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# If x is not equal to 1, is x^2/(x-1) greater than x?

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If x is not equal to 1, is x^2/(x-1) greater than x? [#permalink]  18 Feb 2011, 10:58
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If x is not equal to 1, is x^2/(x-1) greater than x?

(1) x is not an integer.
(2) x is positive.
[Reveal] Spoiler: OA
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Re: 217. If x is not equal to 1, is greater than x? (1) x is [#permalink]  18 Feb 2011, 11:18
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banksy wrote:
217. If x is not equal to 1, is x^2/(x-1) greater than x?
(1) x is not an integer.
(2) x is positive.

Is $$\frac{x^2}{x-1}>x$$ --> is $$\frac{x^2}{x-1}-x>0$$? --> is $$\frac{x^2-x^2+x}{x-1}>0$$ --> is $$\frac{x}{x-1}>0$$ --> is $$x<0$$ or $$x>1$$?

(1) x is not an integer --> clearly insufficient: for example if $$x=1.5$$ the the answer will be YES but if $$x=0.5$$ then the answer will be NO.

(2) x is positive. Also not sufficient.

(1)+(2) The values of $$x$$ from (1) also satisfy (2) (x can be positive fraction from the range (0,1) or some non integer more than 1) thus even taken together statements are not sufficient.

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Re: If x is not equal to 1, is x^2/(x-1) greater than x? [#permalink]  06 Feb 2014, 12:42
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Re: 217. If x is not equal to 1, is greater than x? (1) x is [#permalink]  10 Apr 2014, 21:43
Bunuel wrote:
banksy wrote:
217. If x is not equal to 1, is x^2/(x-1) greater than x?
(1) x is not an integer.
(2) x is positive.

Is $$\frac{x^2}{x-1}>x$$ --> is $$\frac{x^2}{x-1}-x>0$$? --> is $$\frac{x^2-x^2+x}{x-1}>0$$ --> is $$\frac{x}{x-1}>0$$ --> is $$x<0$$ or $$x>1$$?

(1) x is not an integer --> clearly insufficient: for example if $$x=1.5$$ the the answer will be YES but if $$x=0.5$$ then the answer will be NO.

(2) x is positive. Also not sufficient.

(1)+(2) The values of $$x$$ from (1) also satisfy (2) (x can be positive fraction from the range (0,1) or some non integer more than 1) thus even taken together statements are not sufficient.

Hi Bunuel,
will you explain this step ?

x/x-1 > 0 translating to x>1 or x<0 ?

thanks.
Math Expert
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Re: 217. If x is not equal to 1, is greater than x? (1) x is [#permalink]  11 Apr 2014, 01:54
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Expert's post
vishalrastogi wrote:
Bunuel wrote:
banksy wrote:
217. If x is not equal to 1, is x^2/(x-1) greater than x?
(1) x is not an integer.
(2) x is positive.

Is $$\frac{x^2}{x-1}>x$$ --> is $$\frac{x^2}{x-1}-x>0$$? --> is $$\frac{x^2-x^2+x}{x-1}>0$$ --> is $$\frac{x}{x-1}>0$$ --> is $$x<0$$ or $$x>1$$?

(1) x is not an integer --> clearly insufficient: for example if $$x=1.5$$ the the answer will be YES but if $$x=0.5$$ then the answer will be NO.

(2) x is positive. Also not sufficient.

(1)+(2) The values of $$x$$ from (1) also satisfy (2) (x can be positive fraction from the range (0,1) or some non integer more than 1) thus even taken together statements are not sufficient.

Hi Bunuel,
will you explain this step ?

x/x-1 > 0 translating to x>1 or x<0 ?

thanks.

$$\frac{x}{x-1}>0$$ --> transition pints are at 0 and 1:

If x is greater than 1, then $$\frac{x}{x-1}=\frac{positive}{positive}=positive$$, thus in this case $$\frac{x}{x-1}>0$$;

If 0<x<1, then $$\frac{x}{x-1}=\frac{positive}{negative}=negative$$, thus in this case $$\frac{x}{x-1}<0$$;

If x is less than 0, then $$\frac{x}{x-1}=\frac{negative}{negative}=positive$$, thus in this case $$\frac{x}{x-1}>$$;

As you can see the inequality holds true for $$x<0$$ and $$x>1$$.

Hope this helps.
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Re: 217. If x is not equal to 1, is greater than x? (1) x is   [#permalink] 11 Apr 2014, 01:54
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# Meaning of Fermi energy in semiconductor

We know from the band theory that in a semiconductor (of course for all metals, insulator) all energy states are not allowed. In a semiconductor there is a valence band and a conduction band, and in between energy states of them i.e energy levels laying in between valence band and conduction band are forbidden. Now when we calculate Fermi energy of the semiconductor we find that it lies in between valence band and conduction band. For intrinsic semiconductor at T=0k, Fermi energy lies exactly half way between valence band and conduction band. But we know energy levels laying in between valence band and conduction band is forbidden, and we also know that Fermi energy is the highest energy level of a material that an electron corresponds to, at T=0 k. So what is the meaning that Fermi energy lies in a forbidden region in semiconductor which corresponds to valid energy level of the material at T=0k ?

• This question is about the Fermi level not the Fermi energy. "The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. " source: wikipedia. – my2cts Apr 21 '20 at 12:25
• @my2cts it is the Fermi energy because he explicitly stated at $T=0K$. – boyfarrell Apr 21 '20 at 14:39
• @boyfarrell No it is the Fermi level. – my2cts Apr 21 '20 at 15:38
• @my2cts I thought that the Fermi-energy is a special case of the Fermi-level when $T=0K$. I know there is a lot of overloading with these terms so happy to know if that’s wrong interpretation in your experience? – boyfarrell Apr 22 '20 at 11:57

The Fermi energy is not the energy of the highest level, but rather en energy characterizing the fact that the states below this energy are filled, whereas the states above this energy are empty, as described by Fermi-Dirac distribution: $$f(E) = \frac{1}{e^{\frac{E-E_F}{k_B T}} +1}.$$ Thus, the position of Fermi energy in the gap energy region reflects the fact that (at zero temperature) all lower energy states (i.e., the states in the valence band) are filled, whereas all the higher energy states (i.e., the states in the conduction band) are empty.
One thing to keep in mind is that in the context of semiconductors one often uses term Fermi energy to mean Fermi level, i.e. the chemical potential. In a free electron gas the two are the same: they designate the position of the Fermi surface at zero temperature in the continuum spectrum. In a semiconductor the notion of Fermi energy is not very useful - the states are filled up to the top of the valence band. The Fermi level (i.e. the chemical potential), entering the Fermi distribution, is meaningful. Note however that adding the last electrons to the valence band costs zero energy, whereas adding the first electrons to the conduction band costs the gap energy, $$E_g$$. This is why the Fermi level is placed in the middle of the gap (for an intrinsic semiconductor).

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Version 1.5
$\newcommand{\lyxlock}{}$
Subsection 3.14.3: Colormap Options Up Section 3.14: FEBio Studio Options Subsection 3.14.5: Lighting Options

3.14.4 Display Options

The display options affect some of the aspects of how the model is displayed on the screen. The following options can be modified:
• Size of nodes: Sets the size (in pixels) of point rendering (e.g. nodes, tags).
• Line width: Sets the thickness (in pixels) for rendering lines.
• Mesh color: sets the mesh color for rendering the mesh.
• Show normals: Show a line on each surface facet that indicates the orientation of the local normal.
• Normals scale factor: Sets the scale factor that is used for rendering the normals.
• Multiview Projection: sets the multiview projection option
• Object transparency mode: Selects under what conditions objects will be rendered as transparent.
• Object color: Sets the color option for rendering objects.
• Default: the object's color is determined by the material assigned to its part. If no material is assigned, the object's color is used.
• Object: always use the object's color.
Subsection 3.14.3: Colormap Options Up Section 3.14: FEBio Studio Options Subsection 3.14.5: Lighting Options


text:
## andijo76 2 years ago Essay; show all work. A simple dartboard has three areas… the main board has a radius of 10 inches, there is a circle with a radius of 7 inches, and the bullseye has a radius of 3 inches. What is the probability of a random dart landing inside the bullseye

1. andijo76

thhere is more

2. andijo76

Round to the nearest thousandth hint use

3. satellite73

it is the area of the bull's eye divided by the total area

4. andijo76

hint use |dw:1328157525876:dw|

5. satellite73

compute the area of the dartboard using the formula you just wrote then compute the area of the bulls eye as well divide the second by the first

6. satellite73

$\frac{\pi\times 3^2}{\pi\times 10^2}=\frac{9}{100}=.09$


text:
## Relations & Their Relatives • Discussion 1

The divisor of relation signified by $x|y$ is a dyadic relation on the set of positive integers $\mathbb{M}$ and thus may be understood as a subset of the cartesian product $\mathbb{M} \times \mathbb{M}.$  It is an example of a partial order, while the less than or equal to relation signified by $x \le y$ is an example of a total order relation.

The mathematics of relations can be applied most felicitously to semiotics but there we must bump the adicity or arity up to three.  We take any sign relation $L$ to be subset of a cartesian product $O \times S \times I,$ where $O$ is the set of objects under consideration in a given discussion, $S$ is the set of signs, and $I$ is the set of interpretant signs involved in the same discussion.

One thing we need to understand is the sign relation $L \subseteq O \times S \times I$ relevant to a given level of discussion may be rather more abstract than what we would call a sign process proper, that is, a structure extended through a dimension of time.  Indeed, many of the most powerful sign relations generate sign processes through iteration or recursion or similar operations.  In that event, the most penetrating analysis of the sign process or semiosis in view is achieved through grasping the generative sign relation at its core.

### 9 Responses to Relations & Their Relatives • Discussion 1

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## Logic: Quantifiers

Some sentences feel an awful lot like statements but aren’t. For example,

$n$ is even

This is not a statement because it doesn’t have a truth value; unless we know what $n$ is, we can’t really do much.

Definition. A sentence with one or more variables, so that supplying values for the variables yields a statement, is called an open sentence.

You can think of an open sentence as a function whose values are statements. In fact we will use function notation to name open sentences.

$S(n)=$ n is even

Just as with ordinary functions, this notation works by substitution. With $S(n)$ defined as above,

$S(3)=$ 3 is even

which happens to be a false statement. $S(14)$, on the other hand, is a true statement.

Along with an open sentence, we have to provide some kind of indication of what sort of thing the variable might be. For our example $S(n)$, it makes most sense to let $n$ be a natural number or possibly an integer. We call possible values for the variable of an open sentence the universe of that sentence.

One thing that cannot be emphasized enough is that variables can represent any type of thing, not just numbers or other mathematical objects. So we could think about the open sentence

$A(r,t)=$ r is t years old

and say that the universe for $r$ is everyone in this class and the universe for $t$ is any whole number between 15 and 60.

Definition. Given an open sentence with one variable $S(x)$, the statement $\forall x, S(x)$ is true when, no matter what value of $x$ we use, $S(x)$ is true; otherwise $\forall x, S(x)$ is false.
Given an open sentence with one variable $S(x)$, the statement $\exists x: S(x)$ is true when there is some value of $x$ for which $S(x)$ is true; otherwise $\exists x:S(x)$ is false.

Terminology. We call $\forall$ the universal quantifier, and we read $\forall x, S(x)$ for all $x$, $S(x)$. We call $\exists$ the existential quantifier, and we read $\exists x:S(x)$ there exists $x$ such that $S(x)$.

Observe that if there are only two possible values in the universe for $S(x)$ (let’s call them $\alpha$ and $\beta$), then $\forall x,S(x)$ is true when both $S(\alpha)$ and $S(\beta)$ are true. But this is the same as $S(\alpha)\wedge S(\beta)$ being true. Similarly, $\exists x:S(x)$ is true when one of $S(\alpha)$ or $S(\beta)$ is true. But this is the same as $S(\alpha)\vee S(\beta)$. So we see that the quantifiers are in some sense a generalization of $\vee$ and $\wedge$. So the following makes sense:

De Morgan’s Laws, quantifier version: For any open sentence $S(x)$ with variable $x$,

1. $\sim \left[\forall x,S(x)\right] \equiv \exists x:\left[\sim S(x)\right]$
2. $\sim \left[\exists x:S(x)\right] \equiv \forall x,\left[\sim S(x)\right]$

For example, a denial of the statement

There is a china teapot floating halfway between the earth and the sun.

is

Every china teapot is not floating halfway between the earth and the sun.

### the universal quantifier, conditionals, and the universe

Quantifiers are most interesting when they interact with other logical connectives. For example, consider the following (true) statement:

Every multiple of 4 is even.

We could choose to take our universe to be all multiples of 4, and consider the open sentence

$E(n)=$ n is even

and translate the statement as $\forall n,E(n)$. But that isn’t very interesting. A much more natural universe for the sentence $n$ is even is the integers. But then we have to do something clever, because if our universe for $n$ is the integers, then $\forall n,E(n)$ is false. The solution is to create another open sentence

$F(k)=$ k is a multiple of 4

How do we use $F$ and $E$ to translate our true statement? We can think of an open sentence as a test–if we plug in a value for its variable(s), we see whether that variable passes the test. Here we have two tests: $E$, a test for evenness, and $F$, a test for multiple-of-4-ness. The statement we are trying to translate says that passing the $F$ test is enough to guarantee passing the $E$ test. That sounds like a conditional. Indeed the correct translation for Every multiple of 4 is even is:

$\forall n,F(n)\Rightarrow E(n)$

Try translating this statement back into English using some of the various translations for $\Rightarrow$ to see that it really does mean the same thing as Every multiple of 4 is even.

Exercise. Translate $\forall n, F(n)\wedge E(n)$ and $\forall n, F(n)\vee E(n)$ into English. Explain why these are false statements. Assume the universe for both $E$ and $F$ is the integers.

In fact, we can always expand the universe by putting in another conditional. If we let $Z(s)$ be the sentence $s$ is an integer and expand our universe to include all mathematical objects encountered in this course, we could translate Every multiple of 4 is even as $\forall s, \left(Z(s)\Rightarrow\left(F(s)\Rightarrow E(s)\right)\right)$.

A Note about Notation. It’s important to keep in mind that, just as for the functions you’ve encountered in calculus and before, the particular symbol we use for a variable is not relevant to the meaning of that variable. The statements

$\forall s,\left(Z(s)\Rightarrow\left(F(s)\Rightarrow E(s)\right)\right)$

and

$\forall p,\left(Z(p)\Rightarrow\left(F(p)\Rightarrow E(p)\right)\right)$

both say the same thing. namely, Every integer which is a multiple of 4 is even. The fact that we called the variable $s$ when we defined $Z$ and $n$ when we defined $E$ does not require us to always use those variables. Notice that in the English translation, no variables appear at all! We could equally well have written

$\forall \odot, \left(Z(\odot)\Rightarrow\left(F(\odot)\Rightarrow E(\odot)\right)\right)$

except that that’s a bit difficult to pronounce.

### The existential quantifier and the universe

We just saw that generally speaking, a universal quantifier should be followed by a conditional. What should an existential quantifier be followed by? Consider the following true statement

There is a multiple of 7 which is even.

How can we represent this symbolically? We could take the universe to be all multiples of 7 and write $\exists n: E(n)$. But as before, that’s not very interesting. So let’s keep our universe as it should be: the integers. As before, we’ll need a test for multiple-of-7-ness: denote by $S(k)$ the sentence $k$ is a multiple of 7.

Now think about what the statement There is a multiple of 7 which is even means. What is the relationship between multiple-of-7-ness and evenness? Just that some number happens to be both. Therefore we can translate:

$\exists n: S(n)\wedge E(n)$

Notice that because $\wedge$ is commutative, our symbolic statement is equivalent to $\exists n:E(n)\wedge S(n)$. But this is just fine, because our statement and the statement

There is an even number which is a multiple of 7

pretty clearly mean the same thing.

Let’s lock in the connection between $\exists$ and $\wedge$ with another example. This time we’ll use De Morgan’s laws and consider the statement

Every multiple of 7 is even.

which happens to be false. Therefore its negation is true. We compute that negation:

$\sim\left[\forall n,\left(S(n)\Rightarrow E(n)\right)\right] \equiv \exists n:\sim\left(S(n)\Rightarrow E(n)\right)$
$\equiv \exists n:\left(S(n)\wedge \sim E(n)\right)$

which we could phrase in English as There is an integer which is a multiple of 7 and not even. Thus we see that the existential quantifier pairs naturally with the connective $\wedge$.

Exercise. Let $E(k)$ stand for $k$ is even, $S(p)$ stand for $p$ is a multiple of 7, and $Z(m)$ stand for $m$ is an integer. Let the universe for all three sentences be the set of all mathematical objects encountered in this course. Write a symbolic translation of There is a multiple of 7 which is even using these open sentences.

Exercise. Translate $\exists n: S(n)\Rightarrow E(n)$ into English. Explain why this is a true statement. Give a useful denial. Assume the universe for both $E$ and $S$ is the integers.

### quantifier order

Many interesting open sentences have more than one variable, such as:

$A(r,t)$=$r$ is $t$ years old

Since there are two variables, we are entitled to ask the question which one? twice. And we may have a different answer each time. There are eight possibilities, of which four are

1. $\forall r,\forall t, A(r,t)$ For any person $r$, for any age $t$, that person is that age.
2. $\forall t,\forall r, A(r,t)$ For any age $t$, for any person $r$, that person is that age.
3. $\exists r: \exists t: A(r,t)$ There is a person $r$ who is some age $t$.
4. $\exists t: \exists r: A(r,t)$ There is an age $t$ so that some person $r$ is that age.

A moment’s thought should make clear that statements 1 and 2 mean the same thing (in our universe, both are false), and statements 3 and 4 mean the same thing (in our universe, both are true if woefully uninformative). This says that we can move existential quantifiers past one another, and move universal quantifiers past one another.

The other four possibilities are

1. $\forall r,\exists t: A(r,t)$ For any person $r$, there is an age $t$ so that that person is that age.
2. $\exists t: \forall r, A(r,t)$ There is an age $t$ so that every person $r$ is that age.
3. $\forall t,\exists r: A(r,t)$ For any age $t$, there is a person who is that age.
4. $\exists r: \forall t, A(r,t)$ There is a person $r$ who is every age.

Quantifier order matters.
Mixed quantifiers do not commute.

Notice that statement 5 is true (in our universe): everyone has an age. But statement 6 says that everyone is the same age, which is false in our universe. So statement 5 and statement 6 mean different things. Similarly, statement 7 is likely true in our universe, whereas statement 8 is false. The lesson is that quantifiers of different flavors do not commute!

## Meta Data

Title: Logic: Quantifiers
Date Posted: July 10, 2018
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Category: Logic
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text:
# Solution of Integral

I want to solve the following integral

$$\frac{\alpha \beta}{2} \int_0^\pi \cos\theta \sec^{2}\theta(\tan(\theta/2))^{-\beta-1} (1+\gamma(\tan(\theta/2))^{-\beta})^{-\frac{\alpha}{\gamma}-1}d\theta$$ after subtituting $(\tan(\theta/2))^{-\beta}=z$

I got this $$\int_0^\infty \frac{{(1+\gamma z)}{}^{-(\frac{\alpha}{\gamma}+1)}}{1+z^{-\frac{2}{\beta}}}dz$$

where $\alpha, \beta and \gamma>0.$ How to solve above integral? Kindly help me in this regards.

• You want that, but what is your question? – Did Mar 12 '15 at 10:49
• @Did I simply need someone help me to solve the stated integral. – SAAN Mar 12 '15 at 10:50
• What have you already tried? What have you already thought of? – Pedro Mar 12 '15 at 10:51
• @Pedro Nice to know. Why? – Did Mar 12 '15 at 10:51
• Have you thought already of some methods? What have you tried to do? People expect that you have shown some effort or that you can present a list of things you might think can help to solve the problem, but you don't know how to implement it. Where did you found the problem? Is it from a book? Is it something you made up yourself? So that people know whether there actually should be a solution and that it is actually solvable. If you don't put this information in your post, your question might be downvoted by people. – Pedro Mar 12 '15 at 11:00

For $B\in\mathbb N$ we have $\quad\displaystyle\int_0^\infty\frac{(1+Ax)^B}{1+x^C}~dx~=~\frac\pi C\cdot\sum_{k=0}^B~{B\choose k}~\frac{A^k}{\sin\bigg((k+1)~\dfrac\pi C\bigg)}\quad$ which can

be easily shown by expanding the numerator using the binomial theorem and letting $t=\dfrac1{1+x^C}$

then recognizing the expression of the beta function in the new integral, and employing Euler's

reflection formula for the $\Gamma$ function to simplify the result.

• But the RHS diverges for every integer C? – Did Mar 12 '15 at 12:25
• @Did: For convergence we must have $C>B+1$, obviously, so $\dfrac{k+1}C$ will always be strictly in between $0$ and $1$, thus avoiding the problematic points $\sin0=\sin\pi=0.$ – Lucian Mar 12 '15 at 20:41
• This should be mentioned clearly in any answer. – Did Mar 12 '15 at 22:35
• @Did: And why is that ? To prevent the reader from God-forbid ever developing any critical thought and analytic skills ? – Lucian Mar 12 '15 at 23:00
• @Lucian For $B=-(\frac{\alpha}{\gamma}+1)$ this solution does not work. – SAAN Mar 13 '15 at 10:12


text:
Lemma 15.19.5. In Situation 15.19.1. Let $I \subset R$ be an ideal. Assume

1. $R$ is a Noetherian ring,

2. $S$ is a Noetherian ring,

3. $M$ is a finite $S$-module, and

4. for each $n \geq 1$ and any prime $\mathfrak q \in V(J + IS)$ the module $(M/I^ n M)_{\mathfrak q}$ is flat over $R/I^ n$.

Then (15.19.1.1) holds for $(R, I)$, i.e., for every prime $\mathfrak q \in V(J + IS)$ the localization $M_{\mathfrak q}$ is flat over $R$.

Proof. Let $\mathfrak q \in V(J + IS)$. Then Algebra, Lemma 10.99.11 applied to $R \to S_{\mathfrak q}$ and $M_{\mathfrak q}$ implies that $M_{\mathfrak q}$ is flat over $R$. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
anonymous one year ago What is the weight (in grams) of a liquid that exactly fills a 163.5 milliliter container if the density of the liquid is 0.492grams over milliliter? Round to the nearest hundredth when necessary and only enter numerical values, which can include a decimal point. (Fill in the Blank)

1. Jack1

ok, cool so density = mass per unit of volume D = m/v in this case Volume = 163.5 mL and Density = 0.492g/mL

2. Jack1

$$\Large D = \frac mV$$ $$\Large 0.492 = \frac m{163.5}$$ so multiply boths sides by 163.5... you'll be left with mass (in grams)

3. anonymous

80.448 grams correct (:

4. Jack1

1 got 80.442... but yeah, perfect just round it off at 80.44 (to the nearest hundredth) and that's it!

5. anonymous

oh ok so the answer would be 80.448 right im just wondering (:

6. Jack1

yep

7. anonymous

oh ok thank you ^-^

8. Jack1

np ;)


text:
+0

# Functions help

0
269
1

If $$f(x)=\dfrac{a}{x+2}$$, solve for the value of a so that $$f(0)=f^{-1}(3a)$$.

May 5, 2018

### 1+0 Answers

#1
+27531
0

To get f-1 rewrite f as:

$$f=\frac{a}{x+2}\\x+2=\frac{a}{f}\\x=\frac{a}{f}-2$$

Now change f into x and x into f-1:

$$f^{-1}=\frac{a}{x}-2$$

We have: $$f(0)=\frac{a}{2}\text{ and }f^{-1}(3a)=\frac{a}{3a}-2\rightarrow-\frac{5}{3}$$

I'm sure you can take it from here!

May 5, 2018


text:
You just can’t knock down this Monster “Variance” —- Part-3

If x & y are two variables, then irrespective of whether you add or subtract them the variance will always add up.

A store wants to know the mean and the variance of sales made by male and female customers in a day. He also wants to see the variance in case sales by both gender are added in pair randomly. Lastly he wants to analyze the mean and variance because of the gender effect (i.e. difference of means and variance). Data of sales in hundred dollars is given below

Using Excel Sheet Mean is calculated by typing formula  =average(array)

Variance is calculated by typing formula  =var.s(array)

Array = column of data, Var.s = variance of sample

But most surprising element is that, irrespective of whether you add or subtract the data, variance always increases. This monster will always raise its head. This is indicated by the resultant variance which is always greater than the individual variances.

In general, the variance always gets added irrespective of whether we are adding or subtracting the individual variances.

where ρ is the correlation coefficient between two variables.

If two random variables are not correlated or they are independent then, ρ = 0 and above formula will get reduced to

Try to calculate the variance for x+y and x-y, are you getting little bit different answer? use correlation coefficient into the equation!

Calculating correlation coefficient (ρ) in excel

Type formula in a cell  =correl(array1, array2)

array1 = column x, array2 = column y

Understanding the Monster “Variance” part-1

Why it is so Important to Know the Monster “Variance”? — part-2

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Why it is so Important to Know the Monster “Variance”? — part-2

Variance occupies the central role in the six-sigma methodology. Any process whether from manufacturing or service industry has many inputs and the variance from each input gets add up in the final product.

Hence variance has an additive property as shown below

Note: you can add two variances but not the standard deviations

Consequence of the variance addition and six sigma

Say if a product/services which is the output of some process, which in turn have many inputs. Then the variance from the input () and from the process () adds up to give the final variance () in the product/services.

DMAIC methodology of 6Sigma try to identify the inputs that contributes maximum towards the variance in the final product and once identified, its effect is studied in detail to minimize the variance from the input. This is done by reducing the variance in the input itself.

Example: if the quality of a input material used to manufacture a product is found to be critical, then steps would be taken to reduce the fluctuation of the quality of that input material from batch to batch either by requesting/threatening the vendor or by performing the rework of the input material at your end.

Related articles:

Understanding the Monster “Variance” part-1

You just can’t knock down this Monster “Variance” —- Part-3

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Understanding the Monster “Variance” part-1

This is one of the ways of calculating the variability in the data set.  Variance helps us in understanding how the data is arranged around the mean. In order to do so, we need to calculate the deviation of each observation from the mean in the data set .

For example: following is the time taken by me during the week to reach the office. The deviation of each  observation from the mean  time is given below.

Now next step is to calculate the average deviation from the mean using well-known formula

Note that the sum of all positive deviations = sum of all negative deviations which indicates that the mean divided the data set in two equal halves. As a result the sum of all deviation becomes zero, hence we need some other way to calculate this average deviation about the mean.

In order to avoid the issue, a very simple idea was used

Negative number → Square of negative number → positive number → square root of this number → parent number

Hence square of all the deviations are calculated and summed-up to give sum of squares (simply SS) [1]. This SS is then divided by total number of observations to give average variance s² around the mean.[2] The square root of this variance gives standard deviation s, the most common measure of variability.

What it physically means is that on an average data is deviating 7.42 units or simply one standard deviation (±1s) in either of the directions in a given data set.

Above discussion about the sample standard deviation represented by s. For population, variance is represented by σ² and standard deviation by σ.

The sample variance s² is the estimator of the population variance σ². The standard deviation is easier to interpret than the variance because the standard deviation is measured in the same units as the data.

[1] Popularly known as sum of squares, this most widely term used in ANOVA and Regression analysis

[2] SS divided by its degree of freedom → mean sum of squares or MSE, these concepts would appear in ANOVA & Regression analysis.

Related articles:

Why it is so Important to Know the Monster “Variance”? — part-2

You just can’t knock down this Monster “Variance” —- Part-3

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Now it’s important to understand the concept of sigma or the standard deviation

We have seen that we need to restrict the width of the car for a given width of the garage. This is analogous to the with of the process (voice of customer, VOP) Vs the width of the customer’s specification (voice of the customer or VOC).
The width of the process is measured in terms of standard deviation denoted by σ (sigma).

The target of the 6sigma methodology is to reduce this variance (width of the car) to such an extent that even by mistake it should not cross the customer’s specification (or should not hit the wall of the garage).

Before we work towards reducing the σ, we should know about this monster very well as we will be encountering him at every step during the 6sigma journey.

There are two very important characteristics of any data set

Location and the spread of the data set.

Location represents the point in the data set where there is maximum clustering of the data –> Mean and median.

Spread represents the variability in the data set, there will be some observations that will be above the mean and there will be some that will be below the mean. Standard deviation σ measures the average spread of the data from the mean in either direction of the mean.

Office arrival time for last 5 days with average time are given below, deviation of each observation from the mean is also captured.

Let’s calculate the average deviation

Note that sum of all positive deviations = sum of all negative deviations which indicates that the mean divided the data in two equal halves.

Sum of all deviation itself becomes zero, hence we need some other way to calculate this average deviation about the mean.

In order to circumvent the issue, a very simple idea was used

Square of negative number → positive number → square root of this number → ±parent number

Hence square of all the deviations are calculated and summed-up to give sum of squares (simply SS) [1]. This SS is then divided by total number of observations to give average variance around the mean.[2] The square root of this variance gives standard deviation s, the most common measure of variability.

What it typically means that “on an average data is 7.42 units (= 1 standard deviation ±1σ) in either direction of the mean in the given data set. Mean of the data set is at ZERO standard deviation.

If process a stabilized and normally distributed then following holds true

i.e. 99.7 % of the observation in the data set would be between ±3σ.

Now we can understand whey we have taken 12σ as the width of the garage and 6σ as the width of the car!

The concept of ‘σ’ is the most important concept in understanding 6sigma. If we can understand it, downstream we wouldn’t be having any problem in understanding other topics. At this moment one important point to be noted here is that the calculation of σ depend on the type of data or data distribution we are handling.

Calculation of mean and σ would be different depending on whether we are dealing with normal distribution, binomial distribution, Poisson distribution etc. The importance of this would be realized when we would be studying the various types of control charts. At that time we just have to remember that “we must calculate mean and σ according to the distribution”.

[1] Popularly known as sum of squares, this most widely term used in ANOVA and Regression analysis

[2] SS divided by its degree of freedom → mean sum of squares or MSE, these concepts would appear in ANOVA & Regression analysis.

Is this information useful to you?


text:
# Distance Formula

• April 8th 2008, 09:23 PM
lilikoipssn
Distance Formula
What is the distance between the points http://homework.math.ucsb.edu/webwor...c617e5de71.png and http://homework.math.ucsb.edu/webwor...bfba8d9dd1.png

I keep getting the square root of 3990, which apparently is wrong?
• April 8th 2008, 09:27 PM
topher0805
The distance formula is:

$
D = \sqrt {(x_2-x_1)^2+(y_2-y_1)^2}$

$
$
$D=67$


text:
# American Institute of Mathematical Sciences

2011, 18: 1-11. doi: 10.3934/era.2011.18.1

## Special functions created by Borel-Laplace transform of Hénon map

 1 Department of Physics, Graduate school of Science and Technology, Ehime University, Bunkyocho 2-5, Matsuyama 790-8577, Japan 2 Department of Mathematics, Graduate school of Science and Technology, Ehime University, Bunkyocho 2-5, Matsuyama 790-8577, Japan

Received  June 2010 Published  January 2011

We present a novel class of functions that can describe the stable and unstable manifolds of the Hénon map. We propose an algorithm to construct these functions by using the Borel-Laplace transform. Neither linearization nor perturbation is applied in the construction, and the obtained functions are exact solutions of the Hénon map. We also show that it is possible to depict the chaotic attractor of the map by using one of these functions without explicitly using the properties of the attractor.
Citation: Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1
##### References:
 [1] M. Hénon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77. doi: doi:10.1007/BF01608556. [2] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130-141. doi: doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. [3] V. Hakim and K. Mallick, Exponentially small splitting of separatrices, matching in the complex plane and Borel summation, Nonlinearity, 6 (1993), 57-70. doi: doi:10.1088/0951-7715/6/1/004. [4] A. Tovbis, Asymptotics beyond all orders and analytic properties of inverse Laplace trnsforms of solutions, Commun. Math. Phys., 163 (1994), 245-255. doi: doi:10.1007/BF02102008. [5] A. Tobvis, M. Tsuchiya and C. Jaffe, Exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the Hénon map as an example, Chaos, 8 (1998), 665-681. doi: doi:10.1063/1.166349. [6] K. Nakamura and M. Hamada, Asymptotic expansion of homoclinic structures in a symplectic mapping, J. Phys. A, 29 (1996), 7315-7327. doi: 10.1088/0305-4470/29/22/025. [7] V. F. Lazutkin, I. G. Schachmannski and M. B. Tabanov, Splitting of separatrices for standard and semistandard mappings, Physica D, 40 (1989), 235-248. doi: doi:10.1016/0167-2789(89)90065-1. [8] M. D. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth, Stud. Appl. math., 85 (1991), 129-181. [9] H. Segur, S. Tanveer and H. Levine (eds), "Asymptotics Beyond All Orders," (Plenum, New York), 1991. [10] A. Voros, The return of quartic oscillator: The complex WKB method, Ann. Inst. H. Poincaré 39 (1983), 211-338. [11] J. Écalle, "Les Fonctions Résurgence vol. 1," (French) [Resurgent functions. Vol. I] Les algèbres de fonctions résurgentes. [The algebras of resurgent functions] With an English foreword. Publications Mathématiques d'Orsay 81 [Mathematical Publications of Orsay 81], 5. Université de Paris-Sud, Département de Mathématique, Orsay, 1981. [12] J. Écalle, "Les Fonctions Résurgence vol. 2," (French) [Resurgent functions. Vol. II] Les fonctions résurgentes appliquées à l'itération. [Resurgent functions applied to iteration] Publications Mathématiques d'Orsay 81 [Mathematical Publications of Orsay 81], 6. Université de Paris-Sud, Département de Mathématique, Orsay, 1981. [13] J. Écalle, "Les Fonctions Résurgence vol. 3," (French) [Resurgent functions. Vol. III] L' équation du pont et la classification analytique des objects locaux. [The bridge equation and analytic classification of local objects] Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], 85-5. Université de Paris-Sud, Département de Math¨¦matiques, Orsay, 1985. [14] B. Y. Sternin and V. E. Shatalov, "Borel-Laplace Transform and Asymptotic Theory," Introduction to Resurgent Analysis, CRC Press, Boca Raton, FL, 1996. [15] V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map, Ann. Inst. Fourier (Grenoble), 51 (2001), 513-67. [16] S. Newhouse and T. Pignataro, On the estimation of topological entropy, J. Stat. Phys., 72 (1993), 1331-1351.

show all references

##### References:
 [1] M. Hénon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77. doi: doi:10.1007/BF01608556. [2] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130-141. doi: doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. [3] V. Hakim and K. Mallick, Exponentially small splitting of separatrices, matching in the complex plane and Borel summation, Nonlinearity, 6 (1993), 57-70. doi: doi:10.1088/0951-7715/6/1/004. [4] A. Tovbis, Asymptotics beyond all orders and analytic properties of inverse Laplace trnsforms of solutions, Commun. Math. Phys., 163 (1994), 245-255. doi: doi:10.1007/BF02102008. [5] A. Tobvis, M. Tsuchiya and C. Jaffe, Exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the Hénon map as an example, Chaos, 8 (1998), 665-681. doi: doi:10.1063/1.166349. [6] K. Nakamura and M. Hamada, Asymptotic expansion of homoclinic structures in a symplectic mapping, J. Phys. A, 29 (1996), 7315-7327. doi: 10.1088/0305-4470/29/22/025. [7] V. F. Lazutkin, I. G. Schachmannski and M. B. Tabanov, Splitting of separatrices for standard and semistandard mappings, Physica D, 40 (1989), 235-248. doi: doi:10.1016/0167-2789(89)90065-1. [8] M. D. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth, Stud. Appl. math., 85 (1991), 129-181. [9] H. Segur, S. Tanveer and H. Levine (eds), "Asymptotics Beyond All Orders," (Plenum, New York), 1991. [10] A. Voros, The return of quartic oscillator: The complex WKB method, Ann. Inst. H. Poincaré 39 (1983), 211-338. [11] J. Écalle, "Les Fonctions Résurgence vol. 1," (French) [Resurgent functions. Vol. I] Les algèbres de fonctions résurgentes. [The algebras of resurgent functions] With an English foreword. Publications Mathématiques d'Orsay 81 [Mathematical Publications of Orsay 81], 5. Université de Paris-Sud, Département de Mathématique, Orsay, 1981. [12] J. Écalle, "Les Fonctions Résurgence vol. 2," (French) [Resurgent functions. Vol. II] Les fonctions résurgentes appliquées à l'itération. [Resurgent functions applied to iteration] Publications Mathématiques d'Orsay 81 [Mathematical Publications of Orsay 81], 6. Université de Paris-Sud, Département de Mathématique, Orsay, 1981. [13] J. Écalle, "Les Fonctions Résurgence vol. 3," (French) [Resurgent functions. Vol. III] L' équation du pont et la classification analytique des objects locaux. [The bridge equation and analytic classification of local objects] Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], 85-5. Université de Paris-Sud, Département de Math¨¦matiques, Orsay, 1985. [14] B. Y. Sternin and V. E. Shatalov, "Borel-Laplace Transform and Asymptotic Theory," Introduction to Resurgent Analysis, CRC Press, Boca Raton, FL, 1996. [15] V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map, Ann. Inst. Fourier (Grenoble), 51 (2001), 513-67. [16] S. Newhouse and T. Pignataro, On the estimation of topological entropy, J. Stat. Phys., 72 (1993), 1331-1351.
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text:
# What exactly is barrier tunneling and potential barrier?

1. Sep 4, 2009

### rozan977

How can we create a potential barrier? And how does it exactly acts as barrier in electron transfer?

2. Sep 4, 2009

### f95toli

By making a "step" in the potential energy. In the real world that usually means making a "sandwich" of 3 materials, the archtypical example being GaAs/AlGaAs/GaAs where the height of the barrier can be adjusted by varying the amount of aluminium in the middle layer.
This is how e.g. blue laser diodes are made.

Another option is to make a three terminal device where the middle terminal can be used as a gate; the height of the barrier is then set by adjusting the voltage applied to that gate.

3. Sep 4, 2009

### rozan977

But how can we set the potential energy of barrier greater than the total energy in it? Also instead of reflection (as we may think in classical mechanics) how does a particle pass through the barrier (the amplitude of wave actually decreases exponentially,how is this possible)?
Answer to mine 2nd question too:
And how does it exactly acts as barrier in particle wave transfer?

4. Sep 4, 2009

### f95toli

I am not sure I understand the question.
Remember that potential energy is always something relative; where you set "zero" is arbitrary so there is nothing preventing us from making a barrier.

It is perhaps worth pointing out that in e.g. the GaAs/AlGaAs/GaAs structure I referred to above the difference in electron energy is just set by properties of the materials (there are tables which relate the amount of aluminium in percent to the energy in eV compared to pure GaAs). The "barrier" is a barrier simply because an electron would need more energy than is available in order to be able to jump up and populate an empty state in the valence band.

The answer to the second questions is: Because it can. Ultimately it is just how nature works simply because there is no such thing as a "classical particle". Everything in nature (including -at least in principle- you and me) have both "particle like" and "wavelike" properties and in structures where tunnelling can occur the latter dominates.

5. Sep 4, 2009

### zenith8

It is impossible to explain the tunneling effect consistently using an interpretation of QM involving the wave function alone. All one can do is predict the probability of the tunneling occurring.

However, this is not so in the Bohm interpretation where there are particles in addition to the objectively existing wave function (with the wave exerting a force on the particles). This extra 'quantum force' can, under appropriate circumstances, boost a particle through the barrier. The force is proportional to the curvature of the amplitude of the wave field - and the rapid spacetime fluctuations of the field in the vicinity of the barrier sometimes mean that this force is very large.

The effective barrier encountered by the particle is thus not $$V$$ (the classical potential) but $$V+Q$$ (where $$Q$$ is the 'quantum potential'). This may be higher or lower than $$V$$ and may vary outside the 'true' barrier.

For more discussion and pretty pictures of trajectories of particle tunneling through barriers see p. 26-28 of lecture 3 of http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" [Broken].

Last edited by a moderator: May 4, 2017
6. Sep 4, 2009

### zenith8

Rozan977 - do you see how the Bohm interpretation is better?

7. Sep 4, 2009

### alxm

Well, I guess in the 'usual' world of solid-state physics ;) The usual for a chemical physicist like me, is that the 'barrier' is the vacuum.

Say, I have an ethylene glycolate cation.. then it's plausible (I don't know if it's actually been measured in this specific case) that the hydroxyl hydrogen atom can 'flip-flop' between the two oxygens, like so:

Now, to get from one oxygen to the other, the hydrogen atom has to pass through the vacuum - weakening its bond before forming a new one. So it goes up in energy and then down to where it was. (since the two structures are equivalent). If you displayed energy in terms of the 1-dimensional coordinate of the hydrogen atom position it'd be two valleys with a hump in-between: The potential barrier. (which'd be something like 0.1 eV in this case, relative the 'valleys' at each oxygen)

But since hydrogen is a very small atom, it can tunnel quantum-mechanically! In other words, it goes through the barrier: from one oxygen to the other without passing any points in-between! This effect is actually big enough to be measurable when it comes to some hydrogen-transfer reactions (they occur faster than if the hydrogen only went 'over' the barrier). Although this specific example is my own invention.

I believe hydrogen and deuterium are the heaviest things tunneling has been observed for, since it's strongly dependent on mass. In electon transfers it's so significant it always has to be taken into account.


text:
# How do you calculate arcsin (-1/ sqrt 2)?

Jul 20, 2015

Use properties of right angled triangle with sides $\frac{1}{\sqrt{2}}$, $\frac{1}{\sqrt{2}}$ and $1$ to find $\arcsin \left(- \frac{1}{\sqrt{2}}\right) = - \frac{\pi}{4}$

#### Explanation:

$\arcsin \left(- \frac{1}{\sqrt{2}}\right) = \alpha$ where $\alpha \in \left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$ such that $\sin \left(\alpha\right) = - \frac{1}{\sqrt{2}}$

$\frac{1}{\sqrt{2}}$ is one side of the right angled triangle with sides:

$\frac{1}{\sqrt{2}}$, $\frac{1}{\sqrt{2}}$ and $1$

(Note ${\left(\frac{1}{\sqrt{2}}\right)}^{2} + {\left(\frac{1}{\sqrt{2}}\right)}^{2} = \frac{1}{2} + \frac{1}{2} = 1 = {1}^{2}$)

This triangle has angles $\frac{\pi}{4}$, $\frac{\pi}{4}$ and $\frac{\pi}{2}$.

So $\sin \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$

Now $\sin \left(- \theta\right) = - \sin \left(\theta\right)$

So $\sin \left(- \frac{\pi}{4}\right) = - \frac{1}{\sqrt{2}}$

So $\arcsin \left(- \frac{1}{\sqrt{2}}\right) = - \frac{\pi}{4}$


text:
# From a Lot of 15 Bulbs Which Include 5 Defective, a Sample of 4 Bulbs is Drawn One by One with Replacement. Find the Probability Distribution of Number of Defective Bulbs. - Mathematics

#### Question

From a lot of 15 bulbs which include 5 defective, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence, find the mean of the distribution.

#### Solution

Let getting a defective bulb in a trial be a success.
We have,

$p = \text{ probability of getting a defective bulb } = \frac{5}{15} = \frac{1}{3} \text{ and}$
$q = \text{ probability of getting non - defective bulb } = 1 - p = 1 - \frac{1}{3} = \frac{2}{3}$
$\text{ Let X denote the number of success in a sample of 4 trials . Then, }$
$\text{ X follows binomial distribution with parameters n = 4 and } p = \frac{1}{3}$
$\therefore P\left( X = r \right) =^{4}{}{C}_r p^r q^\left( 4 - r \right) = ^{4}{}{C}_r \left( \frac{1}{3} \right)^r \left( \frac{2}{3} \right)^\left( 4 - r \right) = \frac{^{4}{}{C}_r 2^\left( 4 - r \right)}{3^4}, \text{ where } r = 0, 1, 2, 3, 4$
$\text{ i . e .}$
$P\left( X = 0 \right) = \frac{^{4}{}{C}_0 2^4}{3^4} = \frac{16}{81},$
$P\left( X = 1 \right) = \frac{^{4}{}{C}_1 2^3}{3^4} = \frac{32}{81},$
$P\left( X = 2 \right) = \frac{^{4}{}{C}_2 2^2}{3^4} = \frac{24}{81},$
$P\left( X = 3 \right) = \frac{^{4}{}{C}_3 2^1}{3^4} = \frac{8}{81},$
$P\left( X = 4 \right) = \frac{^{4}{}{C}_4 2^0}{3^4} = \frac{1}{81}$

So, the probability distribution of X is given as follows:

 X: 0 1 2 3 4 P(X): $\frac{16}{81}$ $\frac{32}{81}$ $\frac{24}{81}$ $\frac{8}{81}$ $\frac{1}{81}$

Now,

$\text{ Mean } , E\left( X \right) = 0 \times \frac{16}{81} + 1 \times \frac{32}{81} + 2 \times \frac{24}{81} + 3 \times \frac{8}{81} + 4 \times \frac{1}{81}$
$= \frac{32 + 48 + 24 + 4}{81}$
$= \frac{108}{81}$
$= \frac{4}{3}$
$\text{ Note: We can also calculate the mean of the binomial distribution by }$
$\text{ Mean } , E\left( X \right) = \text{ np } = 4 \times \frac{1}{3} = \frac{4}{3}$

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Solution for Mathematics for Class 12 (Set of 2 Volume) (2018 (Latest))
Chapter 33: Binomial Distribution
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### 统计代写|R代写project|Virus Protein ODE/PDE Models

R是一个用于统计计算和图形的自由软件环境。

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|R代写project|ODE/PDE model for a single virus protein

The ODE/PDE variables are listed in Table 1.1.
Table 1.1 ODE/PDE model variables
$V_{1}(x, t)$ virus protein concentration in the cell membrane
$C_{1}(t)$ virus protein concentration in the cell interior
$x$ position within the membrane
time
The movement of the virus protein in the cell membrane is modeled with the diffusion equation
$$\frac{\partial V_{1}(x, t)}{\partial t}=D_{V 1} \frac{\partial^{2} V_{1}(x, t)}{\partial x^{2}}$$
where $D_{V 1}$ is a diffusivity.
Eq. (1.1-2) is second order in $x$ and therefore requires two boundary conditions (BCs).
$$D_{V 1} \frac{\partial V_{1}\left(x=x_{i t}, t\right)}{\partial x}=k_{1 u}\left(V_{1 s}(t)-V_{1}\left(x=x_{u}, t\right)\right)$$
BC (1.1-2) equates the rate of diffusion of the protein at the membrane outer boundary $x=x_{u},-D_{V 1} \frac{\partial V_{1}\left(x=x_{u}, t\right)}{\partial x}$ (Fick’s first law), to the rate of mass transfer at the outer boundary, $k_{1 u}\left(V_{1 s}(t)-V_{1}\left(x=x_{l}, t\right)\right.$ ), where $k_{1 u}$ is a mass transfer coefficient to be specified. $V_{1 s}(t)$ is the concentration of viral genetic material outside the cell.
$$-D_{V 1} \frac{\partial V_{1}\left(x=x_{l}, t\right)}{\partial x}=k_{1 l}\left(C_{1}(t)-V_{1}\left(x=x_{l}, t\right)\right)$$
$B C$ (1.1-3) equates the rate of diffusion of the protein at the membrane inner boundary $x=x_{l},-D_{V 1} \frac{\partial V_{1}\left(x=x_{l}, t\right)}{\partial x}$ (Fick’s first law), to the rate of mass transfer at the inner boundary, $k_{1 l}\left(C_{1}(t)-V_{1}\left(x=x_{l}, t\right)\right)$, where $k_{1 l}$ is a mass transfer coefficient to be specified.

Eq. (1.1-1) is first order in $t$ and requires one initial condition (IC)
$$V_{1}(x, t=0)=V_{10}(x)$$
where $V_{10}(x)$ is a function to be specified, and is generally taken as the zero function, that is, no virus protein initially in the cell membrane.

The concentration of the protein in the cell interior, $C_{1}(t)$, is modeled with an ODE
$$\frac{d C_{1}(t)}{d t}=-k_{1 l}\left(C_{1}(t)-V_{1}\left(x=x_{l}, t\right)\right)$$
that equates the temporal derivative $\frac{d C_{1}(t)}{d t}$ to the rate of transfer of the protein from the inner boundary of the membrane, $-k_{l l}\left(C_{1}(t)-V_{1}\left(x=x_{l}, t\right)\right)$, to the cell interior.
The IC for eq. (1.1-6) is
$$C_{1}(t=0)=C_{10}$$
where $C_{10}$ is a prescribed constant.
Eqs. (1.1) constitute the ODE/PDE model for a single virus protein. The model is next extended so that the cell interior produces a protein that then diffuses out of the cell.

## 统计代写|R代写project|ODE/PDE model for a second virus protein

The variables and parameters for the second protein are designated with subscript 2. The diffusion equation is a direct analog of eq. (1.1-1).
$$\frac{\partial V_{2}(x, t)}{\partial t}=D_{V 2} \frac{\partial^{2} V_{2}(x, t)}{\partial x^{2}}$$
The $\mathrm{BC}(1.2-2)$ at the outer boundary $x=x_{u}=1$ equates the rate of diffusion, $D_{V 2} \frac{\partial V_{2}\left(x=x_{l}, t\right)}{\partial x}$, to the mass transfer rate in response to an ambient protein concentration $V_{2 a}, k_{2 u}\left(V_{2 a}-V_{2}\left(x=x_{u}, t\right)\right)$, where $k_{2 u}$ is a mass transfer coefficient for the second protein.
$$D_{V 2} \frac{\partial V_{2}\left(x=x_{u}, t\right)}{\partial x}=k_{2 u}\left(V_{2 a}-V_{2}\left(x=x_{u}, t\right)\right)$$

$B C(1.2-3)$ equates the rate of diffusion of the protein at the membrane inner boundary $x=x_{l}, D_{V 2} \frac{\partial V_{2}\left(x=x_{l}, t\right)}{\partial x}$, to the rate of mass transfer at the inner boundary, $-k_{2 l}\left(C_{2}(t)-V_{2}\left(x \stackrel{\partial x}{=} x_{l}, t\right)\right)$, where $k_{2 l}$ is a mass transfer coefficient for the second protein.
$$D_{V 2} \frac{\partial V_{2}\left(x=x_{l}, t\right)}{\partial x}=-k_{2 l}\left(C_{2}(t)-V_{2}\left(x=x_{l}, t\right)\right)$$
The initial condition for $V_{2}(x, t)$ is
$$V_{2}(x, t=0)=V_{20}(x)$$
where $V_{20}(x)$ is a function to be specified (usually taken as the zero function).
The concentration of the second protein in the cell interior is modeled with an ODE
$$\frac{d C_{2}(t)}{d t}=-k_{2 l}\left(C_{2}(t)-V_{2}\left(x=x_{l}, t\right)\right)+k_{r 2} C_{1}^{n_{2}}$$
that equates the temporal derivative $\frac{d C_{2}(t)}{d t}$ to the sum of (1) the rate of transfer of the second protein from the cell interior to the membrane inner boundary, $-k_{2 l}\left(C_{2}(t)-V_{2}\left(x=x_{l}, t\right)\right)$, and (2) the rate of production of the second protein from the first protein, $+k_{r 2} C_{1}^{n_{2}}$. For (2), an $n_{2}^{t h}$ order reaction is assumed, but this rate can be modified to reflect another rate of production of the second protein.
The IC for eq. $(1.2-5)$ is
$$C_{2}(t=0)=C_{20}$$
where $C_{10}$ is a prescribed constant.
Eqs. (1.1), (1.2) constitute the ODE/PDE models implemented in the R routines discussed in Chapter $2 .$

## 统计代写|R代写project|R routines for the ODE/PDE models

• $21 t$ output points as the first dimension of the solution matrix out from lsodes as programmed in the main program of Listing $2.1$ (with nout=21).
• The solution matrix out returned by lsodes has 23 elements as a second dimension. The first element is the value of $t$. Elements 2 to 23 are $V_{1}(x, t), C_{1}(t)$ from eqs. (1.1) (for each of the 21 output points).
16
2 Implementation of the ODE/PDE Models
• The solution is displayed for $t=0,240 \star 4 / 20=48, \ldots, 240$ as programmed in Listing $2.1$ (every fourth value of $t$ is displayed as explained previously).
• ICs (1.1-4, 1.1-6) are confirmed ( $t=0)$.
• $V_{1}\left(x=x_{u}=1, t\right)$ tracks $V_{1}(s)$ according to $\mathrm{BC}$ (1.1-2).
• $C_{1}(t)$ tracks the solution $V_{1}\left(x=x_{l}=0, t\right)$ as defined by $\mathrm{BC}(1.1-3)$ (at $x=0$ ) and the RHS of eq. (1.1-5).
• The computational effort as indicated by ncall $=235$ is modest so that sodes computed the solution to eqs. (1.1) efficiently.
The graphical output is in Figs. 2.1.

## 统计代写|R代写project|ODE/PDE model for a single virus protein

ODE/PDE 变量列于表 1.1。

C1(吨)细胞内部的病毒蛋白浓度
X在膜

∂在1(X,吨)∂吨=D在1∂2在1(X,吨)∂X2

D在1∂在1(X=X一世吨,吨)∂X=ķ1在(在1s(吨)−在1(X=X在,吨))
BC (1.1-2) 等于蛋白质在膜外边界的扩散速率X=X在,−D在1∂在1(X=X在,吨)∂X（菲克第一定律），外边界的质量传递速率，ķ1在(在1s(吨)−在1(X=Xl,吨)）， 在哪里ķ1在是要指定的传质系数。在1s(吨)是细胞外病毒遗传物质的浓度。
−D在1∂在1(X=Xl,吨)∂X=ķ1l(C1(吨)−在1(X=Xl,吨))

dC1(吨)d吨=−ķ1l(C1(吨)−在1(X=Xl,吨))

C1(吨=0)=C10

## 统计代写|R代写project|ODE/PDE model for a second virus protein

∂在2(X,吨)∂吨=D在2∂2在2(X,吨)∂X2

D在2∂在2(X=X在,吨)∂X=ķ2在(在2一种−在2(X=X在,吨))

D在2∂在2(X=Xl,吨)∂X=−ķ2l(C2(吨)−在2(X=Xl,吨))

dC2(吨)d吨=−ķ2l(C2(吨)−在2(X=Xl,吨))+ķr2C1n2

C2(吨=0)=C20

## 统计代写|R代写project|R routines for the ODE/PDE models

• 21吨输出点作为解矩阵的第一维从 lsodes 输出，如清单的主程序中编程的那样2.1（与nout = 21）。
• lsodes 返回的解矩阵 out 有 23 个元素作为第二维。第一个元素是值吨. 元素 2 到 23 是在1(X,吨),C1(吨)从等式。(1.1)（对于 21 个输出点中的每一个）。
16
2 ODE/PDE 模型的实现
• 解决方案显示为吨=0,240⋆4/20=48,…,240如清单中所编程2.1（每四个值吨显示如前所述）。
• IC（1.1-4、1.1-6）已确认（吨=0).
• 在1(X=X在=1,吨)轨道在1(s)根据乙C (1.1-2).
• C1(吨)跟踪解决方案在1(X=Xl=0,吨)定义为乙C(1.1−3)（在X=0) 和等式的 RHS。(1.1-5)。
• ncall 表示的计算量=235是适度的，因此 sodes 计算出 eqs 的解。(1.1) 有效。
图形输出如图。2.1。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。


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# Quotes for essay titles

Quotation marks in titles a couple of generations ago, it was the custom to enclose all titles in quotation marks: titles of books, titles of poems, titles of films, titles of newspapers, and so on this usage, however, has now largely disappeared, and the modern custom is to write most titles in italics but in academic circles,. As for enclosing titles in quotation marks or italicizing them, you can get pretty far by following the “big/heavy equals italics” (like books) and “small/light equals quotes” (like poems) generalizations, but associated press style doesn't italicize nothin' and chicago style has layers of specificity and if-then statements fun. Do you italicize book titles underline them put book titles in quotes here's the real scoop on how to handle styling book titles in your writing. Rules for using italics and quotation marks in titles there are several different writing style guides: the modern language association (mla) is the style generally used in arts and humanities papers the american psychological association (apa) is used for social sciences the associated press stylebook ( ap) is. Essay quotes from brainyquote, an extensive collection of quotations by famous authors, celebrities, and newsmakers.

In handwriting, you should underline where you would normally italicize on a computer (eg for a book title) you would use quotes instead when mentioning a poem, short story, or other work that would be placed within quotation marks when typing. Titles of works the titles of certain works are indicated with quotation marks, others with italics, and yet others with regular type the style presented here is consistent with the chicago manual of style (16th ed) and the mla handbook for writers of research papers (7th ed), and is appropriate for most academic and. In a serious essay or academic paper i would put any book title, whether fiction or non-fiction, or the title of any journal article, in italics the name of the author should not be italicised publishing date and details usually go in brackets so in the body of the text you could say something like: in his study of. Prior to computers, people were taught to underline titles of books and plays and to surround chapters, articles, songs, and other shorter works in quotation marks however, here is what the a familiar event to the reader i'm referring to a charity event, and writing about something that happened there in a personal essay.

Titles for quote marks/italics/nothing chicago manual of style 16th edition rule albums, italics, 8192 art, italics, 8193 blog entries, quotes, 8187 blogs, italics, 8187 books, italics, 8166 conferences, nothing, 869 departments, nothing, 8175 drawings, italics, 8193 essays, quotes, 8175 exhibitions, italics. Do not capitalize articles, prepositions, or conjunctions when they fall in the middle of a title separate a subtitle with a colon and a space italicize titles if the source is self-contained and independent titles of books, periodicals, databases , and web sites are italicized place titles in quotation marks if the.

Adding quotes and underlines is always wrong: a great book is “the grapes of wrath” no a great book is $\underline{\text{the grapes of wrath}}$ no a great book is the grapes of wrath yes the rule, from the chicago manual of style, is. Formatting titles gives some writers a headache should the title of songs, stories, movies, books, screenplays, etc be in italics or quotes when you're trying to remember if you're supposed to use underlining or italics or quotation marks for titles, here are a few simple rules from writer's relief remember.

Punctuating titles: when to use italics, underlining, and quotation marks 1) short works and parts of long works are usually in quotation marks title of a novel ex: young goodman brown ex: the scarlet letter 3) title of an essay title of a collection or anthology of essays ex: the fiction of langston hughes. We've been writing quite a bit lately about ebooks and their potential as a distribution mechanism (and maybe even revenue driver) for journalism whether it's foreign policy, the new york times, propublica, or n+1, lots of news organizations are interested in the medium as a place for work tha.

Quotes for essay titles
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text: Frequency divided by the speed of light is Î½/c, which from the above equation is 1/Î». It states the mathematical relationship between the speed (v) of a wave and its wavelength (λ) and frequency (f). Use the equation: ν = 1 /L = f / v. To calculate the spatial wavenumber (ν), noting that L (lambda) means wavelength, f means frequency and v means the speed of the wave. The good news is that there is a simple formula for the wavenumber, and you need only very basic information about the wave to calculate it. The wavenumber of a particle, given by k = {2\pi p\over h} = {p\over\hbar}. ~ But avoid … Asking for help, clarification, or responding to other answers. w = V / f. Where w is the wavelength (m) where k0 is the free-space wavenumber, as above. screencasts, that are short lessons on physics concepts. Physics Formulas - Get list of all Physics formulas here at Vedantu.com prepared by subject expert teachers. This number is called the wave number of the spectrum line. It includes all formulas from the chapters on light and electricity. Rearranging the equation yields a new equation of the form: Speed = Wavelength • Frequency. It still needs to be converted from J to kJ for the final answer: energy = (2.394 x 10 5 J) (1 kJ / 1000 J) energy = 2.394 x 10 2 kJ or 239.4 kJ. Physics Formulas Physics Formulas . The wavenumber is obtained by dividing the true frequency by the speed of light. If I have an Energy spectrum in frequency can I always obtain an Energy spectrum in wavenumber using the formula that I wrote in the first post? See wavepacket for discussion of the case when these quantities are not constant. The units will be the inverse of wavenumber, for example, if your units for wavenumber is nm-1, the units for wavelength is nm. Angular momentum, in physics, is a property that characterizes the rotatory inertia of an object in motion about the axis that may or may not pass through the specified object. Physics Formulas Wavelength Formula. Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. If the attenuation constant is positive, then the wave amplitude decreases as the wave propagates in the x direction. where p is the momentum, h is Planck's constant, and \hbar is h-bar. {\displaystyle \mu _{r}} David Morin, morin@physics.harvard.edu The waves we’ve looked at so far in this book have been \dispersionless" waves, that is, waves whose speed is independent of! Click here to go to our page on characteristic impedance. The physics formulas for class 10 are based on NCERT science textbook. ε Angular frequency ω = rad/s = x10^ rad/s. In the physical sciences, the wavenumber (also wave number) is the spatial frequency of a wave, either in cycles per unit distance or radians per unit distance. {\displaystyle {\tilde {\nu }}} The above equation is known as the wave equation. For quantum mechanical waves, wavenumber multiplied by Planck’s constant is the canonical momentum. Wave numbers are usually measured in units of reciprocal metres (1/m, or m-1) and reciprocal centimetres (1/cm, or cm-1). The frequency of a wave is denoted by v and is equal to c (speed of light)/ƛ(wavelength). Wave number, a unit of frequency in atomic, molecular, and nuclear spectroscopy equal to the true frequency divided by the speed of light and thus equal to the number of waves in a unit distance. Also Download the Chapter wise Important Maths Formulas and Equations to Solve the Problems Easily and Score More Marks in Your CBSE Board Exams. D. McMahon (2008). It’s All Numbers ... Also, it's becoming more common to have students solve physics problem by creating and coding their own programs— and I think that is a good thing. Depending on the type of wave, wavelength can be measured in meters, centimeters, or nanometers (1 m = 10 9 nm). n. The number of waves per unit distance in a series of waves of a given wavelength; the reciprocal of the wavelength. The wavelength is the property of a wave which is the distance between identical points between two successive waves. Wavenumber, as used in spectroscopy and most chemistry fields, is defined as the number of wavelengths per unit distance, typically centimeters (cm ): The dependence of the wavenumber on the frequency (or more commonly the frequency on the wavenumber) is known as a dispersion relation. ~ Sources I’ve already found explain why it was chosen rather than energy of waves, but the conversion from wavelength to wavenumber is never explained. , where νs is a frequency in hertz. The units will be the inverse of wavenumber, for example, if your units for wavenumber is nm-1, the units for wavelength is nm. Our editors will review what youâve submitted and determine whether to revise the article. , relative permeability Cheers Vivek . Blackbody Calculator. In the physical sciences, the wavenumber (also wave number or repetency[1]) is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. In a traveling plane wave, wavenumber is the spatial equivalent of frequency. Modern Physics: Quantum Physics: Quantum Mechanics: Early Quantum Mechanics: de Broglie Wavenumber : The wavenumber of a particle, given by Wavelength, distance between corresponding points of two consecutive waves. Also, surfers like to ride on them. John Murray. Wavenumber is also used to define the group velocity. {\displaystyle {\tilde {\nu }}} The problem seems to be that if the peak is at wavelength X, and a wave of wavelength X has a wavenumber of 1/X by definition, then Planck's law predicts the same exact physical thing has a different value depending on if it is measured one way or the other. The frequency , represented by the Greek letter nu ( ν ), is the number of waves that pass a certain point in a specified amount of time. ... converting from wavelength to wavenumber is a very simple equation. Omissions? P.M. Whelan, M.J. Hodgeson (1978). Updates? First one is the number of wavelengths per 2π meters. In theoretical physics: It is the number of radians present in the unit distance. If the wave has amplitude A = m. and initial phase φ = degrees = radians. We provide free learning opportunities for everyone around the world. [8], A wavenumber in inverse cm can be converted to a frequency in GHz by multiplying by 29.9792458 (the speed of light in centimeters per nanosecond). f = 4.2 ×1014 Hz. See also: de Broglie Wavelength for one mole, the energy is 2.394 x 10 5 J. These three remaining wavenumber gradient equations are derived similarly, by taking ∂/∂ y of Eq. Formulas or Code? In a sense, the wave number is like a spatial analogue of frequency. Using the formula above, we find the wave number to be .10 m^-1. In spectroscopy, "wavenumber" Wavenumber equation is mathematically expressed as the number of the complete cycle of a wave over its wavelength, given by – $$k=\frac{2\pi }{\lambda }$$ Where, k is the wavenumber; is the wavelength of the wave; Measure using rad/m ; Wavenumber Definition. In physics a wave can be thought of as a disturbance or oscillation that travels through space-time, accompanied by a transfer of energy. = 3×108 / 700×10-9. In multidimensional systems, the wavenumber is the magnitude of the wave vector. The imaginary part of the wavenumber expresses attenuation per unit distance and is useful in the study of exponentially decaying evanescent fields. For this example we will say the wave length is 10m. Frequently used equations in physics. Wavenumber can be used to specify quantities other than spatial frequency. Corrections? s Wavenumber, as used in spectroscopy and most chemistry fields, is defined as the number of wavelengths per unit distance, typically centimeters (cmâ1): where Î» is the wavelength. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wave number, a unit of frequency in atomic, molecular, and nuclear spectroscopy equal to the true frequency divided by the speed of light and thus equal to the number of waves in a unit distance. IGCSE Physics equation. A spectroscopic wavenumber can be converted into energy per photon E by Planck's relation: It can also be converted into wavelength of light: where n is the refractive index of the medium. Determine its frequency. We derived it here for a transverse wave, but it is equally important when investigating longitudinal waves. The former quantity can be called angular wavenumber (in analogy with angular frequency) to avoid confusion, but that term is not very common. Wavelength Formula Questions: 1) The speed of sound is about 340 m/s.Find the wavelength of a sound wave that has a frequency of 20.0 cycles per second (the low end of human hearing).. Answer: The wave velocity v = 340 m/s, and the frequency f = 20.0 cycles/s.The wavelength ? List of Basic Physics Formulas. However, spectroscopic data kept being tabulated in terms of spectroscopic wavenumber rather than frequency or energy. Next, determine the wave number. c 2π radians = 360 degrees. In multidimensional systems, the wavenumber is the magnitude of the wave vector. The RydbergâRitz combination principle of 1908 was also formulated in terms of wavenumbers. First, determine the wavelength. [1] It equals the spatial frequency. then at x = m and time t = s the wave can be described by = There are two main wavenumber measurements. Feb 29, 2020 #16 What is the exact difference between wavenumber and propagation constant in an electromagnetic wave propagating in a medium such as a transmission line, cause i am a bit confused. and refraction index n as:[5]. Provide details and share your research! Mostly algebra based, some trig, some calculus, some fancy calculus. It can be envisaged as the number of waves that exist over a specified distance (analogous to frequency being the number of cycles or radians per unit time). It consists of a large number of equally spaced parallel slits.” Its working principle is based on the phenomenon of diffraction.The space between lines acts as slits and these slits diffract the light waves thereby producing a large number of beams that interfere in such a way to produce spectra. Mc Graw Hill (USA). The wavenumber of a particle, given by k = {2\pi p\over h} = {p\over\hbar}. [4] The angular wavenumber may be expressed in radians per meter (rad⋅mâ1), or as above, since the radian is dimensionless. ... {\lambda}[/tex] as the "wavenumber", whereas in physics, the fraction $$\frac{2\pi}{\lambda}$$ is called the wavenumber. Thanks for contributing an answer to Physics Stack Exchange! The same exact wave has a fixed wavenumber no matter what Planck's law says! Physics Formulas. A few years later spectral lines could be understood in quantum theory as differences between energy levels, energy being proportional to wavenumber, or frequency. Essentially, the equations are the same except the angular wavenumber uses 2π as the numerator, because this is the number of radians in a whole circle (equivalent to 360°). In physics, the wave number is a quantity that is inversely related to wavelength, having inverse meters as its SI unit (/m or m-1). Convert wave number to wavelength formula. , so often that such spatial frequencies are stated by some authors "in wavenumbers",[7] incorrectly transferring the name of the quantity to the CGS unit cmâ1 itself. ISBN 978-0-470-03294-7. By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. Click here to go to our main page on transmission lines. Login to create quizzes If you are not registered user register here to login . For example, the spectroscopic wavenumbers of the emission spectrum of atomic hydrogen are given by the Rydberg formula: where R is the Rydberg constant, and ni and nf are the principal quantum numbers of the initial and final levels respectively (ni is greater than nf for emission). But that cannot be right. Conventionally, inverse centimeter (cmâ1) units are used for Units of Wavenumbers (click on equations to view enlarged) Yet a third spectral unit, commonly used in spectroscopy, is wavenumber, the number of waves per cm: σ = ν /100c cm-1.Converting (1) to these units gives … Essential Principles of Physics (2nd ed.). Wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium —that is, with little or no associated mass transport. The term wave number refers to the number of complete wave cycles of an electromagnetic field that exist in one meter (1 m) of linear space.Wave number is expressed in reciprocal meters (m-1).. New for December 2007! Unfortunately, there are different definitions of the wavenumber of light in the literature. ... 10^-9 metres in wavelength does not necessarily contain the same energy as an interval in wave number of 1/10^-9 = 10^9 m-1. A typical spectral line in the visible region of the spectrum has a wavelength of 5.8 Ã 10-5 cm; this wavelength corresponds to a frequency (Î½) of 5.17 Ã 1014 Hz (hertz equals one cycle per second) obtained from the equation. ISBN 978-0-07-154382-8. In theoretical physics, a wave number defined as the number of radians per unit distance, sometimes called "angular wavenumber", is more often used:[2], When wavenumber is represented by the symbol ν, a frequency is still being represented, albeit indirectly. But, the thing that people do not know is that there is a formula which we use to measure wavelength. Another definition for wavelength can be given using the phase. Wavenumber is another very important property of a wave. The Earth’s rotation and revolution are the best real-life examples of angular momentum. The circumference of a circle = π times its diameter. v = f • λ ν Everybody has seen waves whether on TV or in real life. … The following formula can be used to calculate a wavelength given the frequency and speed (velocity). Wavelength, phase velocity, and skin depth have simple relationships to the components of the wavenumber: Here we assume that the wave is regular in the sense that the different quantities describing the wave such as the wavelength, frequency and thus the wavenumber are constants. They’re posted here for you to enjoy … and it’s my conviction that a big part of enjoyment is learning … really learning, so you have a gut feel for what you’ve learned. Manchester Physics Series, John Wiley & Sons. In a sense, the wave number is like a spatial analogue of frequency. For the special case of an electromagnetic wave in a vacuum, in which the wave propagates at the speed of light, k is given by: where E is the energy of the wave, Ä§ is the reduced Planck constant, and c is the speed of light in a vacuum. Please be sure to answer the question. Wave Motion in One Dimension • The shape of the disturbance at one instance in time is called the wave profile • If the wave moves with constant velocity, then ˝, =˚ − – Positive, the wave moves to the right – Negative, the wave moves to the left – Sometimes we will write ˝, =˚ ± … Physics is one of the most feared subjects in class 12 because of its complex theories and physics formulas list. The space of wave vectors is called reciprocal space. Typically, wave number is taken to be 2π times the number of wavelengths per unit of distance, which is the number of radians for each unit of distance as well. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Because of this, wavenumbers are used as a unit of energy in spectroscopy. Now when the radius equals 1, C = 2π. G. Woan (2010). Below are two relations from Wikipedia, which explain how it can be used in equations, but in all of these cases, $\lambda$ seems to be a choice that’s easier to work with. In some areas of physics (such as geophysics), you might encounter an “angular wavenumber.” This is very similar to wavenumber, except it’s multiplied by 2π, so it describes rotations or oscillations. Basic physics.html math formulas and equations are listed here. The frequency, symbolized by the Greek letter nu ( ν ), of any wave equals the speed of light, c, divided by the wavelength λ: thus ν = c / λ. = 10,000,000 / y nm wavenumber rather than frequency or energy /ƛ wavelength. But it is sometimes called the wave that are short lessons on physics concepts people not. 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text:
## Pool

• 문제 ID
• 시간 제한
• 메모리 제한
• 제출 횟수
• 정답 횟수 (비율)
• 출처
• 분류

#### 문제

Billiards, also commonly known as “pool,” is a popular game in North America. The game is played on a rectangular table with six pockets—one at each corner and one in the middle of each of the two longer sides of the table. The object of the game is to strike a cue ball so that it collides with other balls, knocking them into the pockets.

The surface of our pool table measures 108” by 54”, and to facilitate computation, we place it on the Cartesian plane with its southwest corner situated at (0, 0), and its northeast corner at (108, 54). Therefore, the centers of the 6 pockets, numbered 1 through 6, will have coordinates of (0, 0), (54, 0), (108, 0), (0, 54), (54, 54), and (108, 54), respectively (see Figure 2). The billiard balls are spherical and measure 2” in diameter.

Figure 2: Diagram of pool table.

Given the location of the cue ball, a target ball, and a number of other balls on the table, your task for this problem is to write a program to determine whether or not you can successfully make a particular pool shot. The cue ball can be struck in any direction in a straight line. We consider collisions between the balls to be perfectly elastic, so that the target ball will always travel in a straight line, away from the point on its surface contacted by the cue ball (see Figure 3). (You may assume that the cue ball disappears immediately after making contact with the target ball.)

A shot is considered possible if the cue ball can be struck so that it collides directly with the target ball, in turn sending the target ball directly into a pocket. Neither ball should collide with any other balls, bounce off the edges of the table (cushions), nor should their centers cross the boundaries of the table. In other words, you are not to consider any bank shots, combination shots, spin shots, or any other trick shots in this problem. Note that the difference between the incoming angle of the cue ball and the outgoing angle of the target ball must be greater than 90◦. The target ball is considered to land in a pocket when its center
coincides with the center of that pocket.

Figure 3: Diagram of pool ball collision.

#### 입력

The input test file will contain multiple cases. The first line of each test case contains four real numbers, xc yc xt yt, where (xc, yc) is the location of the cue ball and (xt, yt) is the location of the target ball. The second line of the test case contains an integer n (where 0 ≤ n ≤ 14), the number of additional balls on the table, followed by n pairs of real numbers, x1 y1 . . . xn yn, where (xi, yi) is the location of the ith additional (possibly obstructing) ball. No two balls overlap, and all balls are strictly in the interior of the table; in particular, all provided coordinates obey 3 < x < 105 and 3 < y < 51.

Input is terminated with a single line containing only the number 0; do not process this line.

#### 출력

For each test case, output on a single line the number(s) of the pocket(s) into which the target ball can be shot, sorted in ascending numerical order. Separate pocket numbers with a single space. If there are no pockets for which a clear shot exists, output the words “no shot”.

#### 예제 입력

30 27 70 24
0
80 27 40 27
2 38 28 38 26
81.0 27.0 54.5 27.0
1 54.0 33.3
81.0 27.0 54.5 25.0
1 54.0 33.3
0

#### 예제 출력

3 6
no shot
1 4
1 2 4