text:
Application of gradient vector in 3D

1. Dec 18, 2012

CAF123

1. The problem statement, all variables and given/known data
The temperature $T$ in a region of Cartesian $(x,y,z)-$ space is given by $$T(x,y,z) = (4 + 3x^2 + 2y^2 + z^2)^{10},$$ and a fly is intially at the point $(-5,6,7)$. Find a vector parametric representation for the curve which the fly should move in order to ensure that the temperature it experiences decreases as rapidly as possible.

3. The attempt at a solution

I have done a 2D analogue of this problem, however, it only asked for the curve in rectangular coordinates and not a vector parametrisation. Furthermore, I was able to get that curve by computing dy/dx which I can't do here.

So, what I have done so far is compute the gradient vector, $\underline{\nabla}T$ and took the negative of this because we want the vector to be pointing in the direction of decreasing values of $T$. I don't really know where to go from here. I am considering using the Implicit Function Theorem to attain $$\frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z}.$$ Is this good? If I do that, then I would just integrate these three expressions. Any ideas? Many thanks

2. Dec 18, 2012

Vargo

You are looking for a curve of the form x(t), y(t), z(t). And you want its time derivative to be proportional to -grad T, evaluated at x(t),y(t),z(t). This is a system of ordinary differential equations for three unknown functions. The key here is that the velocity does not have to be the negative gradient, it only needs to be proportional to it. You can use that to make the differential equations very easy to solve.

3. Dec 18, 2012

CAF123

So the velocity is in the same direction as the negative gradient vector. So I have: $$\frac{dx}{dt} = k(\underline{T} \cdot \hat{x}), k \,\in \,\mathbb{R}$$ and similar expressions for y and z?

4. Dec 18, 2012

Vargo

Not quite. The factor k need not be a constant scalar. The only requirement is that for all t, the velocity vector is parallel to the negative gradient. The positive scalar k could (and should if you want a simple solution) vary with time.

Also, I am not familiar with the underline notation, but I assume it means negative gradient.

If you have a system of ODE such as
$\frac{d\vec{r}}{dt} = F(\vec{r})$
where F(r) is a vector field, then the solution curves are not changed if you multiply F by a scalar function. The parametrizations change, but the image curves do not. In other words, the velocity with which the curve is traversed changes, but the curve itself does not change. The reason is that the curves have to be everywhere tangent to the vector field. That requirement in itself determines the curves and is unchanged by multiplying F by a scalar field as long as the scalar field (function) is never 0. If your scalar field is negative everywhere, it reverses the direction of the curves, but still gives you the same actual curves.

Last edited: Dec 18, 2012
5. Dec 18, 2012

CAF123

Ok, so something like: $$\frac{dx}{dt} = k(t)(-\underline{\nabla}T \cdot \hat{x})$$
I made an error on notation in the previous post. It is fixed now.

6. Dec 18, 2012

Vargo

Yes, something like that. Saying k depends on time might have been misleading on my part though. It would have been better to say it depends on time through its dependence on position. k = k(x(t),y(t),z(t)). So really, your k is a function of position.

7. Dec 18, 2012

CAF123

I suppose this makes sense - as the curve is traced out, the velocity may change in magnitude. However, the resulting differential eqn looks messy: $$\frac{dx}{dt} = - k(x(t))(10(4+3x^2+2y^2+z^2)^{9} \cdot 6x)$$ Should I do this by separation of variables?

8. Dec 18, 2012

Vargo

No,

$\frac{dx}{dt} = - k(x,y,z)*20(4+3x^2+2y^2+z^2)^9*3x$

k depends on all three variables x,y,z (which in turn all depend on t). I am suppressing the dependence on t in the right hand side to make it more readable. Now write down the corresponding equations for y' and z', and then find a function k(x,y,z) that is the most convenient possible choice for making all three equations simpler.

Essentially, we are replacing -grad T with -k(x,y,z)grad T in order to make the vector field simpler.

9. Dec 18, 2012

CAF123

$$\frac{dy}{dt} = - k(x,y,z)*20(4+3x^2+2y^2+z^2)^9*2y$$ and $$\frac{dz}{dt} = - k(x,y,z)*20(4+3x^2+2y^2+z^2)^9*z$$ I am not sure what you mean by finding a function k(x,y,z) to make the vector field simpler. By introducing this k, how does that simplify things?
Thanks.

10. Dec 18, 2012

Vargo

What I mean is choose a function k(x,y,z) that simplifies the expression. Any choice (as long as it is never 0) will not affect the integral curves. So you are free to substitute anything you like for k(x,y,z). For example, you could try
$k(x,y,z) = 1+|x|^5\cos(y)+z^4.$
but that would be a poor choice because it would only make your equations more complicated.

11. Dec 18, 2012

CAF123

Ok, so I could have something like k(x,y,z) = xyz or k(x,y,z) = 1? As long as it's some scalar (function). Or what about $k(x,y,z) = (4 + 3x^2 + 2y^2 + z^2)^{-9}?$

12. Dec 18, 2012

Vargo

Now you're catching on.

13. Dec 18, 2012

CAF123

By doing the integration, I got $x = (Ae^t)^{1/60}, y= (Be^t)^{1/40}, z = (Ce^t)^{1/20}$?. So $\underline{r} = x \hat{i} + y\hat{j} + z \hat{k}$

14. Dec 18, 2012

Vargo

Looks close. But you lost a minus sign somewhere. Your solutions should decay. Also, you have an initial value of (-5,6,7), so you can solve for A,B,C. You could simplify this more by including a factor of 1/20 in your function k.

15. Dec 18, 2012

CAF123

In the end, I have $$\underline{r} = 7e^{-2t}\hat{i} + 6e^{-3t}\hat{j} -5e^{-t}\hat{k}$$


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MathSciNet bibliographic data MR823246 (87e:11112) 11N60 Erdős, Paul; Pomerance, Carl On the normal number of prime factors of $\phi(n)$$\phi(n)$. Number theory (Winnipeg, Man., 1983). Rocky Mountain J. Math. 15 (1985), no. 2, 343–352. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.

format: plain


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2.1.16.1.5 grnd

Description

normally distributed random number with mean=0 and sd=1, nSeed > 0 to set random seed (returns 0)

Syntax

double grnd( int nSeed = 0 )

Parameters

nSeed
(optional) pass any nonzero value to set the seed for random number generation, or 0 to use the current seed.

Return

A normally distributed random number.

Examples

EX1

// The example displays 10 normally distributes random numbers with mean=0 and
// standard deviation = 1.
void    grnd_ex1()
{
for (int ii = 0; ii < 10; ii++)
{
double    rr = grnd();
out_double("value = ", rr);
}
}

Remark

On every call it generates a normally distributed random double value with mean = 0. and

standard deviation = 1.

origin.h


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# Perl Weekly Challenge 117: Missing Row

by Abigail

## Challenge

You are given text file with rows numbered 1-15 in random order but there is a catch one row in missing in the file.

11, Line Eleven
1, Line one
9, Line Nine
13, Line Thirteen
2, Line two
6, Line Six
8, Line Eight
10, Line Ten
7, Line Seven
4, Line Four
14, Line Fourteen
3, Line three
15, Line Fifteen
5, Line Five

Write a script to find the missing row number.

## Solution

This is fairly trivial. The sum of the number $$1$$ to $$15$$ equals $$\sum_{i=1}^{15} i = 120$$. So, if we add the sum of the line numbers in the file, and subtract this from $$120$$ we get the missing row number.

### Perl

First we import the sum method from List::Util:

use List::Util qw [sum];

Then we can get the answer in a one liner:

say 120 - sum <>;

We take advantage of the fact that if we use a string as a number, Perl will look at the beginning of the string, and tries to interpret that a number. It does issue a warning if there is some trailing stuff, but hey, if you don't turn on warnings, you won't see them.

Find the full program on GitHub.

We could also have used a one-liner, not requiring another module, nor having the number of lines hard coded:

perl -nE '$;+=$.-$_;END{say$.+1+$;}'  ### AWK Just like in Perl, AWK looks at the beginning of a string if it's treated as a number. It won't issue a warning. So we get: BEGIN {SUM_15 = 120} {sum +=$1}
END   {print SUM_15 - sum}

Find the full program on GitHub.

### Bash

For the Bash solution, we split the input. Bash does this already be default, splitting on white space. But we want to split on commas:

IFS=","

Now it's a matter of summing the first column, and subtracting the result from 120:

SUM_15=120
while read number tail; do ((sum += number)); done
echo \$((SUM_15 - sum))

Find the full program on GitHub.

### C

In C, we use the atol function to get the number at the beginning of a string.

# define SUM_15 120

int main (void) {
char *  line    = NULL;
size_t  len     = 0;
size_t  str_len;
int     sum     = 0;

while ((str_len = getline (&line, &len, stdin)) != -1) {
sum += atol (line);
}
printf ("%d\n", SUM_15 - sum);
free (line);

return (0);
}

Find the full program on GitHub.

### Go

In Go, we use the method ReadString in an bufio object to read lines of input. On EOF, this method returns an error, which we use to break the loop. Go does have an Atoi method, but that doesn't work like atoi in C: it balks if there are non-numbers following the number at the beginning of the string. So, we're going to use Sscanf to parse the number.

func main () {
var i      int
var sum    int = 0
var SUM_15 int = 120

for {
if (err1 != nil) {
break
}
var n, err2 = fmt . Sscanf (text, "%d", &i)
if (n < 1 || err2 != nil) {
continue
}
sum += i
}
fmt . Printf ("%d\n", SUM_15 - sum)
}

Find the full program on GitHub.

### Java

In Java, we use a scanner object to read lines of input. Using Google, I found a couple of methods to convert string to integers, but all of them require the string to contain exactly an integer. No atoi equivalent.

So, we're just parsing the beginning of the string manually. We're making use of the charAt method on string objects: given an argument n, it returns the ASCII value of the nth character in the string.

import java.util.*;

public class ch1 {
public static void main (String [] args) {
Scanner scanner = new Scanner (System . in);
int sum    = 0;
int SUM_15 = 120;
try {
while (true) {
String line = scanner . nextLine ();
int i, n = 0;
for (i = 0; i < line . length (); i ++) {
if (line . charAt (i) < "0" . charAt (0) ||
line . charAt (i) > "9" . charAt (0)) {
break;
}
n *= 10;
n += line . charAt (i) - "0" . charAt (0);
}
sum += n;
}
}
catch (Exception e) {
System . out . println (SUM_15 - sum);
}
}
}

Find the full program on GitHub.

### Lua

To turn the beginning of a string into a number, we capture the leading digits, and use the function tonumber to make it a number:

local SUM_15 = 120
local sum = 0
for line in io . lines () do
sum = sum + tonumber (line : match ("%d+"))
end
print (SUM_15 - sum)

Find the full program on GitHub.

### Node.js

Here, we split the lines of input on a comma, and use the unary + to change a string into a number:

let SUM_15 = 120
let sum = 0

. createInterface ({input: process . stdin})
. on              ('line', line => {
let [num] = line . split (/,/)
sum +=+ num
})
. on              ('close', () => console . log (SUM_15 - sum))

Find the full program on GitHub.

### Python

Splitting on commas is the way to go in Python as well, and we use int to turn the string into an integer:

import fileinput

sum = 0
SUM_15 = 120

for line in fileinput . input ():
sum = sum + int ((line . split (",")) [0])

print (SUM_15 - sum)

Find the full program on GitHub.

### Ruby

Another split in Ruby, using the to_i method to turn the result into an integer:

sum = 0
SUM_15 = 120
ARGF . each_line do
|line|
sum += ((line . split (/,/)) [0]) . to_i
end

puts (SUM_15 - sum)

Find the full program on GitHub.


text:
Found 2 Documents (Results 1–2)

100
MathJax

No rational nonsingular quartic curve $${\mathcal C}_ 4\subset {\mathbb{P}}^ 3$$ can be set-theoretic complete intersection on a cubic surface. (English)Zbl 0706.14031

Reviewer: M.Herrmann
MSC:  14M10 14H50 14M20
Full Text:

The curve $${\tilde{\mathcal C}}_ 4=(\lambda^ 4,\lambda^ 3\mu,\lambda\mu^ 3,\mu^ 4)\subset {\mathbb{P}}^ 3_ k$$, is not set- theoretic complete intersection of two quartic surfaces. (English)Zbl 0613.14025

Reviewer: Ngo Viet Trung
MSC:  14H45 14N05 14M10
Full Text:


text:
• Share

# Solve Combined Stresses in Mohr’s Circle – Problems for GATE & SSC JE Exams (Part III)

9 months ago .

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Mohr’s contribution extended the use of graphical representation for both two- and three-dimensional stresses and developed a failure criterion based on the stress circle. In the previous article, we got to solve the Mohr Circle Quiz and also got to know about the study notes on Biaxial Stresses. In this article, we will focus on how to solve Combined Stresses in Mohr’s Circle. You can read the previous article with the link given below & take the Mohr’s Circle Quiz at the end of the article. Aspirants of &  as well as SSC JE ME & SSC JE CE, will benefit from this article.

Also check Part I & Part II of Mohr Circle by clicking the link below;

## Mohr’s Circle – An Introduction

• The transformation equations for plane stress can be represented in graphical form by a plot known as Mohr’s Circle. This graphical representation is extremely useful because it enables you to visualize the relationships between the normal and shear stresses acting on various inclined planes at a point in a stressed body. Using Mohr’s circle, you can also calculate principal stresses, maximum shear stresses and stresses on inclined planes.
• Stress analysis is a fundamental part of engineering. It is critical that engineers understand stresses caused by different loading conditions and determine the location of maximum stress. Most often, a structural member is subjected to different types of stresses that acts simultaneously.
• The stresses σx and σy may be compressive or tensile and may be the result of direct forces or as a result of bending. The shear stresses may be as shown or completely reversed and occur as a result of either shear force or torsion.

### Combined Stresses in Mohr’s Circle – Steps to Draw

• θ on the stress element = 2θ in Mohr’s circle
• Draw a set of coordinate axes with σx as positive to the right and τxy as positive upward.
• Locate point A, representing the stress conditions on the x face of the element by plotting its coordinates σx and τxy. Note that point A on the circle corresponds to θ = 0°.
• Locate point B, representing the stress conditions on the y face of the element by plotting its coordinates σy and -τxy. Note that point B on the circle corresponds to θ = 90°.
• Draw a line from point A to point B, a diameter of the circle passing through point O (centre of circle). Points A and B are at opposite ends of the diameter (and therefore 180° apart on the circle).
• Using point O as the centre, draw Mohr’s circle through points A and B. This circle has radius R. The centre of the circle c at the point having coordinates σavg and τxy.

### Combined Stresses in Mohr’s Circle – Diameter

Here’s how to calculate diameter of Mohr’s Circle

### Combined Stresses in Mohr’s Circle: Steps to draw Mohr’s Circle on inclined Plane (θ):

●        On Mohr’s circle, point A corresponds to θ = 0. Thus it’s the reference point from which angles are measured.

●        The angle 2θ locates the point M on the circle, which has coordinates σ and τ . M represents the stresses on the xl face of the inclined element.

●        Principle stresses are stresses that act on a principle surface. This surface has no shear force components (that means τxy = 0).

●        Planes of maximum shear stress occur at 45° to the principal planes.

●        The maximum shear stress is equal to one half the difference of the principal stresses.

## Solve Combined Stresses in Mohr’s Circle Problems for GATE Quiz

Now take this quiz on Combined Stresses in Mohr’s Circle:

Mohr’s Circle for Principal Stress and Maximum Shear Stress_Combined Stress
Que. 1

When a body is subjected to a direct tensile stress (σx), in one plane accompanied by a simple shear stress (τxy), the maximum shear stress is

1.
$$\frac{{{\sigma _x}}}{2}\; + \;\frac{1}{2}\sqrt {\sigma _x^2\; + \;4\tau _{xy}^2}$$
2.
$$\frac{{{\sigma _x}}}{2} – \frac{1}{2}\sqrt {\sigma _x^2\; + \;4\tau _{xy}^2}$$
3.
$$\frac{{{\sigma _x}}}{2}\; + \;\frac{1}{2}\sqrt {\sigma _x^2 – 4\tau _{xy}^2}$$
4.
$$\frac{1}{2}\sqrt {\sigma _x^2\; + \;4\tau _{xy}^2}$$

Que. 2

For the stress state (in MPa) as shown in figure, the major principal stress is 15 MPa. The shear stress, τ is

1.
0
2.
5 MPa
3.
10 MPa
4.
15 MPa

Que. 3

If at a point in a body σx = 70 MPa, σy = 60 MPa and τxy = – 5 MPa then the radius of the Mohr’s circle is equal to

1.
$$5\sqrt 5 MPa$$
2.
$$2\sqrt 5 MPa$$
3.
$$5\sqrt 2 MPa$$
4.
25 MPa

Que. 4

Determine the value of maximum principal stress (in kPa)

Que. 5

The radius of Mohr’s circle of stress of a strained element is 20 N/mm2 and minor principal tensile stress is 20 N/mm2. The major principal stress is

1.
30 N/mm2
2.
50 N/mm2
3.
60 N/mm2
4.
100 N/mm2

As we all know, practice is the key to success. Therefore, boost your preparation by starting your practice now.

Furthermore, chat with your fellow aspirants and our experts about Combined Stresses in Mohr’s Circle to get your doubts cleared on Testbook Discuss:

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text:
## Elementary Geometry for College Students (6th Edition)

$A: 0,0 \\ B: 2b,2c \\ C: 2a,0$
An ideally placed triangle, which we will call triangle ABC, should have point A placed at the origin and side AC be on the x-axis. Calling the length of AC "2a" and the coordinate of point B (2b,2c), we obtain: $A: 0,0 \\ B: 2b,2c \\ C: 2a,0$ Note, there are now twos in front of the numbers, for the book asks for midpoints, meaning that we must double all of the values to get the length of the side as opposed to the lengths of half of a side.


text:
# Bounded Natural Functors With Covariance and Contravariance

April 24, 2018

### Abstract

Bounded natural functors (BNFs) provide a modular framework for the construction of (co)datatypes in higher-order logic. Their functorial operations, the mapper and relator, are restricted to a subset of the parameters, namely those where recursion can take place. For certain applications, such as free theorems, data refinement, quotients, and generalised rewriting, it is desirable that these operations do not ignore the other parameters. In this article, we formalise the generalisation BNFCC that extends the mapper and relator to covariant and contravariant parameters. We show that
1. BNFCCs are closed under functor composition and least and greatest fixpoints,
2. subtypes inherit the BNFCC structure under conditions that generalise those for the BNF case, and
3. BNFCCs preserve quotients under mild conditions.
These proofs are carried out for abstract BNFCCs similar to the AFP entry BNF Operations. In addition, we apply the BNFCC theory to several concrete functors.


text:
# I like science and math but I hate working in industry

1. Feb 16, 2016

### OrangeDog

I normally would have put this in the grad school section, but the question is more pertinent to choosing a career path.

So basically at one point in time I was in a graduate program. I left pretty early due to funding issues as well as some personal issues that I needed to resolve. I ended up in industry - where I have been for the last two years.

I did pretty good in undergrad: 3.44 cum, 3.65 major GPA plus a time consuming extra curricular activity involving UAVs (not quadcopters, we built actual planes!). I found that I really enjoyed the academic environment. School was the first place that I really felt like I fit in - I was "riding the wave" so to speak. My life had never gone smoother.

Unfortunately grad school didn't go as smooth, so I left. That was a long story in itself so Ill spare you the details. This year I decided to try again and apply to PhD opportunities. There is a very strong chance Ill have an offer with good funding. This offer will be working on nonlinear dynamics and control of UAVs - right up my alley!

So, my concerns:
1) I don't want to be disillusioned. I absolutely hate sitting at a desk all day. I hate being swamped in paperwork doing mostly uncreative garbage. I hate corporate culture, I hate having a strict 9 to 5 routine, and I really hate having so little time to pursue what I enjoy. Yes, financially I am doing ok, but without having the time to enjoy my work, whats the point? And I hate being watched/having no privacy all the time (currently work in a cube farm). What I am worried about is going to grad school, doing my PhD, and ending up in a similar spot that I am now.

2) Ill admit I am pretty scared that grad school wont go so well again. I left because the biweekly stipend was too low (600/month) and would require me to take out loans to pay for living expenses. Also, I had some personal problems with PTSD which made me question why I was going in the first place, and that spiraled into depression.

3) I am wondering if I should even stay in engineering at all. Like I said, I love math and sciences but I cant stand engineers. I think that is the worst part of my job actually. I hate being around anti social people. A lot of them are very boring or lack life experience - making them naïve. These aren't malicious attributes, but as someone who likes to have fun, is outgoing, and has been down some interesting roads in life, I find the engineering social circle miserable. I've thought about being a bar tender and working on my projects in my free time. Or opening up my own gym and continuing my studies in my free time. Both require more money than I have at the moment.

4) the student debt is a bit of a concern. My loan balance will balloon due to interest, but if I get a higher paying job my debt to income ratio might actually be lower, so I should theoretically be in a better financial position. This is contingent on getting a good paying job though or by some miracle the debt disappears.

So..what should I do haha??

2. Feb 16, 2016

### StatGuy2000

Here is a question I would like to pose to you before I, or anyone else, can respond with the rest of your questions. Based on your current salary at your workplace, how long would you estimate it would take to fully pay off your current student debt?

3. Feb 16, 2016

Based on the professors and students I talked to, my income if I went into industry could be (starting) between $90000 and$110000. My debt (if I just stopped paying it) would balloon to about $77617. Now there are some tricks I could do to bring that balance lower, but that isn't the topic of this thread. 5. Feb 16, 2016 ### Ben Espen Maybe you just need to try working somewhere else? The office culture of different companies differs quite a bit, even within engineering. 6. Feb 16, 2016 ### OrangeDog Ive thought about it. I worked at another company for my internship and felt the same way. Granted, the people were a little better to work with and it was a smaller group which was also nice, but I still got really annoyed at sitting at a desk all day. I need to move and recharge! Maybe this is an unrealistic expectation? 7. Feb 16, 2016 ### mfb ### Staff: Mentor There are some academic engineering-related* positions that don't require a PhD, but my experience is mainly about Europe, where you do bachelor -> master -> PhD in that order (the US PhD is similar to master+PhD together). On the other hand, you can have two years of related work experience... And there are academic engineering-related* PhD positions with a reasonable salary, but those usually require a master degree. I don't know if two years of work experience would count here, the universities often have strict admission rules for PhD candidates. *including positions that involve going to a lab and assembling stuff. They still require work on a computer as nearly every STEM job, but not exclusively. 8. Feb 16, 2016 ### Dale ### Staff: Mentor Has that changed? 9. Feb 17, 2016 ### OrangeDog If I get accepted into this one program I am guaranteed funding of at least$1600/month. Additionally, in interviewing with the program in question it was strongly suggested that I be considered for a university fellowship. The fellowship will raise my funding for two (maybe 3, I cant remember) years to $2500 a month. I emailed the university and they stated that I will be receiving an email of my acceptance shortly. The way I look at it though is until I have a signed contract stating my funding and my acceptance then I will not consider myself "in" 10. Feb 17, 2016 ### mfb ### Staff: Mentor$2500 a month should allow to reduce the loan significantly. One more comment on them, wanted to write it but forgot it in the previous post:
It is certainly useful to have some money available quickly, but does that amount have to increase by \$300 per month? I guess you don't get 6.4% interest rate on that.

11. Feb 17, 2016

### Ben Espen

That isn't an unrealistic expectation. There is a wide variety of technical work available, with varying degrees of desk work. You ought to be able to find something. Out of curiosity, do you not spend most of your time in grad school at a desk, or is that not your experience?

12. Feb 17, 2016

### OrangeDog

I like having cash on hand in case anything comes up. Maybe a lucrative job offer, maybe a car accident. I know many people without savings (also engineers) who have to resort to credit cards. To me, more debt is a mistake. No debt = freedom. I will lower it when I feel like I have a comfortable amount in my long short and intermediate term savings.

The difference is that when I was in school I could disengage from my work if I needed too. Granted, that did bother me sitting all day, but I felt like if I needed to walk away and think I could do so. Plus the work was more engaging so time went by much faster and I felt more fulfilled. This is a good point you bring up - I do remember being very ant-sy sitting a desk.

13. Feb 17, 2016

### Ben Espen

This is something important to know about yourself. I also dislike sitting for a long time. I have the ability to get up and walk most anytime I feel like it, so I try to do that multiple times every day. When the weather is nice, I go outside. I walk, gather my thoughts, and then plow back into whatever I am doing. Different places probably have different expectations about this is, but it is possible.

14. Feb 17, 2016

### Student100

Just to add what others are saying and asking here, don't ever stop putting money into your 401K- you'd just be throwing money away if they do any matching. When you leave your job, you can take the money out (with the match) and pay the penalties if you find you really need funds. Otherwise, keep it in the retirement account and roll it over in your next job.

15. Feb 18, 2016

### OrangeDog

That is what I liked about school. If it was a nice day out I could sit outside and work or at least go to a quiet area by a nice window and get my stuff done while relaxed. Not possible in a cube farm at a very conservative company. I got yelled at once because I ate my lunch (I was off the clock) near the water on company property (eating company food I might add)... Really absurd. What Ive noticed in the working world is most of the control freaks (or those who obsess about minutia) end up in middle management.

One of the reason why I want to do the PhD is because I want to teach and I really thrive on being intellectually engaged. But I have done my research and know that a teaching position at a University is very hard to come by today. There are other opportunities that come with higher education. I suppose I am torn between my love for the subject but not having the personality of a scientist. I am tired too, but that is another issue.

16. Feb 22, 2016

### Kilo Vectors

My relative graduated from engineering and one of the best universities, has nearly the same job as you..I think for me, even being able to have a job with a salary like yours would be great, let alone being able to actually work in a field I want! so would be willing to take anything.

best option is to stick it out a little while applying for other companies. 2 years isn't a long time, you can surely change or find a different company that would suit your interests somewhat more. In engineering people always say (especially on here) that degree and level of education is less important, work experience is everything. You're already getting some experience, you just need to stick it out. No, I would not do Phd right now. but thats my opinion.getting a life and stability is also important, a bachelors in engineering is enough for that.

Last edited: Feb 22, 2016


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# An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for The first officer on the spacecraft measures that the searchlight is on for (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earth expressed as a fraction of the speed of light c?

## $a ) t=12 m s$$b ) u=0.998 c$

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### Video Transcript

So here we have the diagram, the space cop as it cools past, that's be you return to be up the proper time, remember, is the time measure in the frame with the two events occur at the same point, and hence the proper time in this situation, which we need to calculate, is the time measured by the first often officer on the spacecraft. But this proper time enabled by doubt he not. And this is equal to 12 maybe sick. It's the time, major, but the light pulse as an observer in the spacecraft. So next up, can we calculate the speed off the space cop you? Well, we know an expression that's a T. The time measured by the observable up is equal to gamma delta t not, And that's down to Do you know what the proper time do you mind it by the square root both one minus use crib a C scream. And so if we do the mathematics and review in this equation, we can solve the speed you and you to see and the square roots off one minus out, not divided by doubts. T, both of which have known all squared And so now we can substitute our values. It's crazy. So we leave. See, as is quantify. Per square root off one minus The proper time is 12 many second. So that's 12 just into the minus three. Divided buying the time measured by the observable on but 0.19 seconds. And this is squared. Indians Uribe Cancer off zero point nine nine eight time. See? So the speed of the spacecraft is actually 99.8 times the speed of light. 99.8 sinned felons.

University of Kwazulu-Natal

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# A non-rotating spherical planet with no atmosphere has a mass M and radius R. A particle is fired...

## Question:

A non-rotating spherical planet with no atmosphere has a mass {eq}M {/eq} and radius {eq}R {/eq}. A particle is fired off from the surface with a speed equal to {eq}\frac{3}{4} {/eq} the escape speed. Calculate the farthest distance it reaches (measured from the center of the planet) if it is fired tangentially.

## Satellite Motion and Orbital Velocity:

An object launched from a given celestial body will orbit that body along an elliptical orbit if its launch velocity is smaller than the escape velocity. In the latter case, an object is not bound and can go to infinity. The motion along a closed bound orbit obeys conservation laws.

We will apply the law of conservation of energy and angular momentum to this problem.

{eq}\begin{align*} &m v_0 R = m v_a r_a &\text (1) \end{align*} {/eq}

This is the law of conservation of angular momentum for the particle;

Here

• {eq}m {/eq} is the mass of the particle;
• {eq}v_0 = \dfrac {3}{4} \sqrt{\dfrac {2GM}{R^2}} {/eq} is the launch velocity;
• {eq}v_a {/eq} is the speed of the particle at the point of farthest separation from the planet;
• {eq}r_a {/eq} is the distance to the aphelion;

At the same time, the total energy is conserved too:

{eq}\begin{align*} &\dfrac {mv^2_0}{2} - \dfrac {G M m}{R} = \dfrac {mv^2_a}{2} - \dfrac {GMm}{r_a} &\text (2) \end{align*} {/eq}

Eliminating the velocity in aphelion, we obtain:

{eq}\dfrac {v^2_0 R^2}{2 r^2_a} - \dfrac {GM}{r_a} = \dfrac {v^2_0}{2} - \dfrac {GM}{R} {/eq}

Substituting the expression for {eq}v_0 {/eq}, we come up with the following equation:

{eq}\dfrac {9}{16} \cdot \dfrac {R}{r^2_a} - \dfrac {1}{r_a} + \dfrac {7}{16 R} = 0 {/eq}

Solving, we get:

{eq}r_a = \boxed{(8 + \sqrt{55})R} {/eq}


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## Running Linux on a the Intel compute stick w5pro

I recently purchased an Intel computer stick W5 pro (4gb/64bg). It comes pre-installed with windows but of course, I want to run lines on it.

The problem is the wireless card isn’t supported out the box with ubuntu 18.04. A there an image/ distr I can use that will just work and install the correct drivers?

## How to compute the convex hull in linear time of the number of points on it?

I must construct 2 functions. The first function is used to add a point passed to it to an arraylist. The second function must take this arraylist, find out the set of points that lie on the convex hull formed from them, and return a linked list of those points.

How can I achieve this in a O(h) time (worst case), given that h is the number of points on the convex hull, NOT n which is the number of points originally present in the arraylist.

The two functions can have different worst case running time but most importantly, non should have a worst case that takes more than O(h).

I made it to O(n), not further though!

## cloud compute, high frequency, bare metal, dedicated cloud?

What is the difference between all these, and which one would be best advisable to use as a developer.

These actually are terms defined by vultr cloud computing

## Polynomial-time Compute the Number of States resulting from the NFA to DFA (greedy) conversion?

The canonical NFA to DFA conversion, starting with an NFA with $$n$$ states, can result in a DFA with $$2^n$$ states. However, in many cases, there are states that are “unnecessary,” such as when minimizing the resulting DFA.

I am interested in the following question: if we have an NFA with $$n$$ states and $$q$$ symbols, is there a $$\mathrm{poly}(n, q)$$-time algorithm to output the number of states that the standard conversion (and minimizing, if possible) to a DFA would produce? Note that producing the DFA itself can take exponential time.

## Compute $H^1(S,\underline{\text{Hom}}(A,\mathbb G_m))$ for a semi-abelian scheme $A$

How can I compute $$H^1(S,\underline{\text{Hom}}(A,\mathbb G_m))$$ (where $$A$$ denotes a semi-abelian scheme over $$S$$, $$\mathbb G_m$$ denotes the multiplicative group over $$S$$ and $$\underline{\text{Hom}}$$ denotes the Hom sheaf)?

## Compute cache miss rate for the given code

Problem Description:

We consider a 128-byte data cache that is 2-way associative ($$E=2$$) and can hold 4 doubles in every cache line. A double is assumed to require 8 bytes.

For the below code we assume a cold cache. Further, we consider an array A of 32 doubles that are cache aligned (that is, $$A[0]$$ is loaded into the first slot of a cache line in the first set). All other variables are held in registers. The code is parameterized by positive integers $$m$$ and $$n$$ that satisfy $$m*n = 32$$ (i.e., if you know one you know the other). Recall that the miss rate is defined as $$\frac{\text{number of misses}}{\text{number of accesses}}.$$

// Code:  float A[32], t = 0; for(int i = 0; i < m; i++)   for(int j = 0; j < n; j++)       t += A[j*m + i];

Problem: Determine the miss rate for when $$m=1,2$$ and $$16.$$

My Attempt:

When $$m=1$$ we access the elements, $$A[0],A[1],A[2],\cdots,A[31]$$ and since each block in the cache can hold $$2$$ doubles, this means that for every miss we get one hit and so the miss rate is $$1/2.$$

When $$m=2$$ we access the elements, $$A[0],A[2],A[4],\cdots,A[30].$$ Here we first miss $$A[0]$$ but then we load the block with $$(A[0],A[1]).$$ However, this does not help since we need to access $$A[2]$$ and therefore we miss every time and thus the miss rate is $$1.$$

When $$m=16$$ we access the elements, $$A[0],A[1],A[2],\cdots,A[31].$$ And so like before this should have a miss rate of $$1/2.$$

However, the answers are: $$m=1, \text{miss rate} = 1/4$$ $$m=2, \text{miss rate} = 1/2$$ $$m=16, \text{miss rate} = 1/4.$$

I am not sure how to obtain these answers and so any help will be much appreciated.

## Remove directory from Google Cloud Compute Engine using serial console [on hold]

The storage on my Google Cloud Compute Engine Instance (inadvertently) got completely used up. I can no longer SSH into my instance. I have increased the space on my instance, but it still will not allow me to SSH. My VM operating system is Ubuntu.

I have tried two things:

1.) I have tried creating a start script to remove a large directory to clear up some space. Under “custom metadata” I have set the key as “startup-script” and the value as

#! /bin/bash

rm /home/myusername/dir-to-rm

I save and reset the instance, click “connect to serial console”, but I still cannot SSH.

2.) I have tried to use a startup script to set a login username and password for the serial console by using the same method, but with the following script:

#! /bin/bash

sudo useradd tmpuser

echo 'tmpuser:password' | chpasswd

Neither of these methods have allowed me to SSH back into the VM.

Any help would be greatly appreciated.

## How can I upgrade the openGL version on a compute engine VM on GCP? [on hold]

GCP Compute Engine Machine info – Windows server 2016 – Xeon (R) 2Ghz 8 cores – RAM 7.2 GB – 1 Nvidia Tesla T4 Virtual workstation GPU – Connection using RDP

I’m trying to install Blender (3d app) which needs a minimum openGL 2.1 but my compute engine has openGL 1.1

Google compute engine VM screenshot:

I’ve installed nvidia drivers for Tesla T4 GPU from the nvidia website as well as updated windows server 2016, still the openGL error.

Please tell me how can I update the openGl version in my compute engine VM.

P.S. If I missed any info, please let me know.

## Randomized algorithm to compute cover radius?

I have self-studied the book “Geometric Approximation Algorithms” by Sariel Har-Peled. I am stuck on a problem and don’t know how to start it.

The problem statement:

Let $$C$$ and $$P$$ be two sets of points in the plane, such that $$k = |C|$$ and $$n = |P|$$.

Let $$M_P=max_{p\in P}$$, and $$m_C=min_{ c \in C }$$

Let $$r = \|m_C-M_P\|$$ be the cover radius of $$P$$ by $$C$$

Give a $$\mathcal{O}(n+k\log n)$$ expected time algorithm that outputs a number $$\alpha$$ such that $$r \leq \alpha \leq 10r$$

Any hints for that?

## How can I find (or compute) a client MAC address when using the strongSwan DHCP plugin?

Server strongswan.conf

libtls {   suites = TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA } charon {   plugins {     dhcp {       # Always use the configured server address.       force_server_address = yes        # Derive user-defined MAC address from hash of IKE identity.       identity_lease = yes        # Interface name the plugin uses for address allocation.       interface = eth0        # Whether to load the plugin. Can also be an integer to increase the       # priority of this plugin.       load = yes        # DHCP server unicast or broadcast IP address.       server = 10.0.2.255     }   } }

Client ipsec.conf

conn %default     ike=aes256gcm16-sha384-modp3072!     esp=aes256gcm16-sha384-modp3072!  conn ikev2     auto=start     leftid=client@my-vpn.com     leftsourceip=%config     leftauth=eap-tls     leftcert=vpn-client.crt     right=my-vpn.com     rightid=my-vpn.com     rightsubnet=0.0.0.0/0     rightauth=pubkey

The client is getting a dynamic IP from isc-dhcp-server but I would like to assign a static one using the following in dhcpd.conf.

host client {   hardware ethernet a1-b2-c3-d4-e5-f6;   fixed-address 10.0.2.2; }

According to https://wiki.strongswan.org/projects/strongswan/wiki/DHCPPlugin, the client MAC address can optionally be based on the IKEv2 identity of the client.


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# Boron arsenide heterostructures: lattice-matched heterointerfaces and strain effects on band alignments and mobility

## Abstract

BAs is a III–V semiconductor with ultra-high thermal conductivity, but many of its electronic properties are unknown. This work applies predictive atomistic calculations to investigate the properties of BAs heterostructures, such as strain effects on band alignments and carrier mobility, considering BAs as both a thin film and a substrate for lattice-matched materials. The results show that isotropic biaxial in-plane strain decreases the band gap independent of sign or direction. In addition, 1% biaxial tensile strain increases the in-plane electron and hole mobilities at 300 K by >60% compared to the unstrained values due to a reduction of the electron effective mass and of hole interband scattering. Moreover, BAs is shown to be nearly lattice-matched with InGaN and ZnSnN2, two important optoelectronic semiconductors with tunable band gaps by alloying and cation disorder, respectively. The results predict type-II band alignments and determine the absolute band offsets of these two materials with BAs. The combination of the ultra-high thermal conductivity and intrinsic p-type character of BAs, with its high electron and hole mobilities that can be further increased by tensile strain, as well as the lattice-match and the type-II band alignment with intrinsically n-type InGaN and ZnSnN2 demonstrate the potential of BAs heterostructures for electronic and optoelectronic devices.

## Introduction

Boron arsenide (BAs) is an attractive electronic material due to its ultra-high thermal conductivity (~1300 W m–1 K–1),1,2,3 native p-type dopability,4 and the availability of millimeter-size single crystals as substrates for thin-film growth.3,5,6 Following the experimental validation1,2,3 of the theoretical prediction7,8 of its ultra-high thermal conductivity, research efforts have focused on its growth and fundamental characterization.9 These results include studies of the thermal, optical, electronic, and structural properties of BAs in both bulk and 2D forms.4,9,10,11,12,13,14,15,16 However, to fully evaluate its potential in electronic and optoelectronic applications, the properties of BAs heterostructures with other semiconductor materials, either as a thin film or as a substrate, must be investigated.

Two important degrees of freedom for thin-film engineering in device architectures are mechanical strain and band alignment. Strain arising from epitaxial mismatch strongly affects the electronic properties of materials, including the band gap,17 band alignments,18 effective masses,19 and mobility.20,21 Though the epitaxial growth of BAs thin films has not been demonstrated yet, it should be feasible under growth conditions that replicate the established chemical vapor transport growth procedure. Candidate tetrahedrally bonded substrate materials for thin-film growth on the (111) plane of BAs are ZnO (3.249 Å22), GaN (3.189 Å23), or GaAs (3.997 Å24), while for (001)-oriented BAs thin films, candidate substrates are (001) oriented rutile TiO2 (4.59 Å25) or MgF2 (4.64 Å26), which result in epitaxial strains ranging from −6% to +15% (Fig. 1a). However, the effects of mechanical strain on the electronic properties of BAs are unknown.

On the other hand, BAs can also be utilized as a high-thermal-conductivity substrate for the epitaxial growth of other semiconductors (Fig. 1b). Specifically, the lattice constant of the (111) plane of BAs (3.380 Å1) is well matched with InGaN alloys (3.189–3.533 Å23,27,28) and disordered ZnSnN2 (~3.383 Å).29 These materials are prominent optoelectronic materials due to their direct band gap in the visible range (and the resulting strong optical absorption and emission), which can be tuned by adjusting the alloy composition (InGaN) or the degree of cation disorder (ZnSnN2),30 but they both face challenges in device applications. On the one hand, the amount of In that can be incorporated into InGaN is limited in part due to the large mismatch with the underlying substrate, typically GaN or sapphire. On the other, ZnSnN2 films are unintentionally heavily n-type in nature and require a partner p-type semiconductor in devices such as light-emitting diodes (LEDs) or solar cells. Both materials could benefit from p-type substrates with minimal lattice misfit for high-quality thin-film growth. BAs has intrinsic p-type conductivity owing to native defects such as BAs and VB as well as unintentional impurities such as C and Si,4,12 and can thus be a potential junction partner to these materials with good lattice match. However, band offsets for these hybrid structures and the assessment of their potential for device applications have not yet been determined.

In this work, we aim to understand the effect of biaxial strain on the band structure, mobility, and absolute band alignments of BAs with density functional theory (DFT) calculations and explore its possible applications in semiconductor heterostructures, either as a thin film, where biaxial strain may enhance its electronic properties, or as a substrate, where it may form advantageous heterojunctions with other materials. We show that, when epitaxially grown with 1% biaxial tensile strain, both electron and hole mobilities are increased by >60%, showing promise for fast-switching and energy-efficient transistors. These results motivate BAs as a promising thin-film semiconductor. We also propose favorable nearly lattice-matched heterojunctions of two important compounds, InGaN and ZnSnN2, with BAs, showing how BAs can be used as a high-thermal-conductivity substrate for type-II band aligned heterojunctions for photovoltaic and photodetector applications.

## Results and discussion

### Strain effects on the band structure and absolute band positions

We first examine the effects of strain on the band structure along the ΓX and ΓZ directions, which include the conduction band minimum (CBM) and valence band maximum (VBM). These results, calculated with the Heyd–Scuseria–Ernzerhof (HSE) hybrid functional and spin–orbit effects, are shown in Fig. 2a, b (Supplementary Fig. 4 shows the local-density approximation (LDA) band structure without spin–orbit effects). For the unstrained structure, the X (in-plane) and Z (out-of-plane) directions are equivalent by symmetry, and the conduction band minima along the ΓX and ΓZ directions are degenerate (Fig. 2a). However, biaxial strain within the (100) plane breaks the cubic symmetry and splits the degeneracy of the CBM valleys near X and Z; the lower CBM occurs along the ΓZ (ΓX) direction under tensile (compressive) strain (Fig. 2b). The broken symmetry under strain also affects the VBM by splitting the topmost two degenerate valence bands (Fig. 2b). These effects are important in explaining the mobility findings we discuss in the following paragraphs. Strain also shifts the absolute band positions as shown in Fig. 2c. We find that the conduction band lowers and the valence band rises in energy with increasing strain (independent of its sign) within both the (100) and (111) planes. This decreases the band gap from 1.78 eV (unstrained) to 0.89 eV (for 4% tensile strain within the (100) plane) or 1.15 eV (for 4% compressive strain within the (100) plane). We find that strain within the (100) plane decreases the band gap more than within the (111) plane. Also, while tensile strain changes the band gap more than compressive strain for the (100) plane, the opposite trend is found for the (111) plane.

### Strain effects on carrier mobility

Next, we investigate the effect of tensile strain on the phonon-limited electron and hole mobility of BAs, i.e., the upper mobility limits in non-defective samples. We first evaluate the converged values by plotting the calculated mobility at 300 K versus the inverse of the number of the Brillouin-zone (BZ) sampling points and extrapolating (Supplementary Fig. 5). For unstrained BAs, we find electron and hole mobility values of 1341 and 1387 cm2 V−1 s−1 at 300 K, respectively. We find excellent agreement between our calculated electron mobilities and the results of Liu et al.13 (1400 cm2 V−1 s−1). However, there are discrepancies for the hole mobility (2110 cm2 V–1 s–1) since our mobility calculations do not include spin–orbit coupling, which results in a threefold (instead of twofold) degenerate VBM (Supplementary Fig. 4). This higher band degeneracy in the absence of spin–orbit effects results in more scattering channels and a proportionately lower mobility. In fact, our result of 1387 cm2 V−1 s−1 is almost exactly 2/3 of the value from Liu et al.13 (2110 cm2 V–1 s–1), providing evidence for the strong impact of band degeneracy on the hole mobility. We note that our calculations for the electron mobility agree well with Liu et al.13 since the CBM is not significantly affected by spin–orbit coupling. Furthermore, while the unique phonon band structure of BAs is responsible for suppressing three-particle phonon–phonon scattering and enabling its ultra-high thermal conductivity,7 we do not expect the mobilities to be similarly affected as they are controlled by a different scattering mechanism (electron–phonon coupling). A possible indirect effect, however, is that the suppression of phonon–phonon scattering may prevent the thermal relaxation of optical phonons, and the resulting higher occupation may amplify carrier scattering by optical phonons and reduce the mobility.

We next examine the mobilities of the 1% tensile-strained structure. From these calculations, we find that the in-plane and out-of-plane mobilities at 300 K for electrons are 2417 and 482 cm2 V−1 s−1, respectively, while the corresponding values for holes are 3550 and 2956 cm2 V−1 s−1. Therefore, in BAs, tensile strain increases both the electron and hole mobilities of BAs, which is consistent with Si, which has a similar band structure.20,21

By examining the band structures in Fig. 2b, we gain useful insight into the different mechanisms contributing to the increased mobility. In the case of electrons, we observe that tensile strain results in the lifting of the threefold degeneracy in the conduction band (Fig. 3a, b), with the out-of-plane minima occurring ~200 meV below the two in-plane minima for 1% strain (Fig. 2b). This energy splitting has a strong effect on the effective mass of electrons. In an unstrained crystal, where all of the conduction valleys are degenerate (Fig. 3a), the average effective mass of electrons is the directional average between the lighter transverse effective mass and the heavier longitudinal effective mass [m* = (2mt* + ml*)/3 = 0.52].11 On the other hand, in strained BAs, similar to strained silicon,21 the majority of electrons occupy the lower-energy out-of-plane valleys (Fig. 3b), so in-plane transport is governed only by the lighter transverse effective mass [m* = 0.24],11 while out-of-plane transport is governed only by the heavier longitudinal effective mass [m* = 1.09].11 This difference in effective mass therefore helps explain not only the elevated in-plane electron mobility but also the suppressed out-of-plane electron mobility. Previous work on strained silicon also found that the changes in effective mass, as opposed to reduced intervalley scattering, are the dominant mechanism affecting the electron mobility.21

In the case of the hole mobility, the effective masses of the topmost three valence bands at Γ are all similar to one another, so the increase of the hole mobility by strain is explained by a different mechanism: the reduction in the number of available hole-scattering channels (Fig. 3c–f). Following from our previous discussion of the unstrained case, the inclusion of tensile strain leads to the topmost valence band becoming non-degenerate, with a separation of ~115 and ~145 meV with and without spin–orbit coupling, respectively. This reduction to only one hole-scattering channel leads to an increase in the hole mobility by approximately a factor of two compared to the unstrained spin–orbit-coupling-corrected results (where there were two scattering channels). We note that our result for the strained hole mobility is not expected to be strongly affected by the inclusion of spin–orbit coupling, since the top valence band of strained BAs is non-degenerate—by a similar degree—whether spin–orbit coupling effects are included (Fig. 3f) or not (Fig. 3e). Unlike electrons, we find that strain also increases the out-of-plane hole mobility due to the same reduction in the number of hole-scattering channels. We attribute the difference between the in-plane and out-of-plane hole-mobility values to the differences of the hole effective masses under strain, as the out-of-plane effective mass (m* = 0.25) is slightly higher than the in-plane mass (m* = 0.22). The qualitative conclusions that the strain increased hole mobility is due to the reduction of interband scattering caused by degeneracy lifting via broken symmetry, while the in-plane and out-of-plane hole-effective mass differences are responsible for the slightly different values along the two orientations are consistent with a similar investigation for strained Si.21

We also examine the temperature dependence of the electron and hole mobilities in the 100–500 K range (Fig. 4a, b). For the unstrained case, we find that, below 200 K, both the electron and hole mobilities follow a power law close to T−1.5 (Fig. 4c, d), indicating a primarily acoustic deformation potential carrier-scattering mechanism. From 200 to 500 K, we find that the temperature dependence of the electron and hole mobilities of unstrained BAs follow a T−2.14 and T−2.41 power law, respectively (Fig. 4c, d), which is similar to the values found by Liu et al.13 and indicates that optical deformation potential scattering becomes the dominant carrier-scattering mechanism.31 In the presence of tensile strain, acoustic deformation potential remains the dominant carrier-scattering mechanism below 200 K. But in the 200–500 K range, the temperature dependence of the mobility changes compared to unstrained BAs—decreasing to ~T−1.8 for electrons and increasing to ~T−3.0 for holes (Fig. 4c, d)—signifying a qualitative change in the dominant electron-scattering mechanism by tensile strain. We attribute the different temperature dependence under tensile strain to the weakening (strengthening) of the optical deformation potential scattering of electrons (holes) due to the reduction of the valley (valence band) degeneracy by tensile strain.

### BAs as a lattice-matched substrate

Next, we considered heterojunctions with other semiconductors where BAs is used as a substrate and their potential device applications. According to Fig. 5a, we determine that BAs can form lattice-matched interfaces with InGaN and ZnSnN2 and can solve several challenges that arise in adapting these materials for optoelectronic devices.

First, InGaN alloys have emerged as prominent optoelectronic materials due to their efficient luminescence, high absorption coefficient, and the broad tunability of their direct band gap across the entire visible spectrum by alloying (from 0.69 eV for InN32 to 3.4 eV for GaN33).23,34,35 However, the difficulty in growing high-quality InGaN films with high In concentrations limits the emission wavelengths of devices to shorter than green and is one of the major challenges for fabricating efficient InGaN-based amber and red LEDs35 for, e.g., full-color displays or efficient solar cells.36 This growth challenge arises mainly due to the lack of lattice-matched substrates. GaN-buffered sapphire (0001) is the most common substrate for InGaN growth, but the lattice mismatch between GaN and InGaN increases with increasing In composition. The calculated critical thickness for the formation of misfit dislocation decreases rapidly with increasing In composition and becomes <1 nm for In0.2Ga0.8N/GaN heterostructures,36 which is too thin for efficient light emission in LEDs and light absorption in solar cells.

Our calculations show that (111)-oriented BAs can be an alternative substrate or heterolayer for epitaxial InGaN growth. The hexagonal lattice constant of the BAs (111) plane (3.379 Å) is 5.62% larger than GaN (3.189 Å) and 4.56% smaller than InN (3.533 Å). Applying Vegard’s law, we find that BAs can serve as an excellent substrate for InxGa1−xN growth with misfit strain <1% for In compositions of ~0.47, with a band gap of 1.75 eV37 that enables red light emission. Thus, by exploiting a BAs substrate or epilayer, the light-emission wavelength range of InGaN-based devices can be expanded to cover the full visible spectrum. In addition, BAs and InxGa1−xN (0.3 < x < 1) have type-II band alignment (Fig. 5b).38 Therefore, BAs substrates or epilayers can serve as a p-type junction for InGaN-based photovoltaic devices. For an In content of 50%, the band offsets between InGaN and BAs are calculated to be 0.42 eV for the CB and 0.30 eV for the VB, which allows for the effective separation and collection of carriers.

Recently, the family of Zn-IV-N2 compounds has been proposed as a next-generation class of solar-absorber materials composed of Earth-abundant elements that can replace the costly and rare In and Ga of conventional group-III nitrides.29 Among their favorable characteristics, the tunability of their direct band gap (between 1.12 and 2.09 eV) at a fixed composition by adjusting the degree of cation disorder is of particular interest.39 Thin films of Zn-IV-N2 compounds and alloys (e.g., ZnGexSn1−xN2) have been successfully synthesized using molecular beam epitaxy40 and sputtering,41,42 and the tuning of their band gap through controlled cation disorder has been demonstrated by adjusting the growth temperature or relative ratio of source elements.39 Despite their attractive properties, there are limited studies concerning device integration and characterization with these materials since intrinsically high electron concentrations from donor point defects (VN, ON, and SnZn) hampers bipolar doping of this material.43 Using alternative methods, a ZnSnN2/p-Si heterojunction was fabricated,44 while NiO, Cu2O, and CuAlO2 have also been proposed as other suitable p-type semiconductors.45 Still, the lattice mismatch for all these materials ranges from 10% to 15%, which is too large to achieve high-crystal-quality epitaxial interfaces.

We propose n-ZnSnN2/p-BAs as an optimal hybrid structure for efficient photovoltaic applications. First, the cation disordered pseudo-wurtzite structure of ZnSnN2 has very close lattice match along the (111) plane of BAs with misfit strain <1%. Also, while ZnSnN2 is intrinsically n-type, BAs is p-type due to B-related acceptor-type point defects and its doping level can also be controlled extrinsically by Si, Ge, and Be acceptors.4 Band alignment of the heterojunction in the intrinsic case is shown in Fig. 5b with band offset energies between BAs and ordered ZnSnN2 of 0.516 eV for the VB and 0.503 eV for the CB. Thus effective separation of electrons and holes into different regions of the device is predicted. Furthermore, we note that the ideal pair of band gaps for a two-gap multi-junction photovoltaic device are 1.1 and 1.8 eV. While both BAs and ZnSnN2 materials have intrinsic gaps close to 1.8 eV, a 1.1-eV gap in ZnSnN2 may be achieved by full cation disorder or a 1.1-eV gap in BAs may be achieved by epitaxial strain up to 4%, as shown in Fig. 2c.

In summary, we examine the potential of BAs for electronic and optoelectronic device applications by predicting important electronic properties of BAs for incorporation into devices as both a substrate material and an epitaxially grown thin film. As a thin film, we find that tensile epitaxial strain increases the in-plane electron and hole mobilities to 2417 and 3550 cm2 V–1 s–1, respectively. These values represent an 80% and 68% increase over the unstrained case, a significant improvement that could lead to faster switching and more energy-efficient transistors. We find that at temperatures below 200 K acoustic deformation potential dominates carrier scattering, while above 200 K optical deformation potential becomes the dominant scattering mechanism. For bulk BAs as a substrate, we find In-rich InGaN and ZnSnN2 are candidate materials that can be epitaxially grown on BAs and form p–n junctions for lasers, LEDs, photodetectors, or photovoltaic cells. These junctions can cover the full visible spectrum, while simultaneously benefiting from the high thermal conductivity of BAs substrates for efficient heat removal.

## Methods

### Band structure and absolute band positions

Our DFT calculations for the band structure and band alignment use the projector augmented wave method46,47 and the HSE0648 hybrid functional as implemented in the Vienna Ab initio Simulation Package.49,50,51 For the band alignment calculation, the mixing parameter of the HSE06 functional was set to 25% for BAs, 25% for InGaN, and 31% for ZnSnN2 to match the experimentally measured band gaps3,23,41 within 0.2 eV. To construct biaxially strained structures, we first relax the cubic unit cell, fix the in-plane lattice constant ($$a_\parallel$$) under −4% to 4% strain, calculate the total energy as a function of the out-of-plane lattice constant ($$a_ \bot$$) for each $$a_\parallel$$, identify the $$a_ \bot$$ that minimizes the total energy (Supplementary Fig. 1), and calculate the band structure of the resulting crystal structure including spin–orbit coupling effects. For band alignment calculations, we build strained and unstrained BAs slabs consisting of 12 layers (48 atoms) along the nonpolar [110] direction and calculate the plane-averaged electrostatic potential without relaxing the ions to determine the band alignment of bulk BAs without the effects of surface dipoles by structural relaxation. We then use the energy difference between the average electrostatic potential inside the slab and in the vacuum region to align the BAs bands to vacuum. For the ZnSnN2 band alignment, we first align the order structure to the vacuum level (Supplementary Fig. 2) using the same method as BAs except that the mixing parameter of HSE06 functional is chosen to be 31%, which is commonly used in previous DFT calculations on ZnSnN2 to match the experimental band gap.30,43 Then the partially disordered ZnSnN2 is aligned to the ordered structure directly using the band eigenvalues from the DFT calculations (Supplementary Fig. 3). The justification and details of this alignment procedure are provided in Supplementary Information.

### Carrier mobility

To calculate the electron and hole mobilities in strained and unstrained BAs, we use the QUANTUM ESPRESSO52 and Electron–phonon Wannier (EPW)53,54 codes. We employ the LDA and the plane-wave norm-conserving pseudopotential method. We first relax the unit cell and subsequently apply a 1% biaxial tensile strain to the (001) plane and allow the out-of-plane (perpendicular) direction to relax to find the strained structure. We also use density functional perturbation theory55 within QUANTUM ESPRESSO to calculate the phonon frequencies and displacements on a coarse 6 × 6 × 6 BZ sampling grid for the strained and unstrained geometries. We then use the EPW code to interpolate the electron–phonon coupling matrix elements to fine BZ sampling meshes (up to 120 × 120 × 120) and determine the electron and hole mobilities at 300 K for the strained and unstrained structures for free-carrier concentrations of 1018 cm−3. We also calculate the temperature dependence of mobility from 100 to 500 K with 50 K steps on fine BZ sampling meshes of 72 × 72 × 72. We did not include spin–orbit coupling effects in the mobility calculations of the strained materials due to the added code complexity and computational cost. We expect that the omission of spin–orbit coupling does not affect the calculated carrier mobilities under strain as the CBM is unaffected by spin–orbit coupling, while the degeneracy splitting of the top valence band by strain (which controls the number of hole-phonon scattering channels) is also not affected by the inclusion of spin–orbit coupling effects.

## Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

We thank Dr. Samuel Poncé and Kelsey Mengle for their assistance with the EPW package and mobility calculations. This work was supported by the Designing Materials to Revolutionize and Engineer our Future (DMREF) Program under Award No. 1534221, funded by the National Science Foundation. K.B. also acknowledges the support of DOE grant DE-SC0020347. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02–05CH11231.

## Author information

Authors

### Contributions

K.B. and S.C. contributed equally to this work and performed first principles calculations, analyzed data, and wrote the paper under the supervision of E.K. Z.D. assisted first principles calculation for alloys and Z.D. and J.T.H. contributed to writing of the paper. E.K. supervised the calculation, analysis, and manuscript preparation. Funding acquisition was done by E.K. All contributors proofread the manuscript.

### Corresponding author

Correspondence to Emmanouil Kioupakis.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Bushick, K., Chae, S., Deng, Z. et al. Boron arsenide heterostructures: lattice-matched heterointerfaces and strain effects on band alignments and mobility. npj Comput Mater 6, 3 (2020). https://doi.org/10.1038/s41524-019-0270-4

• Accepted:

• Published:

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text:
Thread: change of variable: divide by zero?

1. change of variable: divide by zero?

Say X is a uniform random var on [-1,1], so $f_X(x)=1/2$ on [-1,1]. I want to find the density of $Y=g(X)=X^3$ over [g(-1),g(1)]=[-1,1]. g is a monotonic function so everything should be kosher. I look up the change of variable formula on wikipedia:

$f_Y(y)=|1/g'(g^{-1}(y))|f_X(g^{-1}(y))$

So I do some algebra:
$g(x)=x^3$
$g'(x)=3x^2$
$g^{-1}(y)=y^{1/3}$
so $g'(g^{-1}(y))=g'(y^{1/3})=3y^{2/3}$
thus
$f_Y(y)=|1/3y^{2/3}|*(1/2)$

But there's a singularity/asymtopes at y=0!... am I doing something wrong? The books I have conveniently choose random vars so there is no divide by zero (for example, if X was on [1,2] or some other interval where g(x) doesn't cross the x-axis, everything works fine).

2. Hello,

X is uniform over [-1,1]. So the probability that it equals 0 is 0. We say that almost surely, X is different from 0. So it's not a problem if there is a singularity at a given point.

3. Ah, ok. In fact, the integral of $f_Y$ on [-1,1] seems to be 1. So it is alright for density functions to have asymptotes, as long as the integral over the interval doesn't explode to infinity?

4. Well actually, there's just one discontinuity, so it doesn't affect the value of the integral.

5. The condition for that equality should be "strict monotocity". For a rigirous solution of that problem you can partition the function x^3 into two (for x>0 and x<0) where strict monotocity holds.


text:
# 100 Roots

Algebra Level 4

Let the 100 roots of the equation $$x^{100}-5x+2=0$$ be $$r_{1}, r_{2}, r_{3},\ldots, r_{100}$$. Find $r_{1}^{100}+r_{2}^{100}+r_{3}^{100}+\cdots+r_{100}^{100} .$

×


text:
Thesis Open Access

Assessment of the utility of the ESA SST CCI product for environmental research using data from meteorological buoys in the North Atlantic

Binner, Hannah

Citation Style Language JSON Export

{
"publisher": "Zenodo",
"DOI": "10.5281/zenodo.1450367",
"language": "eng",
"title": "Assessment of the utility of the ESA SST CCI product for environmental research using data from meteorological buoys in the North Atlantic",
"issued": {
"date-parts": [
[
2018,
3,
16
]
]
},
"abstract": "
This study assessed the accuracy and utility of the European Space Agency’s Sea Surface Temperature Climate Change Initiative (ESA SST CCI) product, which was established through the Global Ocean Data Assimilation Experiment (GODAE) and the GODAE High-Resolution SST Pilot-Project (GHRSST-PP). The product, the GHRSST Multi-product ensemble (GMPE), delivered Sea Surface Temperature (SST) measurements as a global daily median observation. These daily observations were aimed to be validated through the use of in-situ data, using four of the Met Éireann meteorological buoys. For the validation, the satellite images were processed in order to obtain the SST measurement that the satellite sensors had taken at the location of each of the four in-situ buoys and compared to the SST measurements taken by the in-situ buoys. Overall, satellite observations were found to have lower SST observations, while the buoy data showed a higher variability. Seasonal variability reflected in the satellite sensor and the buoy data. It was validated, that the satellite sensor data was accurate within 0.18 °C for the M3 buoy, 0.22 °C for the M6 buoy, 0.28 °C for the M5 buoy and 0.85 °C for the M4 buoy. It was also shown that the data was more reliable for summer months at the M3 and M5 buoy locations and least reliable for the M4 buoy location, which was attributed to a potential calibration error. Results of the M6 buoy location were attributed with a large uncertainty due to outliers.
\n\n
The utility of the SST variable GMPE data for use in Irish waters is suggested for applications in fisheries and aquaculture.
",
"author": [
{
"family": "Binner, Hannah"
}
],
"version": "1.0",
"type": "thesis",
"id": "1450367"
}
64
50
views


text:
# Showing a set is open using boundary.

Given a subset $B$ of a metric space $(M, d)$, show that $B$ is open iff $\partial B = \overline{B}\setminus B$.

I feel like that this question shouldn't be too hard to prove, but I'm not sure how to do it.

Attempt: I know how to prove in left to right direction. Suppose $B$ is an open set. That means $B=\operatorname{int}(B)$. We know $\partial B=\overline B\setminus\operatorname{int}(B)$. However, $B$ is just $\operatorname{int}(B)$ in this case. Thus $\partial B=\overline B\setminus B$.

However, I don't know how to prove this in the opposite direction.

(Weak attempt): Assume $\partial B=\overline B\setminus B$. We also know from a known fact that $\partial B=\overline B\setminus\operatorname{int}(B)$. If $\partial B=\overline B\setminus B$ and $\partial B=\overline B\setminus\operatorname{int}(B)$ this would imply $B=\operatorname{int}(B)$, which would imply $B$ is open.

If $B$ is not open, then take $x\in B$ such that $x\notin\mathring{B}$. Then every neighborhood of $x$ intersects $B^\complement$ and therefore, $x\in\overline{B^\complement}$. Since $x\in B$, then, in particular, $x\in\overline B$. So, $x\in\overline B\cap\overline{B^\complement}=\delta B$. But $x\notin\overline B\setminus B$, and this proves that $\delta B\neq\overline B\setminus B$.


text:
# zbMATH — the first resource for mathematics

## Yi, Lijun

Compute Distance To:
 Author ID: yi.lijun Published as: Yi, Li-Jun; Yi, Lijun Homepage: http://teacher.shnu.edu.cn/slxy/ylj/list.htm
 Documents Indexed: 27 Publications since 2006 Reviewing Activity: 2 Reviews
all top 5

#### Co-Authors

 3 single-authored 7 Wang, Zhongqing 6 Guo, Benqi 4 Sun, Tao 3 Wang, Lina 2 Meng, Tingting 1 Guo, Yuling 1 Li, Shuangshuang 1 Liang, Honghui 1 Liang, Zi-Qiang 1 Meng, Lingxiong 1 Wei, Yichen 1 Zhu, Qiding
all top 5

#### Serials

 3 Journal of Scientific Computing 2 Mathematical Methods in the Applied Sciences 2 Journal of Computational and Applied Mathematics 2 Applied Numerical Mathematics 2 International Journal of Computer Mathematics 2 Numerical Mathematics: Theory, Methods and Applications 1 Mathematics of Computation 1 Calcolo 1 SIAM Journal on Numerical Analysis 1 Mathematics in Practice and Theory 1 Numerical Methods for Partial Differential Equations 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 Advances in Computational Mathematics 1 Discrete and Continuous Dynamical Systems. Series B 1 International Journal of Numerical Analysis and Modeling 1 Journal of Hunan University of Arts and Science. Natural Science Edition 1 Science China. Mathematics 1 Communication on Applied Mathematics and Computation 1 East Asian Journal on Applied Mathematics

#### Fields

 26 Numerical analysis (65-XX) 8 Partial differential equations (35-XX) 6 Integral equations (45-XX) 4 Ordinary differential equations (34-XX) 4 Approximations and expansions (41-XX)


text:
# nLab division algebra and supersymmetry

Contents

supersymmetry

## Applications

#### Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Idea

There is a close relationship between

This is based on the fact that in certain dimensions, spin group representations are naturally identified with a $\mathbb{K}^n$, for $\mathbb{K}$ one of the normed division algebras, see at spin group The exceptional isomorphisms.

For detailed exposition see

exceptional spinors and real normed division algebras

Lorentzian
spacetime
dimension
$\phantom{AA}$spin groupnormed division algebra$\,\,$ brane scan entry
$3 = 2+1$$Spin(2,1) \simeq SL(2,\mathbb{R})$$\phantom{A}$ $\mathbb{R}$ the real numberssuper 1-brane in 3d
$4 = 3+1$$Spin(3,1) \simeq SL(2, \mathbb{C})$$\phantom{A}$ $\mathbb{C}$ the complex numberssuper 2-brane in 4d
$6 = 5+1$$Spin(5,1) \simeq$ SL(2,H)$\phantom{A}$ $\mathbb{H}$ the quaternionslittle string
$10 = 9+1$Spin(9,1) ${\simeq}$SL(2,O)$\phantom{A}$ $\mathbb{O}$ the octonionsheterotic/type II string

The structure of the normed division algebras also governs the existence of the brane scan and the super-∞-Lie algebras such as the supergravity Lie 3-algebra. By the D'Auria-Fre formulation of supergravity the ∞-Lie algebra valued forms with values in these constitute the field content of (11-dimensional) supergravity.

Combining this, one finds that supergravity coupled to super Yang-Mills theory (super Einstein-Yang-Mills theories) are parameterized by triples of real normed division algebras, forming a “magic pyramid”.

## References

The relation between supersymmetry and division algebras was gradually established by a variety of authors, including

Corresponding discussion of twistor space:

Streamlined proof and exposition is in

The case of 2-component quaternionic spinors in 6d (see at SL(2,H)) is discussed in more detail in:

Discussion with an emphasis on super Yang-Mills theory and U-duality in supergravity and the Freudenthal magic square is in

The relationship in string theory via octonion algebra between the NRS spinning string and the Green-Schwarz superstring sigma-models is discussed in

• Rafael I. Nepomechie, Nonabelian bosonization, triality, and superstring theory Physics Letters B Volume 178, Issues 2-3, 2 October 1986, Pages 207-210

• Itzhak Bars, D. Nemschansky and S. Yankielowicz, SLACPub-3758.

• H. Tachibana, K. Imeda, Octonions, superstrings and ten-dimensional spinors , Il nuovo cimento, Vol 104 B N.1

The relation of the division algebras to ordinary (Lie algebraic) extensions of the super Poincare Lie algebra is discussed in

• Jerzy Lukierski, Francesco Toppan, Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory (arXiv:hep-th/0203149, pdf)

• A. Anastasiou, L. Borsten, Mike Duff, L. J. Hughes, S. Nagy, An octonionic formulation of the M-theory algebra (arXiv:1402.4649)

Normed division algebras are used to describe the construction of Lie 2-algebra extensions of the super Poincare Lie algebra in

This is made use of in the homotopy theoretic description of M-theory in:

Last revised on July 23, 2022 at 17:50:11. See the history of this page for a list of all contributions to it.


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# Why is Lagrangian important?

In physics Lagrangian is a formulation for dynamic systems and we can deduce equation of motion using the Lagrange equation. But, mathematicaly, does lagrangian system means something? Which are the application of this? Is it linked somehow with variational formulations?

Wikipedia, in fact, does not clarify anything.


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The current GATK version is 3.3-0

#### Howdy, Stranger!

It looks like you're new here. If you want to get involved, click one of these buttons!

Service Notice: Due to the blizzard currently hammering the US Northeast, the Broad is shut down and the GATK forum will be mostly unattended while we hunker down and sip hot cocoa with marshmallows. Assuming the power stays on and we're able to dig ourselves out of the snow when it's all over, normal service should resume Wednesday or Thursday.

# Haplotype Caller -dcov flag

Posts: 6Member

Is there any way to override/re-implement the -dcov flag (which has been removed from the haplotype caller). I have a region that is almost 700 base pairs deep and continually causes a memory overload error. How can I restrict the coverage depth effectively?

Thanks, -Paul

Tagged:

Hi Paul,

What version are you using? The -dcov option is available in the current version, which is 2.6-4.

Geraldine Van der Auwera, PhD

• Posts: 6Member

I am using GATK v2.5-2, maybe it is just the online documentation that is missing the -dcov discription, but I was unable to discern whether or not -dcov refers to the number of reads starting at a single location (i.e. -dcov = 2 would only allow 2 reads to begin at the same place), or if it refers to the number of reads at any location (i.e. -dcov =2 would only allow 2 reads to appear at a specific location in the genome).

Thanks, -Paul

• Posts: 6Member

Perfect, thanks for the quick responses!


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# Identity with Ramanujan's generalized continued fraction

Let $F(x,q)=\sum_{n\geq 0}x^n\dfrac{q^{n^2}}{(q)_n}$, where $(q)_n=(1-q)(1-q^2)\dots(1-q^n)$. Then: $$H(x,q)=\frac{F(-xq,q)}{F(-x,q)}=\dfrac{1}{1-\dfrac{qx}{1-\dfrac{q^2x}{1-\dots}}}$$ is the generalized continued fraction of Ramanujan.

Does anybody know any nice formula for $H(x,q)+\dfrac{1}{H(x,q)}-1$?

• Excuse my notational ignorance but in the opening statement, what do you mean by $(q)_n$? And, in the definition of $H(x)$, $q$ looks like an ordinary parameter; in that case, why did you not call this $H(x,q)$ or something like $H(x;q)$? – Mark Fischler Feb 24 '16 at 16:31
• Excuse my notational ignorance, but in the opening statement, what is the meaning of $(q)_n$? And in the definition of $H(x)$, it appears that $q$ is a numerical parameter; in that case, why is $H(x)$ not called $H(x,q)$ or possibly $H(x;q)$? – Mark Fischler Feb 24 '16 at 16:34
• Why is it essential to have $-1$ is the last expression? Won't "1+nice formula" be a nice formula itself? – Max Alekseyev Feb 24 '16 at 20:51
• Max, -1 is not essential. – Adam Gyenge Feb 25 '16 at 21:43
• @Adam Gyenge :Perhaps this post is in the spirit of your question – Nicco Aug 25 '17 at 1:56


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Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Hierarchical Image Retrieval by Multi-Feature Fusion

Version 1 : Received: 26 April 2017 / Approved: 26 April 2017 / Online: 26 April 2017 (18:51:42 CEST)

How to cite: Lu, X.; Wang, J.; Hou, Y.; Yang, M.; Wang, Q.; Zhang, X. Hierarchical Image Retrieval by Multi-Feature Fusion. Preprints 2017, 2017040174 (doi: 10.20944/preprints201704.0174.v1). Lu, X.; Wang, J.; Hou, Y.; Yang, M.; Wang, Q.; Zhang, X. Hierarchical Image Retrieval by Multi-Feature Fusion. Preprints 2017, 2017040174 (doi: 10.20944/preprints201704.0174.v1).

## Abstract

Aiming at the problems that are poor generalization performance, low retrieval accuracy and large time consumption of existing content-based image retrieval system, the hierarchical image retrieval method based on multi feature fusion is proposed in this paper. The retrieval accuracy rates on Corel5K, UKbeach and Holidays are 68.23(Top 1), 3.73(N-S) and 88.20(mAp), respectively. The experimental results show that the method proposed in this paper can effectively improve the deficiency of single feature retrieval and save time significantly in the premise of a small amount of loss of accuracy.

## Keywords

Hierarchical search; Image retrieval; Multi-feature fusion

## Subject

MATHEMATICS & COMPUTER SCIENCE, Information Technology & Data Management

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# Different definitions of configuration space

Configuration space has been mentioned in many different areas of physics and from my personal experience I have found the definition has been different from topic to topic.

There are two examples to demonstrate my confusion. In standard lagrangian mechanics, the generalised coordinates span the configuration space. For example a particle in a newtonian setting has a configuration space $$Q=\mathbb{R}^3$$. The particles trajectory is then a particular curve in this configuration space. The configuration space for this example is finite dimensional.

Now consider classical field theory. We are told the configuration space for a field (with $$n$$ components) that depends on position and time, $$\phi(t,x,y,z)$$, is infinite dimensional with each point in configuration space representing a different function $$\phi: \mathbb{R}^4 \rightarrow \mathbb{R}^n$$, that is, a particular configuration is an element of configuration space. The feynman functional integral over configurations is a functional integrated over configuration space.

These seem like two different definitions to me. Following the first example then by analogy surely the configuration space would just be $$\mathbb{R}^n$$ and a particular configuration would be represented by a 'path' (function $$f:\mathbb{R}^4$$ to configuration space) in said space?

They are different definitions.

Take your first definition where $$Q=\mathbb R^3$$ for a Newtonian particle. According to that, the configuration space in classical field theory per your example is the space of functions $$\mathbb R^3\to\mathbb R^n\,.$$ So for each point in time $$t$$, you get a function that depends on the coordinates $$(x,y,z)\in\mathbb R^3$$: $$\phi(t,x,y,z)\,.$$

Of course in relativistic field theory splitting time from space like this is not always convenient, hence your second definition. Perhaps it's best to think of them like this: in the first case, the configuration space is the space of snapshots, while in the second case it is the space of histories.

For example a particle in a newtonian setting has a configuration space 𝑄=ℝ3. The particles trajectory is then a particular curve in this configuration space.

Right. A configuration is a point in $$Q=\mathbb R^3$$, and the particle evolves along a curve $$\gamma:t \mapsto \gamma(t)\in Q$$.

These seem like two different definitions to me. Following the first example then by analogy surely the configuration space would just be ℝ𝑛 and a particular configuration would be represented by a 'path' (function 𝑓:ℝ4 to configuration space) in said space?

This doesn't make sense to me. The configuration in point mechanics is an element of $$Q=\mathbb R^n$$, not a path. The "trajectory" in field theory is a path $$\gamma: t \mapsto \gamma(t)\in Q$$ as before, but now $$\gamma(t)$$ is a field configuration, i.e. a function from $$\mathbb R^n \rightarrow \mathbb R$$ (for an $$\mathbb R$$-valued field, obviously).


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# A polynomial algorithm for $P | p_j = 1, r_j, outtree | \Sigma C_j$ and $P | p_j = 1,r_j, outtree, pmtn | \Sigma C_j$

Peter Brucker, Johann L. Hurink, Sigrid Knust

Research output: Book/ReportReportOther research output

### Abstract

A polynomial algorithm is proposed for two scheduling problems for which the complexity status was open. A set of jobs with unit processing times, release dates and outtree precedence relations has to be processed on parallel identical machines such that the total completion time $\sum C_j$ is minimized. It is shown that the problem can be solved in $O(n^2)$ time if no preemption is allowed. Furthermore, it is proved that allowing preemption does not reduce the optimal objective value, which verifies a conjecture of Baptiste and Timkovsky.
Original language English Enschede University of Twente, Faculty of Mathematical Sciences Published - 2001

### Publication series

Name Memorandum / Department of Applied Mathematics University of Twente, Faculty of Mathematical Sciences 1566 0169-2690

• MSC-90B35
• IR-65753
• EWI-3386

### Cite this

Brucker, P., Hurink, J. L., & Knust, S. (2001). A polynomial algorithm for $P | p_j = 1, r_j, outtree | \Sigma C_j$ and $P | p_j = 1,r_j, outtree, pmtn | \Sigma C_j$. (Memorandum / Department of Applied Mathematics; No. 1566). Enschede: University of Twente, Faculty of Mathematical Sciences.


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# Can the permutation group of a matrix be generated by the subset of row permutation and column permutation?

Consider a 3*3 matrix $$\left( \begin{array}{*{20}{c}} {{m_{11}}}&{{m_{12}}}&{{m_{13}}}\\ {{m_{21}}}&{{m_{22}}}&{{m_{23}}}\\ {{m_{31}}}&{{m_{32}}}&{{m_{33}}} \end{array} \right)$$ and its permutation group, which consists of all the possible permutation over its element such as $$\left( {{m_{22}},{m_{33}}} \right) \cdot \left( {{m_{11}},{m_{21}},{m_{32}}} \right)$$. I wonder whether all these permutations can be expressed as a composition of several row permutation and column permutation? If so, does it still hold when we consider a non-square matrix?

• If I understand your question correctly, the answer is no. As you have defined it, there are $9!$ elements in your permutation group, but there are only $6$ permutation matrices. Apr 23, 2019 at 9:50

Of course not, for the following reason. When applying either a row or a column permutation, then the set of products $$m_{1\sigma(1)}\cdots m_{n\sigma(n)},\qquad\sigma\in\frak{S}_n$$ is unchanged. On the contrary, if you apply another permutation of the entries, this becomes false. For instance, if you apply the transposition $$(m_{22},m_{33})$$, then the product $$m_{12}m_{21}m_{33}$$ becomes $$m_{12}m_{21}m_{22}$$.