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# Frictional Force 2

INTRODUCTION:
This is a problem from my Introduction to Physical Science class using "Conceptual Physics" 10th Ed.by Paul G. Hewitt

EXACT PROBLEM:
"A hockey puck slides 20.0m across a frozen pond in a time of 8.0 seconds before coming to a rest."

PROBLEMS FACED:
Find the coefficient of kinetic friction between the ice and the puck.

MY THOUGHTS:
I don't know what to do here really. I am not sure which formulas to use.

Last edited:

you use the equation x= x0 + vt +at^2 to determine how quickly the puck deaccelerates and plug that rate into the friction equation.

you use the equation x= x0 + vt +at^2 to determine how quickly the puck deaccelerates and plug that rate into the friction equation.

That sounds like a plan! So I need to use one of these friction equations?
https://www.physicsforums.com/showpost.php?p=920984&postcount=4

you use the equation x= x0 + vt +at^2 to determine how quickly the puck deaccelerates and plug that rate into the friction equation.

so how do I rewrite this to solve for (de)acceleration?

Does this look correct?
a = (x-x0-v0t)/(.5t)

Do the X variables cancel out?

Last edited:
hage567
Homework Helper
"you use the equation x= x0 + vt +at^2 to determine how quickly the puck deaccelerates and plug that rate into the friction equation."

I don't think this the proper equation to use, at least not by itself. V is this equation represents the initial velocity, which we don't know. So you can start by getting an expression for the initial velocity (Vo, by using another kinematic equation), and then put that into this equation for V.

If you bring arrange the equation so (X-Xo) = Vo + 0.5at^2, you can see that the (X-Xo) is just the displacement of the puck (20 - 0 = 20 m).

"you use the equation x= x0 + vt +at^2 to determine how quickly the puck deaccelerates and plug that rate into the friction equation."

I don't think this the proper equation to use, at least not by itself. V is this equation represents the initial velocity, which we don't know. So you can start by getting an expression for the initial velocity (Vo, by using another kinematic equation), and then put that into this equation for V.

If you bring arrange the equation so (X-Xo) = Vo + 0.5at^2, you can see that the (X-Xo) is just the displacement of the puck (20 - 0 = 20 m).

how can I figure out the velocity if I don't know the acceleration?

hage567
Homework Helper
You can get an equation for Vo in terms of a by using one equation. Then you put that into the other equation you have with Vo and a, and you will only have "a" left over to solve for. It is a two equations, two unknowns kind of thing.

You can get an equation for Vo by using one equation. Then you put that into the other equation you have with Vo and a, and you will only have "a" left over to solve for. It is a two equations, two unknowns kind of thing.

To me, this is like telling a lost person to just go North, when the lost person doesn't even know which direction north is.

I need a point of reference and a starting point. I know the forum rules about how you aren't supposed to do my work, but how are you helping me if you only give me a small fraction of advice every time?

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Well, that's because we want you to try to work it out on your own. Unfortunately people who are lost see these pieces of advice as cryptic hints. But, you should probably know these two equations:

$$x = x_0 + v_0t + \frac{1}{2}gt^2$$

$$v = v_0 + at$$

Then think about what the problem gives you. The problem says the puck is initially traveling at some speed v0 (which you don't know) and it is decelerating because of friction. It travels a certain distance from the point x0 = 0 to x = 20 in t = 8 seconds. See if you can do it from there!

Well, that's because we want you to try to work it out on your own. Unfortunately people who are lost see these pieces of advice as cryptic hints. But, you should probably know these two equations:

$$x = x_0 + v_0t + \frac{1}{2}gt^2$$

$$v = v_0 + at$$

Then think about what the problem gives you. The problem says the puck is initially traveling at some speed v0 (which you don't know) and it is decelerating because of friction. It travels a certain distance from the point x0 = 0 to x = 20 in t = 8 seconds. See if you can do it from there!

I am familiar with these equations, but why did you replace variable a with variable g? Gravity, I take it?

Oh crap, yes. That's just a typo, I'm so used to using g there it just slipped out :X But yes, the same equation, just use a instead of g.

Oh crap, yes. That's just a typo, I'm so used to using g there it just slipped out :X But yes, the same equation, just use a instead of g.

so does that mean it is:

x = x0 + v0t + (1/2)gt^2
20 = 0 + v0(8) + (1/2)(9.8)(8^2)
20 = v0(8) + 313.6
-293.6/8 = v0
v0 = 36.7 m/s

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No, it looks like this typo is causing more trouble than it was worth :X Your acceleration is not g, you aren't even considering gravity in this problem. The equations I meant to write are:

$$x = x_0 + v_0t + \frac{1}{2}at^2$$

$$v = v_0 + at$$

No, it looks like this typo is causing more trouble than it was worth :X Your acceleration is not g, you aren't even considering gravity in this problem. The equations I meant to write are:

$$x = x_0 + v_0t + \frac{1}{2}at^2$$

$$v = v_0 + at$$

superb. idk then.

Alright, we have those two equations, and the problem told us:

x0 = 0 m
x = 20.0 m
t = 8.0 s
v = 0 m/s (if the object comes to rest, it doesn't have velocity!)
v0 = ?
a = ?

We have two equations and two unknowns, so we need to solve a system of equations! Go ahead and write those numbers into those equations and you'll probably see how to solve it. It's just algebra from here!

Alright, we have those two equations, and the problem told us:

x0 = 0 m
x = 20.0 m
t = 8.0 s
v = 0 m/s (if the object comes to rest, it doesn't have velocity!)
v0 = ?
a = ?

We have two equations and two unknowns, so we need to solve a system of equations! Go ahead and write those numbers into those equations and you'll probably see how to solve it. It's just algebra from here!

Ok then:

V = v0 + at
0 = v0 + a8
v0 = -a8

x = x0 + v0t +(.5)a(t^2)
20 = 0 +(-a8) + (.5)a(8^2)
20 = -a64 + a32
.625 = -a64 +a
.0097 = -a +a
.0097 = 0 ???

well now, maybe that algebra isn't so simple after all...

Try to do it without the numbers first, solve for it in terms of variables only. All the numbers is confusing you. When you have the eqn already, then plug in the numbers.

Ok then:

V = v0 + at
0 = v0 + a8
v0 = -a8

x = x0 + v0t +(.5)a(t^2)
20 = 0 +(-a8) + (.5)a(8^2)
20 = -a64 + a32
So far, so good...

.625 = -a64 +a
Here is where you made a mistake. If 20 = -a*64 + a*32, isn't this the same as
20 = -64*a + 32*a
20 = -32*a

So far, so good...

Here is where you made a mistake. If 20 = -a*64 + a*32, isn't this the same as
20 = -64*a + 32*a
20 = -32*a

OHHHHHH! Stupid mistake there!

Ok then:

V = v0 + at
0 = v0 + a8
v0 = -a8

x = x0 + v0t +(.5)a(t^2)
20 = 0 +(-a8) + (.5)a(8^2)
20 = -a64 + a32
20 = -32a
a = -.625

v0 = -a8
v0 = .625(8)
v0 = 5

so here's all of the variables that we know now:
x0 = 0 m
x = 20.0 m
t = 8.0 s
v = 0
v0 = 5
a = -.625

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Alright! So we have our acceleration on the puck. Now, remember Newton's second law, F = ma. We can ignore forces in the y-direction because they are mg and N, and they cancel exactly in this case. So, we are only concerned about forces along the x-axis. And that force acting on the puck is...?

Alright! So we have our acceleration on the puck. Now, remember Newton's second law, F = ma. We can ignore forces in the y-direction because they are mg and N, and they cancel exactly in this case. So, we are only concerned about forces along the x-axis. And that force acting on the puck is...?

Friction?
Ffrictional = (Muk)(Fn) = (Muk)mg ???

That's exactly right. Now, F = ma, and we have the only force acting on our puck (friction = mu*m*g), we have the mass, and we have the acceleration...let's plug it all in.

That's exactly right. Now, F = ma, and we have the only force acting on our puck (friction = mu*m*g), we have the mass, and we have the acceleration...let's plug it all in.

Force = ma
Force = (20)(5)
Force = 100N

I don't see where Force plugs into (friction = mu*m*g).

Not quite. When you know the forces which are acting on the object, you can add up the forces (to obtain the NET force acting on an object) and plug them in for the F in F=ma.

Just as a sidenote, unrelated to this problem, the way you might do this is the following: Say you have a force A pulling an object to the right, and a force B pulling the object to the left. If you say the +x direction is to the right, we can say that the NET force F = A - B = ma. Then, if you know A and B and m, you can figure out how the object will accelerate, or if you know m and a and B, you can figure out A, and so on.

So, in this case, since there is only one force acting on the object, friction, that friction formula is your F in F = ma.

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Not quite. When you know the forces which are acting on the object, you can add up the forces (to obtain the NET force acting on an object) and plug them in for the F in F=ma.

Just as a sidenote, unrelated to this problem, the way you might do this is the following: Say you have a force A pulling an object to the right, and a force B pulling the object to the left. If you say the +x direction is to the right, we can say that the NET force F = A - B = ma. Then, if you know A and B and m, you can figure out how the object will accelerate, or if you know m and a and B, you can figure out A, and so on.

So, in this case, since there is only one force acting on the object, friction, take the formula for the force of friction and put it in for F in F = ma.

I realized my 20 is 20 METERS.

(mu)mg = ma
(mu)m(9.8) = m(5)
(mu) = [(m)(9.8)]/[(m)(5)]

if this is correct, I still don't have a mass of the hockey puck.

First, where did 5 come from? Shouldn't a = -.625 (the value for acceleration we figured out earlier)?

And, look at your equation: m cancels!

First, where did 5 come from? Shouldn't a = -.625 (the value for acceleration we figured out earlier)?

And, look at your equation: m cancels!

I used the v0 value of 5 instead of the a value of .625.

(mu)mg = ma
(mu)m(9.8) = m(.625)
(mu) = (9.8)/(.625)
mu = 15.68N

is it Newtons?

I did that wrong as well...

(mu)mg = ma
(mu)m(9.8) = m(.625)
(mu) = (.625)/(9.8)
mu = .0637N

There you go, you've got it! It's not Newton's; actually, $\mu$ is a dimensionless value, so it has no units. Can you see why? In your equation, you have mu = .625 m/s^2 / 9.8 m/s^2. The m/s^2 cancel, leaving you with just a pure number, with no units.

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It's not Newton's; actually, $\mu$ is a dimensionless value, so it has no units. Can you see why? In your equation, you have mu = 9.8 m/s^2 / .625 m/s^2. The m/s^2 cancel, leaving you with just a pure number, with no units.

that makes sense.

So final answer to the problem is:
$\mu$ = .0637


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# How to change the background color of a plot created by using plot function in R?

R ProgrammingServer Side ProgrammingProgramming

To change the focus of a plot we can do multiple things and one such thing is changing the background of the plot. If the background color of a plot is different than white then obviously it will get attention of the readers because this is unusual as most of the times the plots have white backgrounds, hence if we want to attract readers on the plot then we might use this technique. It can be done by using par(bg= "color_name").

## Example

Creating a simple histogram −

Live Demo

> x<-rnorm(100)
> hist(x)

## Example

Creating histogram with different background colors −

> par(bg="green")
> hist(x)

## Output

> par(bg="yellow")
> hist(x)

## Output

par(bg="blue")
> hist(x)

## Output

Published on 04-Sep-2020 07:38:10

format: plain


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# Infinite series vs compact representation

I understand the attractiveness and usefulness of infinite-series expansions such as Taylor expansions, but I wonder if they sometimes hide important aspects of the described system.

For example, take $$F(x) = e^x$$, whose Taylor expansion is $$F(x) = \sum_n \frac{x^n}{ n!}.$$ Sure, we can approximate $$F(x)$$ using the first few terms of the Taylor expansion for low values of $$x$$, and that is often useful. However, $$F(x)$$ is actually an exponential function, so has properties that are not easily evident if F(x) is approximated by truncating a Taylor expansion-- such as the beautiful fact that $$\frac{d}{dx}F(x) = F(x)$$.

Similarly, a truncated Fourier series representation of the delta function can obscure the fact that a delta function has precisely zero width.

Though chaos theory isn't in my skill set, I imagine that a truncated infinite series representation of a chaotic system might fail to reveal the fact that infinitesimally narrow domains exist in which the system is actually stationary.

It seems that quantum field theory (QFT) makes extensive use of the Euler-Heisenberg effective Lagrangian, which uses only terms up to the second order in the field invariants. One of the dreams of some field theorists has been to show that massive particles are stationary structures that "condense" out of vacuum fields due to nonlinearities; but it seems that the existence of such structures might be completely hidden if the field Lagrangian is not complete "all the way to the end" of an infinite series representation.

I would like to know if there exists an exact Lagrangian for QFT. Presumably it would not be an infinite power series expansion; instead it would be a relatively compact expression like $$L = e^{H(f(E^2 – B^2), g(E \cdot B))}$$ where $$H$$ is a simple function and $$f$$ and $$g$$ are low-order functions, or “natural” functions like $$cos$$ or $$tanh$$, and would contain only a small handful of constants whose values need to be adjusted to fit constraints set by experimental results. A Lagrangian like this could still be represented as an infinite series, but could also be represented in a very compact non-series form; and in either case it would have only a finite number of adjustable constants.

• That seems quite broad. There are many QFTs, some of which are approximations of other QFTs. Do you have anything specific in mind? – knzhou Feb 26 at 15:39
• Are you aware that the actual Euler-Heisenberg Lagrangian for QED is a complicated integral involving hyperbolic cosines of square roots of the field invariants, not just a power series? See en.wikipedia.org/wiki/Euler–Heisenberg_Lagrangian But this compact representation is still just an approximation which takes only one-loop perturbative quantum effects into account. It has no adjustable constants because it is calculated from QED. – G. Smith Feb 26 at 17:48
• Are you asking if there are any exact results in QFT? – Qmechanic Feb 26 at 18:35
• @Qmechanic, No, I'm pretty sure that there can be exact solutions to equations that, themselves, are not exact. But when a Lagrangian is first represented as an arbitrary infinite-series function and then truncated, we can be quite sure a priori that the resulting field equations are only an approximation to reality, and usually only applicable in the case of weak fields. I'd like to know if there are plausible QFT Lagrangians that span the range from extremely weak to extremely strong fields. – S. McGrew Feb 27 at 2:30
• @DanYand, please expand your "in continuous space without relying on any series expansion" comment. – S. McGrew Feb 27 at 2:34


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# Comparing Contact Structures: What do we Mean when we Say that two Contact Structures are Homotopic/Eliashbergs Class. of OT structures

Please forgive me if this is too simple for MO; most of my posts on anything contact-structure-related in Math Stack, other sites, have barely received answers (maybe because I'm not an expert in the area and my questions are not clear-enough).

1)What do we mean when we say two contact structures $(M,\xi); (M,\xi')$ are homotopic to each other ( as plane fields), and, 2) what is the precise meaning of Eliashberg's classification of O.T contact structures up to homotopy classes of plane fields?

For the first, naively, I would say that there is a pointwise homotopy of planes $\xi_p \rightarrow \xi'_p$, and the homotopy takes place in the ambient space M. But if this was the case, then there would only be , e.g., one class in $\mathbb R^3$ . Do we then consider the orientation of the planes relative to the orientation of the ambient space? This seems too complicated.

For the second, according to results by Eliashberg, the classification of overtwisted structures is equivalent to the classification of homotopy classes of plane fields. Do we mean that homotopy classes of planes coincide with contactomorphic classes of OT structures, i.e., if $(M^3, \xi )$ is overtwisted and H is a homotopy between the contact planes $\xi(p), \xi'(p)$ , then $\xi'$ is contactomorphic with $\xi$?.

Thanks for any help, refs., etc.

I hope I'm not mucking up some definitions below, as I am by no means an expert, but let me attempt to answer your questions. No doubt you have a copy, but just for others, here is a link to the relevant paper of Eliashberg, by the way.

You might also find chapter 4 of Geiges's book "An Introduction to Contact Topology" useful.

(1) Recall that a contact structure on a $(2n+1)$-dimensional manifold is a codimension 1 tangent distribution that can be locally defined by the kernel of a 1-form $\alpha$ satisfying $\alpha\wedge(d\alpha)^n\neq0$. Two rank $k$ distributions (e.g. two contact structures) are said to be homotopic as distributions if there is a smooth family of rank $k$ distributions beginning at the first and ending at the second. Being homotopic as plane fields is just the special case for rank 2 distributions.

If you look at section 1.5 of Eliashberg's paper you'll see that he phrases "homotopic as plane fields" in terms of contact structures lying in the same connected component of the space of 2-plane distributions.

This notion is to be distinguished from being homotopic as contact structures which requires that the family of distributions must satisfy the contact condition throughout. Eliashberg talks about two contact structures lying in the same connected component of the space of contact structures.

As you point out, there is only one homotopy class of plane fields on $\mathbb{R}^3$, however another paper of Eliashberg proves that there are an infinite number of distinct classes of contact structures pairwise non-homotopic as contact structures. Thus the two notions of homotopy are generally quite different.

(2) Eliashberg's theorem on the classification of overtwisted contact structures states that despite the difference in definitions above, when we look at the subset of contact structures that are overtwisted (whose definition I won't give here) the two notions coincide!

Let me try to rephrase your last paragraph. If $(M^3,\xi_0)$ and $(M^3,\xi_1)$ are both overtwisted contact structures and are homotopic as plane fields (i.e. we can find a smooth family of plane fields on $M$ which go from $\xi_0$ to $\xi_1$), then indeed, they are contactomorphic, and we can even find a smooth family of contact structures $(M^3,\xi_t)$ which connect them.

• Is this homotopy a pointwise one, i.e., for fixed p in $M^3$, we have $\xi_0 (p)$ homotopic to $\xi_1 (p)$ , for all p in $M^3$? I would think obviously yes, but I've been wrong way to often when I thought something was "obviously" – Kontakt Mar 30 '14 at 23:38
• Sorry for taking so long to get back to you. Yes, that is indeed true; indeed these homotopies depend smoothly on $p$ as well. – j.c. Apr 4 '14 at 9:39


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# 1 function to call when setInterval has failed you

March 15, 2017

From time to time, we have to check if something is finished, or a particular script has been loaded successfully in the browser, we would make use of the setInterval() function in the vanilla Javascript. This function can be traced all the way back when Javascript first came out. We should be all familiar, but seldom use the function due to the advancement of Javascript libraries and framework. The function is useful, but it has 1 problem. The problem is that in a low latency requirement, this function cannot meet the demand of periodically to run, as soon as possible. Even if you call like this,

setInterval(function() {
//...impl
}, 1);

The function does not guarantee to run every 1ms.

Hold up Nick, actually it is not confusing at all. All can be explained by the lowest design of how browsers are rendering webpages. Webpages is nothing different with the canvas in any game engines. In the canvas, all the rendering logic is put into an endless loop. The loop is something like:

while(true) {
//...
render();
}

The loop is run everytime the 1/60 s at most. 1/60 s for one call equals 60fps. 60fps is also the refresh rate of the screen. But setInterval() DOES NOT guarantee to run on each refresh. How about a function that guarantee to run in each refresh? requestAnimationFrame(fn) is the new-ish function that can put fn into the refresh loop, so it is guarantee to run as much as possible. The refresh loop becomes:

while(true) {
//...
fn();
render();
}

Nothing runs as tight as the refresh loop in the context of browser so you got a win here.

1 function to call when setInterval has failed you - March 15, 2017 - Gilbert Wat (屈振鵬)


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+0

# fraction Dividing

0
116
1

why do you multiply fractions instead of dividing? cause my teacher always said to invert and multiply and i was confused why.

Guest Jun 13, 2017

#1
+4747
+2

It is easier to multiply fractions.

For example:

$$\frac34\div\frac57=\frac34\,*\,\frac75$$

Then, just multiply straight across.

$$=\frac{3*7}{4*5}=\frac{21}{20}$$

It  is  true that

$$\frac34\div\frac57=\frac{3\div5}{4\div7}=\frac{21}{20}$$

But how are you going to evaluate   3 / 5    and   4 / 7    without using a calculator or long division?

And...to see why inverting and multiplying is the same as dividing, take a simple example:

$$10\div2=10\,*\,\frac12=\frac{10}{1}\,*\,\frac12=\frac{10}{2}$$

hectictar  Jun 14, 2017
Sort:

#1
+4747
+2

It is easier to multiply fractions.

For example:

$$\frac34\div\frac57=\frac34\,*\,\frac75$$

Then, just multiply straight across.

$$=\frac{3*7}{4*5}=\frac{21}{20}$$

It  is  true that

$$\frac34\div\frac57=\frac{3\div5}{4\div7}=\frac{21}{20}$$

But how are you going to evaluate   3 / 5    and   4 / 7    without using a calculator or long division?

And...to see why inverting and multiplying is the same as dividing, take a simple example:

$$10\div2=10\,*\,\frac12=\frac{10}{1}\,*\,\frac12=\frac{10}{2}$$

hectictar  Jun 14, 2017

### 13 Online Users

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text:
Browse Questions

# Which are the species in which sulphur undergoes $sp^3$ hybridisation ?

$\begin{array}{1 1}(1)\;SF_4&(2)\;SCl_2\\(3)\;SO_4^{2-}&(4)\;H_2S\end{array}$

$\begin{array}{1 1}(a)\; 1\; and\; 3&(b)\; 2,3\; and\; 4\\(c)\; 1,2\; and\; 4&(d)\; 3 \;and\; 4\end{array}$

Toolbox:
• HOW TO FIND HYBRIDIZATION OF CENTRAL ATOM & SHAPE OF MOLECULE?

STEP-1: Write the Lewis structure
$\quad \quad$ It is better to write the Lewis structural formula to get a rough idea about the structure of molecule and bonding pattern. Use the valence concept to arrive at this structure. Concentrate on the electron pairs and other atoms linked directly to the concerned atom.

STEP-2: Calculate the number of sigma (σ) bonds
Number of σ-bonds formed by the atom in a compound is equal to the number of other atoms with which it is directly linked to.

STEP-3: Calculate the number of lone pairs
Number of lone pairs $=\Large\frac{v-b-c}{2}$
where,
v = no. of valence electrons in the concerned atom in free state (i.e. before bond formation).
b = no. of bonds (including both σ & π bonds) formed by concerned atom.
c = charge on the atom (take care: it may not be the charge on entire molecule or ionic species).

STEP-4: Calculate the steric number
Steric number = no. of σ-bonds + no. of lone pairs

STEP-5: Assign hybridization and shape of molecule
Now, based on the steric number, it is possible to get the type of hybridization of the atom.

table,th,td { border:1px solid black; border-collapse:collapse; } th,td { padding:5px; }
Steric number Hybridization Structure
2 $sp$ linear
3 $sp_2$ trigonal planar
4 $sp_3$ tetrahedral
5 $sp_3d$ trigonal bipyramidal
6 $sp_3d_2$ octahedral
7 $sp_3d_3$ pentagonal bipyramidal
2,3 and 4
Hence (b) is the correct answer.
edited Mar 31, 2014


text:
## Announcements:

Continue reading Section 4.2 for next class. Work through recommended homework questions.

Tutorials: Quiz next week: 3.7, 4.1 and some of 4.2.

Office hour: Monday, 1:30-2:30, MC103B.

Help Centers: Monday-Friday 2:30-6:30 in MC 106.

## Brief summary of Section 4.1: Eigenvalues and eigenvectors

Definition: Let $A$ be an $n \times n$ matrix. A scalar $\lambda$ (lambda) is called an eigenvalue of $A$ if there is a nonzero vector $\vx$ such that $A \vx = \lambda \vx$. Such a vector $\vx$ is called an eigenvector of $A$ corresponding to $\lambda$.

Question: Why do we only consider square matrices here?

Example A: Since $$\bmat{rr} 1 & 2 \\ 2 & -2 \emat \coll 2 1 = \coll 4 2 = 2 \coll 2 1 ,$$ we see that $2$ is an eigenvalue of $\bmat{rr} 1 & 2 \\ 2 & -2 \emat$ with eigenvector $\coll 2 1$.

In general, the eigenvectors for a given eigenvalue $\lambda$ are the nonzero solutions to $(A - \lambda I) \vx = \vec 0$.

Definition: Let $A$ be an $n \times n$ matrix and let $\lambda$ be an eigenvalue of $A$. The collection of all eigenvectors corresponding to $\lambda$, together with the zero vector, is a subspace called the eigenspace of $\lambda$ and is denoted $E_\lambda$. In other words, $$E_\lambda = \null(A - \lambda I) .$$

We worked out many examples, and used an applet to understand the geometry.

### Finding eigenvalues

Given a specific number $\lambda$, we know how to check whether $\lambda$ is an eigenvalue: we check whether $A - \lambda I$ has a nontrivial null space. (And we can find the eigenvectors by finding the null space.)

By the fundamental theorem of invertible matrices, $A - \lambda I$ has a nontrivial null space if and only if it is not invertible. For $2 \times 2$ matrices, we can check invertibility using the determinant!

Example: Find all eigenvalues of $A = \bmat{rr} 1 & 2 \\ 2 & -2 \emat$.

Solution: We need to find all $\lambda$ such that $\det(A-\lambda I) = 0$. \kern-6ex \begin{aligned} \det(A-\lambda I) &= \det \bmat{cc} 1-\lambda & 2 \\ 2 & -2-\lambda \emat \\ &= (1-\lambda)(-2-\lambda)-4 = \lambda^2 + \lambda - 6 , \end{aligned} so we need to solve the quadratic equation $\lambda^2 + \lambda - 6 = 0$. This can be factored as $(\lambda - 2)(\lambda + 3) = 0$, and so $\lambda = 2$ or $\lambda = -3$ are the eigenvalues.

So now we know how to handle the $2 \times 2$ case. To handle larger matrices, we need to learn about their determinants, which is Section 4.2.

## New material: Section 4.2: Determinants

Recall that we defined the determinant of a $2 \times 2$ matrix $A = \bmat{rr} a & b \\ c& d \emat$ by $\det A = a d - b c$. We also write this as $$\kern-6ex \det A = |A| = \bdmat{rr} a & b \\ c& d \edmat = a d - b c .$$

For a $3 \times 3$ matrix $A$, we define $$\kern-9ex \det A = \bdmat{rrr} \red{a_{11}} & \red{a_{12}} & \red{a_{13}} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \edmat = \red{a_{11}} \bdmat{rr} a_{22} & a_{23} \\ a_{32} & a_{33} \edmat - \red{a_{12}} \bdmat{rr} a_{21} & a_{23} \\ a_{31} & a_{33} \edmat + \red{a_{13}} \bdmat{rr} a_{21} & a_{22} \\ a_{31} & a_{32} \edmat$$ If we write $A_{ij}$ for the matrix obtained from $A$ by deleting the $i$th row and the $j$th column, then this can be written $$\kern-9ex \det A = a_{11} \det A_{11} - a_{12} \det A_{12} + a_{13} \det A_{13} = \sum_{j=1}^{3} (-1)^{1+j} \, a_{1j} \det A_{1j}$$ We call $\det A_{ij}$ the $(i,j)$-minor of $A$.

Example: On board.

Example 4.9 in the book shows another method, that doesn't generalize to larger matrices.

### Determinants of $n \times n$ matrices

Definition: Let $A = [a_{ij}]$ be an $n \times n$ matrix. Then the determinant of $A$ is the scalar \kern-9ex \begin{aligned} \det A = |A| &= a_{11} \det A_{11} - a_{12} \det A_{12} + \cdots + (-1)^{1+n} a_{1n} \det A_{1n} \\ &= \sum_{j=1}^{n} (-1)^{1+j} \, a_{1j} \det A_{1j} . \end{aligned}

This is a recursive definition!

Example: $A = \bdmat{rrrr} 2 & 0 & 1 & 0 \\ 3 & 1 & 0 & 0 \\ 1 & 0 & 2 & 3 \\ 2 & 0 & 4 & 5 \edmat$, on board.

The computation can be very long if there aren't many zeros! We'll learn some better methods.

Note that if we define the determinant of a $1 \times 1$ matrix $A = [a]$ to be $a$, then the general definition works in the $2 \times 2$ case as well. So, in this context, $|a| = a$ (not the absolute value!)

It will make the notation simpler if we define the $(i,j)$-cofactor of $A$ to be $$C_{ij} = (-1)^{i+j} \det A_{ij} .$$ Then the definition above says $$\det A = \sum_{j=1}^{n} \, a_{1j} C_{1j} .$$ This is called the cofactor expansion along the first row. It turns out that any row or column works!

Theorem 4.1 (The Laplace Expansion Theorem): Let $A$ be any $n \times n$ matrix. Then for each $i$ we have $$\kern-6ex \det A = a_{i1} C_{i1} + a_{i2} C_{i2} + \cdots + a_{in} C_{in} = \sum_{j=1}^{n} \, a_{ij} C_{ij}$$ (cofactor expansion along the $i$th row). And for each $j$ we have $$\kern-6ex \det A = a_{1j} C_{1j} + a_{2j} C_{2j} + \cdots + a_{nj} C_{nj} = \sum_{i=1}^{n} \, a_{ij} C_{ij}$$ (cofactor expansion along the $j$th column).

The book proves this result at the end of this section, but we won't cover the proof.

The signs in the cofactor expansion form a checkerboard pattern: $$\bmat{ccccc} + & - & + & - & \cdots \\ - & + & - & + & \cdots \\ + & - & + & - & \cdots \\ - & + & - & + & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \emat$$

Example: Redo the previous $4 \times 4$ example, saving work by expanding along the second column. On board. Note that the $+-$ pattern for the $3 \times 3$ determinant is not from the original matrix.

Example: A $4 \times 4$ triangular matrix, on board.

A triangular matrix is a square matrix that is all zero below the diagonal or above the diagonal.

Theorem 4.2: If $A$ is triangular, then $\det A$ is the product of the diagonal entries.

### Better methods

Laplace Expansion is convenient when there are appropriately placed zeros in the matrix, but it is not good in general. It produces $n!$ different terms (explain). A supercomputer would require $10^{30}$ times the age of the universe just to compute a $50 \times 50$ determinant in this way. And that's a puny determinant for real-world applications.

So how do we do better? Like always, we turn to row reduction! These properties will be what we need:

Theorem 4.3: Let $A$ be a square matrix.

a. If $A$ has a zero row, then $\det A = 0$.
b. If $B$ is obtained from $A$ by interchanging two rows, then $\det B = - \det A$.
c. If $A$ has two identical rows, then $\det A = 0$.
d. If $B$ is obtained from $A$ by multiplying a row of $A$ by $k$, then $\det B = k \det A$.
e. If $A$, $B$ and $C$ are identical in all rows except the $i$th row, and the $i$th row of $C$ is the sum of the $i$th rows of $A$ and $B$, then $\det C = \det A + \det B$.
f. If $B$ is obtained from $A$ by adding a multiple of one row to another, then $\det B = \det A$.

All of the above statements are true with rows replaced by columns.

Explain verbally, making use of: $$\kern-6ex \det A = a_{i1} C_{i1} + a_{i2} C_{i2} + \cdots + a_{in} C_{in} = \sum_{j=1}^{n} \, a_{ij} C_{ij}$$ The following will help explain how (f) follows from (d) and (e): $$\kern-8ex A = \collll {\vr_1}{\vr_2}{\vr_3}{\vr_4},\quad B = \collll {\vr_1}{5 \vr_4}{\vr_3}{\vr_4},\quad B' = \collll {\vr_1}{\vr_4}{\vr_3}{\vr_4},\quad C = \ccollll {\vr_1}{\vr_2 + 5 \vr_4}{\vr_3}{\vr_4}$$ $$\kern-9ex \det C = \det A + \det B = \det A + 5 \det B' = \det A + 5 (0) = \det A$$

The bold statements are the ones that are useful for understanding how row operations change the determinant.

Example: Use row operations to compute $\det A$ by reducing to triangular form, where $A = \bmat{rrrr} 2 & 4 & 6 & 8 \\ 1 & 4 & 1 & 2 \\ 2 & 2 & 12 & 8 \\ 1 & 2 & 3 & 9 \emat$. On board.


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# How does one go about using the elimination method to solve an equation?

## I would love a step by step process please! I just can't seem to understand it the way i was taught. Thank you!!! Here is an example: Solve the system using the elimination method. x+y−2z=5 −x+2y+z=2 2x+3y−z=9

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

1
Feb 9, 2018

$x = 1$, $y = 2$ and $z = - 1$

#### Explanation:

Given the system:

$\left\{\begin{matrix}x + y - 2 z = 5 \\ - x + 2 y + z = 2 \\ 2 x + 3 y - z = 9\end{matrix}\right.$

We can eliminate $z$ from the first equation by adding twice the second equation to it to get:

$- x + 5 y = 9$

We can eliminate $z$ from the second equation by adding the third equation to it to get:

$x + 5 y = 11$

We can eliminate $y$ from this last equation by subtracting the previous equation to get:

$2 x = 2$

Then we can divide both sides by $2$ to find:

$\textcolor{b l u e}{x = 1}$

We can substitute this value of $x$ into the previous equation to find:

$1 + 5 y = 11$

Subtracting $1$ from both sides, we get:

$5 y = 10$

Then dividing both sides by $5$ we find:

$\textcolor{b l u e}{y = 2}$

Substituting these values for $x$ and $y$ into the second equation we were given, we find:

$- \left(1\right) + 2 \left(2\right) + z = 2$

That is:

$3 + z = 2$

Subtracting $3$ from both sides, we find:

$\textcolor{b l u e}{z = - 1}$

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3 years ago

# Protoplanetary Disc Response to Distant Tidal Encounters in Stellar Clusters.

C. J. Clarke, R. A. Booth, A. J. Winter, G. Rosotti

The majority of stars form in a clustered environment. This has an impact on the evolution of surrounding protoplanetary discs (PPDs) due to either photoevaporation or tidal truncation. Consequently, the development of planets depends on formation environment. Here we present the first thorough investigation of tidally induced angular momentum loss in PPDs in the distant regime, partly motivated by claims in the literature for the importance of distant encounters in disc evolution. We employ both theoretical predictions and dynamical/hydrodynamical simulations in 2D and 3D. Our theoretical analysis is based on that of Ostriker (1994) and leads us to conclude that in the limit that the closest approach distance $x_{min} \gg r$, the radius of a particle ring, the fractional change in angular momentum scales as $(x_{min}/r)^{-5}$. This asymptotic limit ensures that the cumulative effect of distant encounters is minor in terms of its influence on disc evolution. The angular momentum transfer is dominated by the $m=2$ Lindblad resonance for closer encounters and by the $m=1$, $\omega = 0$ Lindblad resonance at large $x_{min}/r$. We contextualise these results by comparing expected angular momentum loss for the outer edge of a PPD due to distant and close encounters. Contrary to the suggestions of previous works we do not find that distant encounters contribute significantly to angular momentum loss in PPDs. We define an upper limit for closest approach distance where interactions are significant as a function of arbitrary host to perturber mass ratio $M_2/M_1$.

Publisher URL: http://arxiv.org/abs/1801.03510

DOI: arXiv:1801.03510v1

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• ### Manipulation and steering of hyperbolic surface polaritons in hexagonal boron nitride(1709.01892)

Sept. 6, 2017 cond-mat.mes-hall
Hexagonal boron nitride (hBN) is a natural hyperbolic material that supports both volume-confined hyperbolic polaritons (HPs) and sidewall-confined hyperbolic surface polaritons (HSPs). In this work, we demonstrate effective excitation, control and steering of HSPs in hBN through engineering the geometry and orientation of hBN sidewalls. By combining infrared (IR) nano-imaging and numerical simulations, we investigate the reflection, transmission and scattering of HSPs at the hBN corners with various apex angles. We show that the sidewall-confined nature of HSPs enables a high degree of control over their propagation by designing the geometry of hBN nanostructures.
• ### Prospects for Discovering Pulsars in Future Continuum Surveys Using Variance Imaging(1708.01992)

Aug. 7, 2017 astro-ph.IM, astro-ph.HE
In Dai et al. (2016) we developed a formalism for computing variance images from standard, interferometric radio images containing time and frequency information. Variance imaging with future radio continuum surveys allows us to identify radio pulsars and serves as a complement to conventional pulsar searches which are most sensitive to strictly periodic signals. Here, we carry out simulations to predict the number of pulsars that we can uncover with variance imaging on future continuum surveys. We show that the Australian SKA Pathfinder (ASKAP) Evolutionary Map of the Universe (EMU) survey can find $\sim30$ normal pulsars and $\sim40$ millisecond pulsars (MSPs) over and above the number known today, and similarly an all-sky continuum survey with SKA-MID can discover $\sim140$ normal pulsars and $\sim110$ MSPs with this technique. Variance imaging with EMU and SKA-MID will detect pulsars with large duty cycles and is therefore a potential tool for finding MSPs and pulsars in relativistic binary systems. Compared with current pulsar surveys at high Galactic latitudes in the southern hemisphere, variance imaging with EMU and SKA-MID will be more sensitive, and will enable detection of pulsars with dispersion measures between $\sim10$ and 100 cm$^{-3}$ pc.
• ### Observing Pulsars with a Phased Array Feed at the Parkes Telescope(1705.08543)

May 23, 2017 astro-ph.IM
During February 2016, CSIRO Astronomy and Space Science and the Max-Planck-Institute for Radio Astronomy installed, commissioned and carried out science observations with a phased array feed (PAF) receiver system on the 64m diameter Parkes radio telescope. Here we demonstrate that the PAF can be used for pulsar observations and we highlight some unique capabilities. We demonstrate that the pulse profiles obtained using the PAF can be calibrated and that multiple pulsars can be simultaneously observed. Significantly, we find that an intrinsic polarisation leakage of -31dB can be achieved with a PAF beam offset from the centre of the field of view. We discuss the possibilities for using a PAF for future pulsar observations and for searching for fast radio bursts with the Parkes and Effelsberg telescopes.
• ### Identification of the white dwarf companion to millisecond pulsar J2317+1439(1705.02593)

May 7, 2017 astro-ph.SR
We report identification of the optical counterpart to the companion of the millisecond pulsar J2317+1439. At the timing position of the pulsar, we find an object with $g=22.96\pm0.05$, $r=22.86\pm0.04$ and $i=22.82\pm0.05$. The magnitudes and colors of the object are consistent with it being a white dwarf. By comparing with white dwarf cooling models, we estimate that it has a mass of $0.39^{+0.13}_{-0.10}$ M$_{\odot}$, an effective temperature of $8077^{+550}_{-470}$ K and a cooling age of $10.9\pm0.3$ Gyr. Combining our results with published constraints on the orbital parameters obtained through pulsar timing, we estimate the pulsar mass to be $3.4^{+1.4}_{-1.1}$ M$_{\odot}$. Although the constraint on the pulsar mass is still weak, there is a significant possibility that the pulsar could be more massive than two solar mass.
• ### Robust Estimation of Scattering in Pulsar Timing Analysis(1703.02108)

March 6, 2017 astro-ph.IM, astro-ph.HE
We present a robust approach to incorporating models for the time-variable broadening of the pulse profile due to scattering in the ionized interstellar medium into profile-domain pulsar timing analysis. We use this approach to simultaneously estimate temporal variations in both the dispersion measure (DM) and scattering, together with a model for the pulse profile that includes smooth evolution as a function of frequency, and the pulsar's timing model. We show that fixing the scattering timescales when forming time-of-arrival estimates, as has been suggested in the context of traditional pulsar timing analysis, can significantly underestimate the uncertainties in both DM, and the arrival time of the pulse, leading to bias in the timing parameters. We apply our method using a new, publicly available, GPU accelerated code, both to simulations, and observations of the millisecond pulsar PSR J1643$-$1224. This pulsar is known to exhibit significant scattering variability compared to typical millisecond pulsars, and we find including low-frequency ($< 1$ GHz) data without a model for these scattering variations leads to significant periodic structure in the DM, and also biases the astrometric parameters at the $4\sigma$ level, for example, changing proper motion in right ascension by $0.50 \pm 0.12$. If low frequency observations are to be included when significant scattering variations are present, we conclude it is necessary to not just model those variations, but also to sample the parameters that describe the variations simultaneously with all other parameters in the model, a task for which profile domain pulsar timing is ideally suited.
• ### Wide-band Profile Domain Pulsar Timing Analysis(1612.05258)

We extend profile domain pulsar timing to incorporate wide-band effects such as frequency-dependent profile evolution and broadband shape variation in the pulse profile. We also incorporate models for temporal variations in both pulse width and in the separation in phase of the main pulse and interpulse. We perform the analysis with both nested sampling and Hamiltonian Monte Carlo methods. In the latter case we introduce a new parameterisation of the posterior that is extremely efficient in the low signal-to-noise regime and can be readily applied to a wide range of scientific problems. We apply this methodology to a series of simulations, and to between seven and nine yr of observations for PSRs J1713$+$0747, J1744$-$1134, and J1909$-$3744 with frequency coverage that spans 700-3600MHz. We use a smooth model for profile evolution across the full frequency range, and compare smooth and piecewise models for the temporal variations in DM. We find the profile domain framework consistently results in improved timing precision compared to the standard analysis paradigm by as much as 40% for timing parameters. Incorporating smoothness in the DM variations into the model further improves timing precision by as much as 30%. For PSR J1713+0747 we also detect pulse shape variation uncorrelated between epochs, which we attribute to variation intrinsic to the pulsar at a level consistent with previously published analyses. Not accounting for this shape variation biases the measured arrival times at the level of $\sim$30ns, the same order of magnitude as the expected shift due to gravitational-waves in the pulsar timing band.
• ### The disturbance of a millisecond pulsar magnetosphere(1608.02163)

Aug. 6, 2016 astro-ph.SR, astro-ph.HE
Pulsar timing has enabled some of the strongest tests of fundamental physics. Central to the technique is the assumption that the detected radio pulses can be used to accurately measure the rotation of the pulsar. Here we report on a broad-band variation in the pulse profile of the millisecond pulsar J1643-1224. A new component of emission suddenly appears in the pulse profile, decays over 4 months, and results in a permanently modified pulse shape. Profile variations such as these may be the origin of timing noise observed in other millisecond pulsars. The sensitivity of pulsar-timing observations to gravitational radiation can be increased by accounting for this variability.
• ### Detecting Pulsars with Interstellar Scintillation in Variance Images(1607.07740)

July 26, 2016 astro-ph.GA, astro-ph.IM
Pulsars are the only cosmic radio sources known to be sufficiently compact to show diffractive interstellar scintillations. Images of the variance of radio signals in both time and frequency can be used to detect pulsars in large-scale continuum surveys using the next generation of synthesis radio telescopes. This technique allows a search over the full field of view while avoiding the need for expensive pixel-by-pixel high time resolution searches. We investigate the sensitivity of detecting pulsars in variance images. We show that variance images are most sensitive to pulsars whose scintillation time-scales and bandwidths are close to the subintegration time and channel bandwidth. Therefore, in order to maximise the detection of pulsars for a given radio continuum survey, it is essential to retain a high time and frequency resolution, allowing us to make variance images sensitive to pulsars with different scintillation properties. We demonstrate the technique with Murchision Widefield Array data and show that variance images can indeed lead to the detection of pulsars by distinguishing them from other radio sources.
• ### From Spin Noise to Systematics: Stochastic Processes in the First International Pulsar Timing Array Data Release(1602.05570)

We analyse the stochastic properties of the 49 pulsars that comprise the first International Pulsar Timing Array (IPTA) data release. We use Bayesian methodology, performing model selection to determine the optimal description of the stochastic signals present in each pulsar. In addition to spin-noise and dispersion-measure (DM) variations, these models can include timing noise unique to a single observing system, or frequency band. We show the improved radio-frequency coverage and presence of overlapping data from different observing systems in the IPTA data set enables us to separate both system and band-dependent effects with much greater efficacy than in the individual PTA data sets. For example, we show that PSR J1643$-$1224 has, in addition to DM variations, significant band-dependent noise that is coherent between PTAs which we interpret as coming from time-variable scattering or refraction in the ionised interstellar medium. Failing to model these different contributions appropriately can dramatically alter the astrophysical interpretation of the stochastic signals observed in the residuals. In some cases, the spectral exponent of the spin noise signal can vary from 1.6 to 4 depending upon the model, which has direct implications for the long-term sensitivity of the pulsar to a stochastic gravitational-wave (GW) background. By using a more appropriate model, however, we can greatly improve a pulsar's sensitivity to GWs. For example, including system and band-dependent signals in the PSR J0437$-$4715 data set improves the upper limit on a fiducial GW background by $\sim 60\%$ compared to a model that includes DM variations and spin-noise only.
• ### The International Pulsar Timing Array: First Data Release(1602.03640)

Feb. 11, 2016 astro-ph.IM
The highly stable spin of neutron stars can be exploited for a variety of (astro-)physical investigations. In particular arrays of pulsars with rotational periods of the order of milliseconds can be used to detect correlated signals such as those caused by gravitational waves. Three such "Pulsar Timing Arrays" (PTAs) have been set up around the world over the past decades and collectively form the "International" PTA (IPTA). In this paper, we describe the first joint analysis of the data from the three regional PTAs, i.e. of the first IPTA data set. We describe the available PTA data, the approach presently followed for its combination and suggest improvements for future PTA research. Particular attention is paid to subtle details (such as underestimation of measurement uncertainty and long-period noise) that have often been ignored but which become important in this unprecedentedly large and inhomogeneous data set. We identify and describe in detail several factors that complicate IPTA research and provide recommendations for future pulsar timing efforts. The first IPTA data release presented here (and available online) is used to demonstrate the IPTA's potential of improving upon gravitational-wave limits placed by individual PTAs by a factor of ~2 and provides a 2-sigma limit on the dimensionless amplitude of a stochastic GWB of 1.7x10^{-15} at a frequency of 1 yr^{-1}. This is 1.7 times less constraining than the limit placed by (Shannon et al. 2015), due mostly to the more recent, high-quality data they used.
• ### Edge and surface plasmons in graphene nanoribbons(1512.06910)

Dec. 21, 2015 cond-mat.mes-hall
We report on nano-infrared (IR) imaging studies of confined plasmon modes inside patterned graphene nanoribbons (GNRs) fabricated with high-quality chemical-vapor-deposited (CVD) graphene on Al2O3 substrates. The confined geometry of these ribbons leads to distinct mode patterns and strong field enhancement, both of which evolve systematically with the ribbon width. In addition, spectroscopic nano-imaging in mid-infrared 850-1450 cm-1 allowed us to evaluate the effect of the substrate phonons on the plasmon damping. Furthermore, we observed edge plasmons: peculiar one-dimensional modes propagating strictly along the edges of our patterned graphene nanostructures.
• ### Versatile Directional Searches for Gravitational Waves with Pulsar Timing Arrays(1510.08068)

Oct. 27, 2015 astro-ph.IM
By regularly monitoring the most stable millisecond pulsars over many years, pulsar timing arrays (PTAs) are positioned to detect and study correlations in the timing behaviour of those pulsars. Gravitational waves (GWs) from supermassive black hole binaries (SMBHBs) are an exciting potentially detectable source of such correlations. We describe a straight-forward technique by which a PTA can be "phased-up" to form time series of the two polarisation modes of GWs coming from a particular direction of the sky. Our technique requires no assumptions regarding the time-domain behaviour of a GW signal. This method has already been used to place stringent bounds on GWs from individual SMBHBs in circular orbits. Here, we describe the methodology and demonstrate the versatility of the technique in searches for a wide variety of GW signals including bursts with unmodeled waveforms. Using the first six years of data from the Parkes Pulsar Timing Array, we conduct an all-sky search for a detectable excess of GW power from any direction. For the lines of sight to several nearby massive galaxy clusters, we carry out a more detailed search for GW bursts with memory, which are distinct signatures of SMBHB mergers. In all cases, we find that the data are consistent with noise.
• ### Timing analysis for 20 millisecond pulsars in the Parkes Pulsar Timing Array(1510.04434)

Oct. 15, 2015 astro-ph.IM, astro-ph.HE
We present timing models for 20 millisecond pulsars in the Parkes Pulsar Timing Array. The precision of the parameter measurements in these models has been improved over earlier results by using longer data sets and modelling the non-stationary noise. We describe a new noise modelling procedure and demonstrate its effectiveness using simulated data. Our methodology includes the addition of annual dispersion measure (DM) variations to the timing models of some pulsars. We present the first significant parallax measurements for PSRs J1024-0719, J1045-4509, J1600-3053, J1603-7202, and J1730-2304, as well as the first significant measurements of some post-Keplerian orbital parameters in six binary pulsars, caused by kinematic effects. Improved Shapiro delay measurements have resulted in much improved pulsar mass measurements, particularly for PSRs J0437-4715 and J1909-3744 with $M_p=1.44\pm0.07$ $M_\odot$ and $M_p=1.47\pm0.03$ $M_\odot$ respectively. The improved orbital period-derivative measurement for PSR J0437-4715 results in a derived distance measurement at the 0.16% level of precision, $D=156.79\pm0.25$ pc, one of the most fractionally precise distance measurements of any star to date.
• ### Gravitational waves from binary supermassive black holes missing in pulsar observations(1509.07320)

Gravitational waves are expected to be radiated by supermassive black hole binaries formed during galaxy mergers. A stochastic superposition of gravitational waves from all such binary systems will modulate the arrival times of pulses from radio pulsars. Using observations of millisecond pulsars obtained with the Parkes radio telescope, we constrain the characteristic amplitude of this background, $A_{\rm c,yr}$, to be < $1.0\times10^{-15}$ with 95% confidence. This limit excludes predicted ranges for $A_{\rm c,yr}$ from current models with 91-99.7% probability. We conclude that binary evolution is either stalled or dramatically accelerated by galactic-center environments, and that higher-cadence and shorter-wavelength observations would result in an increased sensitivity to gravitational waves.
• ### Tunneling Plasmonics in Bilayer Graphene(1508.07883)

Aug. 31, 2015 cond-mat.mes-hall
We report experimental signatures of plasmonic effects due to electron tunneling between adjacent graphene layers. At sub-nanometer separation, such layers can form either a strongly coupled bilayer graphene with a Bernal stacking or a weakly coupled double-layer graphene with a random stacking order. Effects due to interlayer tunneling dominate in the former case but are negligible in the latter. We found through infrared nano-imaging that bilayer graphene supports plasmons with a higher degree of confinement compared to single- and double-layer graphene, a direct consequence of interlayer tunneling. Moreover, we were able to shut off plasmons in bilayer graphene through gating within a wide voltage range. Theoretical modeling indicates that such a plasmon-off region is directly linked to a gapped insulating state of bilayer graphene: yet another implication of interlayer tunneling. Our work uncovers essential plasmonic properties in bilayer graphene and suggests a possibility to achieve novel plasmonic functionalities in graphene few-layers.
• ### Pulsar Observations of Extreme Scattering Events(1506.07948)

June 26, 2015 astro-ph.SR, astro-ph.HE
Extreme scattering events (ESEs) in the interstellar medium (ISM) were first observed in regular flux measurements of compact extragalactic sources. They are characterized by a flux variation over a period of weeks, suggesting the passage of a "diverging plasma lens" across the line of sight. Modeling the refraction of such a lens indicates that the structure size must be of order AU and the electron density of order 10s of cm^{-3}. Similar structures have been observed in measurements of pulsar intensity scintillation and group delay. Here we report observations of two ESEs showing increases in both intensity scintillation and dispersion made with the Parkes Pulsar Timing Array (PPTA). These allow us to make more complete models of the ESE, including an estimate of the "outer-scale" of the turbulence in the plasma lens. These observations show clearly that the ESE structure is fully turbulent on an AU scale. They provide some support for the idea that the structures are extended along the line of sight, such as would be the case for a scattering shell. The dispersion measurements also show a variety of AU scale structures which would not be called ESEs, yet involve electron density variations typical of ESEs and likely have the same origin.
• ### Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial(1501.06956)

June 23, 2015 cond-mat.mes-hall
Hexagonal boron nitride (h-BN) is a natural hyperbolic material, for which the dielectric constants are the same in the basal plane (epsilon^t = epsilon^x = epsilon^y) but have opposite signs (epsilon^t*epsilon^z < 0) from that in the normal plane (epsilon^z). Due to this property, finite-thickness slabs of h-BN act as multimode waveguides for propagation of hyperbolic phonon polaritons - collective modes that originate from the coupling between photons and electric dipoles in phonons. However, control of these hyperbolic phonon polaritons modes has remained challenging, mostly because their electrodynamic properties are dictated by the crystal lattice of h-BN. Here we show by direct nano-infrared imaging that these hyperbolic polaritons can be effectively modulated in a van der Waals heterostructure composed of monolayer graphene on h-BN. Tunability originates from the hybridization of surface plasmon polaritons in graphene with hyperbolic phonon polaritons in h-BN, so that the eigenmodes of the graphene/h-BN heterostructure are hyperbolic plasmon-phonon polaritons. Remarkably, the hyperbolic plasmon-phonon polaritons in graphene/h-BN suffer little from ohmic losses, making their propagation length 1.5-2.0 times greater than that of hyperbolic phonon polaritons in h-BN. The hyperbolic plasmon-phonon polaritons possess the combined virtues of surface plasmon polaritons in graphene and hyperbolic phonon polaritons in h-BN. Therefore, graphene/h-BN structures can be classified as electromagnetic metamaterials since the resulting properties of these devices are not present in its constituent elements alone.
• ### Subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolic material(1502.04094)

Uniaxial materials whose axial and tangential permittivities have opposite signs are referred to as indefinite or hyperbolic media. In such materials light propagation is unusual, leading to novel and often non-intuitive optical phenomena. Here we report infrared nano-imaging experiments demonstrating that crystals of hexagonal boron nitride (hBN), a natural mid-infrared hyperbolic material, can act as a "hyper-focusing lens" and as a multi-mode waveguide. The lensing is manifested by subdiffractional focusing of phonon-polaritons launched by metallic disks underneath the hBN crystal. The waveguiding is revealed through the modal analysis of the periodic patterns observed around such launchers and near the sample edges. Our work opens new opportunities for anisotropic layered insulators in infrared nanophotonics complementing and potentially surpassing concurrent artificial hyperbolic materials with lower losses and higher optical localization.
• ### A Study of Multi-frequency Polarization Pulse Profiles of Millisecond Pulsars(1503.01841)

March 6, 2015 astro-ph.GA, astro-ph.HE
We present high signal-to-noise ratio, multi-frequency polarization pulse profiles for 24 millisecond pulsars that are being observed as part of the Parkes Pulsar Timing Array (PPTA) project. The pulsars are observed in three bands, centred close to 730, 1400 and 3100 MHz, using a dual-band 10 cm/50 cm receiver and the central beam of the 20 cm multibeam receiver. Observations spanning approximately six years have been carefully calibrated and summed to produce high S/N profiles. This allows us to study the individual profile components and in particular how they evolve with frequency. We also identify previously undetected profile features. For many pulsars we show that pulsed emission extends across almost the entire pulse profile. The pulse component widths and component separations follow a complex evolution with frequency; in some cases these parameters increase and in other cases they decrease with increasing frequency. The evolution with frequency of the polarization properties of the profile is also non-trivial. We provide evidence that the pre- and post-cursors generally have higher fractional linear polarization than the main pulse. We have obtained the spectral index and rotation measure for each pulsar by fitting across all three observing bands. For the majority of pulsars, the spectra follow a single power-law and the position angles follow a $\lambda^2$ relation, as expected. However, clear deviations are seen for some pulsars. We also present phase-resolved measurements of the spectral index, fractional linear polarization and rotation measure. All these properties are shown to vary systematically over the pulse profile.
• ### A 24-Hour Global Campaign To Assess Precision Timing of the Millisecond Pulsar J1713+0747(1408.1694)

Dec. 17, 2014 astro-ph.IM, astro-ph.HE
The radio millisecond pulsar J1713+0747 is regarded as one of the highest-precision clocks in the sky, and is regularly timed for the purpose of detecting gravitational waves. The International Pulsar Timing Array collaboration undertook a 24-hour global observation of PSR J1713+0747 in an effort to better quantify sources of timing noise in this pulsar, particularly on intermediate (1 - 24 hr) timescales. We observed the pulsar continuously over 24 hr with the Arecibo, Effelsberg, GMRT, Green Bank, LOFAR, Lovell, Nancay, Parkes, and WSRT radio telescopes. The combined pulse times-of-arrival presented here provide an estimate of what sources of timing noise, excluding DM variations, would be present as compared to an idealized root-N improvement in timing precision, where N is the number of pulses analyzed. In the case of this particular pulsar, we find that intrinsic pulse phase jitter dominates arrival time precision when the S/N of single pulses exceeds unity, as measured using the eight telescopes that observed at L-band/1.4 GHz. We present first results of specific phenomena probed on the unusually long timescale (for a single continuous observing session) of tens of hours, in particular interstellar scintillation, and discuss the degree to which scintillation and profile evolution affect precision timing. This paper presents the data set as a basis for future, deeper studies.
• ### Searching for gravitational wave memory bursts with the Parkes Pulsar Timing Array(1410.3323)

Anisotropic bursts of gravitational radiation produced by events such as super-massive black hole mergers leave permanent imprints on space. Such gravitational wave "memory" (GWM) signals are, in principle, detectable through pulsar timing as sudden changes in the apparent pulse frequency of a pulsar. If an array of pulsars is monitored as a GWM signal passes over the Earth, the pulsars would simultaneously appear to change pulse frequency by an amount that varies with their sky position in a quadrupolar fashion. Here we describe a search algorithm for such events and apply the algorithm to approximately six years of data from the Parkes Pulsar Timing Array. We find no GWM events and set an upper bound on the rate for events which could have been detected. We show, using simple models of black hole coalescence rates, that this non-detection is not unexpected.
• ### An all-sky search for continuous gravitational waves in the Parkes Pulsar Timing Array data set(1408.5129)

We present results of an all-sky search in the Parkes Pulsar Timing Array (PPTA) Data Release 1 data set for continuous gravitational waves (GWs) in the frequency range from $5\times 10^{-9}$ to $2\times 10^{-7}$ Hz. Such signals could be produced by individual supermassive binary black hole systems in the early stage of coalescence. We phase up the pulsar timing array data set to form, for each position on the sky, two data streams that correspond to the two GW polarizations and then carry out an optimal search for GW signals on these data streams. Since no statistically significant GWs were detected, we place upper limits on the intrinsic GW strain amplitude $h_0$ for a range of GW frequencies. For example, at $10^{-8}$ Hz our analysis has excluded with $95\%$ confidence the presence of signals with $h_0\geqslant 1.7\times 10^{-14}$. Our new limits are about a factor of four more stringent than those of Yardley et al. (2010) based on an earlier PPTA data set and a factor of two better than those reported in the recent Arzoumanian et al. (2014) paper. We also present PPTA directional sensitivity curves and find that for the most sensitive region on the sky, the current data set is sensitive to GWs from circular supermassive binary black holes with chirp masses of $10^{9} M_{\odot}$ out to a luminosity distance of about 100 Mpc. Finally, we set an upper limit of $4 \times 10^{-3} {\rm{Mpc}}^{-3} {\rm{Gyr}}^{-1}$ at $95\%$ confidence on the coalescence rate of nearby ($z \lesssim 0.1$) supermassive binary black holes in circular orbits with chirp masses of $10^{10}M_{\odot}$.
• ### The Role of FAST in Pulsar Timing Arrays(1407.0435)

July 2, 2014 astro-ph.IM, astro-ph.HE
The Five-hundred-meter Aperture Spherical Telescope (FAST) will become one of the world-leading telescopes for pulsar timing array (PTA) research. The primary goals for PTAs are to detect (and subsequently study) ultra-low-frequency gravitational waves, to develop a pulsar-based time standard and to improve solar system planetary ephemerides. FAST will have the sensitivity to observe known pulsars with significantly improved signal-to-noise ratios and will discover a large number of currently unknown pulsars. We describe how FAST will contribute to PTA research and show that jitter- and timing-noise will be the limiting noise processes for FAST data sets. Jitter noise will limit the timing precision achievable over data spans of a few years while timing noise will limit the precision achievable over many years.
• ### Limitations in timing precision due to single-pulse shape variability in millisecond pulsars(1406.4716)

June 18, 2014 astro-ph.SR, astro-ph.IM
High-sensitivity radio-frequency observations of millisecond pulsars usually show stochastic, broadband, pulse-shape variations intrinsic to the pulsar emission process. These variations induce jitter noise in pulsar timing observations; understanding the properties of this noise is of particular importance for the effort to detect gravitational waves with pulsar timing arrays. We assess the short-term profile and timing stability of 22 millisecond pulsars that are part of the Parkes Pulsar Timing Array sample by examining intra-observation arrival time variability and single-pulse phenomenology. In 7 of the 22 pulsars, in the band centred at approximately 1400MHz, we find that the brightest observations are limited by intrinsic jitter. We find consistent results, either detections or upper limits, for jitter noise in other frequency bands. PSR J1909-3744 shows the lowest levels of jitter noise, which we estimate to contribute $\sim$10 ns root mean square error to the arrival times for hour-duration observations. Larger levels of jitter noise are found in pulsars with wider pulses and distributions of pulse intensities. The jitter noise in PSR J0437-4715 decorrelates over a bandwidth of $\sim$2 GHz. We show that the uncertainties associated with timing pulsar models can be improved by including physically motivated jitter uncertainties. Pulse-shape variations will limit the timing precision at future, more sensitive, telescopes; it is imperative to account for this noise when designing instrumentation and timing campaigns for these facilities.
• ### Electronic and plasmonic phenomena at graphene grain boundaries(1311.6827)

Nov. 26, 2013 cond-mat.mes-hall
Graphene, a two-dimensional honeycomb lattice of carbon atoms, is of great interest in (opto)electronics and plasmonics and can be obtained by means of diverse fabrication techniques, among which chemical vapor deposition (CVD) is one of the most promising for technological applications. The electronic and mechanical properties of CVD-grown graphene depend in large part on the characteristics of the grain boundaries. However, the physical properties of these grain boundaries remain challenging to characterize directly and conveniently. Here, we show that it is possible to visualize and investigate the grain boundaries in CVD-grown graphene using an infrared nano-imaging technique. We harness surface plasmons that are reflected and scattered by the graphene grain boundaries, thus causing plasmon interference. By recording and analyzing the interference patterns, we can map grain boundaries for a large area CVD-grown graphene film and probe the electronic properties of individual grain boundaries. Quantitative analysis reveals that grain boundaries form electronic barriers that obstruct both electrical transport and plasmon propagation. The effective width of these barriers (~10-20 nm) depends on the electronic screening and it is on the order of the Fermi wavelength of graphene. These results uncover a microscopic mechanism that is responsible for the low electron mobility observed in CVD-grown graphene, and suggest the possibility of using electronic barriers to realize tunable plasmon reflectors and phase retarders in future graphene-based plasmonic circuits.


text:
# Home

Score : $100$ points

### Problem Statement

The magnitude of an earthquake is a logarithmic scale of the energy released by the earthquake. It is known that each time the magnitude increases by $1$, the amount of energy gets multiplied by approximately $32$.
Here, we assume that the amount of energy gets multiplied by exactly $32$ each time the magnitude increases by $1$. In this case, how many times is the amount of energy of a magnitude $A$ earthquake as much as that of a magnitude $B$ earthquake?

### Constraints

• $3\leq B\leq A\leq 9$
• $A$ and $B$ are integers.

### Input

Input is given from Standard Input in the following format:

$A$ $B$

### Output

Print the answer as an integer.

### Sample Input 1

6 4

### Sample Output 1

1024

$6$ is $2$ greater than $4$, so a magnitude $6$ earthquake has $32\times 32=1024$ times as much energy as a magnitude $4$ earthquake has.

### Sample Input 2

5 5

### Sample Output 2

1

Earthquakes with the same magnitude have the same amount of energy.


text:
# Innovation, Logistics, and Lean

Amonth ago, a reader asked Michael Ballé “If lean really is about innovation, why does so much of it seem to be about logistics, with truck preparation areas, leveling boxes, small trains, kanbans and so on?” His short answer “because logistics is the way into innovation” is a head scratcher and I fail to see any support for this assertion in the rest of his response.

While TPS and, more generally, the Toyota Way are innovative in the management and technology of operations, discussions of innovation are usually about products. Even in the car industry, which companies come to mind today about product innovation? Which ones would you want to learn from? Most likely not Toyota but Tesla for its electric cars and Alphabet/Google’s subsidiary Waymo for self-driving cars, both based in Silicon Valley.

According to The Lean Strategy (p. 227), “The Lean theory of innovation is fundamentally different from the superficial ‘disruption’ narrative. The Lean starting point is the intent to solve a problem that customers have now, rather than looking for new things that a new technology could do for them.”

Sakichi Toyoda spent 30 years making incremental improvements to one of the oldest machines known to humans. Yes, it was innovative and yes, this philosophy has served Toyota well. It is, however, complementary to more radical innovation and not a substitute for it.

Having spent most of my life in Silicon Valley, I can testify that there is nothing superficial about the disruption from innovation and that it has not taken the form of “looking for new things that a new technology could do.” Technology has usually come first.

When personal computers first arrived in the early 1980s, their makers had no clue what uses they would be put to, and the same was true for tablets and smartphones 30 years later. A key feature of innovation as practiced in Silicon Valley is that it is not driven by market research, focus groups, or customer surveys. Instead, technology is made available for users to invent useful applications.

The next part of the question was about why so much of Lean is about logistics, which, as the author of the first book on Lean Logistics, I have to comment on. According to Crispin Vincenti-Brown, manufacturing operations have the following four dimensions:

1. Engineering
2. Logistics and Production Control
3. Organization and People
4. Performance management and accountability

When you study TPS and the Toyota Way, you find principles, concepts, and tools covering all four dimensions. In Lean, on the other hand, you find almost nothing about engineering, and what is said on the management topics — organization, people, and performance management — is largely warmed-up ideas from American business schools with the “Lean” label slapped on. This leaves logistics and production control as the only dimension with a substantive technical content in Lean.

When I first got involved with production planning and scheduling — in the semiconductor industry — my boss warned me that everybody was an expert on that subject. Those who are not familiar with the key points of a process and the operations of the machines and lines that execute it usually do not get involved in production engineering discussions. On the other hand, cases, pallets, trucks, planning boards, etc., are similar across industries and look easy to understand and are an attractive topic.

The reason Logistics is overrepresented in Lean when compared to, say, Engineering is not that it is “the way to innovation” but that the loudest voices in Lean have no engineering background. Instead, they are political scientists, economists, social scientists, and MBAs.

It doesn’t mean that engineers would do a better job because they might instead focus the discussion exclusively on jigs, fixtures, control systems, etc., at the expense of everything else. Regardless of its focus, an imbalanced approach won’t work any better than Lean has over the past 25 years. What is needed instead is paying attention to all the dimensions. Mastering all of them may be too much to ask of an individual, but not of a team.


text:
Two hemispherical surfaces, 1 and 2, of respective radii and , are centered at a point charge and are facing each otherso that their edges define an annular ring (surface 3), as shown.The field at position due to the point charge is:

where is a constant proportional to the charge, , and is the unit vector in the radial direction.
A. What is the electric flux through the annular ring, surface 3?


text:
# Brief Summary of SVM - ANN Equivalence

The paper considers only a shallow network architecture with one hidden layer.

Very short and rough summary: The paper establishes the following correspondence:

• activation functions correspond to kernels
• hidden nodes correspond to support vectors

A kernel method fixes the activation functions but learns the number of hidden nodes. Neural net learning fixes the number of hidden nodes but learns the activation functions (kernel).

More detailed version: Fix n datapoints.

1. Given an SVM, define a neural network where there is a hidden unit for each support vector. The activation function of a hidden unit is the value of the kernel applied to the input data point and the support vector represented by the hidden unit. The weights from hidden units to the single output unit are derived from the SVM weights obtained in the dual formulation.

2. Consider a neural network with n hidden units, one for each datapoint. The error function is squared error plus a regularization term. From the regularization operator we can derive Green functions. The Green functions define the activation functions for the hidden units. They also determine the weights from hidden to output units via linear equations.

3. The main theorem now states, roughly, that for every kernel, there is a corresponding regularization operator, and vice versa, s.t. solving the SVM problem for the kernel is equivalent to finding weights that minimize the regularized squared error criterion. Rough summary follows.

1. Mapping from the kernel, the regularization term becomes the weighted average of entries in the kernel matrix, using the support vector weights. So the more of these weights are 0, the smaller the complexity of the neural network.
2. Mapping from the regularization term, think of a vector space of activation functions. A given input vector maps to a member of this space. Define the kernel for two data points as the norm of their activation functions. This turns out to satisfy Mercer's conditions, hence is a valid kernel.
Updated Tue Jan. 01 2019, 16:38 by oschulte.


text:
Proving that a mermorphic function with three linearly independent (over the rationals) periods is constant

I would like to understand and answer the following question from Serge Lange's Introduction to Complex Analysis at a graduate level.

I understand how is one supposed to use the hint to prove the desired result but I do not understand how one proves the hint. (I think once the hint is proven we have basically shown that about every point the function is locally constant, even about the poles, so the poles are removable singularities and hence the function is constant).

I tried setting up equations of the lines $$aw_1+bw_2+cw_3=\epsilon$$ in complex and vector forms but it was a dead end as I wasn't able to prove that such $$a,b,c$$ exist for all epsilon not equal to zero.

Does anybody have any ideas has what the construction of such proof would look like and moreover, how should one think about this problem intuitively?

• You need to show that $S=\{b_1w_1+b_2w_2+b_3w_3: b_1,b_2,b_3\in\Bbb Z; b_1,b_2,b_3 \text { not all } 0\}$ has members arbitrarily close to $0$. Note $0\not\in S$ because $w_1, w_2,w_3$ are linearly independent over $\Bbb Q$, so, a fortiori, they are lin. indep. over $\Bbb Z.$ Also if one of the ratios $w_i/w_j$ (with $i\ne j$) is real, it is irrational and we can apply an old result to obtain $b_i,b_j\in \Bbb Z$ \ $\{0\}$ with $b_iw_i+b_jw_j$ arbitrarily close to $0$. I have not yet got an A for the case where $w_1/w_2,w_2/w_3,w_3/w_1\not\in \Bbb R$. Jul 16 at 15:37
• The "old result" I mentioned is that if $r=w_i/w_j\in\Bbb R$ \ $\Bbb Q$ then there are members of $\{nr: n\in\Bbb Z^+\}$ that are arbitrarily close to integers. Jul 16 at 15:43
• We might need to use the following: For $x\in\Bbb R$ let $[x]$ denote the largest integer not exceeding $x.$ If $r\in\Bbb R$ \ $\Bbb Q$ then $T=\{nr-[nr]: n\in\Bbb Z^+\}$ is dense in $[0,1].$ In particular $\inf \,T=0.$ Jul 16 at 15:52
• @DanielWainfleet thank you for the comments. I am too still puzzled by this. To be honest this just looks like an awful question given the fact that there has been no discussion for such methods before this exercise. Jul 16 at 19:43
• It's more like a Number Theory problem with an immediate corollary in Analysis. Jul 17 at 4:16

Case 1: $$w_1/w_2 \not\in \Bbb R.$$ Consider $$\Bbb C$$ as a 2-dimensional vector space over the field $$\Bbb R.$$ So $$\{w_1,w_2\}$$ is a vector-space basis for $$\Bbb C.$$ So there exist $$r_1,r_2\in\Bbb R$$ such that $$r_1w_1+r_2w_2=w_3.$$

Definition. For $$x\in \Bbb R$$ let $$d(x,\Bbb Z)=\min \{|x-z|:z\in\Bbb Z\}.$$

Lemma. For any finite $$S\subset \Bbb R$$ and any $$\epsilon >0$$ there exist infinitely many $$n\in\Bbb N$$ such that $$\forall x\in S\,(d(nx,\Bbb Z)<\epsilon).$$

We only need this lemma when $$S$$ has 1 or 2 members, and we do not need infinitely many $$n\in \Bbb N$$. Just one $$n$$ will be needed for a given $$\epsilon$$.

Return the the 1st paragraph. By the Lemma, with $$S=\{w_1,w_2\}$$, for any $$\epsilon >0$$ there exists $$n_{\epsilon}\in\Bbb N$$ such that $$n_{\epsilon}r_j=m_j+\delta_j$$ for $$j\in \{1,2\}$$, with $$m_j\in\Bbb Z$$ and $$|\delta_j|<\epsilon.$$ Hence $$|m_1w_1+m_2w_2-n_{\epsilon}w_3|=|n_{\epsilon}(r_1w_1+r_2w_2-w_3)-(w_1\delta_1+w_2\delta_2)|=$$ $$=|0-(w_1\delta_1+w_2\delta_2)|\le$$ $$\le (|w_1|+|w_2|)\epsilon$$ which can be arbitrarily small but not $$0$$, because if $$m_1w_1+m_2w_2-n_{\epsilon}w_3=0$$ then $$w_1,w_2,w_3$$ would be linearly dependent over $$\Bbb Q.$$

Case 2. $$w_1/w_2\in \Bbb R.$$ By the Lemma with $$S=\{w_1/w_2\},$$ for any $$\epsilon>0$$ there exists $$n_{\epsilon}\in \Bbb N$$ such that $$n_{\epsilon}(w_1/w_2)=m+\delta$$ with $$m\in\Bbb Z$$ and $$|\delta|<\epsilon.$$ Hence $$|n_{\epsilon}w_1-mw_2|=|\delta|\cdot |w_2|\le\epsilon |w_2|$$ which can be arbitrarily small but not $$0$$, because if $$n_{\epsilon}w_1-mw_2=0$$ then $$w_1,w_2$$ would be linearly dependent over $$\Bbb Q.$$

• I can give a proof of the Lemma if needed. Jul 18 at 8:04
• Hi thank your for the answer. A proof of the lemma would be great as I have not seen anything like this. Jul 18 at 17:17
• I will add a proof of the lemma soon. Jul 30 at 1:53


text:
Velocity Reviews > JavaScript. Array "extras" in detail.

# JavaScript. Array "extras" in detail.

Dmitry A. Soshnikov
Guest
Posts: n/a

 02-23-2011
A new article written specially for Opera software:

JavaScript. Array "extras" in detail.

http://dev.opera.com/articles/view/j...ras-in-detail/

Dmitry.

Dr J R Stockton
Guest
Posts: n/a

 02-24-2011
In comp.lang.javascript message <ik3a4c$sng$(E-Mail Removed)-
september.org>, Wed, 23 Feb 2011 18:46:05, Dmitry A. Soshnikov
<(E-Mail Removed)> posted:

>A new article written specially for Opera software:
>
>JavaScript. Array "extras" in detail.
>
>http://dev.opera.com/articles/view/j...ras-in-detail/

Interesting.

Reminds me of <http://en.wikipedia.org/wiki/Jensen%27s_Device>.

increase my chances of finding it again.

--
(c) John Stockton, nr London, UK. ?@merlyn.demon.co.uk Turnpike v6.05.
Website <http://www.merlyn.demon.co.uk/> - w. FAQish topics, links, acronyms
PAS EXE etc. : <http://www.merlyn.demon.co.uk/programs/> - see in 00index.htm
Dates - miscdate.htm estrdate.htm js-dates.htm pas-time.htm critdate.htm etc.

Dmitry A. Soshnikov
Guest
Posts: n/a

 02-25-2011
On 24.02.2011 23:22, Dr J R Stockton wrote:
> In comp.lang.javascript message<ik3a4c$sng$(E-Mail Removed)-
> september.org>, Wed, 23 Feb 2011 18:46:05, Dmitry A. Soshnikov
> <(E-Mail Removed)> posted:
>
>> A new article written specially for Opera software:
>>
>> JavaScript. Array "extras" in detail.
>>
>> http://dev.opera.com/articles/view/j...ras-in-detail/

>
> Interesting.
>
> Reminds me of<http://en.wikipedia.org/wiki/Jensen%27s_Device>.
>

Yeah, actually roots of this idea (of the parametrized list handling)
went to also to math. There, mathematician are used to operate with the
concept of _a sum_ (regardless the sum of _what_ exactly). They have a
special higher-order function (HOF) of summation with the known sign --
∑ (∑).

My example with a generic sum is exactly a direct reflection of this
math concept. And as a consequence, higher-order functions can be
applied for any manipulation on the sequences/lists/arrays. That's the
reason they are available in all functional programming langauges
(including JS which supports this style of programming, since all
functions here are "first-class" and can be passed to HOFs).

> increase my chances of finding it again.
>

Yeah, thanks.

Dmitry.

Michael Haufe (\TNO\)
Guest
Posts: n/a

 02-25-2011
On Feb 25, 5:30*am, "Dmitry A. Soshnikov" <(E-Mail Removed)>
wrote:
[...]
> My example with a generic sum is exactly a direct reflection of this
> math concept. And as a consequence, higher-order functions can be
> applied for any manipulation on the sequences/lists/arrays. That's the
> reason they are available in all functional programming langauges
> (including JS which supports this style of programming, since all
> functions here are "first-class" and can be passed to HOFs).

[...]

It is unfortunate that JavaScript couldn't be a bit less clunky in the
functional style:

["1","2","3"].map(parseInt); //[1, NaN, NaN]

Lasse Reichstein Nielsen
Guest
Posts: n/a

 02-25-2011
"Michael Haufe (\"TNO\")" <(E-Mail Removed)> writes:

> It is unfortunate that JavaScript couldn't be a bit less clunky in the
> functional style:
>
> ["1","2","3"].map(parseInt); //[1, NaN, NaN]

Don't blame your tools. The map function passes three arguments to
the function, and parseInt expects two. What you want is probably:
["1","2","3"].map(function(x) { return parseInt(x, 10); });

/L
--
Lasse Reichstein Holst Nielsen
'Javascript frameworks is a disruptive technology'

Michael Haufe (\TNO\)
Guest
Posts: n/a

 02-25-2011
On Feb 25, 11:05*am, Lasse Reichstein Nielsen <(E-Mail Removed)>
wrote:
> "Michael Haufe (\"TNO\")" <(E-Mail Removed)> writes:
>
> > It is unfortunate that JavaScript couldn't be a bit less clunky in the
> > functional style:

>
> > ["1","2","3"].map(parseInt); //[1, NaN, NaN]

>
> Don't blame your tools. The map function passes three arguments to
> the function, and parseInt expects two. What you want is probably:
> *["1","2","3"].map(function(x) { return parseInt(x, 10); });

Alternatively for this example: ["1","2","3"].map(Number);

The point is that it is often the case that solutions like yours are
necessary in these types of methods which I find clunky.

Ry Nohryb
Guest
Posts: n/a

 02-26-2011
On Feb 25, 7:03*pm, "Michael Haufe (\"TNO\")"
<(E-Mail Removed)> wrote:
> On Feb 25, 11:05*am, Lasse Reichstein Nielsen <(E-Mail Removed)>
> wrote:
>
> > "Michael Haufe (\"TNO\")" <(E-Mail Removed)> writes:

>
> > > It is unfortunate that JavaScript couldn't be a bit less clunky in the
> > > functional style:

>
> > > ["1","2","3"].map(parseInt); //[1, NaN, NaN]

>
> > Don't blame your tools. The map function passes three arguments to
> > the function, and parseInt expects two. What you want is probably:
> > *["1","2","3"].map(function(x) { return parseInt(x, 10); });

>
> Alternatively for this example: ["1","2","3"].map(Number);
>
> The point is that it is often the case that solutions like yours are
> necessary in these types of methods which I find clunky.

Not clunky, no, if you put garbage in you get garbage out.
--
Jorge.

Michael Haufe (\TNO\)
Guest
Posts: n/a

 02-26-2011
On Feb 26, 4:50*am, Ry Nohryb <(E-Mail Removed)> wrote:

> Not clunky, no, if you put garbage in you get garbage out.

If a construct is not intuitive and requires extra boilerplate to make
it work, it can be considered clunky. Having to use "function()
{ return ... }" or similar to massage parameters is not something to
smile at. \be's shorter function syntax will help a bit in this area
"#(x) { x * x }", but even this only dulls the pain of having to do it
in the first place. Compare code similar to the above with other
functional languages such as Miranda, Haskell, OCaml, SML, F#, popular
Lisp languages and others, and you'll see what I mean. These array
"extras" and "Array generics" go quite far to help enable a more
functional programming style in JS but with the choices made in their
signatures it has made it painful to perform one of the most important
important aspects of developing in the functional style : composition.
irt being non-intuitive, if you've used one or more of the above
languages little things like this are probably going to bite you more
than once. I'm not trying to rally against these new array methods,
but simply showing that there is another side of the coin.

Lasse Reichstein Nielsen
Guest
Posts: n/a

 02-27-2011
"Michael Haufe (\"TNO\")" <(E-Mail Removed)> writes:

> On Feb 26, 4:50*am, Ry Nohryb <(E-Mail Removed)> wrote:
>
>> Not clunky, no, if you put garbage in you get garbage out.

>
>
> If a construct is not intuitive and requires extra boilerplate to make
> it work, it can be considered clunky.

What part of the construct is it that is not intuitive here?

My guess is the fact that map actually passes three arguments, not
just one, which differs from, e.g., Lisp's map-car or Scheme's map.
But that's actually very much a Javascript-like feature.

> Having to use "function()
> { return ... }" or similar to massage parameters is not something to
> smile at. \be's shorter function syntax will help a bit in this area
> "#(x) { x * x }", but even this only dulls the pain of having to do it
> in the first place. Compare code similar to the above with other
> functional languages such as Miranda, Haskell, OCaml, SML, F#, popular
> Lisp languages and others, and you'll see what I mean.

Notice that neither of those languages allow you to call a function with
fewer or more arguments than what you expect. That means that you won't
even typecheck (for the statically typed langauges) if your map passed
three arguments and you function expected two.

The same function would fail in Lisp and Scheme, if map passed three
arguments.

If you had a map function passing three arguments in SML, you would
still have to write
map myList (fn (x,i,l) => myUnaryFunction(x))
to use it with an unary function.

Javascript is unique among the functional languages you mention in that
you can pass too few or too many arguments to a function. That's why they
allowed themselves to let map pass three arguments instead of just one.

If you have a function expecting just one argument, then it still works.

What you did here was to pass a function that expected two arguments
(but would work with only one) and you got predictably incorrect
results to a function calling it with three arguments, where the second
did not work as a second argument for the first function.

> These array "extras" and "Array generics" go quite far to help
> enable a more functional programming style in JS but with the
> choices made in their signatures it has made it painful to perform
> one of the most important important aspects of developing in the
> functional style : composition.

If your functions don't allow a variable number of arguments, then
there is no problem - it will ignore any surplus arguments.

The way Javascript is designed, that's sadly not something you can
rely on.

> irt being non-intuitive, if you've used one or more of the above
> languages little things like this are probably going to bite you more
> than once. I'm not trying to rally against these new array methods,
> but simply showing that there is another side of the coin.

The Javascript array functions differ from similar list-functions in
other languages. That's both good and bad - they actually fit
Javascript well, since the language allows optional arguments so
easily, but they bite you when you combine two different ways of being
optional (passing potentially unnecessary extra arguments to a
function expecting optional arguments, where the user thinks just of
the one-argument case).

/L 'Never call parseInt with only one argument'.
--
Lasse Reichstein Holst Nielsen
'Javascript frameworks is a disruptive technology'

Michael Haufe (\TNO\)
Guest
Posts: n/a

 02-27-2011
On Feb 27, 6:16*am, Lasse Reichstein Nielsen <(E-Mail Removed)>
wrote:
> "Michael Haufe (\"TNO\")" <(E-Mail Removed)> writes:
>
> > On Feb 26, 4:50 am, Ry Nohryb <(E-Mail Removed)> wrote:

>
> >> Not clunky, no, if you put garbage in you get garbage out.

>
> > If a construct is not intuitive and requires extra boilerplate to make
> > it work, it can be considered clunky.

>
> What part of the construct is it that is not intuitive here?
>
> My guess is the fact that map actually passes three arguments, not
> just one, which differs from, e.g., Lisp's map-car or Scheme's map.

Yes

> But that's actually very much a Javascript-like feature.
>
> > Having to use "function()
> > { return ... }" or similar to massage parameters is not something to
> > smile at. \be's shorter function syntax will help a bit in this area
> > "#(x) { x * x }", but even this only dulls the pain of having to do it
> > in the first place. Compare code similar to the above with other
> > functional languages such as Miranda, Haskell, OCaml, SML, F#, popular
> > Lisp languages and others, and you'll see what I mean.

>
> Notice that neither of those languages allow you to call a function with
> fewer or more arguments than what you expect.

Fewer is allowed, but not more. (currying) Unless you are referring to
Lisp/Scheme only by saying "neither"?

> That means that you won't
> even typecheck (for the statically typed langauges) if your map passed
> three arguments and you function expected two.
>
> The same function would fail in Lisp and Scheme, if map passed three
> arguments.
>
> If you had a map function passing three arguments in SML, you would
> still have to write
> * *map myList (fn (x,i,l) => myUnaryFunction(x))
> to use it with an unary function.

Of course since mutability is avoided, the third argument would have
little reason to exist. For the second it's a little meaningless since
the use case are hard to find. In fact I don't think there is even an
(indexof foo) style function in the standard lib.

> Javascript is unique among the functional languages you mention in that
> you can pass too few or too many arguments to a function.

Not quite... <http://mlton.org/Fold>

Unless you want to compare arbitrary size tuples to JS arguments.

> That's why they
> allowed themselves to let map pass three arguments instead of just one.
>
> If you have a function expecting just one argument, then it still works.
>
> What you did here was to pass a function that expected two arguments
> (but would work with only one) and you got predictably incorrect
> results to a function calling it with three arguments, where the second
> did not work as a second argument for the first function.

Yes. It looks like a similar discussion as this one is going on here:
<http://www.wirfs-brock.com/allen/posts/166>.
Also here <http://whereswalden.com/2011/02/26/t...r-way-to-call-

> > These array "extras" and "Array generics" go quite far to help
> > enable a more functional programming style in JS but with the
> > choices made in their signatures it has made it painful to perform
> > one of the most important important aspects of developing in the
> > functional style : composition.

>
> If your functions don't allow a variable number of arguments, then
> there is no problem - it will ignore any surplus arguments.
>
> The way Javascript is designed, that's sadly not something you can
> rely on.

The identity crises strikes again.

> > irt being non-intuitive, if you've used one or more of the above
> > languages little things like this are probably going to bite you more
> > than once. I'm not trying to rally against these new array methods,
> > but simply showing that there is another side of the coin.

>
> The Javascript array functions differ from similar list-functions in
> other languages. That's both good and bad - they actually fit
> Javascript well, since the language allows optional arguments so
> easily, but they bite you when you combine two different ways of being
> optional (passing potentially unnecessary extra arguments to a
> function expecting optional arguments, where the user thinks just of
> the one-argument case).

In one of the links above there was a suggestion of adding
Function.prototype.only(...) as a solution. Though I find that just as
bad since in the case of parseInt you still have to know there was a
problem in the first place requiring this type of solution.