type="box") # pspec <- add.constraint(portfolio=pspec, type="long_only") #### 4.2.2.3 Group Constraints Group constraints allow the user to specify constraints per groups, such as industries, sectors or geography.3 These groups can be randomly defined, below we will set group constraints for the sectors as specified above. The input arguments are the following: groupslist of vectors specifying the groups of the assets, group_labels character vector to label the groups (e.g. size, asset class, style, etc.), group_min and group_max specifying minimum and maximum weight per group, group_pos to specifying the number of non-zero weights per group (optional). pspec <- add.constraint(portfolio=pspec, type="group", groups=list(pspec$category_labels$Information Technology, # list of vectors specifying the groups of the assets pspec$category_labels$Consumer Discretionary, pspec$category_labels$Health Care), group_min=c(0.1, 0.15,0.1), group_max=c(0.85, 0.55,0.4), group_labels=pspec$category_labels)
# print(pspec)

#### 4.2.2.4 Position Limit Constraint

The position limit constraint allows the user to specify limits on the number of assets with non-zero, long, or short positions. Its arguments are: max_pos which defines the maximum number of assets with non-zero weights and max_pos_long/ max_pos_short that specify the maximum number of assets with long (i.e. buy) and short (i.e. sell) positions.4

pspec <- add.constraint(portfolio=pspec, type="position_limit", max_pos=3)
# pspec <- add.constraint(portfolio=pspec, type="position_limit", max_pos_long=3, max_pos_short=3))
# print(pspec)

#### 4.2.2.5 Diversification Constraint

The diversification constraint enables to set a minimum diversification limit by penalizing the optimizer if the deviation is larger than 5%. Diversification is defined as $$\sum_{i=1}^{N}w_i^2$$ for $$N$$ assets.5 Its only argument is the diversification taregt div_target.

pspec <- add.constraint(portfolio=pspec, type="diversification", div_target=0.7)
# print(pspec)

#### 4.2.2.6 Turnover Constraint

The turnover constraint allows to specify a maximum turnover from a set of initial weights that can either be given or are the weights initially specified for the portfolio object. It is also implemented as an optimization penalty if the turnover deviates more than 5% from the turnover_target.6

pspec <- add.constraint(portfolio=pspec, type="turnover", turnover_target=0.2)
# print(pspec)

#### 4.2.2.7 Target Return Constraint

The target return constraint allows the user to target an average return specified by return_target.

pspec <- add.constraint(portfolio=pspec, type="return", return_target=0.007)
# print(pspec)

#### 4.2.2.8 Factor Exposure Constraint

The factor exposure constraint allows the user to set upper and lower bounds on exposures to risk factors. We will use the factor exposures that we have calculated in @(#sss_3FactorExposure). The major input is a vector or matrix B and upper/lower bounds for the portfolio factor exposure. If B is a vector (with length equal to the number of assets), lower and upper bounds must be scalars. If B is a matrix, the number of rows must be equal to the number of assets and the number of columns represent the number of factors. In this case, the length of lower and upper bounds must be equal to the number of factors. B should have column names specifying the factors and row names specifying the assets.

B <- stocks.factor_exposure %>% as.data.frame() %>% column_to_rownames("symbol")
B=B,
lower=c(0.8,0,-1),
upper=c(1.2,0.8,0))
# print(pspec)

#### 4.2.2.9 Transaction Cost Constraint

The transaction cost constraint enables the user to specify (porportional) transaction costs.7 Here we will assume the proportional transation cost ptc to be equal to 1%.

pspec <- add.constraint(portfolio=pspec, type="transaction_cost", ptc=0.01)
# print(pspec)

#### 4.2.2.10 Leverage Exposure Constraint

The leverage exposure constraint specifies a maximum level of leverage. Below we set leverage to 1.3 to create a 130/30 portfolio.

pspec <- add.constraint(portfolio=pspec, type="leverage_exposure", leverage=1.3)
# print(pspec)

#### 4.2.2.11 Checking and en-/disabling constraints

Every constraint that is added to the portfolio object gets a number according to the order it was set. If one wants to update (enable/disable) a specific constraints this can be done by the indexnum argument.

summary(pspec)
# To get an overview on the specs, their indexnum and whether they are enabled I suggest the follwoing
consts <- plyr::ldply(pspec$constraints, function(x){c(x$type,x$enabled)}) consts pspec$constraints[[which(consts$V1=="box")]] pspec <- add.constraint(pspec, type="box", min=0, max=0.5, indexnum=which(consts$V1=="box"))
pspec$constraints[[which(consts$V1=="box")]]
# to disable constraints
pspec$constraints[[which(consts$V1=="position_limit")]]
pspec <- add.constraint(pspec, type="position_limit", enable=FALSE, # only specify argument if you do enable the constraint
indexnum=which(consts$V1=="position_limit")) pspec$constraints[[which(consts$V1=="position_limit")]] ### 4.2.3 Objectives For an optimal portfolio there first has to be specified what optimal in terms of the relevant (business) objective. Such objectives (target functions) can be added to the portfolio object with add.objective. With this function, the user can specify the type of objective to add to the portfolio object. Currently available are ‘return’, ‘risk’, ‘risk budget’, ‘quadratic utility’, ‘weight concentration’, ‘turnover’ and ‘minmax’. Each type of objective has additional arguments that need to be specified. Several types of objectives can be added and enabled or disabled by specifying the indexnum argument. #### 4.2.3.1 Portfolio Risk Objective Here, the user can specify a risk function that should be minimized. We start by adding a risk objective to minimize portfolio variance (minimum variance portfolio). Another example could be the expected tail loss with a confidence level 0.95. Whatever function (even user defined ones are possble, the name must correspond to a function in R), necessary additional arguments to the function have to be passed as a named list to arguments. Possible functions are: pspec <- add.objective(portfolio=pspec, type='risk', name='var') pspec <- add.objective(portfolio=pspec, type='risk', name='ETL', arguments=list(p=0.95), enabled=FALSE) # print(pspec) #### 4.2.3.2 Portfolio Return Objective The return objective allows the user to specify a return function to maximize. Here we add a return objective to maximize the portfolio mean return. pspec <- add.objective(portfolio=pspec, type='return', name='mean') # print(pspec) #### 4.2.3.3 Portfolio Risk Budget Objective The portfolio risk objective allows the user to specify constraints to minimize component contribution (i.e. equal risk contribution) or specify upper and lower bounds on percentage risk contribution. Here we specify that no asset can contribute more than 30% to total portfolio risk. See the risk budget optimization vignette for more detailed examples of portfolio optimizationswith risk budgets. pspec <- add.objective(portfolio=pspec, type="risk_budget", name="var", max_prisk=0.3) pspec <- add.objective(portfolio=pspec, type="risk_budget", name="ETL", arguments=list(p=0.95), max_prisk=0.3, enabled=FALSE) # for an equal risk contribution portfolio, set min_concentration=TRUE pspec <- add.objective(portfolio=pspec, type="risk_budget", name="ETL", arguments=list(p=0.95), min_concentration=TRUE, enabled=FALSE) print(pspec) ## ************************************************** ## PortfolioAnalytics Portfolio Specification ## ************************************************** ## ## Call: ## portfolio.spec(assets = stocks.selection$symbol, category_labels = stocks.selectionsector) ## ## Number of assets: 10 ## Asset Names ## [1] "AAPL" "MSFT" "AMZN" "CSCO" "NVDA" "ORCL" "AMGN" "ADBE" "QCOM" "GILD" ## ## Category Labels ## Information Technology : AAPL MSFT CSCO NVDA ORCL ADBE QCOM ## Consumer Discretionary : AMZN ## Health Care : AMGN GILD ## ## ## Constraints ## Enabled constraint types ## - full_investment ## - box ## - group ## - position_limit ## - diversification ## - turnover ## - return ## - factor_exposure ## - transaction_cost ## - leverage_exposure ## ## Objectives: ## Enabled objective names ## - var ## - mean ## - var ## Disabled objective names ## - ETL ### 4.2.4 Solvers Solvers are the workhorse of our portfolio optimization framework, and there are a variety of them available to us through the portfolioAnalytics-package. I will briefly introduce the available solvers. Note that these solvers can be specified through optimize_method in the optimize.portfolio and optimize.portfolio.rebalancing method. #### 4.2.4.1 DEOptim This solver comes from the R package DEoptim and is a differential evolution algorithm (a global stochastic optimization algorithm) developed by (???). The help on ?DEoptim gives many more references. There is also a nice vignette("DEoptimPortfolioOptimization") on large scale portfolio optimization using the portfolioAnalytics-package. #### 4.2.4.2 Random Portfolios There are three methods to generate random portfolios contained in portfolioAnalytics:8 1. The most flexible but also slowest method is ‘sample’. It can take leverage, box, group, and position limit constraints into account. 2. The ‘simplex’ method is useful to generate random portfolios with the full investment and min box constraints (values for min_sum/ max_sum are ignored). Other constraints (box max, group and position limit constraints will be handled by elimination) which might leave only very few feasible portfolios. Sometimes it will also lead to suboptimal solutions. 3. Using grid search, the ‘grid’ method only satisfies the min and max box constraints. #### 4.2.4.3 pso The psoptim function from the R package pso (Bendtsen. 2012) and uses particle swarm optimization. #### 4.2.4.4 GenSA The GenSA function from the R package GenSA (Gubian et al. 2018) and is based on generalized simmulated annealing (a generic probabilistic heuristic optimization algorithm) #### 4.2.4.5 ROI The ROI (R Optimization Infrastructure) is a framework to handle optimization problems in R. It serves as an interface to the Rglpk package and the quadprog package which solve linear and quadratic programming problems. Available methods in the context of the portfolioAnalytics-package are given below (see section @(#sss_4Objectives) for available objectives. 1. Maxmimize portfolio return subject leverage, box, group, position limit, target mean return, and/or factor exposure constraints on weights. 2. Globally minimize portfolio variance subject to leverage, box, group, turnover, and/or factor exposure constraints. 3. Minimize portfolio variance subject to leverage, box, group, and/or factor exposure constraints given a desired portfolio return. 4. Maximize quadratic utility subject to leverage, box, group, target mean return, turnover, and/or factor exposure constraints and risk aversion parameter. (The risk aversion parameter is passed into optimize.portfolio as an added argument to the portfolio object). 5. Minimize ETL subject to leverage, box, group, position limit, target mean return, and/or factor exposure constraints and target portfolio return. #### 4.2.4.6 Multicore capabilities For large sets of assets and solvers that can parallelize, the doParallel-package provides multicore capabilities to the portfolioAnalytics package. # require(doParallel) # registerDoParallel() ## 4.3 Optimization examples Modern portfolio theory suggests how rational investors should optimize their portfolio(s) of risky assets to take advantage of diversification effects. This is possible, because risk as opposed to return is not additive and depends very much on the pairwise comovement (correlation) between the risky assets. The concepts of modern portfolio theory go back to Markowitz (1952) considers asset returns as random variables and estimates their risk and return plainly as mean and standard deviation within the available data sample. Usually we do not introduce any constraints except the full investment constraint that constrains the portfolio weights $$w$$ to sum to $$1$$, but allow for unlimited shortselling. The two basic portfolio problems are either to minimize the portfolio risk $$w'\Sigma w$$ given a target return $$\bar{r}$$ (4.1) or maximize the portfolio return $$w'\mu$$ given a target level of risk $$\bar{\sigma}$$ (4.2): \begin{align} & \underset{w}{\text{min}} & & w'\hat{\Sigma}w \\ & \text{subject to} & & w'\hat{\mu}=\bar{r}, \\ &&& w'\mathbf{1}=1. \tag{4.1} \end{align} and \begin{align} & \underset{w}{\text{max}} & & w'\hat{\mu}\\ & \text{subject to} & & w'\hat{\Sigma}w=\bar{\sigma}, \\ &&& w'\mathbf{1}=1. \tag{4.2} \end{align} The solutions to these two problems is a hyperbola that depicts the efficient frontier in the $$\mu$$-$$\sigma$$-diagram. We can plot the efficient frontier using create.EfficientFrontier9 if we specify a return (mean) and a risk (sd) objective:10 pspec <- portfolio.spec(assets = stocks.selectionsymbol,
category_labels = stocks.selection$sector) pspec <- add.constraint(portfolio=pspec, type="full_investment") # weights sum to 1 p <- add.constraint(portfolio=p, type="box", min = -0.5, max = +0.5) # to create the efficient frontier pspec <- add.objective(portfolio=pspec, type="return", name="mean") # mean pspec <- add.objective(portfolio=pspec, type="risk", name="var") # uses sd meansd.ef <- create.EfficientFrontier( R = returns, portfolio = pspec, type = "mean-sd", n.portfolios = 25, ) # summary(meansd.ef, digits=2) # to print the whole efficient frontier # meansd.ef$frontier[1:2,] # shows the first two portfolios
chart.EfficientFrontier(meansd.ef,
match.col="StdDev", # which column to use for risk
type="l",
RAR.text="Sharpe Ratio",
tangent.line = FALSE,
chart.assets=TRUE,
labels.assets=TRUE,xlim=c(0.03,0.20),ylim=c(0,0.055))


text:
+0

# Question in math

0
121
2

how would you solve this 3/5 of__=24? Because i don't know what it means by that?

Feb 23, 2022

#1
+675
+1

3/5 of x = 24 solve for x

24 : 3/5 = 24/1 : 3/5 = 24/1 * 5/3 = 40.

so x = 40 - 3/5 of 40 = 24.

Feb 23, 2022
#2
+13890
+1

How would you solve this 3/5 of__=24?

Hello Guest!

That means

$$\frac{3}{5}\cdot x =24\ |\ \cdot \frac{5}{3}\\ x=24\cdot \frac{5}{3}\\ \color{blue}x=40$$

$$3/5\ of\ \underline{40}=24$$

!

Feb 23, 2022


text:
# principle of inclusion-exclusion

The principle of inclusion-exclusion provides a way of methodically counting the union of possibly non-disjoint sets.

Let $C=\{A_{1},A_{2},\dots A_{N}\}$ be a finite collection of finite sets. Let $I_{k}$ represent the set of $k$-fold intersections of members of $C$ (e.g., $I_{2}$ contains all possible intersections of two sets chosen from $C$).

Then

 $\left|\bigcup_{i=1}^{N}A_{i}\right|=\sum_{j=1}^{N}\left((-1)^{(j+1)}\sum_{S\in I% _{j}}|S|\right)$

For example:

 $|A\cup B|=|A|+|B|-|A\cap B|$
 $|A\cup B\cup C|=|A|+|B|+|C|-(|A\cap B|+|A\cap C|+|B\cap C|)+|A\cap B\cap C|$

The principle of inclusion-exclusion, combined with de Morgan’s laws, can be used to count the intersection of sets as well. Let $A$ be some universal set such that $A_{k}\subseteq A$ for each $k$, and let $\overline{A_{k}}$ represent the complement of $A_{k}$ with respect to $A$. Then we have

 $\left|\bigcap_{i=1}^{N}A_{i}\right|=\left|\overline{\bigcup_{i=1}^{N}\overline% {A_{i}}}\right|$

thereby turning the problem of finding an intersection into the problem of finding a union.

Title principle of inclusion-exclusion PrincipleOfInclusionexclusion 2013-03-22 12:33:25 2013-03-22 12:33:25 Mathprof (13753) Mathprof (13753) 9 Mathprof (13753) Theorem msc 05A99 inclusion-exclusion principle SieveOfEratosthenes2 BrunsPureSieve


text:
# Global existence of timelike minimal surface of general co-dimension in Minkowski space time

Yinxia Wang1* and Jingzi Liu2

Author Affiliations

1 School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China

2 College of Electric Power, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China

For all author emails, please log on.

Boundary Value Problems 2014, 2014:32  doi:10.1186/1687-2770-2014-32

 Received: 13 November 2013 Accepted: 23 January 2014 Published: 7 February 2014

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we prove that the global existence of solutions to timelike minimal surface equations having arbitrary co-dimension with slow decay initial data in two space dimensions and three space dimensions, provided that the initial value is suitably small.

MSC: 35L70.

##### Keywords:
timelike minimal surface; global existence; slow decay initial value

### 1 Introduction

The theory of minimal surfaces has a long history, originating with the papers of Lagrange (1760) and the famous Plateau problem; we refer to the classical papers by Calabi [1] and by Cheng and Yau [2]. Timelike minimal submanifolds may be viewed as simple but nontrivial examples of D-branes, which play an important role in string theory, and the system under consideration here thus has natural generalizations motivated by string theory. The case of timelike surfaces has been investigated by several authors (see [3-5] and [6]). Huang and Kong [7] studied the motion of a relativistic torus in the Minkowski space (). They derived the equations for the motion of relativistic torus in the Minkowski space (). This kind of equation also describes the three dimensional timelike extremal submanifolds in the Minkowski space . They showed that these equations can be reduced to a dimensional quasilinear symmetric hyperbolic system and the system possesses some interesting properties, such as nonstrict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, and the strong null condition (see [8] and [9]). They also found and proved the interesting fact that all plane wave solutions to these equations are lightlike extremal submanifolds and vice versa, except for a type of special solution. For small initial data with compact support, the global existence problem for timelike minimal hypersurfaces has been considered by Brendle [10] and Lindblad [11].

Paul et al.[12] investigated timelike minimal submanifolds of dimension , , of Minkowski spacetimes of dimension , . The authors considered an embedding of into Minkowski spacetime given by the graph of a map . Let Greek indices take values in and let uppercase Latin indices take values in . Introduce cartesian coordinates on and on . The induced metric is

(1.1)

where , and is the Minkowski metric. By variational principles (see [13]), they derived the Euler-Lagrange equations

(1.2)

Moreover for a small initial value with compact support, they also proved the global existence of classical solutions for (1.2).

In this paper, we consider (1.2) with the initial data

(1.3)

where () satisfying

where is a constant and is a small parameter. The aim of this paper is to prove that the Cauchy problem (1.2), (1.3) has a global classical solution, provided that the initial value is sufficiently small and satisfy , (, ). We reduce the restriction on compact support of the initial data to some decay. In other words, we show the global existence of solutions to timelike minimal surface in two space dimensions and three space dimensions, provided that the initial value is suitably small.

To study (1.2), we note that (1.2) can be written in divergence form

(1.4)

where is the Minkowski wave operator and , as well as in the form

(1.5)

where

(1.6)

We raise and lower Greek (intrinsic) indices using and its inverse, while Latin (extrinsic) indices are raised and lowered using the identity and its inverse. From (1.6), it follows that has the symmetries

(1.7)

Due to the symmetries, an energy estimate and local well posedness holds for the system (1.5).

The plan of this paper is as follows. In Section 2, we cite some estimates and prove some estimates on the solution of linear wave equations. The global existence of solutions to timelike minimal surface equations with slow decay initial value in two space dimensions and three space dimensions will be proved in Section 3 and Section 4, respectively.

### 2 Preliminaries

Following Klainerman [14], we introduce a set of partial differential operators

(2.1)

where

(2.2)

(2.3)

(2.4)

and

(2.5)

denotes a product of of the vector fields (2.2), (2.3), (2.4), and (2.5). is a multi-index, , σ is the number of partial differential operators in and

(2.6)

It is easy to prove Lemma 2.1 (see [15]).

Lemma 2.1For any multi-index, we have

(2.7)

and

(2.8)

wherestands for the Poisson bracket, are multi-indices, □ is the wave operator, and, , andare constants.

We need the following lemma that is basically established in [16] and [17]. For completeness, the proof will also be sketched here.

Lemma 2.2Letand satisfy

Assume thatis a solution to the following Cauchy problem:

(2.9)

Then we have

(2.10)

Remark 2.1 Under the condition that

Tsutaya [18] has showed that the solution of the Cauchy problem (2.9) satisfies

Obviously, Lemma 2.2 improves the result in [18].

Proof The solution of (2.9) is given

(2.11)

First, we make an estimate for ; switching to polar coordinates, we have

(2.12)

where and , and χ is the characteristic function of positive numbers.

Let be a continuous function on and . Define

and

where, as before, φ is given by

We will use the following proposition, which is proved in Kovalyov [19].

Proposition 2.1 (I) Ifand, thensatisfies

(2.13)

(II) Ifand, then

(2.14)

whereχis the characteristic function of positive numbers.

We next continue to make an estimate for (2.12); we make an estimate for the right-hand side of (2.12) by dividing into two cases.

Case 1. .

By (2.14), we get

(2.15)

We subdivide into three cases again.

(i) .

Note that

and

Thus,

(2.16)

(ii) .

From (2.15), we have

(2.17)

(iii) .

It follows from (2.15) that

Note that ; we get

Hence

(2.18)

In other words, if and , from (2.16)-(2.18), we get

(2.19)

In what follows, we prove that (2.19) also holds if and . In this case, we also subdivide into two cases.

(1) .

By changing variables, , we obtain

(2.20)

(2) .

Note that and , thus we obtain . From Cases (i)-(iii), we get

(2.21)

In other words, when , from (2.19)-(2.21), we have

(2.22)

Case 2. .

From (2.12), we get

(2.23)

where

and

In what follows, we make an estimate for I and II, respectively, when .

It follows from (2.14) that

(2.24)

By changing variables ,

(2.25)

By (2.13), we get

Let , then

In what follows, we make estimate II by dividing into three cases.

(i) .

(ii) .

(iii) .

In other words, when , we get

(2.26)

For , by changing variables , we obtain

(2.27)

Combining (2.25)-(2.27) gives

(2.28)

Thus (2.22) and (2.28) imply that

(2.29)

By Tsutaya [18], we obtain

(2.30)

and

(2.31)

Equation (2.10) follows from (2.29)-(2.31), and (2.11) immediately. Then we have completed the proof of lemma.  □

The following lemma plays a key role in our main results. It is basically established in [20] and [21].

Lemma 2.3Letand satisfy

Assume thatuis a solution to the following Cauchy problem:

(2.32)

Then

(2.33)

Lemma 2.4Letsatisfy

and assume thatϕdecays to 0 at infinity. If

It follows forthat

(2.34)

where.

For the proof of Lemma 2.4, see Klainerman [22].

Using Lemmas 2.2, 2.3 and the estimate of the linear wave equation with zero initial data, it is not difficulty to prove the following.

Lemma 2.5Suppose that. Letbe the solution to the Cauchy problem

(2.35)

Then

(2.36)

By Lemmas 2.2 and 2.3, we can prove the following lemma.

Lemma 2.6Assume thatandis the solution to the Cauchy problem

(2.37)

where the coefficients () are constants. Then we have

(2.38)

where

anddepends on, and.

### 3 Global existence in three space dimensions

Theorem 3.1Suppose thatand satisfy

whereis a constant. Then there existssuch that forthe Cauchy problem (1.2), (1.3) has a global classical solution for all.

Proof The local existence argument follows from the method of Picard iteration [23] (see also [24] and [12]). In what follows, we will prove the global existence of the classical solutions by a continuous induction, or a bootstrap argument. Let , we set

(3.1)

and

To set up the bootstrap argument, we assume that there is a positive constant K so that on we have the following estimates for the norms defined in (3.1):

(3.2)

and

To close the bootstrap, we can prove that we can in fact choose K sufficiently large and ε suitably small so that the above inequalities hold independent of T with K replaced by .

From Lemma 2.1 and (1.5), we obtain

(3.3)

It follows from Lemma 2.4 and (3.2), (3.3) for that

(3.4)

if K is sufficiently large and is suitably small.

Note that ; from (1.4) and Lemma 2.1, we have

(3.5)

where again at most one of the can satisfy .

Applying Lemma 2.6 to (3.5), we obtain

(3.6)

if K is sufficiently large and is suitably small.

Since , (3.3) may also be written as

(3.7)

where .

Using Lemma 2.5, (3.2), and (3.7), we get

if K is sufficiently large and is suitably small.

So

(3.8)

if K is sufficiently large and is suitably small.

From (3.1), we know that the estimate for implies the desired estimate for . We have completed the proof of Theorem 3.1. □

### 4 Global existence in two space dimensions

Theorem 4.1Suppose thatand satisfy

whereis a constant. Then there existssuch that forthe Cauchy problem (1.2), (1.3) has a global classical solution for all.

Proof The local existence argument follows from the method of Picard iteration [23] (see also [24] and [12]). In what follows, we will prove global existence of classical solutions by a continuous induction, or bootstrap argument. Let , we set

(4.1)

and

To set up the bootstrap argument, we assume that there is a positive constant K so that on we have following estimates for the norms defined in (4.1),

(4.2)

and

where is a fixed, arbitrary constant.

To close the bootstrap, we can prove that we can in fact choose K sufficiently large and ε suitably small so that the above inequalities hold independent of T with K replaced by .

It follows from Lemma 2.4 and (3.3) for that

(4.3)

if K is sufficiently large and is suitably small.

Applying Lemma 2.6 to (3.5), we obtain

(4.4)

if K is sufficiently large and is suitably small.

In what follows, we make an estimate for . In order to make this estimate for , define the following null forms:

(4.5)

and

(4.6)

Let Q symbolically stand for any of the full forms (4.5) and (4.6). Then

(4.7)

for some constants .

Let Q be one of null form in (4.5)-(4.7), we have

(4.8)

Note that the Lagrangian associated to the volume element of the induced metric is . For small , we have

and thus the Euler-Lagrange equations take the form

For small , we obtain

So we have

(4.9)

By Lemma 2.1, (4.9), we have

(4.10)

where .

From Lemma 2.5 and (4.1), (4.2), (4.8), (4.10) when , i.e., , we get

(4.11)

if K is sufficiently large and is suitably small and since .

From (4.1), we know that the estimate for implies the desired estimate for . We have completed the proof of the theorem. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed to each part of this work equally and read and approved the final manuscript.

### References

1. Calabi, E: Examples of Bernstein problems for some nonlinear equations. Proc. Sysmpos. Pure Math., pp. 223–230. Am. Math. Soc., Providence (1970)

2. Cheng, S, Yau, S: Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces. Ann. Math.. 104, 407–419 (1976). Publisher Full Text

3. Kong, D, Sun, Q, Zhou, Y: The equation for time-like extremal surfaces in Minkowski space . J. Math. Phys.. 47, (2006) Article ID 01303

4. Zhou, Q: The dynamics of relativistic strings moving in the Minkowski space . Commun. Math. Phys.. 269, 153–174 (2007)

5. Kong, D, Zhang, Q: Solution formula and time-periodicity for the motion of relativistic strings in the Minkowski space . Phys. D, Nonlinear Phenom.. 238, 902–922 (2009). Publisher Full Text

6. Milnor, T: Entire timelike minimal surfaces in . Mich. Math. J.. 37, 163–177 (1990)

7. Huang, S, Kong, D: Equations for the motion of relativistic torus in the Minkowski space . J. Math. Phys.. 48, (2007) Article ID 103508

8. Christodoulou, D: Global solutions of nonlinear hyperbolic equations for small initial data. Commun. Pure Appl. Math.. 39, 267–282 (1986). Publisher Full Text

9. Klainerman, S: The Null Condition and Global Existence to Nonlinear Wave Equations, pp. 293–326. Am. Math. Soc., Providence (1986)

10. Brendle, S: Hypersurfaces in Minkowski space with vanishing mean curvature. Commun. Pure Appl. Math.. 28, 1249–1279 (2002)

11. Lindblad, H: A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time. Proc. Am. Math. Soc.. 132, 1095–1102 (2004). Publisher Full Text

12. Paul, A, Lars, A, James, I: Timelike minimal submanifolds of general co-dimension in Minkowski space time. J. Hyperbolic Differ. Equ.. 3, 691–700 (2006). Publisher Full Text

13. Katsouleas, G, Maroulas, J: Variational characterizations for eigenfunctions of analytic self-adjoint operator functions. Opusc. Math.. 33(2), 307–321 (2013). Publisher Full Text

14. Klainerman, S: Uniform decay estimates and the Lorentz invariance of the classical wave equations. Commun. Pure Appl. Math.. 38, 321–332 (1985). Publisher Full Text

15. Li, T, Zhou, Y: Life-span of classical solution to fully nonlinear wave equations. II. Nonlinear Anal., Theory Methods Appl.. 19, 833–853 (1992). Publisher Full Text

16. Wang, Y: Global existence of classical solutions to the minimal surface equation in two space dimensions with slow decay initial value. J. Math. Phys.. 50, (2009) Article ID 103506

17. Kong, D, Liu, K, Wang, Y: Life-span of classical solutions to hyperbolic geometric flow in two space variables with slow decay initial data. Commun. Partial Differ. Equ.. 36, 162–184 (2011). Publisher Full Text

18. Tsutaya, K: Global existence theorem for semilinear wave equations with non-compact data in two space dimensions. J. Differ. Equ.. 104, 332–360 (1993). Publisher Full Text

19. Kovalyov, M: Long-time behaviour of solutions of systems of nonlinear wave equations. Commun. Partial Differ. Equ.. 12, 471–501 (1987). Publisher Full Text

20. Asakura, F: Existence of a global solution to a semilinear wave equation with slowly decreasing initial data in three space dimensions. Commun. Partial Differ. Equ.. 11, 1459–1487 (1986). Publisher Full Text

21. Wang, Y: Global existence of classical solutions to the minimal surface equation with slow decay initial value. Appl. Math. Comput.. 216, 576–583 (2010). Publisher Full Text

22. Klainerman, S: Global existence for nonlinear wave equation. Commun. Pure Appl. Math.. 33, 43–101 (1980). Publisher Full Text

23. Hörmander, L: Lectures on Nonlinear Hyperbolic Differential Equations, Springer, Berlin (1997)

24. Sogge, C: Lecures on Nonlinear Wave Equations, International Press, Somerville (1995)


text:
# Arithmetic/Knowledge Step-By-Step Number Puzzle! (What Is The Final Answer?) [closed]

1. Start with the number of U.S. ounces in a U.S. gallon.
2. Subtract the atomic number for Samarium.
3. Add the amount of days in February on a leap year.
4. Multiply this by 6 factorial.
5. Divide this by pi rounded to the nearest whole.
6. Divide this by xx+x if x=4
7. Add xxxxxx if x=y and y = 2z and z=3
8. Multiply the amount of prime numbers under 20
10. Divide this by 1001110001000000000 in the base 10 system (It's in binary there)
12. Divide this by 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 5 * 5 * 5 * 5 * 5 * 5 * 5

• Hi! Welcome to Puzzling Beta! Please read help center and tour to find out more to create puzzles of high quality, and especially this post: puzzling.stackexchange.com/questions/5880/… As commented by Joe Z., this isn't so much a puzzle as it is a list of instructions. – justhalf Apr 9 '15 at 7:09
• I was going to comment that this doesn't seem much worse than this puzzle with a positive score, but in fact maybe it does. There isn't much to look up that people won't know already (except the Samarium thing), and it's not really very challenging provided you have a big calculator. Maybe it's even more of a maths problem than a maths puzzle - see this meta post. – Rand al'Thor Apr 9 '15 at 8:33

This isn't so much a puzzle as it is a list of instructions to get a secret number, but I'll humour you.

1. Start with the number of U.S. ounces in a U.S. gallon.

There are 8 ounces in a cup, 2 cups in a pint, 2 pints in a quart, and 4 quarts in a gallon, for a total of 8 × 2 × 2 × 4 = 128 ounces in a gallon.

1. Subtract the atomic number for Samarium.

Samarium has an atomic number of 62, so our result is 128 - 62 - 66.

1. Add the amount of days in February on a leap year.

February has 29 days in a leap year, so 66 + 29 = 95.

1. Multiply this by 6 factorial.

6 factorial is 720, so 95 × 720 is 68400.

1. Divide this by pi rounded to the nearest whole.

Pi rounded to the nearest whole is 3, so 68400 ÷ 3 = 22800.

1. Divide this by xx+x if x=4

44 + 4 = 48, and 22800 divided by 48 is 475.

1. Add xxxxxx if x=y and y = 2z and z=3

If z = 3, then y = 6 and x = 6, so we add 666666 to get 667141.

1. Multiply the amount of prime numbers under 20

There are eight prime numbers under 20 (2, 3, 5, 7, 11, 13, 17, 19), so we multiply by 8 to get 5337128.

20 factorial is 2432902008176640000 (Wolfram Alpha), so 5337128 plus that is 2432902008181977128.

1. Divide this by 1001110001000000000 in the base 10 system (It's in binary there)

1001110001000000000 in base 2 is 320000 in base 10, so our result is 7602818775568.678525.


text:
# Plotting Flows with riverplot

August 14, 2014
By

(This article was first published on Exegetic Analytics » R, and kindly contributed to R-bloggers)

I have been looking for an intuitive way to plot flows or connections between states in a process. An obvious choice is a Sankey Plot, but I could not find a satisfactory implementation in R… until I read the riverplot post by January Weiner. His riverplot package does precisely what I am need.

Getting your data into the right format is a slightly clunky procedure. However, my impression is that the package is still a work in progress and it’s likely that this process will change in the future. For now though, here is an illustration of how a multi-level plot can be constructed.

## Assembling the Data

The plan for this example is to have four nodes at each of six layers, with flows between layers. The data are a little contrived, but they illustrate the procedure quite nicely and they produce a result which is not dissimilar to the final plot I was after. We have to create data structures for both nodes and edges. I will start with the edges and then use these data to extract the nodes.

The edges data frame consists of records with a “from” node (N1) and a “to” node (N2) as well as a value for the flow between them. Here I systematically construct a grid of random flows and remove some records to break the symmetry.

> edges = data.frame(N1 = paste0(rep(LETTERS[1:4], each = 4), rep(1:5, each = 16)),
+                    N2 = paste0(rep(LETTERS[1:4], 4), rep(2:6, each = 16)),
+                    Value = runif(80, min = 2, max = 5) * rep(c(1, 0.8, 0.6, 0.4, 0.3), each = 16),
+                    stringsAsFactors = F)
>
> edges = edges[sample(c(TRUE, FALSE), nrow(edges), replace = TRUE, prob = c(0.8, 0.2)),]
N1 N2  Value
1  A1 A2 2.3514
4  A1 D2 2.2052
5  B1 A2 3.0959
7  B1 C2 2.8756
9  C1 A2 4.5099
10 C1 B2 4.1782

The names of the nodes are then extracted from the edge data frame. Horizontal and vertical locations for the nodes are calculated based on the labels. These locations are not strictly necessary because the package will work out sensible default values for you.

> nodes = data.frame(ID = unique(c(edges$N1, edges$N2)), stringsAsFactors = FALSE)
> #
> nodes$x = as.integer(substr(nodes$ID, 2, 2))
> nodes$y = as.integer(sapply(substr(nodes$ID, 1, 1), charToRaw)) - 65
> #
> rownames(nodes) = nodes$ID > head(nodes) ID x y A1 A1 1 0 B1 B1 1 1 C1 C1 1 2 D1 D1 1 3 A2 A2 2 0 B2 B2 2 1  Finally we construct a list of styles which will be applied to each node. It’s important to choose suitable colours and introduce transparency for overlaps (which is done here by pasting “60″ onto the RGB strings). > library(RColorBrewer) > # > palette = paste0(brewer.pal(4, "Set1"), "60") > # > styles = lapply(nodes$y, function(n) {
+   list(col = palette[n+1], lty = 0, textcol = "black")
+ })
> names(styles) = nodes\$ID

## Constructing the riverplot Object

Now we are in a position to construct the riverplot object. We do this by joining the node, edge and style data structures into a list and then adding “riverplot” to the list of class attributes.

> library(riverplot)
>
> rp <- list(nodes = nodes, edges = edges, styles = styles)
> #
> class(rp) <- c(class(rp), "riverplot")

Producing the plot is then simple.

> plot(rp, plot_area = 0.95, yscale=0.06)

## Conclusion

I can think of a whole host of applications for figures like this, so I am very excited about the prospects. I know that I am going to have to figure out how to add additional labels to the figures, but I’m pretty sure that will not be too much of an obstacle.

The current version of riverplot is v0.3. Incidentally, when I stumbled on a small bug in v0.2 of riverplot, January was very quick to respond with a fix.


text:
## Brackets in representation algebras of Hopf algebras

For any graded bialgebras $A$ and $B$, we define a commutative graded algebra $A_B$ representing the functor of $B$-representations of $A$. When $A$ is a cocommutative graded Hopf algebra and $B$ is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in $A_B$ from a Fox pairing in $A$ and a balanced biderivation in $B$. Our construction is inspired by Van den Bergh's non-commutative Poisson geometry, and may be viewed as an algebraic generalization of the Atiyah--Bott--Goldman Poisson structures on moduli spaces of representations of surface group

Publication Date:
Jun 11 2018
Date Submitted:
Jun 28 2019
Citation:
Journal of Noncommutative Geometry
12
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External Resources:

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