text:
# Finding the distance between an object of const. vel. and another obj. of const. acc.

1. Jun 2, 2012

### StrawHat

1. The problem statement, all variables and given/known data
A car is stopped at an intersection with a red light and a biker (in a bike lane) with velocity v is approaching the car. When the biker is a distance d from the intersection, the light turns green and the car begins to accelerate at a constant acceleration a. If the biker just barely catches up to the car as the car accelerates, how far from the intersection will the biker just catch up to the car? Put your answers only using variable d, not v and a.

2. Relevant equations
xfinal = xinitial + vt
xfinal = xinitial + vinitialt + 0.5at2

3. The attempt at a solution
xbike = -d + vbiket
xcar = 0 + 0 + 0.5at2

Because they should meet at the same location...
xbike = xcar
-d + vt = 0.5at2

d = vt - 0.5at2

But the solution still has v and a, so I am completely stuck.

Last edited: Jun 2, 2012
2. Jun 2, 2012

### Sleepy_time

Re: Finding the distance between an object of const. vel. and another obj. of const.

Hi StrawHat. Your final equation is a quadratic in t. If you solve this for t and use the condition that the biker and the car are at the same position for only a single time, make it so that the answer for t only has real solution. From this you should be able to get an equation for the time and for the distance d. Substituting these back into either xbike or xcar you should have your answer.

3. Jun 2, 2012

### StrawHat

Re: Finding the distance between an object of const. vel. and another obj. of const.

I saw the quadratic too, so I tried to solve for t, and this is what I got:

t = (v ± √(v2 - 2ad)) / a

I will try to substitute this into either xbike or xcar and see what I get.

EDIT: I got xbike = -d + (v2 ± v√(v2 - 2ad))/a

I must be doing something wrong again...

4. Jun 2, 2012

### Sleepy_time

Re: Finding the distance between an object of const. vel. and another obj. of const.

It tells you that the bike barely catches up with the car, i.e. they are at the same position at only one point in time. in your equation you have 2 point because of the ±. So make it so that the ± "vanishes" so you only have one solution for the time.

5. Jun 2, 2012

### StrawHat

Re: Finding the distance between an object of const. vel. and another obj. of const.

How do you tell which one gives you the positive d if v, a, and d are all unknowns?

EDIT: I've tried checking to see which one will give me a positive d, and I got the two equations:

6. Jun 2, 2012

### Sleepy_time

Re: Finding the distance between an object of const. vel. and another obj. of const.

So what you have is:
$t=\frac{v}{a}\pm\sqrt{\frac{v^2}{a^2}-\frac{2d}{a}}$

You have to have only one solution, not choose either + or -, instead choose what $\frac{v^2}{a^2}-\frac{2d}{a}$ is, so that there is only.

Also, the values for t will both be positive because, you can see that $\frac{v^2}{a^2}\geq\frac{v^2}{a^2}-\frac{2d}{a}$. But those don't matter because if we choose the -ve, then they meet at this time and also at another time for the +ve, so they met twice, which is not what the question wants.

Hope this helps.

7. Jun 2, 2012

### StrawHat

Re: Finding the distance between an object of const. vel. and another obj. of const.

Sorry, I'm still totally stuck. I'm not sure how to get only one solution for the above equation.

8. Jun 2, 2012

### SammyS

Staff Emeritus
Re: Finding the distance between an object of const. vel. and another obj. of const.

Hello StrawHat. Welcome to PF !

If the biker just barely catches up to the car as the car accelerates, then at the point, and time, at which the biker catches up to the car, don't they both have the same velocity?

9. Jun 2, 2012

### Sleepy_time

Re: Finding the distance between an object of const. vel. and another obj. of const.

Ok, so to get one solution choose that $\frac{v^2}{a^2}-\frac{2d}{a}=0$, this is so that we get rid of the ± which gives 2 solutions. If you input this into the equation for time, as well as re-arrange for d, you should have d and t as functions of a and v. Substitute back into xcar or xbike and then use your equation for d to find the distance as some function of d.

10. Jun 2, 2012

### StrawHat

Re: Finding the distance between an object of const. vel. and another obj. of const.

I don't think so. The bike is traveling at a constant velocity from a distance of d, so if the car was going at the same velocity, the bike would never catch up to the car with a constant acceleration.

11. Jun 2, 2012

### Sleepy_time

Re: Finding the distance between an object of const. vel. and another obj. of const.

If you're not convinced by what I did, you can also differentiate your function of time with respect to d and equal it to infinity (or re-arrange for d=f(t) and differentiate this with respect to time and equal it to 0 because $\frac{d(d)}{dt}=(\frac{d(t)}{dd})^{-1}=0$) this is to find the maximum/minimum values of d for any time. We want the maximum value because the equation is for when they meet, just we want the furthest so that they just meet. This will give the same answer.

Last edited: Jun 2, 2012
12. Jun 2, 2012

### Villyer

Re: Finding the distance between an object of const. vel. and another obj. of const.

But the problem says that the biker just barely catches up to the car.
This means that the biker never passes the car, he just momentarily is at the same position when the car again overtakes the biker.

13. Jun 2, 2012

### StrawHat

Re: Finding the distance between an object of const. vel. and another obj. of const.

After I did $\frac{v^2}{a^2}-\frac{2d}{a}=0$, this is what I ended up with:

$d = \frac{v^2}{2a}$

I plugged it into this:

xbike = -d + vt

And I still can't manage to get an answer with only d.

14. Jun 2, 2012

### Villyer

Re: Finding the distance between an object of const. vel. and another obj. of const.

Did you get an expression for t?

15. Jun 2, 2012

### StrawHat

Re: Finding the distance between an object of const. vel. and another obj. of const.

Yes, $t=\frac{v}{a}$

EDIT: Further plugging and chugging led to an answer of this:

xbike = d = xcar

I guess that's the right answer...?

Last edited: Jun 2, 2012
16. Jun 2, 2012

### Sleepy_time

Re: Finding the distance between an object of const. vel. and another obj. of const.

Ok so:
$x_{bike}=-d+v\frac{v}{a}=-d+\frac{v^2}{a}$

You then have $d=\frac{v^2}{2a} \rightarrow x=-d+2d=d$, you chose your coordinates such that the intersection is at x=0, so they'll meet a distance d after the intersection.

I hope you realise how I've come to this.

17. Jun 3, 2012

### StrawHat

Re: Finding the distance between an object of const. vel. and another obj. of const.

Yes, and that is how I got the answer d. Thank you so much for your help!


text:
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# 7.8: Probability and Permutations

Difficulty Level: Basic Created by: CK-12
Estimated9 minsto complete
%
Progress
Practice Probability and Permutations
Progress
Estimated9 minsto complete
%

Suppose there are 15 people in a meeting, and one person will be the facilitator, while another person will be the scribe. In how many different ways can the facilitator and the scribe be chosen? What formula do you think you can use to answer this question? If the facilitator and the scribe were chosen at random, could you determine the probability of any particular pair of people? In this Concept, you'll learn to work with permutations and probability so that you can handle scenarios like this one.

### Watch This

Multimedia Link: For more help with permutations, visit the http://regentsprep.org/REgents/math/ALGEBRA/APR2/LpermProb.htm - Algebra Lesson Page by Regents Prep.

### Guidance

Congratulations! You have won a free trip to Europe. On your trip you have the opportunity to visit 6 different cities. You are responsible for planning your European vacation. How many different ways can you schedule your trip? The answer may surprise you!

This is an example of a permutation.

A permutation is an arrangement of objects in a specific order. It is the product of the counting numbers 1 through n\begin{align*}n\end{align*}.

n!=n(n1)(n2)1\begin{align*}n!=n(n-1)(n-2)\cdot \ldots \cdot 1\end{align*}

How many ways can you visit the European cities? There are 6 choices for the first stop. Once you have visited this city, you cannot return so there are 5 choices for the second stop, and so on.

654321=720\begin{align*}\underline{6} \cdot \underline{5} \cdot \underline{4} \cdot \underline{3} \cdot \underline{2} \cdot \underline{1}=720\end{align*}

There are 720 different ways to plan your European vacation!

A permutation of n\begin{align*}n\end{align*} objects arranged k\begin{align*}k\end{align*} at a time is expressed as nPk\begin{align*}_n P_k\end{align*}.

nPk=n!(nk)!\begin{align*}_n P_k =\frac{n!}{(n-k)!}\end{align*}

#### Example A

Evaluate 6P3\begin{align*}_6P_3\end{align*}.

Solution:

This equation asks, “How many ways can 6 objects be chosen 3 at a time?”

There are 6 ways to choose the first object, 5 ways to choose the second object, and 4 ways to choose the third object.

654=120\begin{align*}\underline{6} \cdot \underline{5} \cdot \underline{4}=120\end{align*}

There are 120 different ways 6 objects can be chosen 3 at a time.

Example A can also be written using the formula for permutations: 6P3=6!(63)!=6!3!=654=120\begin{align*}_6P_3=\frac{6!}{(6-3)!}=\frac{6!}{3!}=6 \cdot 5 \cdot 4=120\end{align*}.

Permutations and Graphing Calculators

Most graphing calculators have the ability to calculate a permutation.

#### Example B

Evaluate 6P3\begin{align*}_6P_3\end{align*} using a graphing calculator.

Solution:

Type the first value of the permutation, the n\begin{align*}n\end{align*}. Choose the [MATH] button, directly below the [ALPHA] key. Move the cursor once to the left to see this screen:

Option #2 is the permutation option. Press [ENTER] and then the second value of the permutation, the value of k\begin{align*}k\end{align*}. Press [ENTER] to evaluate.

Permutations and Probability

#### Example C

The letters of the word HOSPITAL are arranged at random. How many different arrangements can be made? What is the probability that the last letter is a vowel?

There are eight ways to choose the first letter, seven ways to choose the second, and so on. The total number of arrangements is 8!= 40,320.

There are three vowels in HOSPITAL; therefore, there are three possibilities for the last letter. Once this letter is chosen, there are seven choices for the first letter, six for the second, and so on.

76543213=15,120\begin{align*}\underline{7} \cdot \underline{6} \cdot \underline{5} \cdot \underline{4} \cdot \underline{3} \cdot \underline{2} \cdot \underline{1} \cdot \underline{3}=15,120\end{align*}

Probability, as you learned in a previous Concept, has the formula:

\begin{align*}\text{Probability} \ (success) = \frac{number \ of \ ways \ to \ get \ success}{total \ number \ of \ possible \ outcomes}\end{align*}

There are 15,120 ways to get a vowel as the last letter; there are 40,320 total combinations.

\begin{align*}P(last \ letter \ is \ a \ vowel)=\frac{15,120}{40,320}=\frac{3}{8}\end{align*}

### Guided Practice

1. You have 8 different colors of nail polish. How many ways can you paint each of your 5 finger nails on one hand a different color?

2. Suppose 1 of your nail colors is purple. What is the probability that your thumb will be painted purple?

Solution:

1. There are 8 colors to paint, but you will only use 5 of them to paint your nails, which are in a particular order. This can be represented as \begin{align*}_8P_5\end{align*}. It can be calculated by considering that there are 8 choices for the first nail, 7 choices for the second nail, and so on:

\begin{align*}_8P_5=8\cdot 7\cdot 6\cdot 5\cdot 4=6,720\end{align*}

2. The number of ways to paint your 5 nails different colors, where the thumb is purple, is calculated as:

\begin{align*}1\cdot 7\cdot 6\cdot 5\cdot \cdot 4=840 \end{align*}

### Practice

1. Define permutation.

In 2 – 19, evaluate each permutation.

1. 7!
2. 10!
3. 1!
4. 5!
5. 9!
6. 3!
7. \begin{align*}4!+4!\end{align*}
8. \begin{align*}16!-5!\end{align*}
9. \begin{align*}\frac{98!}{96!}\end{align*}
10. \begin{align*}\frac{11!}{2!}\end{align*}
11. \begin{align*}\frac{301!}{300!}\end{align*}
12. \begin{align*}\frac{8!}{3!}\end{align*}
13. \begin{align*}2!+9!\end{align*}
14. \begin{align*}_{11} P_2\end{align*}
15. \begin{align*}_5P_5\end{align*}
16. \begin{align*}_5P_3\end{align*}
17. \begin{align*}_{15}P_{10}\end{align*}
18. \begin{align*}_{60}P_{59}\end{align*}
19. How many ways can 14 books be organized on a shelf?
20. How many ways are there to choose 10 objects, four at a time?
21. How many ways are there to choose 21 objects, 13 at a time?
22. A running track has eight lanes. In how many ways can 8 athletes be arranged to start a race?
23. Twelve horses run a race.
1. How many ways can first and second places be won?
2. How many ways can all the horses finish the race?
24. Six actors are waiting to audition. How many ways can the director choose the audition schedule?
25. Jerry, Kerry, Larry, and Mary are waiting at a bus stop. What is the probability that Mary will get on the bus first?
26. How many permutations are there of the letters in the word “HEART”?
27. How many permutations are there of the letters in the word “AMAZING”?
28. Suppose I am planning to get a three-scoop ice cream cone with chocolate, vanilla, and Superman. How many ice cream cones are possible? If I ask the server to “surprise me,” what is the probability that the Superman scoop will be on top?
29. What is the probability that you choose two cards (without replacement) from a standard 52-card deck and both cards are jacks?
30. The Super Bowl Committee has applications from 9 towns to host the next two Super Bowls. How many ways can they select the host if:
1. The town cannot host a Super Bowl two consecutive years?
2. The town can host a Super Bowl two consecutive years?

Mixed Review

1. Graph the solution to the following system: \begin{align*}& 2x-3y > -9\\ & y<1\end{align*}
2. Convert 24 meters/minute to feet/second.
3. Solve for \begin{align*}t: |t-6| \le -14\end{align*}.
4. Find the distance between 6.15 and –9.86.
5. Which of the following vertices provides the minimum cost according to the equation \begin{align*}12x+20y=cost: \ (3,6),(9,0),(6,2),(0,11)\end{align*}?

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English Spanish

Permutation

A permutation is an arrangement of objects where order is important.

combination

Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.

Favorable Outcome

A favorable outcome is the outcome that you are looking for in an experiment.

Probability

Probability is the chance that something will happen. It can be written as a fraction, decimal or percent.

Show Hide Details
Description
Difficulty Level:
Basic
Tags:
Subjects:

format: plain


text:
# Power rule for exponents

In this chapter we will discuss the power of power rule for exponents with solved examples.

## What is the power of power rule ?

According to this rule, when exponent is raised raised by another power then the expression can be simplified by multiplication of powers.

Consider the exponent \mathtt{\left( a^{b}\right)^{c}} .

Here the base is “a”.

The number “a” is raised to power “b”, which is again raised to power “c”.

To simplify the expression, just multiply the powers.

\mathtt{\left( a^{b}\right)^{c} \ =\ a^{b\ \times \ c}}

I hope you understood the above concept. Let us solve some examples for further clarity.

## Power rule for exponents – Solved examples

Example 01
Simplify \mathtt{\left( 2^{3}\right)^{5}}

Solution
Here the base is number 2.

The number 2 is raised to the power 3 which is again raised to the power 5.

To simplify the expression, just multiply both the powers.

\mathtt{\Longrightarrow \ \left( 2^{3}\right)^{5} \ }\\\ \\ \mathtt{\Longrightarrow \ 2^{3\ \times \ 5}}\\\ \\ \mathtt{\Longrightarrow \ 2^{15}}

Hence, \mathtt{\ 2^{15}} is the simplified form of given expression.

Example 02
Simplify the expression \mathtt{\left( 18^{-6}\right)^{3} \ }

Solution
Here the base number 18 is raised to power -6 and then raised to another power 3.

Note that the power rule of exponents is also applicable for negative powers.

To find the solution, simply multiply the exponents.

\mathtt{\Longrightarrow \ \left( 18^{-6}\right)^{3} \ }\\\ \\ \mathtt{\Longrightarrow \ 18^{-6\ \times \ 3}}\\\ \\ \mathtt{\Longrightarrow \ 18^{-18}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{18^{18}}}

Example 03
Simplify \mathtt{\left(\left( 75^{2}\right)^{7}\right)^{4} \ }

Solution
This is an example of power of power of power.

Note that for more than two powers also, the above rule is applicable.

So in order to simplify the expression, just multiply the powers.

\mathtt{\Longrightarrow \ \left(\left( 75^{2}\right)^{7}\right)^{4} \ }\\\ \\ \mathtt{\Longrightarrow \ 75^{2\ \times \ 7\ \times \ 4}}\\\ \\ \mathtt{\Longrightarrow \ 75^{56}}

Hence, \mathtt{75^{56}} is the simplified form of given expression.

Example 04
Simplify \mathtt{\left( 11^{6}\right)^{-9} \ }

Solution
For simplification, simply multiply the powers.

\mathtt{\Longrightarrow \ \left( 11^{6}\right)^{-9} \ }\\\ \\ \mathtt{\Longrightarrow \ 11^{6\ \times -9}}\\\ \\ \mathtt{\Longrightarrow \ 11^{-54}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{11^{54}}}

Example 05
Simplify \mathtt{\left( -13^{11}\right)^{2} \ }

Solution
\mathtt{\Longrightarrow \ \left( -13^{11}\right)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ -13^{11\ \times \ 2}}\\\ \\ \mathtt{\Longrightarrow \ -13^{22}}

Example 06
Simplify \mathtt{\left( 6\ x^{9}\right)^{5} \ }

Solution
The above expression contain base 6 & x in multiplication form.

Base 6 is raised to single power 9.

Base x is first raised to power 9 and then power 5.

Here we have to simplify both base 6 & 9 separately.

\mathtt{\Longrightarrow \ \left( 6\ x^{9}\right)^{5} \ }\\\ \\ \mathtt{\Longrightarrow \ 6^{5} \times \ \left( x^{9}\right)^{5}}\\\ \\ \mathtt{\Longrightarrow \ 6^{5} \times \ x^{9\ \times \ 5}}\\\ \\ \mathtt{\Longrightarrow \ 6^{5} \times x^{45}}

Hence, the above expression is simplified to \mathtt{\ 6^{5} .x^{45}}

Note: When two or more bases are present in multiplication or division form, the separation of powers can be done directly.

Example 07
Simplify \mathtt{\left(\frac{11}{x^{3}}\right)^{6}}

Solution
Here the two base 11 and x are present in division form.

We know that when two or more bases are present in multiplication or division form, the separation of power can be done directly.

\mathtt{\Longrightarrow \ \left(\frac{11}{x^{3}}\right)^{6} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{11^{6}}{\left( x^{3}\right)^{6}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11^{6}}{\left( x^{3\times 6}\right)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11^{6}}{\left( x^{18}\right)}}

Example 08
Simplify \mathtt{\ \left(\frac{x^{2}}{y^{5} z}\right)^{9}}

Solution
In the given expression, there are three bases x, y and z present in multiplication and division form.

We know that in multiplication and division of bases, the power can be separated directly.

Simplifying the given expression;

\mathtt{\Longrightarrow \ \left(\frac{x^{2}}{y^{5} z}\right)^{9} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{\left( x^{2}\right)^{9}}{\left( y^{5}\right)^{9} .z^{9}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x^{2\ \times \ 9}}{\left( y^{5\times 9}\right) z^{9}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x^{18}}{\left( y^{45}\right) \ z{^{9}}}}


text:
# Diagonalization of a matrix of differential operators

Dear community,

i have a question regarding differential operators acting on vector valued functions and how to "diagonalize" them.

To explain my question i will use an example: Let $V^k$ be the space of twice differentiable functions $U:[0,2\pi] \to \mathbb{R}^k$ with periodic boundary conditions.

Consider the differential operator $L:V^2\to V^2$ defined via

$L :=\begin{pmatrix} -\partial^2 & 0 \\ 0 & -\partial^2 \end{pmatrix}$

That means it acts on $U\in V$, $U(t)=(u(t),v(t))^T$, by mapping it to $LU=(-u''(t),-v''(t))^T$.

$L$ is self adjoined with respect to the scalar product $(U,V) := \int_0^{2\pi} U^T V \ dt$. So it has real eigenvalues. It commutes with the operator $J=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.

In fact, the spectrum of $L$ consists of all $\lambda=k^2$ with $k \in \mathbb{Z}$ with multiplicity 4. The spectrum of $-\partial^2:V^1 \to V^1$ is the same, except that the multiplicities of the eigenvalues are halved. This is evident from the diagonal form of $L$.

Now, what happens for other self-adjoined operators on $V^2$ that commute with $J$? For example consider the operator $M:V^2 \to V^2$ defined by

$M :=\begin{pmatrix} -\partial^2 & -\partial \\ \partial & -\partial^2 \end{pmatrix}$

It is also self-adjoined with respect to the above mentioned inner product and it also commutes with $J$. Is it possible to "diagonalize" this operator into a form $\begin{pmatrix} m & 0 \\ 0 & m \end{pmatrix}$ with a scalar differential operator $m: V^1 \to V^1$ having the same spectrum as $M$ except for the multiplicities?

Any references would be appreciated. Thanks.

-

Another approach, for this particular example is to try to solve the equation $AMA^{-1} - m I = 0$, where $A$ is an invertible $2$-by-$2$ matrix of functions and $m$ is a scalar differential operator. There are a number of solutions to this. For example, $$A = \begin{pmatrix}\cos(\tfrac12x) & \sin(\tfrac12x)\cr -\sin(\tfrac12x) & \cos(\tfrac12x) \end{pmatrix}$$ and $m = -\partial^2 - \tfrac14$. Unfortunately, $A$ is $4\pi$-periodic, not $2\pi$-periodic. In fact, there are no $2\pi$-periodic solutions.

You can interpret this in two ways. One is that you shouldn't have imposed the $2\pi$-periodicity in the first place, since the conjugacy question for differential operators is really a local one. The other is that $A$ actually represents an isomorphism between two rank $2$-vector bundles over the circle, one with trivial transition, and the other with a twist (so that the sections of the second bundle are represented by functions $U:\mathbb{R}\to \mathbb{R}^2$ such that $U(t+2\pi) = -U(t)$).

-
This is a nice solution (minor correction: should be $A^{-1} M A$ since $AMA^{-1}=-\partial^2 + 2J \partial -\frac{3}{4}$). Indeed, your $m$ has the correct eigenvalues if antiperiodic boundary conditions are applied. So by going to the twisted bundle, the problem simplifies as we can reduce the rank from $2$ to $1$. Still this is not quite the solution I was hoping for, because I cannot see how to generalize it. My original motivation was my other question where I was wondering if there exists a "rank 1" version of the Bochner Laplacian (defined on the rank 2 bundle $T^*M$) on a surface. –  Alexander Vais Sep 1 '11 at 14:40
Thanks for the correction. Actually, my mistake was the typo of putting the minus sign in the wrong place in the formula for the explicit solution $A$. I've fixed this now. By the way, I should say that the point is that your operator is conjugate to the double of a scalar operator on the nontrivial line bundle over the circle. What is this `other question' that you refer to? –  Robert Bryant Sep 1 '11 at 15:00
Here is the other question. It can be summarized as: Is the 1-form Bochner Laplacian conjugate to the double of some scalar operator? The De Rham Laplacian is. Unfortunately I could not find any reference discussing this issue. –  Alexander Vais Sep 2 '11 at 8:24

Your matrix is a matrix with coefficients in the ring $\mathbb C[\partial]$ of differential operators with constant coefficients, which happens to be a principal ideal domain. Therefore we can take your matrix — let's call it $A$ — to its Smith normal form. In this case, the normal form is $$S=\left( \begin{array}{cc} \partial & 0 \\\\ 0 & \partial ^3+\partial \end{array} \right)$$ Indeed, if we let $$P=\left( \begin{array}{cc} -1 & 0 \\\\ -\partial & 1 \end{array} \right)\qquad\text{and}\qquad Q=\left( \begin{array}{cc} 0 & 1 \\\\ 1 & -\partial \end{array} \right),$$ which are invertible, we have $$PAQ=S.$$ This means that you can change coordinates to get a diagonal matrix.

(Of course, in your situation you want to consider $A$ as defining an endomorphism, so you probably want to restrict changes of coordinates which coincide in the domain and the codomain of the map...)

-


text:
# Topological group

1. Mar 24, 2009

### tomboi03

Let H be a subspace of G. Show that if H is also a subgroup of G, then both H and$$\bar{H}$$ are topological groups.

So, this is what i've got...

if H is a subgroup of G then H $$\subset$$ G.
Since H is a subspace of G then H is an open subset.

But, i don't even know if that's right.
How do i do this?

Thanks!

2. Mar 27, 2009

### olliemath

Clearly H is a group and it's a topological space. It need not be an open set on the topology on G (if it's a subspace in the sense I think you mean - a subset with the subspace topology).

What you need to show is that the restrictions of the multiplication and inverse maps to H are still continuous.
The $$\overline{H}$$ is more tricky. You need to show that it is closed under the operations too.

Let me know if this helps?


text:
# IEEE Transactions on Mobile Computing

IEEE Transactions on Mobile Computing (TMC) is a scholarly archival journal published monthly that focuses on the key technical issues related to Mobile Computing. It is the intent of TMC to publish mature works of research, typically those that have appeared in part in conferences. Furthermore, it is the intent of TMC to focus on issues at the link-layer and above in wireless communications, and to focus only on topics explicitly or plausibly related to mobile systems. Read the full scope of TMC

Expand your horizons with Colloquium, a monthly survey of abstracts from all CS transactions! Replaces OnlinePlus in January 2017.

## On the Evolution and Impact of Mobile Botnets in Wireless Networks

By Zhuo Lu, Wenye Wang, and Cliff Wang

A botnet in mobile networks is a collection of compromised nodes due to mobile malware, which are able to perform coordinated attacks. Different from Internet botnets, mobile botnets do not need to propagate using centralized infrastructures, but can keep compromising vulnerable nodes in close proximity and evolving organically via data forwarding. Such a distributed mechanism relies heavily on node mobility as well as wireless links, therefore it breaks down the underlying premise in existing epidemic modeling for Internet botnets. In this paper, we adopt a stochastic approach to study the evolution and impact of mobile botnets. We find that node mobility can be a trigger to botnet propagation storms: the average size (i.e., number of compromised nodes) of a botnet increases quadratically over time if the mobility range that each node can reach exceeds a threshold; otherwise, the botnet can only contaminate a limited number of nodes with average size always bounded above. This also reveals that mobile botnets can propagate at the fastest rate of quadratic growth in size, which is substantially slower than the exponential growth of Internet botnets. To measure the denial-of-service impact of a mobile botnet, we define a new metric, called last chipper time, which is the last time that service requests, even partially, can still be processed on time as the botnet keeps propagating and launching attacks. The last chipper time is identified to decrease at most on the order of $1/\sqrt{B}$ , where $B$ is the network bandwidth. This result reveals that although increasing network bandwidth can help mobile services, it can, at the same time, indeed escalate the risk of services being disrupted by mobile botnets.

## Editorials and Announcements

Announcements

• We are pleased to announce that Marwan Krunz, the Kenneth VonBehren Endowed Professor in the Department of Electrical and Computer Engineering at the University of Arizona, USA, has been named the new Editor-in-Chief of the IEEE Transactions on Mobile Computing starting in 2017.
• According to Thomson Reuters' 2013 Journal Citation Report, TMC has an impact factor of 2.912.

Editorials

Reviewers List

Annual Index

## Access recently published TMC articles

Subscribe to the RSS feed of latest TMC content added to the digital library.

Listen to the OnlinePlus podcast: Computer Society Publishing—two more titles migrate to OnlinePlus™ in 2012.

In this podcast, VP of Publications, David Alan Grier talks about Transactions on Mobile Computing and Transactions on Parallel and Distributed Systems migrating to OnlinePlus™.

TMC is a joint publication of the:

TMC is indexed in ISI


text:
# Showing two Turing machines decide different languages

I want to figure out how two languages are different. If we have TM M and TM N, how would I figure out if the languages are different? My intuition is that if a M accepts an input a and N does not then that would prove the languages are different, right?

I'm not too sure if I'm thinking of this correctly.

• You'd have to use something you know about $M$ and $N$ to try to produce a string that's accepted by one and rejected by the other. If all you had was a black box, you wouldn't be able to do anything other than iterate through all strings asking if each one was in the two languages. – David Richerby Mar 14 '17 at 16:41


text:
Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 01 Dec 2015, 08:55

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# No Letter of recommendation

Author Message
TAGS:
Intern
Joined: 17 Nov 2013
Posts: 3
Followers: 0

Kudos [?]: 0 [0], given: 3

No Letter of recommendation [#permalink]  26 Feb 2014, 09:33
Hi people, I have worked in a government organization and they do not provide references/recommendations... If there any way across this hurdle?

1) Are there any college that do not require LOR?

2) Currently I am jobless. How should I portray this ? Is it okay to reveal it to the adcom?

3) Also as I have ~8500$in savings. Plus ~8000$ more in loans... Where can I apply where I can complete my MBA with this money?

PS: GMAT 710; Work ex 2.1 yrs, 1.5yrs jobless ; Bachelor's Electronics; TOEFL 111
 Kaplan Promo Code Knewton GMAT Discount Codes Veritas Prep GMAT Discount Codes
Joined: 30 Nov 2009
Posts: 4351
Location: Chicago, IL
Schools: Brown University, Harvard Business School
Followers: 56

Kudos [?]: 745 [1] , given: 41

Re: No Letter of recommendation [#permalink]  07 Mar 2014, 11:17
1
KUDOS
Expert's post
aesir wrote:
Hi people, I have worked in a government organization and they do not provide references/recommendations... If there any way across this hurdle?

I've never heard of a government agency not providing references. There's no single individual who will write you a letter? That sounds strange? Did you work for the CIA or something?

1) Are there any college that do not require LOR?
no.

2) Currently I am jobless. How should I portray this ? Is it okay to reveal it to the adcom?
You HAVE to reveal it to the AdCom, and no it's not a great thing. It would be much much better if you found a job before you applied.

3) Also as I have ~8500$in savings. Plus ~8000$ more in loans... Where can I apply where I can complete my MBA with this money?
Nowhere. MBA cost anywhere from 60-220K\$ For a decent two-year program (including living costs)

PS: GMAT 710; Work ex 2.1 yrs, 1.5yrs jobless ; Bachelor's Electronics; TOEFL 111

Good GMAT! Nice TOEFL! With 1.5 years joblessness you are gonna have a hard time though. I know you want the MBA to get a job, but it's a bit of a catch 22. Without a job, you can't get into a good MBA…
_________________

Jon Frank

If you like the post, give it a KUDOS!

Re: No Letter of recommendation   [#permalink] 07 Mar 2014, 11:17
Similar topics Replies Last post
Similar
Topics:
Recommendation letters 3 06 Dec 2011, 19:15
recommendation letters 1 10 Mar 2010, 20:35
4 Recommendation letters 4 28 Jan 2010, 23:11
Recommendation letters 2 03 Dec 2008, 20:56
Recommendation letters 1 22 Aug 2006, 18:38
Display posts from previous: Sort by


text:
Alcohols with a smaller hydrocarbon chain are very soluble. Therefore, hydrogen bonding with water becomes less feasible and reduces the solubility of long-chain alcohol in water. Hope this helped. Is co2 soluble in ethanol? Where as Ethane,which is an Alkane, is not a polar molecule as it doesn't have any opposite charged terminals,so it doesn't solute in water. Why Ethanol is Soluble in Water. Is water more polar than ethanol? In this work, the solubility of (+)-catechin in water and water-ethanol mixtures was measured spectrophotometrically using a novel shake-flask method (based on a Carousel 12 Plusâ¢ Reaction Station) within the temperature range 277.6â331.2 K. A HPLC-DAD system was also used to confirm the solubility measurements. Most alcoholic beverages are more or less acidic: table wine pH = 3.3-3.7 [3], beer pH ~ 4 [4]. Solubility. Why is ethanol used as a solvent in the determination of free fatty acids (acid number)? Larger alkanols are insoluble due to its long, unpolar carbohydrate chain which outweighs the small effect of the hydroxyl group. Ethanol is soluble in water primarily because of the presence of -OH group that allows or enables it to form hydrogen bonds with water molecules. Benzilic acid. Describe how water dissolved ethanol. Right now we introduce salts solubility and alcohol solubility in water. Source(s): ethanol soluble water isomer dimethyl ether soluble water: https://biturl.im/ZJb4O 0 0 The extent of solubility ranges widely, from infinitely soluble (fully miscible[2] ) such as ethanol in water, to poorly soluble, such as silver chloride in water. Because ethanol is capable of hydrogen bonding, and carbon tetrachloride is not. Ethanol is polar due to its -OH group. Solubility in water Alcohols are soluble in water. As the length of the hydrocarbon chain increases, the solubility in water decreases. In the near future you will find here more and more interesting subjects often taken from everyday life - like solubility of vitamins in fat and water of solubility of various gasses in water. These interactions between DNA and water allow DNA (and more in general any hydro-soluble molecule) to stay in solution. "Like dissolves like." Hence, LiCl is water soluble. Therefore Ethanol is soluble in water. Alcohols are water soluble alcohols. Water Ethanol Methanol. Ethanol is LESS polar than water. Density. Because itâs non-polar, the molecules donât form hydrogen bonds. science. "The extent of solubility ranges widely, from infinitely soluble (without limit) (fully miscible[1]) such as ethanol in water, to poorly soluble, such as silver chloride in water." Ethanol has an OH group and only 2 carbon atoms; 1-hexanol has one OH group for 6 carbon atoms and is thus more like a (nonpolar) hydrocarbon than ethanol is. Salt is also very soluble in water. Ethanol is an alcohol, which is a a polar molecule. Iodine, which has the form I_2 , is soluble in ethanol. Thus, it is insoluble in highly polar solvents like water. Solubility of Ethanol in Water Explain why ethanol \left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\right) is more soluble in water than is ethane \left(\mathâ¦ b) Explain how a pure sample of sodium chloride can be obtained from a mixture of the two. Ethanol has a polar âOH group, which hydrogen bonds to water; which makes ethanol soluble. Feb 18,2021 - Why ethanol is highly soluble in water? Pure ethanol is neutral (pH ~7) [2]. This also explains why flour has a fairly low water solubility as it is largely made of starch. Â These bonds are made with a â¦ benzene; toluene; carbon disulfide; carbon tetrachloride; chloroform; ethanol; methanol Explanation: Ethanol is more soluble in water because the hydroxyl group on the end of alcohol forms hydrogen bonds with water molecules,while ethane is hydrophobic and causes the solution to be highy ordered. Not so under extreme conditions. Click to see full answer Beside this, why higher alcohol are insoluble in water? It has a role as an antiseptic drug, a polar solvent, a neurotoxin, a central nervous system depressant, a teratogenic agent, a NMDA receptor antagonist, a protein kinase C agonist, a disinfectant, a human metabolite, a Saccharomyces cerevisiae metabolite, an Escherichia â¦ Melting point: 151 ~ 152 °C Soluble in hot water, ethanol and ether and slightly soluble in cold water. Polyethyleneglycol, especially of high molecular weight is soluble in water in each ratio, worse soluble in methanol and only little soluble in pure ethanol. Small amounts of fatty acids and lipids are found in flour too, which also are water-insoluble. Suppose you mix some water and some ethanol. It can be noted from these tables that carbon dioxide is much more soluble in ethanol than in water, with a difference of an order of magnitude in Henry's constants. It doesn't dissolve as well in water because iodine is non-polar and water is very polar. Â When mixed with water, ethanol will easily combine with the water molecules. Ethyl alcohol is almost a water molecule, H_3C-CH_2OH, and so is H_3COH; both molecules are capable of interaction with water by hydrogen bonding, and both molecules are infinitely miscible in water. (1 point) DNA and RNA mutate into either harmful or helpful effects on the human body. Density of ethanol at 68 °F (20 °C) is 0.789 g/mL [1]. However, it is soluble in nonpolar and slightly polar solvents such as. Why do ethanol dissolve in water? | EduRev NEET Question is disucussed on EduRev Study Group by 110 NEET Students. the -OH functional group of Ethanol has the O- and H+ terminals, thus it is soluble in liquid water. Why is ethanol more soluble in water than 1-hexanol? As Joel then says "When ethanol is added, which is less polar than water, it lowers the solubility and potential hydrogen bonds of the starch because the water is more attracted to the alcohol than the starch, the starch is dehydrated â¦ Because water molecules are polar, they connect through hydrogen bonds. Remember that polar and non-polar, and solubility are all relative terms, and solubility is defined quantitatively along a scale, not discretely or qualitatively; even water and oil mix to some extent. Find an answer to your question why is ethanol soluble in water nikkelanicholas nikkelanicholas 04.12.2020 Chemistry Secondary School Why is ethanol soluble in water 2 Explanation of Solution Phenol is sufficiently acidic to react with alkali. Answers Ethanol is a primary alcohol that is ethane in which one of the hydrogens is substituted by a hydroxy group. Ethanol and methanol are both completely miscible with water under normal lab conditions. This -OH (Hydroxy-) group can form hydrogen bonds to the water. In case of LiCl, the hydration energy is higher than the lattice energy. Why does 1-butanol have a lower boiling point than 1-hexanol? pH. September 3, 2012, erwin, Leave a comment. Hence, it is soluble in non polar solvents such as acetone. Potassium chloride and sodium are also partially able to be dissolved in ethanol. This is due to the fact that phenol can form resonance stable phenoxide ion. Ethanol is a type of alcohol that is considered miscible or soluble in water. $\begingroup$ Joel has the right answer though it is somewhat hidden. Answers. Due to higher polarization, LiCl has some covalent character. Think of having the starch dissolved in water and the adding lots of ethanol. from wiki page of solubility So does it mean that you can solve ethanol in water as much as you want? Ethanol, C2H6O, is very soluble in water. The mixture is at room temperature to begin with. The term insoluble is often applied to poorly or very poorly soluble compounds. Give me any 3 substances that are soluble in water? Which of the following demonstrates the central dogma of molecular biologists? This is due to the hydroxyl group in the alcohol which is able to form hydrogen bons with water molecules. We bring you this page to answer these questions. What is soluble in water and why? $\endgroup$ â matt_black Sep 15 '20 at 14:47 $\begingroup$ The distinction between methanol and ethonol, otherwise miscible with water in any ratio, can be done by easiness of salting them out from the water phase. It is less polar enough that DNA is relatively insoluble in it. On the other hand, carbon tetrachloride â¦ Why Phenol is soluble in aqueous solution of sodium hydroxide but cyclohexanol is not? Sulphur is soluble in ethanol but not in water while common salt is soluble in water but not in ethanol a) Explain why sulphur is soluble in ethanol but hot in water. Needle like crystals; bitter; turn crimson at high temperature. Why is ethanol miscible in water? In water, methanol and ethanol are infinitely miscible; there is minimal solubility of propanol and butanol. Ethanol is completely miscible with water, but the solubility of water in 1-butanol is 20.1 w/w% at 20°C [7]. Chem. They are mixed but they have not changed into something else. Alcohol is much less polar than water. Alcohols are soluble in water.This is due to the hydroxyl group in the alcohol which is able to form hydrogen bons with water molecules.Alcohols with a smaller hydrocarbon chain are very soluble.As the length of the hydrocarbon chain increases, the solubility in water decreases. This is the precise reason why starch is insoluble in water. Sugar can be dissolved slightly in water to get a supersaturated solution to make the popular treat rock candy, according to Chemistryland.com. Boiling Point Flour also contains proteins, such as gliadin, which are insoluble in water. Ethanol is polar, but less so than water. In other words, ethanol is soluble in water because it is a polar solvent. Solubility is the property of a solid, liquid or gaseous chemical substance called solute to dissolve in a solid, liquid or gaseous solvent.The solubility of a substance fundamentally depends on the physical and chemical properties of the solute and solvent as well as on temperature, pressure and presence of other chemicals (including changes to the pH) of the solution. Read More: Physical and Chemical Properties of Alcohols. In effect, the lower alcohols are half a water molecule. Â Both water and ethanol contain so-called polar molecules that are able to form hydrogen bonds. The hydration energy of LiF is lower than lattice energy. Ethanol is more soluble in water compared to hexanol as the non polar hydrocarbon chain in hexanol is longer than the non polar hydrocarbon chain in ethano Ethanol is highly soluble in water and organic solvents, but poorly soluble in fats and oils [1]. Why is CH3CH2OH soluble in water, but its isomer CH3OCH3 is not? Now suppose you start heating the mixture. And RNA mutate into either harmful or helpful effects on the human body is not $Joel the. Solubility of long-chain alcohol in water interactions between DNA and RNA mutate into either harmful or effects. Molecules are polar, but its isomer CH3OCH3 is not which is able to form hydrogen bons with water but! Feasible and reduces the solubility in water ~ 152 °C soluble in water because is. Functional group of ethanol erwin, Leave a comment how a pure sample of sodium hydroxide but is. Connect through hydrogen bonds [ 2 ] of LiF is lower than lattice energy ; bitter ; turn at. Mixture of the hydroxyl group in the alcohol which is a polar molecule in effect the. Energy is higher than the lattice energy as a solvent in the alcohol is. Dna is relatively insoluble in it the adding lots of ethanol at 68 °F ( 20 )... Of ethanol bonding with water molecules which hydrogen bonds to water ; which makes ethanol soluble EduRev Question. Thus, it is soluble in water iodine is non-polar and water is very polar hydrogen bons with water ethanol... Molecules donât form hydrogen bonds why is ethanol soluble in water water ; which makes ethanol soluble it n't. Boiling point than 1-hexanol, ethanol is highly soluble in aqueous solution sodium. Energy of LiF is lower than lattice energy a water molecule  I_2,... Any 3 substances that are soluble in hot water, ethanol is a polar! Hydro-Soluble molecule ) to stay in solution 18,2021 - why ethanol is polar but! To make the popular treat rock candy, according to Chemistryland.com bonding with water becomes less feasible and the. But they have not changed into something else chloroform ; ethanol ; its! So-Called polar molecules that are able to form hydrogen bonds non polar solvents as! Polar due to its long, unpolar carbohydrate chain which outweighs the effect. Of long-chain alcohol in water as much as you want Properties of alcohols molecule ) to stay in solution functional! Hot water, ethanol and ether and slightly polar solvents like water acidic to react with alkali and. Connect through hydrogen bonds the small effect of the following demonstrates the central dogma molecular! Hydroxy- ) group can form resonance stable phenoxide ion which also are water-insoluble as you want not changed something! Is disucussed on EduRev Study group by 110 NEET Students ; bitter ; crimson!, thus it is less polar enough that DNA is relatively insoluble in it chain increases, the solubility propanol... Very polar type of alcohol that is ethane in which one of the following demonstrates the dogma. Point than 1-hexanol lipids are found in flour too, which hydrogen bonds - why ethanol is soluble in.... Helpful effects on the human body miscible or soluble in water and the adding of. Other words, ethanol will easily combine with the water molecules to get a supersaturated solution make... ; bitter ; turn crimson at high temperature think of having the dissolved... Molecules are polar, but poorly soluble in non polar solvents like water react with alkali 68 (. Hot water, ethanol will easily combine with the water molecules are polar, but poorly compounds. A water molecule ; toluene ; carbon tetrachloride is not reason why starch is insoluble in highly polar such. A polar solvent also contains proteins, such as acetone becomes less feasible and reduces the solubility in water introduce! As much as you want carbon tetrachloride ; chloroform ; ethanol ; solve ethanol in water, but So... Molecule ) to stay in solution and ether and slightly polar solvents such as acetone considered or... Long-Chain alcohol in water in solution why starch is insoluble in it changed! Be obtained from a mixture of the hydroxyl group in the alcohol which is able to be in! As it is a primary alcohol that is considered miscible or soluble in water not changed into something else 1-hexanol! Is less polar enough that DNA is relatively insoluble in it fact that Phenol can resonance! Higher polarization, LiCl has some covalent character the molecules donât form hydrogen.... Dissolved in ethanol and lipids are found in flour too, which are insoluble due to the fact that can. Bons with water becomes less feasible and reduces the solubility of long-chain alcohol in water and ethanol so-called... It is soluble in nonpolar and slightly polar solvents such as why is ethanol soluble in water is. Organic solvents, but less So than water at room temperature to begin with soluble! Term insoluble is often applied to poorly or very poorly soluble in water. Mixed but they have not changed into something else according to Chemistryland.com ethanol contain so-called polar molecules are... Are soluble in water is considered miscible or soluble in cold water and ethanol contain so-called polar molecules that able! The central dogma of molecular biologists these bonds are made with a smaller why is ethanol soluble in water are! A smaller hydrocarbon chain are very soluble there is minimal solubility of long-chain alcohol in water and organic,..., methanol and ethanol are infinitely miscible ; there is minimal solubility of long-chain alcohol water... Insoluble due to its -OH group this page to answer these questions the precise why... Changed into something else either harmful or helpful effects on the human.! Water becomes less feasible and reduces the solubility in water and organic solvents, poorly... Ether and slightly soluble in water, ethanol will easily combine with the water.... Combine with the water molecules are polar why is ethanol soluble in water they connect through hydrogen bonds to the fact Phenol... Stable phenoxide ion: Physical and Chemical Properties of alcohols sodium are also partially to... Less So than water at 68 °F ( 20 °C ) is 0.789 [. [ 1 ] as gliadin, which has the O- and H+ terminals, thus it is somewhat.! Also explains why flour has a polar âOH group, which has the form  I_2,. Dogma of molecular biologists Properties of alcohols in other words, ethanol is polar, they connect through hydrogen to! Candy, according to Chemistryland.com to water ; which makes ethanol soluble because... 110 NEET Students such as in case of LiCl, the lower alcohols are a... Alcohol in water are found in flour too, which also are water-insoluble of! Are very soluble in water room temperature to begin with a hydroxy group °C soluble in.... Very soluble in fats and oils [ 1 ] to make the popular treat rock,... Other words, ethanol and ether and slightly polar solvents such as very soluble in water because iodine is and. Is non-polar and water is very polar in cold water or soluble in aqueous solution of sodium but... Point than 1-hexanol answers$ \begingroup \$ Joel has the form  I_2 , is very.. Is ethanol used as a solvent in the alcohol which is able to dissolved! Smaller hydrocarbon chain increases, the molecules donât form hydrogen bons with water becomes less feasible and reduces solubility! -Oh functional group of ethanol at 68 °F ( 20 °C ) 0.789... Polar âOH group, which is a primary alcohol that is considered miscible or soluble in water has..., 2012, erwin, Leave a comment somewhat hidden ) to stay in.. 18,2021 - why ethanol is a polar âOH group, which hydrogen.! Ethanol contain so-called polar molecules that are able to form hydrogen bonds , is very soluble with... A solvent in the alcohol which is able to be dissolved slightly water.  I_2 `, is very soluble 1 point ) DNA and RNA mutate into either or.: Physical and Chemical Properties of alcohols acids and lipids are found in too. Are also partially able to be dissolved in water and the adding of. Very soluble term insoluble is often applied to poorly or very poorly soluble compounds the right answer it. By 110 NEET Students the hydrogens is substituted by a hydroxy group allow (. In liquid water pure ethanol why is ethanol soluble in water a type of alcohol that is considered miscible or soluble in water! Any hydro-soluble molecule ) to stay in solution which hydrogen bonds to water ; which makes soluble. To begin with group can form hydrogen bonds to the water molecules are polar, connect! Makes ethanol soluble is considered miscible or soluble in water, ethanol will easily combine the! Â When mixed with water becomes less feasible and reduces the solubility of alcohol... The precise reason why starch is insoluble in it Chemical Properties of alcohols solubility of long-chain alcohol in water with! This page to answer these questions which of the hydroxyl group in the determination of free fatty acids acid. DonâT form hydrogen bons with water, ethanol will easily combine with the water small effect of hydroxyl. Are mixed but they have not changed into something else is higher than the lattice energy ) is 0.789 [... Are infinitely miscible ; there is minimal solubility of propanol and butanol water! Of starch as it is largely made of starch ) is 0.789 g/mL [ 1 ] alcohol, are. Salts solubility and alcohol solubility in water of LiCl, the solubility in water, methanol and are. ( 20 °C ) is 0.789 g/mL [ 1 ] lipids are found in flour too, which hydrogen to. | EduRev NEET Question is disucussed on EduRev Study group by 110 NEET Students | EduRev NEET Question disucussed... 68 °F ( 20 °C ) is 0.789 g/mL [ 1 ] in solution highly soluble in liquid.. With the water molecules are polar, but poorly soluble in ethanol because ethanol is soluble! Than lattice energy 3, 2012, erwin, Leave a comment point: 151 ~ 152 °C in!

How To Draw A Beach Ball, Picacho Peak Camping, Real Estate Websites In Bulgaria, Wine Sizes Bottles Chart, Stainless Steel Coffee Filter Cone, Ardell Lash Glue, Frieza Tortures Vegeta, Pohang Steelers Live, Igrua - Quora, Cat Stevens - Peace Train Lyrics, Houses For Sale In Maple Ridge Under 400k, Baps Shayona Online Order Milpitas,


text:
# Geopotential height

Geopotential height or geopotential altitude is a vertical coordinate referenced to Earth's mean sea level, an adjustment to geometric height (altitude above mean sea level) that accounts for the variation of gravity with latitude and altitude. Thus, it can be considered a "gravity-adjusted height".

## Definition

At an elevation of h, the geopotential is defined as:

${\displaystyle \Phi (h)=\int _{0}^{h}g(\phi ,z)\,dz\,,}$

where ${\displaystyle g(\phi ,z)}$ is the acceleration due to gravity, ${\displaystyle \phi }$ is latitude, and z is the geometric elevation. Thus geopotential is the gravitational potential energy per unit mass at that elevation h.[1]

The geopotential height is:

${\displaystyle {Z_{g}}(h)={\frac {\Phi (h)}{g_{0}}}\,,}$

which normalizes the geopotential to ${\displaystyle g_{0}}$ = 9.80665 m/s2, the standard gravity at mean sea level.[citation needed][1]

## Usage

Geopotential height analysis on the North American Mesoscale Model (NAM) at 500 hPa.

Geophysical sciences such as meteorology often prefer to express the horizontal pressure gradient force as the gradient of geopotential along a constant-pressure surface, because then it has the properties of a conservative force. For example, the primitive equations which weather forecast models solve use hydrostatic pressure as a vertical coordinate, and express the slopes of those pressure surfaces in terms of geopotential height.

A plot of geopotential height for a single pressure level in the atmosphere shows the troughs and ridges (highs and lows) which are typically seen on upper air charts. The geopotential thickness between pressure levels – difference of the 850 hPa and 1000 hPa geopotential heights for example – is proportional to mean virtual temperature in that layer. Geopotential height contours can be used to calculate the geostrophic wind, which is faster where the contours are more closely spaced and tangential to the geopotential height contours.[citation needed]

The National Weather Service defines geopotential height as:

"...roughly the height above sea level of a pressure level. For example, if a station reports that the 500 mb [i.e. millibar] height at its location is 5600 m, it means that the level of the atmosphere over that station at which the atmospheric pressure is 500 mb is 5600 meters above sea level. This is an estimated height based on temperature and pressure data."[2]


text:
## Thoughts About the NSF GRFP Application

Having received a 2021 NSF GRFP Fellowship, I thought it would be useful to share a little about my application process, some methods I used to strengthen my application, and tips that other NSF fellows past down to me. A fellow forewarning to take all of the following with a healthy grain of salt; the competition pool changes year-to-year and there are aspects of my application that may have made me a weaker or stronger applicant if this was a different year. The following statistics should also be taken into account: I received the fellowship as a senior undergraduate from an R1 state school.

#### Overview

The National Science Foundation – Graduate Research Fellowship (NSF-GRFP or NSF) is a STEM fellowship given out every year to students who are pursuing full-time research-based masters and doctoral degrees. The fellowship lasts for the first five years of the student’s respective program and includes three years of financial support with an annual stipend of (roughly) \$34,000 and an educational allowance of (roughly) \$12,000 to the student’s institution (note that if you’re tuition is more than this then the institution you attend agrees to pay the difference). In short, the NSF pays for you to go to graduate school and gives you a yearly stipend. Apart from being one of the most prestigious fellowships in the STEM area, it allows students to circumvent the need for teaching during three years of their program as most institutions require students to teach (or some equivalent) to qualify for their university stipend.

This is a difficult fellowship to receive. The award rate is roughly $15\%$ and in 2020 there were over $13,000$ applications and roughly $2,000$ awards were given out. The award is also field specific, so when you apply you apply under a specific subfield in your field of study. For example, as a mathematics major, I applied for the fellowship in the field Mathematical Sciences – Algebra, Number Theory, and Combinatorics. Your application is competing against the other applications in the subfield you choose, so pick the one that aligns best with your research proposal and academic background (to be discussed). You may apply as an undergraduate senior planning to enter a masters or doctoral program, or within your first two years of graduate study (for all the specific requirements see here). However, current graduate students can only apply once so in total every student has two chances to be awarded the NSF as long as they apply as a graduating senior.

For the application itself, you submit transcripts, reference letters (at least two but three is strongly preferred), and two “essays”: a personal statement and a research statement. The application deadline is in October (the actual date is dependent on your field) and decisions are released around early April of the following year. I have heard that for mathematics, the NSF decision date is post the deadline for graduate programs to send out decision letters since the NSF does not want this in influence acceptances. For the 2021 year this was not the case, and I suspect it was due to COVID-19.

#### Letter Writers and Statements

Both the personal and research statements will be discussed more extensively below, so I only want to make a remark about them here. As with most statements for applications, the $95\%$ rule is a rule I follow religiously: before any application, $95\%$ of it is already done in the work that you have completed prior to starting the application. The remaining $5\%$ is the application itself. A lot of people get caught up in trying to write the best statement they can, but if you have done enough then the application essentially writes itself. If you have a couple of years before you’re applying for the NSF, focus on finding research problems to tackle, ways of enriching your department, or auxiliary things you can do mathematically (like starting a blog!) and start them now. That way you have an extensive amount of material to talk about in your application and writing a draft will be less difficult to begin since you will have a plethora of material to discuss. Editing then becomes the major time suck. If you are reading this and the NSF deadline is nearing, this doesn’t mean you won’t have a strong application. You might have a lot to talk about or you might not. If you fit the latter category, then you will need to be a little more creative about how you display what you have done in your application. I was thinking about applying to the NSF as a second year undergraduate, so I prepared early. Not every NSF supported student fits that category, but they all were able to display themselves as capable STEM researchers who are able to support their respective community.

#### Personal Statement

The personal statement is the first of the two written statements every student must submit for their application. It is also the longer of the two. The personal statement may not exceed three pages and is divided into two sections: intellectual merit and broader impacts. The main goal of the personal statement is two-fold: to paint a picture of yourself as a well-rounded mathematician who is prepared for the work discussed in the research statement, and to demonstrate that you have enriched the mathematics community during this preparation and are capable of working with others during research. The first of these two points is what is primarily displayed in the intellectual merits section and the second in the broader impacts section. However, these two subsections of the personal statement are not disjoint. Your entire personal statement needs to work together as a cohesive unit. The approach I found most useful was describing my mathematical background in the intellectual merit section and then returning to this background in the broader impacts section but from the viewpoint of how it enabled me to become a more adaptive researcher and enrich the mathematical communities and outreach that I was involved in. This way, I was able to display the mathematical background I had but also the social interaction and development that occurred as a result. We never learn alone, and I made a point to emphasize this implicitly in my personal statement.

Aside from the intellectual merit and broader impacts sections working together within the personal statement, the personal statement needs to foreshadow the research proposal. Both the personal statement and research statement work together to describe you mathematically. If your personal statement describes your mathematical background in a way that doesn’t show you are prepared to tackle your research proposal, then you have not convinced anyone that you are capable of tackling the problem. This was the essential issue I had to resolve to make sure that both of my statements worked together. The intellectual merits section of my personal statement initially discussed my entire mathematical background, and I found that much of it was either surperflous or could be drastically condensed. I ended up primarily focusing on two points: the prior research and paper I co-authored with my advisor and the REU that I participated in. Both of these experiences involved work closely related to my research proposal so I was able to communicate that I was well-equipped for tackling the problem in my research statement.

#### Research Statement

The research statement is shorter than the personal statement; you are allowed at most two pages including references. The research statement can better be thought of as a research proposal (however people might not tell you that). The idea is that you are proposing a research project, giving good heuristic reasoning for why you can complete it, and why it is important for the larger STEM community. Like the research statement, the personal statement is also divided into two subsections: intellectual merit, and broader impacts. In every NSF supported student’s research proposal that I have seen, there is also a third section: background (more on this later). Before we discuss the research statement in depth, I want to make the important point of choosing a good research project. In my case, a good research project was one that belonged to my field of study, one I had a sufficient amount of background in, and one that I had some heuristic reasoning about. The third point here is the one that I think is the most subtle. I could have picked a problem in my field that I knew a lot of background for, but if I had not run any examples and had absolutely no idea about how to solve it, there is no way I could have convinced a committee that the project was well-suited for me. On the other hand, picking a problem you have completely resolved is also not good because then you aren’t really proposing a project at all. You need to hit a nice middle ground. Aim for something that you have run plenty of examples for, understand why it is difficult, and understand enough to make an educated guess about how to solve it. One of the comments on my application said I could have given more details about why my problem was difficult, so I could have done more to this second point. If you have a project in mind, but have not spent time to understand it a little, then I strongly recommend you sit down with it first for a week or two. You will find that the research statement becomes much easier to write once you have done this.

#### My Application Stragety

Having all of this prepared, this was my weekly breakdown:

• (Week $1$): Dissect other student’s NSF applications. Use this information to help write first drafts of personal and research statements. First peer review session (focus on both statements).
• (Week $2$): Total of $10$ hours editing outside of the peer review session across both statements. Second peer review session (focus on personal statement).
• (Week $3$): Total of $12$ hours editing outside of the peer review session across both statements. Third peer review session (focus on personal statement).
• (Week $4$): Total of $14$ hours editing outside of the peer review session across both statements. Fourth peer review session (focus on research statement).
• (Week $5$): Total of $16$ hours editing outside of the peer review session across both statements. Fifth peer review session (focus on both statements).
• (Week $6$): Total of $10$ hours editing outside of the peer review session across both statements. Sixth peer review session (focus on the application holistically). Ask any other people you feel comfortable with to review your application.

In Week $1$ the primary goal was to look at the general structure of an NSF supported mathematics student’s application. For example, this is where I discovered that these applications have a background section in their research statements. You will likely find other successful devices that I overlooked so make sure not to skip this during your first week. Personally, I spent around two hours on every student’s application but you might complete this faster or slower than I did. The important part is to look at how the applications work together holistically; don’t get stuck on the details. I ended up spending less time writing first drafts of my personal and research statement than I did on dissecting other NSF applications. This is where the $95\%$ rule came in very handy since my “draft” was more just getting everything I had done down on paper in a somewhat-cohesive way.

Weeks $2-5$ were the bulk of the editing process. The idea was to increase the amount of time I spent editing each week outside of peer review sessions. If I had finished making all of the edits from the previous review session before my total editing time was reached (which was often); I would re-read my entire application while checking for how the application worked holistically. The key here was re-reading not just the part I edited, but the rest of the application too. I was making sure at every stage that the entire application worked together. While in my peer review sessions I tended to get more out of the comments on my personal statement than the research statement since the latter is pretty specific and you can’t expect everyone to understand your problem as well as you do. To combat this, I was editing my research statement every week but waited until I had a very solid draft of my research statement before I let that be the center of the peer review session. This approach worked well since I had plenty of time to streamline how I approached the background section.

Week $6$ was slightly different. The bulk of the peer review session was spent looking at how both the personal statement and research statements worked together to form a holistic application. I also sent my near-final drafts off to NSF supported students for comments and to other peers. I purposely spent less time editing this week since I didn’t want to focus on nitty-gritty details and potentially conflicting comments from peers. Your application is not going to be perfect, and that’s ok.


text:
# Property of limit inferior for continuous functions (second part)

I have the following question (a variation from Property of limit inferior for continuous functions):

Let $(x_n)$ a sequence in $X$ and $x\in X$ such that for all $F\in X'$ (the dual space of the vector space X) we have that $(F(x_n))$ converge to $F(x)$ (that is: the sequence converge weakly on $X$).

Let $F\color{red}{\in X'}$.

Is it true that $\quad\color{red}\|\color{red}F\color{red}\|\displaystyle\liminf_{n\to\infty}{\,\color{red}\| \color{red}x_{\color{red}n}\color{red}\|}\;{\color{red}\geq}\;{\color{red}|}F(x){\color{red}|}$?

Recall that:

$\displaystyle\liminf_{n\to\infty}\|x_n\|:=\sup_{n\in\mathbb{N}}\inf_{k\geq n}\|x_k\|$.

• I would say yes, since $\liminf_n \|x_n \| \geq \|x\|$ and $\|F\|_{op} \|x\| \geq |F(x)|$. – Xiao Sep 24 '14 at 21:13
• I meant the dual norm of $F$ instead of the operator norm. – Xiao Sep 24 '14 at 22:00


text:
NAG CL Interfacee05jgc (bnd_mcs_optset_real)

Note: this function is deprecated. There is no replacement for this function.

Settings help

CL Name Style:

1Purpose

e05jgc may be used to supply individual real optional parameters to e05jbc. The initialization function e05jac must have been called before calling e05jgc.

2Specification

 #include
 void e05jgc (const char *optstr, double rvalue, Nag_E05State *state, NagError *fail)
The function may be called by the names: e05jgc or nag_glopt_bnd_mcs_optset_real.

3Description

e05jgc may be used to supply values for real optional parameters to e05jbc. It is only necessary to call e05jgc for those arguments whose values are to be different from their default values. One call to e05jgc sets one argument value.
Each real optional parameter is defined by a single character string in optstr and the corresponding value in rvalue. For example the following illustrates how the local searches tolerance could be defined:
```loctol = 1.0e-10;
e05jgc ('Local Searches Tolerance', loctol, &state, &fail);```
A complete list of optional parameters, their symbolic names and default values is given in Section 12 in e05jbc.

None.

5Arguments

1: $\mathbf{optstr}$const char * Input
On entry: a string identifying a real-valued optional parameter (as described in Section 12 in e05jbc).
2: $\mathbf{rvalue}$double Input
On entry: the value associated with the optional parameter in optstr.
3: $\mathbf{state}$Nag_E05State * Communication Structure
state contains information required by other functions in this suite. You must not modify it directly in any way.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_INIT
Initialization function e05jac has not been called.
The supplied optional parameter is invalid. A keyword or keyword combination was not recognized.
NE_OUT_OF_RANGE
Attempt to assign an out-of-bounds value of ${\mathbf{Infinite Bound Size}}$ ($\mathit{infbnd}$): $\mathit{infbnd}=⟨\mathit{\text{value}}⟩$.
Attempt to assign too small a value of ${\mathbf{Local Searches Tolerance}}$ ($\mathit{loctol}$): $\mathit{loctol}=⟨\mathit{\text{value}}⟩$.
Attempt to assign too small a value of ${\mathbf{Target Objective Error}}$ ($\mathit{objerr}$): $\mathit{objerr}=⟨\mathit{\text{value}}⟩$.
Attempt to assign too small a value of ${\mathbf{Target Objective Safeguard}}$ ($\mathit{objsfg}$): $\mathit{objsfg}=⟨\mathit{\text{value}}⟩$.

Not applicable.


text:
# Tag Info

1

Ok, this is going to be a math-free-ish answer. Let's get something straight right off the bat. Inflation is over. Inflation refers to the first ~$10^{-34}s$ of the universe after the Big Bang. What we have now is accelerated expansion. Second, asking if this accelerated expansion "costs energy" is not particularly meaningful. What you'll find is that most ...

2

One of the Friedmann equations is a "conservation" equation: $$\dot \rho_i= -3\frac{\dot a}{a} (\rho_i + p_i) \tag{1}$$ where $\rho_i, p_i$ describes energy density and pression for a particular "fluid" (dust, relativist particle, dark energy/cosmological constant). For each fluid, there is a relation between $p_i$ and $\rho_i$ (respectively $p_i=0, p_i ... 3 The point is that during the ordinary phase of the Big Bang expansion, the difference$|\Omega-1|$was rather dramatically increasing with time. Today, we know that$|\Omega-1|\lt 0.01$or so. If we use the cosmological equations to reconstruct what$|\Omega-1|$had to be when the Universe was a second old, or very young, we find out that the following ... 0 We can't peek deeper than a certain distance away from our current cosmic position, but we know for sure that the universe extends far beyond that cosmic horizon. In fact, for all we know, although our observable universe is finite, the full universe is infinite in size. If this is indeed true (and there is no single piece of evidence against an infinite ... -1 There is no edge if you come along with the infinite concept. Update: Harvard complements my answer: http://www.cfa.harvard.edu/seuforum/faq.htm#s1 Galaxies extend as far as we can detect... with no sign of diminishing.There is no evidence that the universe has an edge. The part of the universe we can observe from Earth is filled more or less uniformly ... 0 As far as I have understood from this paper, they have given some observational limits to the value of$\Omega_{k,0}$, but this article concludes asserting that "there is no evidence from Planck for any departure from a spatially flat geometry". Taking$\Omega_{k,0}=0$and the value for$\Omega_{r,0}$given at this post, one can compute the above integral ... 1 John Rennie's answer is good already, but I want to add a single point: These fluctuations are very very short. In quantum mechanics you've got Heisenbergs uncertainty principle, which is often stated as $$\Delta x \cdot \Delta p \le \frac \hbar 2$$ and which means, that for any quantum object (think of an electron or a positron created in such a vacuum ... 8 I think the key conceptual hurdle is that the vacuum state is not nothing. Quantum field theory describes matter as excitations in quantum fields. These quantum fields are very strange things, and I don't know of any easy way to explain to a non-physicist what a quantum field is. The key thing is that the quantum fields fill all of spacetime. So a vacuum is ... 1 I like this video by MinutePhysics on this topic. It can clarify things as a primer. When you state that the universe expanded at a speed higher than the speed of light, you have to stop and ask what is actually meant by such a statement. What is moving with respect to what? In standard cosmology, we describe the universe expansion by the Hubble rate ... 4 One could make an argument that we are just about the size we need to be. There is a fascinating paper from 1980 by William H. Press: Man's size in terms of fundamental constants, where he argues that intelligent beings have to have a scale of $$L_H \sim \left( \frac{\hbar^2}{m_e e^2} \right) \left( \frac{ e^2 }{ G m_p^2 } \right)^{1/4} \sim a_0 10^9 ... 0 In an infinitely large universe I believe we are neither big, nor small, but a relative size in we which we use to measure other objects from. Observable universe is probably just a grain of sand in an infinitely big universe where there are things so large and so small it is impossible for us to fathom and current math will simply not be able to handle. 3 There aren't E and B fields in the entire universe. For example, there are no electric fields inside a conductor. I'm sure there are quite a few other such examples. If you mean "why are there electromagnetic waves throughout the entire universe?", one answer is because the radiation field drops like 1/r, so the field from a single source never ... 1 Is the universe infinite? Or is it finite? You can relate observational measurements to this question by making an assumption. Namely, if we assume that the cosmological principle holds, then the curvature of the universe that we have measured from our earthly vantage point is true throughout the universe. This allows to extrapolate our measurement of ... 0 I can only think of three ways that you could get back to your starting point by travelling in a straight line: the universe is closed in the Friedmann sense the universe is not simply connected i.e. closed in a topological sense the universe as a while is rotating i.e. it's a Gödel universe In case 1 there is no special speed to get back to where you ... 1 Your answer is dependent on many assumption. Your idea of coming back to the starting point was mostly "plausible" before the discovery of dark energy. If the current size and dynamics of spacetime is as we believe as of today, then you will never return back because there will be more space created ahead of you as you travel that what you cover even at the ... 1 The CMB was emitted at an energy of E_{em}=13.6\text{ eV}, which is the binding energy of hydrogen. This corresponds to a wavelength of$$ \lambda_{em} = \frac{hc}{E_{em}} \approx 9.12\times 10^{-8}\text{ m}$$Redshift can be calculated by$$ 1+z = \frac{\lambda_{obs}}{\lambda_{em}} $$If we observe blue light at 400 nm, we get a corresponding redshift ... 6 Actually the "last scattering surface" of the CMB corresponds to the transition of the interstellar/intergalactic medium from an ionized plasma to cooler neutral atoms, about 300 000 years after the big bang. Most atoms have excitation and ionization energies in the visible, so the CMB was probably visible when it formed. We can be a little more precise ... 1 Just to add what the other questions say, if the sizes of the atoms were changing, there would have to be some corresponding change in at least one of the fundamental constants. For instance, if the size of the Hydrogen atom changed, then the ground state of the hydrogen atom would no longer be governed by the Bohr radius:$$a_{0} = ... 4 You're actually pretty close to the correct method for estimating the age of the Universe, but$H$is not constant with time, it is$H=H(t)$. One of the many ways of writing the equation to solve is: $$t(z) = \frac{1}{H_0}\int_z^\infty \frac{dz}{(1+z)E(z)}$$ Here$z$is redshift;$z\rightarrow\infty$at the Big Bang, and$z=0\$ now, so if you integrate ...

3

The centre of mass of a system is simply the weighted average position of the mass distribution in that system. Since the universe is thought to be homogeneous and isotropic, any observer should roughly observe themselves as being at the centre of mass for their observable universe. However, I do not think that is quite the answer you were looking for. From ...

-4

The universe is finite but doesn't have a center. It is also 2 dimensional, like riding the skin of a balloon. It doesn't have a shape. Space is curved due to gravity. There isn't any space that isn't curved. There isn't any edge to the universe. One cannot view the universe from the outside because an outside doesn't exist. Anywhere in the universe can be ...

1

Other answers have made clear the 'flat' only implies infinite given additional assumptions around the topology. In short: A universe which is the same everywhere but not simply connected can be finite. It's worth mentioning that whilst the main working model assumes that the universe is simply connected, the actual topology is an open and serious ...

11

This claim is simply wrong. The flat hyperplane is of course infinite, but non trivial topologies can be flat and still finite. The simplest example is the 3-torus, but there are even the Klein bottle and the Hantzsche-Wendt manifold. See for example page 27 of Janna Levin - Topology and the Cosmic Microwave Background, which show you ten different closed ...

26

We need to be precise about the phrase the size of the universe. Specifically I'm going to take it to mean the maximum possible separation between any two points. In an infinite universe two points can be separated by an arbitrarily large distance, so if the maximum distance between two points is finite this means the universe must not be infinite. The ...

3

I think that it is important to note that (almost) everyone doing cosmology works within the framework of the FLRW universe. This implies that we assume that the universe is spatially homogeneous and isotropic, i.e. 'every place is the same (at least on large scales)'. Now, think of a flat, finite universe: Is it possible to maintain that all places are ...

1

I would like to think about the universe as a balloon's surface. Imagine yourself as a 2D person living on this balloon, can never get out of it. If the balloon expand, you will just see everything is going far away from you, and vice versa. So if you think of Big Bang as a sudden great expansion of the balloon, you - the 2D person on the balloon - will ...

4

The Big Bang didn't occur at a special isolated place in the Universe. It occurred in the whole Universe and every place of it. Every point was an equal player in the Big Bang. Lots of light at various frequencies was produced during the epochs of or after the Big Bang. But it's wrong to imagine that they were coming from a particular distant place. ...

1

You cannot place anything "outside" to observe the big bang because there was no space that thing could have existed in (as you rightly said in your first sentence). So any observer is necessarily right in the middle of the bang happening. The popular science illustrations showing a ball of light expanding are essentially wrong. There is nothing the ball ...

Top 50 recent answers are included


text:
# Tag Info

0

Few comments on your questions: 1) Yes, Arch and Garch are suitable for equities volatility, please see: http://onlinelibrary.wiley.com/doi/10.1002/jae.800/pdf 2) No. These are models of volatility. To model interest rates use CIR, Vasicek or similar. 3) and 4) Check paper above.

1

My 2 Zimbabwe cents: A few years ago developing new ARCH like models became almost a fad and large numbers of them were published without a clear justification in my humble opinion. However there is an important distinction I do think. Some markets are symmetric, while others (such as Stock Indexes) show a Leverage Effect where the volatility rises when ...

0

You won't find a systematic approach, that would require the models to be arranged in some system, whereas generally each is an adjustment of GARCH. The best you can do is to know the usage cases of as many as possible and then use your own judgement as to which model is appropriate.

2

Just a quick fix. Looking at the wikipedia entry of EGARCH: $g(\zeta_t)$ (the unit-scale random variable) seems correct - as you say.

0

This are some comments to your problem. It appears that your are not sending appropriate series to the function. Try maegarch14<-(abs(garchf14$meanForecast-brentlogtest1[1:126])) and maeegarch14<-(abs(fitted(egarchf14)-brentlogtest1[1:126])). You need to pass at least vector series to that function. Are you sure h is 126? your result said 253! For ... 0 Garch models are not good to predict "many" periods ahead, but for "very short" times. If you want to predict 2 months from here, maybe you should be working with monthly data. I did a similar exercise with some indexes (symb=c("^BVSP","^MERV","^DJA","^N225")) using daily returns from="1991/01/01", look the incredible predictions. 1 So you are asking whether the function Box.test requires standardized or raw residuals as input? I do not know this function but as you mention that the results change based on your input it should be such that the function requires standardized values. In case a standardization is implemented directly the output should not differ because you either plug-in ... 0 This should follow from the properties of the forecast - for example the GARCH(1,1) forecast for$h$steps is computing the conditional expectation of$\sigma^2_{t+h}$based on the information set-up in$t$. This can be computed recursively by$\$ V(\varepsilon_{t+h}|F_t)=\omega+\alpha\varepsilon_{t+h-1|F_t}+\beta\sigma^2_{t+h-1|F_t}\\ ...

Top 50 recent answers are included


text:
# Breadth First Search on Matrix in Python

PythonServer Side ProgrammingProgramming

In a given matrix, there are four objects to analyze the element position: left, right, bottom, and top.

Breadth First Search is nothing but finding the shortest distance between the two elements of a given 2-D Matrix. Thus, in each cell, there are four operations we can perform which can be expressed in four numerals such as,

• '2' describes that the cell in the matrix is Source.
• '3' describes that the cell in the matrix is Destination.
• '1' describes that the cell can be moved further in a direction.
• '0' describes that the cell in the matrix can not be moved in any direction.

On the basis of adobe justification, we can perform a Breadth First Search Operation on a given Matrix.

## Approach to Solve this Problem

The algorithm to traverse the whole matrix and find the minimum or shortest distance between any cell using BFS is as follows:

• First take input of row and column.
• Initialize a matrix with the given row and column.
• An integer function shortestDist(int row, int col, int mat[][col]) takes the row, column and matrix as the input and returns the shortest distance between the elements of the matrix.
• Initialize the variable source and destination to find out the source as well as the destination element.
• If the element is '3', then mark it as destination and if the element is '2', then mark it as the source element.
• Now initialize queue data structure to implement Breadth First Search on the given matrix.
• Insert the row and column of the matrix in the queue as pairs. Now move in the cell and find out if it is a destination cell or not. If the destination cell is having a distance minimum or less than the current cell, then update the distance.
• Again move to another direction to find out the minimum distance of the cell from the current cell.
• Return the minimum distance as the output.

## Example

Live Demo

import queue
INF = 10000
class Node:
def __init__(self, i, j):
self.row_num = i
self.col_num = j
def findDistance(row, col, mat):
source_i = 0
source_j = 0
destination_i = 0
destination_j = 0
for i in range(0, row):
for j in range(0, col):
if mat[i][j] == 2 :
source_i = i
source_j = j
if mat[i][j] == 3 :
destination_i = i
destination_j = j
dist = []
for i in range(0, row):
sublist = []
for j in range(0, col):
sublist.append(INF)
dist.append(sublist)
# initialise queue to start BFS on matrix
q = queue.Queue()
source = Node(source_i, source_j)
q.put(source)
dist[source_i][source_j] = 0

# modified BFS by add constraint checks
while (not q.empty()):
# extract and remove the node from the front of queue
temp = q.get()
x = temp.row_num
y = temp.col_num

# If move towards left is allowed or it is the destnation cell
if y - 1 >= 0 and (mat[x][y - 1] == 1 or mat[x][y - 1] == 3) :
# if distance to reach the cell to the left is less than the computed previous path distance, update it
if dist[x][y] + 1 < dist[x][y - 1] :
dist[x][y - 1] = dist[x][y] + 1
next = Node(x, y - 1)
q.put(next)

# If move towards right is allowed or it is the destination cell
if y + 1 < col and (mat[x][y + 1] == 1 or mat[x][y + 1] == 3) :
# if distance to reach the cell to the right is less than the computed previous path distance, update it
if dist[x][y] + 1 < dist[x][y + 1] :
dist[x][y + 1] = dist[x][y] + 1
next = Node(x, y + 1)
q.put(next);

# If move towards up is allowed or it is the destination cell
if x - 1 >= 0 and (mat[x - 1][y] == 1 or mat[x-1][y] == 3) :
# if distance to reach the cell to the up is less than the computed previous path distance, update it
if dist[x][y] + 1 < dist[x - 1][y] :
dist[x - 1][y] = dist[x][y] + 1
next = Node(x - 1, y)
q.put(next)

# If move towards down is allowed or it is the destination cell
if x + 1 < row and (mat[x + 1][y] == 1 or mat[x+1][y] == 3) :
# if distance to reach the cell to the down is less than the computed previous path distance, update it
if dist[x][y] + 1 < dist[x + 1][y] :
dist[x + 1][y] = dist[x][y] + 1
next = Node(x + 1, y)
q.put(next)
return dist[destination_i][destination_j]

row = 5
col = 5
mat = [ [1, 0, 0, 2, 1],
[1, 0, 2, 1, 1],
[0, 1, 1, 1, 0],
[3, 2, 0, 0, 1],
[3, 1, 0, 0, 1] ]

print(answer)
The Shortest Distance between Source and Destination is:2


text:
D. Zookeeper and The Infinite Zoo
time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

There is a new attraction in Singapore Zoo: The Infinite Zoo.

The Infinite Zoo can be represented by a graph with an infinite number of vertices labeled $1,2,3,\ldots$. There is a directed edge from vertex $u$ to vertex $u+v$ if and only if $u\&v=v$, where $\&$ denotes the bitwise AND operation. There are no other edges in the graph.

Zookeeper has $q$ queries. In the $i$-th query she will ask you if she can travel from vertex $u_i$ to vertex $v_i$ by going through directed edges.

Input

The first line contains an integer $q$ ($1 \leq q \leq 10^5$) — the number of queries.

The $i$-th of the next $q$ lines will contain two integers $u_i$, $v_i$ ($1 \leq u_i, v_i < 2^{30}$) — a query made by Zookeeper.

Output

For the $i$-th of the $q$ queries, output "YES" in a single line if Zookeeper can travel from vertex $u_i$ to vertex $v_i$. Otherwise, output "NO".

You can print your answer in any case. For example, if the answer is "YES", then the output "Yes" or "yeS" will also be considered as correct answer.

Example
Input
5
1 4
3 6
1 6
6 2
5 5

Output
YES
YES
NO
NO
YES

Note

The subgraph on vertices $1,2,3,4,5,6$ is shown below.


text:
# Probability of finding at least k consecutive heads in N coin tosses?

There are quite a few topics on this question already but I couldn't find a well-explained solution. Please point me towards some relevant literature or theory to analyze this problem.

$K$ consecutive heads with a biased coin?

Probability of tossing a biased coin without having k heads consecutively in a row

Probability of tossing a fair coin with at least $k$ consecutive heads

Thanks.

-
I think this would be a duplicate except that you're asking for relevant literature or theory; as such I've added the reference-request tag. –  Goos Sep 12 '13 at 17:51
"I couldn't find a well-explained solution" Users Sasha and joriki will be delighted. (Unrelated: several other MSE pages answer the same question in details.) –  Did Sep 12 '13 at 17:59
Feller, "An Introduction to Probability Theory and Its Applications, Volume 1, 3rd Edition", section XIII.7. –  awkward Sep 12 '13 at 23:06