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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Concept. Go to the latest version.

# Photoelectric Effect

## Light acts like both a particle and a wave.

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Photoelectric Effect

#### Are you a fan of science fiction?

In old science fiction stories (1950s), one of the space travel themes was the use of solar sails for propulsion. The idea was that the photon pressure from the sun would push the sail (like the wind sails) and move the space craft. What once was science fiction is now reality as solar sails are being developed and tested for modern space travel.

### Photoelectric Effect and the Particle Nature of Light

In 1905 Albert Einstein (1879-1955) proposed that light be described as quanta of energy that behave as particles. A photon is a particle of electromagnetic radiation that has zero mass and carries a quantum of energy. The energy of photons of light is quantized according to the  $E=hv$ equation. For many years light had been described using only wave concepts, and scientists trained in classical physics found this wave-particle duality of light to be a difficult idea to accept. A key experiment that was explained by Einstein using light’s particle nature was called the photoelectric effect.

The photoelectric effect is a phenomenon that occurs when light shined onto a metal surface causes the ejection of electrons from that metal. It was observed that only certain frequencies of light are able to cause the ejection of electrons. If the frequency of the incident light is too low (red light, for example), then no electrons were ejected even if the intensity of the light was very high or it was shone onto the surface for a long time. If the frequency of the light was higher (green light, for example), then electrons were able to be ejected from the metal surface even if the intensity of the light was very low or it was shone for only a short time. This minimum frequency needed to cause electron ejection is referred to as the threshold frequency .

Classical physics was unable to explain the photoelectric effect. If classical physics applied to this situation, the electron in the metal could eventually collect enough energy to be ejected from the surface even if the incoming light was of low frequency. Einstein used the particle theory of light to explain the photoelectric effect as shown in the Figure below .

Low frequency light (red) is unable to cause ejection of electrons from the metal surface. At or above the threshold frequency (green) electrons are ejected. Even higher frequency incoming light (blue) causes ejection of the same number of electrons but with greater speed.

Consider the  $E=hv$ equation. The  $E$ is the minimum energy that is required in order for the metal’s electron to be ejected. If the incoming light’s frequency, $v$ , is below the threshold frequency, there will never be enough energy to cause electron to be ejected. If the frequency is equal to or higher than the threshold frequency, electrons will be ejected. As the frequency increases beyond the threshold, the ejected electrons simply move faster. An increase in the intensity of incoming light that is above the threshold frequency causes the number of electrons that are ejected to increase, but they do not travel any faster. The photoelectric effect is applied in devices called photoelectric cells , which are commonly found in everyday items such as a calculator which uses the energy of light to generate electricity.

Photoelectric cells convert light energy into electrical energy which powers this calculator.

#### Summary

• Light has properties of both a wave and a particle.
• The photoelectric effect is produced by light striking a metal and dislodging electrons form the surface of the metal.

#### Practice

Use the link below to answer the following questions:

1. What is the emitting material in this photocell?
2. What causes photoelectrons to be emitted?
3. Does the current increase or decrease with an increase in light intensity?
4. What happens if light does not strike the cathode?

#### Review

1. What are the properties of a photon?
2. What does the photoelectric effect show about the properties of light?
3. How does the frequency of light affect the release of photons?


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# New version of Anki doesn't seem to support Mathjax macros definitions

I have recently upgrade Anki to 2.1.43, and latex rendered with mathjax doesn’t work anymore with macros.
Previously, if I included

<script>
MathJax.Hub.Config({
TeX: {
Macros: {
d: "\\mathrm{d}"
}
}
})
</script>

in the template, I could use \d in a mathjax source, and it would show a right d.

After the update, it just shows \d in red, as if it was an undefined command…

Linked from the change notes: Upgrade to Mathjax 3 (ankitects/help-wanted#19) by hgiesel · Pull Request #809 · ankitects/anki · GitHub

Ok, so MathJax has been upgraded to Mathjax 3, and I have tried converting my configuration using this tool, but it didn’t really work out. With the new configuration, the cards simply don’t render. Here is the configuration in my template, after migrating to Mathjax 3.

<script data-name="MathJax Macros" data-version="v0.3">
window.MathJax = {
tex: {
macros: {
d: "\\mathrm d",
// ...
}
}
};
</script>

I couldn’t find any other informations to make it work in the PR discussion.

Maybe you need to tell MathJax to reload things, as described on Configuring and Loading MathJax — MathJax 3.1 documentation which is linked from the previous link.

Yes, I saw the link and I read the documentation but it didn’t change anything. I have tried adding MathJax.startup.getComponents(); at the end of the script (which is supposed to regenerate every object, taking into account the configuration), but it still doesn’t work.
However, I don’t think it’s an issue of Mathjax not taking into account configuration. Here is the minimal code that leads to this issue:

<script>
MathJax = {};
</script>

I am not a JavaScripte programmer, so I be wrong, but I think that I am overwriting MathJax, which seems to be a critical variable… (nonetheless this is the way it is supposed to be configured). To patch that, I tried instead of setting the configuration by setting attributes instead of overwriting the whole object:

<script>
MathJax.tex.macros = {
d: "\\mathrm d",
// ...
};
</script>

This allows the page to be rendered, but it stills doesn’t define macros (I get the same result as in the beginning).
Hence, I tried what was told in the documentation

<script>
MathJax.tex.macros.options = { /* ... */ };
</script>

which is supposed to work for certain fields. Oc, it didn’t work. So I tried the hard way, telling MathJax to reload every object

<script>
MathJax.tex.macros = { /* ... */ };
MathJax.startup.getComponents();
</script>

Didn’t work…

At this point, I just tried every possible combination of the aforementioned “solutions”.

This one worked…

<script>
MathJax.config.tex.macros = { /* ... */ };
MathJax.startup.getComponents();
</script>

that took some time and the answer was in the first post you sent me, I feel like an idiot. An idiot that took way too much time trying to read the doc and too few reading what an other user did. Hope it helps anyone else that has the same problem.

3 Likes

Any suggestions for changing the delimiters? I know it’s not recommended but it works well for my setup.

I tried a couple methods here but haven’t gotten it to work. Thanks!

format: plain


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How to get rid of the name overlay on this vector control?

I can draw the control shown below on a panel using:

col = self.layout.column(align=True)
col.prop(bpy.context.scene.display, "light_direction")

But, the minute I try and add any sort of param, the whole control disappears. How can I remove the "Light Direction" label on the sphere?

Set the text argument.

As with all properties in blender, can set the text argument to display something other than the property name, (the default)

def draw(self, context):
scene = context.scene
col = self.layout.column(align=True)
col.prop(scene.display, "light_direction", text="")

Also note the usage of context as passed to the draw method, rather than using bpy.context

• Thanks so much! Is context.scene different from bpy.context.scene ? Aug 2 '21 at 8:11
• Thankyou. Re context: No. It is generally the case that the context passed is same. IMO simply good practice to use the argument provided, just in case there are changes in the future. Outside scope of this question, the possibility for future as I see it is , can pass an overridden context to an operator. Would be handy to know from within the operator what is "real" context and what is "fake". Aug 2 '21 at 8:20


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## A PhD thesis, LaTeX and me

For writing a PhD thesis in LaTeX, there are many package choices to make and a whole lot of little issues and pitfalls to handle. This starts with choosing the document class and … goes on and on. Here, I am collecting some points which took me more than a few moments, so I don’t have to go through the whole process again. And maybe my little list can help others to make their choices a little faster or avoid some of the mistakes I made.

# Choosing the document class

Even though there are alternatives, for me, the main contenders for the document class were KOMA-Script class ( scrbook  or scrreprt ) and memoir. In the end I chose KOMA-Script as it seems to be less bloated (i.e., memoir emulates many well established packages instead of using them directly) and there really seems to be no reason not to use it 😉 However, as you can see, I am not going into minute details as the articles mentioned above were enough to sway me.

As for the subcategory I use scrbook . It seems that from the book, report or article classes (koma-script or not) there seems to be an awful lot of features which fit a PhD thesis more than the report or article class (see this discussion). Just to name a few, such features include two-page layout or more flexible numbering with regard to front, main and back matter. Also this article explicitly states that scrbook  is for dissertations.

# How to handle subfigures

An interesting choice was which package to use for subfigures. I chose the subcaption package. As to why: the subfigure package seems to be old and is superseded by subfig. While the newest most flexible option seems to be a combination of the caption and the floatrow package. However, the subcaption package supposedly was a case study for the interfaces provided by the latter two packages and provides a more easy to use functionality. Indeed, the introduction of the subfigure package on CTAN specifically describes itself as “superseded […], […] and users may find the more recent subcaption package more satisfactory.” Discussions on StackExchange further support this notion. I think overall, it boils down to subfig supposedly having issues with hyperref, and subcaption being more simple than using a combination of caption and floatrow. Thus in the end subcaption seems to be the best choice.


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## McGurk Effect: Meaning And Application

Updated on 14-Dec-2022 13:00:58
The McGurk effect is a perceptual phenomenon that illustrates how hearing and vision interact when people perceive speech. For example, visual sensory data presented along with a sound can influence how individuals perceive it. The McGurk effect, which occurs when speech sounds are dubbed over a film of people pronouncing a distinct sound, is probably the most well-known example. A person experiences the McGurk effect when they believe that the movement of another person's lips does not correspond to what that person is saying. In other words, the perception of speech is an illusion that results from the combination of ... Read More

## Basic Concepts of Emf and Cell Notation

Updated on 14-Dec-2022 12:52:12
Introduction The Emf or electromotive force is a term used to depict the potential difference and is not a term related to force and it is measured in volt. It was invented by Alessandro Volta when he first invented the battery. And also he mentioned the contact of electrode and electrolyte is responsible for the emf not mentioned chemical reactions. In 1830 Michael Faraday ascertained that the chemical reactions taking place at the electrode and electrolyte interface are responsible for the emf. It is a characteristic of all the energy sources that have a circuit in them. The electromotive force ... Read More

## Base Meaning

Updated on 14-Dec-2022 12:39:58
Introduction There are different definitions for the base. But the one definition that is common in all three definitions is bases react with acid to form a neutral compound or salts and water. And is given by G.-F. Rouelle. It is the ion that is present in this compound that makes it a characteristic property. Arrhenius proposed that the presence of hydroxyl ion, $\mathrm{OH^{-}}$ leads to the characteristic property of a base. According to Bronsted Lowry's theory of bases, a base is a substance that can accept protons or Hydrogen ions, $\mathrm{H^{+}}$. And in lewis's base theory base is a ... Read More

## C++ Program to Copy All the Elements of One Array to Another Array

Updated on 14-Dec-2022 12:22:48
The array data structures are used to store homogeneous data in a consecutive memory location to access them in a sequential manner. Arrays are linear data structure so that the basic operations on array can be performed in linear time. In this article we will see how to copy elements from one array to another new array in C++. The new array will be of same type since array elements are homogeneous. After creating another array of same size, we simply copy the elements from first array to the second one by one. Let us see the algorithm and the ... Read More

## Haskell program to check whether the input number is a Prime number

Updated on 14-Dec-2022 12:21:57
This tutorial discusses writing a program to check whether the input number is a Prime number in the Haskell programming language. Haskell is a Declarative, Strongly Typed, and Functional programming Language. The computations in Haskell are mathematical functions. A prime number is a number having only 1 and the number itself as a factor. Note 1 is neither a prime nor a composite number because it has only one factor. For example, 3 is a prime number because it has only two factors 1 and the number 3 itself. In this tutorial we see, Program to check whether a ... Read More

## C++ Program to Sort the Elements of an Array in Ascending Order

Updated on 14-Dec-2022 12:21:17
To solve some problems effectively, it's important to arrange the data items in the correct sequence. One of the most popular arranging problems is the element sorting problem. This article will demonstrate how to arrange array members in C++ in ascending order (according to rising values). To arrange either numerical or non-numerical elements in a specific order, a wide variety of sorting algorithms are available in this field. Only two straightforward sorting techniques will be covered in this article. the selection sort and the bubble sort. Let's examine each one individually using appropriate techniques and C++ implementation code. Sort array ... Read More

## Haskell Program to Print the Multiplication Table in Triangular Form

Updated on 14-Dec-2022 12:20:00
This tutorial discusses writing a program to print the multiplication table in the triangular form in the Haskell programming language. For example, the multiplication table for the number 10 in the triangular format is 1 2 4 3 6 9 4 8 12 16 5 10 15 20 25 6 12 18 24 30 36 7 14 21 28 35 42 49 8 16 24 32 40 48 56 64 9 18 27 36 45 54 63 72 81 10 20 30 40 50 60 70 80 90 100 Algorithm Steps Take input or initialize an integer variable ... Read More

## C++ Program to Sort the Elements of an Array in Descending Order

Updated on 14-Dec-2022 12:18:13
Arranging data items in a proper form is an essential task while solving some problems in an efficient way. The element sorting problem is one of the most commonly discussed arranging problem. In this article we will see how to arrange the array elements in descending order (decreasing order of their values) in C++. There are many different sorting algorithms present in this domain to sort numeric or nonnumeric elements in a given order. In this article we will see only two simple methods of sorting. The bubble sort and the selection sort. Let us see them one by one ... Read More

## Barium Sulfate

Updated on 14-Dec-2022 12:15:20
Introduction The word barium is derived from the Greek word 'Barys' which means heavy. Barium Sulphate is an inorganic compound. It is alkaline earth metal. The compound is found on earth in the form of mineral barite also known as barytes. Barites are pale yellow, red, grey or green but barium sulphate is a white solid soft crystalline compound. It is a water-insoluble and odourless compound. It is formed by the reaction of barium carbonate and sulphuric acid. It is a non-toxic compound and used for many medicinal purposes like a root canal, x-rays and CT- scans. It is soluble ... Read More

## C++ Program to Convert Set of String to Array of String

Updated on 14-Dec-2022 12:15:06
Sets in C++ re containers that contain unique values of a specific type. Arrays or the array container in std C++ is a fixed-size container that contains elements of a specific size. Arrays are like vectors, but the main difference is that arrays are of a fixed size whereas vectors can be dynamic. The array container in C++ is a wrapper for the standard arrays that are available in both C and C++. There is a problem in this conversion though, std arrays do not have support for the common insertion methods that are available with the other containers. So, ... Read More


text:
# Math Help - Are my answers correct to these problems?

1. ## Are my answers correct to these problems?

Write an equation for the given line:

1. P(2,3), m=2
2. P(1,0), no slope
3. P (-1,2), m=-1/2
4. P(2,3), m = 0

1.y=2(x-2)+3
2. y=(x-1)
3. y= -1/2(x+1)+2
4. y=(x-2)+3

2. no slope or m=0, means the line is horizontal, having the same y-value for all x

so it doesnt depends of x

2 and 4 arent correct

3. Originally Posted by yitsongg
Write an equation for the given line:

1. P(2,3), m=2
2. P(1,0), no slope
3. P (-1,2), m=-1/2
4. P(2,3), m = 0

1.y=2(x-2)+3
2. y=(x-1)
3. y= -1/2(x+1)+2
4. y=(x-2)+3
The point-slope form of the equation to a line is given by $(y_2-y_1)=m(x_2-x_1)$.

The answer you have for the first problem is correct. In the second problem, however, your are asked to find the equation to a line that goes through the point $(1,0)$ that has no slope. No slope implies that the line is vertical. Thus, the equation is $x=1$.

Number 3 is correct.

Number 4 is wrong. If I use the formula that I have written at the beginning og this post, I have

$y-3=0(x-2)$
$y-3=0$
$y=3$

This is a horizontal line, as is expected given a slope of 0.

4. Thanks


text:
# Skoda cars

There were 16 passenger cars in the parking. It was the 10 blue and 10 Skoda cars. How many are blue Skoda cars in the parking?

Correct result:

n =  4

#### Solution:

$n=10+10-16=4$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Please write to us with your comment on the math problem or ask something. Thank you for helping each other - students, teachers, parents, and problem authors.

Showing 0 comments:

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text:
### Home > APCALC > Chapter 11 > Lesson 11.2.3 > Problem11-72

11-72.

Multiple Choice: No calculator! A company predicts its revenue to grow at a rate of $R^\prime = 5 -\frac { 1 } { 2 }(t − 3)^2$ where time $t$ is measured in months. In how many months should they see a maximum revenue?

1. $0$

1. $1$

1. $3$

1. $5$

1. $6$

Since the equation given represents a derivative, solve:

$0=5-\frac{1}{2}(t-3)^2$


text:
# Compactness of Set of Probability Measures

I'm currently studying Information Theory from Csiszar and Korner, "Information Theory: Coding theorems for Discrete Memoryless Sources". There are several questions pertaining to the set of all probability measures on the underlying space. I figured if I prove the following lemma, it will make things easier:

Let $\mathcal{X}$ be a finite set. Let $\mathcal{P}=\mathcal{P}(\mathcal{X})$ be the set of all probability measures that can be defined on $\mathcal{X}$. Under the total variation metric i.e. for any two measures $P,Q \in \mathcal{P}$ $$d_{TV}(P,Q) := \sum_{x \in \mathcal{X}}|P(x)-Q(x)|$$

Prove that $(\mathcal{P},d_{TV})$ is compact.

I'm actually looking for a simple proof of this. For instance, I wanted to use the result that a metric space is compact iff it is complete and totally bounded. Completeness was easy as I could just use the completeness of $\mathbb{R}$ in a pointwise fashion and show the pointwise limit is a measure. Finiteness of $\mathcal{X}$ helps greatly here.

So all that is left is Totally Boundedness. Unfortunately, while I did start by assuming an $\epsilon$-cover of $\mathcal{P}$, I could not find a way to pick finitely many that would cover the whole space. I'm sure that finiteness of $\mathcal{X}$ would help here as well.

The other idea, in fact the property I wanted to use, was to show sequential compactness since this is equivalent to compactness. But even here I'm not sure how to go about things.

Kindly help me in this regard in the form of subtle hints and tips. If I get it, I'll post my solution and credit the most helpful hint.

Update: It turns out I could use the fact that $\mathcal{P}$ is convex. Since any measure is a convex combination of degenerate distributions... Could I use this I wonder?

When $X$ is finite, you can treat $\mathcal P$ as a subset of $\Bbb R^X$. Total variation is the norm there, and all the norms over finite-dimensional spaces are equivalent, so the induced metric is equivalent to the Eucledian one. Embedded $\mathcal P$ is bounded and closed in Eucledian metric, hence compact.

• This is also a very good and simple proof... probably simpler than mine. Thanks. – Gautam Shenoy Nov 25 '15 at 4:38

Ok. I got it. The Lemma is true. Given a sequence of probability measures $P_n$, we consider $P_n(x)$ for every $x \in \mathcal{X}$. Using Weierstrass Bolzano theorem, noting that $P_n(x)$ is a bounded sequence of real numbers, there exists a convergent subsequence. We do this iteratively: WLOG, assume $\mathcal{X} = \{1,2,...,N\}$.

Step $1$: Find a convergent subsequence in $P_n(1)$. Let the limit be $\hat{P}(1)$. Note that the measures are applied to $1$ here.

Step $2$: For $i \geq 2$, repeat step $1$ but for the subsequence obtained in step $i-1$ and the measure applied to $i$. Stop after step $N$.

Thus we get a final subsequence, $P_{n_{k}}$ which converges in Total Variation to $\hat{P}$. To see that $\hat{P}$ is a prob measure, firstly $\hat{P}(x) \in[0,1]$ for every $x$. Also we have $$\sum_{x \in \mathcal{X}} \hat{P}(x) = \sum_{x \in \mathcal{X}} \lim_{k \to \infty} P_{n_k}(x) = \lim_{k \to \infty} \sum_{x \in \mathcal{X}} P_{n_k}(x)=1$$

Thus $\hat{P}$ is a valid prob. measure and hence $\hat{P} \in \mathcal{P}$.

Thus $\mathcal{P}$ is sequentially compact and hence is compact as it is a metric space. $\blacksquare$


text:
## Concluding remarks

Now that you’ve made it to this point in the book, we suspect that you know a thing or two about how to work with data in R! You’ve also gained a lot of knowledge about how to use simulation-based techniques for statistical inference and how these techniques help build intuition about traditional theory-based inferential methods like the $$t$$-test.

The hope is that you’ve come to appreciate the power of data in all respects, such as data wrangling, tidying datasets, data visualization, data modeling, and statistical inference. In our opinion, while each of these is important, data visualization may be the most important tool for a citizen or professional data scientist to have in their toolbox. If you can create truly beautiful graphics that display information in ways that the reader can clearly understand, you have great power to tell your tale with data. Let’s hope that these skills help you tell great stories with data into the future. Thanks for coming along this journey as we dove into modern data analysis using R and the tidyverse!


text:
# Template talk:Infobox official post

Jump to: navigation, search
WikiProject Politics (Rated Template-class)
This template is within the scope of WikiProject Politics, a collaborative effort to improve the coverage of politics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Template  This template does not require a rating on the project's quality scale.
WikiProject Organizations (Rated Template-class)
This template is within the scope of the WikiProject Organizations. If you would like to participate please visit the project page, where you can join the project and see a list of open tasks.
Template  This template does not require a rating on the project's quality scale.

## 1

Shouldn't it be something like "first" rather than "conceiver"?- J Logan t: 14:06, 10 December 2007 (UTC)

I have no idea.. I just searched for a seemingly appropriate word in english dictionaries. - $\mathrm{S}$. $\mathrm{Solberg}$ $\mathrm{J}$. 15:50, 10 December 2007 (UTC)
Okay, well if no one minds I'll change it to a word a tad more common for the post, inaugural. - J Logan t: 17:10, 10 December 2007 (UTC)

## 2

Could we add an option to add a link to the corisponding "list of..." article for the political post? I think that many people who come looking for an article about a title would want to know who its past holders were, and as of now they have to scroll down to the 'see also' section to find the link. --Arctic Gnome (talkcontribs) 01:40, 6 April 2008 (UTC)

## Agency name

Can we add a parameter for the agency led by this post? For example, United States Secretary of State is responsible for United States Department of State. It will be useful if this department can also appear in the Secretary's post template. And can we also add some blank parameters for some special items such as the presidential line of succession in United States Secretary of State? Thanks. --Neo-Jay (talk) 00:45, 11 April 2008 (UTC)

## Incumbent since field

I noticed the "incumbent since" field doesn't seem to work - the fields are filled but do not display anything. I've done some experimenting and suggest that the following adjustment be made:

! colspan=2 style="line-height: 1.1em; text-align:center"|Incumbent:{{{incumbent}}}{{#if:{{{incumbentsince|}}}{{!}}
 Took office: {{{incumbentsince}}}}}

Become:

! colspan=2 style="line-height: 1.1em; text-align:center"|Incumbent:{{{incumbent}}}{{#if:{{{incumbentsince|}}}|
 Took office: {{{incumbentsince}}}}}

I'd go ahead and make the changes myself but I'm still fairly new here and don't want to act unilaterally, but will be happy to do so if no one objects?--ZedderZulu (talk) 16:51, 20 August 2008 (UTC)

## Change "since" to "X—present"?

Example, for George W. Bush, it could be:

January 20, 2001—present

instead of

since January 20, 2001

Anyone agree?. — CRAZY(lN)SANE 02:24, 11 October 2008 (UTC)

## Box mess

Something has broken this box, I think around the Successor {{{isuccessor}}}}}, but I can't see how to revert the change. It's messing up a lot of articles. Timrollpickering (talk) 05:49, 5 November 2008 (UTC)

Can people please stop trying to insert a successor option just so they can add post-elects in individual positions? The additions are often mucking up the code and making other pages that don't need them ugly. Please a) discuss it here; and b) only have people with full code experience make the changes. Timrollpickering (talk) 14:26, 5 November 2008 (UTC)

## Inauguration link

Currently the text "Inaugural" in "Inaugural holder" is linked to Inauguration. But, that article is not really relevant, as it talks about a "a formal ceremony to mark the beginning of a leader's term of office". We should simply unlink this text. --Rob (talk) 03:43, 26 September 2009 (UTC)

• I agree, I have unlinked it. Snappy (talk) 20:41, 8 October 2009 (UTC)

## Multiple fields for position name

While the system of "post" "body" divided by the word "of" which is part of a template typically works fine some posts simply don't contain the word "of" might someone more versed in infobox syntax than I be able to possibly make one, singular "post" field that doesn't require the word "of" to be used. Another solution might be to make the "Nativename" field to, when filled, not require the word "of" if the other two fields are left blank.

Thanks to anyone who can fix this, Aricci526 (talk) 04:55, 5 November 2009 (UTC)

## Deputy

Can we have a field to change the name "deputy"? So we can have it as "Vice-President:" or whatever is applicable for the article?- J.Logan`t: 09:19, 11 December 2009 (UTC)

## extra 's

I noticed some extra ' marks appearing around John Robert's photograph on Chief Justice of the United States, before the word "Incumbent". I took a quick look at the template but couldn't find a cause. Any ideas? Cmprince (talk) 05:40, 26 December 2009 (UTC)

also see: Lord Mayor of London. Cmprince (talk) 05:47, 26 December 2009 (UTC)
This was caused when insigniacaption was blank. I have added an if statement to fix this. Snappy (talk) 14:24, 26 December 2009 (UTC)

## The "of" in the template

The "of" in the template is problematic because it appears even when body parameter is empty or not present. For examples, see United States Secretary of Transportation. I attempted to fix, but I couldn't.—Chris!c/t 00:32, 10 January 2010 (UTC)

Pending a fix, there is a simple workaround, which I have done for United States Secretary of Transportation. Snappy (talk) 20:42, 10 January 2010 (UTC)
I have added a fix for the 'of' issue. Snappy (talk) 22:07, 11 January 2010 (UTC)
Thanks.—Chris!c/t 22:57, 11 January 2010 (UTC)

## US linking

Would it be possible for someone to add the option to make a US-specific succession link? It currently links to Order of succession, but for a US Cabinet post it would be more appropriate linking to United States presidential line of succession, so could there be a way to specify this? Thanks, Reywas92Talk 18:32, 16 January 2010 (UTC)

Is 'Order of Succession' used much in non US infoboxes? If not, then 'Order of Succession' could be unlinked and 'United States presidential line of succession' could be linked to the number as in United States Secretary of State. Snappy (talk) 19:57, 17 January 2010 (UTC)
Good idea. Of the pages the template is used on, it seems only the US Cabinet ones would use that parameter. Reywas92Talk 02:08, 18 January 2010 (UTC)
I have unlinked succession. I'll leave it up to you (or others) to updates the other US Cabinet posts as per the United States Secretary of State. Rgds, Snappy (talk) 17:58, 18 January 2010 (UTC)

## Appointer parameter

This is an infobox meant to provide a summary of information about political posts—not the posts' holders. As a result, I find the behaviour of the appointer and nominator parameter puzzlings. Take Leader of the House of Commons. The words "appointed by" displayed in the infobox should be taken to refer to the office, not the current incumbent; that is especially true because the appointer information is in a separate section of the infobox from the incumbent. You would expect "the Leader of the House of Commons is appointed by" to be followed by "the Prime Minister", not "David Cameron (as Prime Minister). I propose therefore that the template be modified to replace the code for |data4 and |data5 such that only the values for appointerpost and nominatorpost are displayed, respectively. -Rrius (talk) 08:27, 20 July 2010 (UTC)

I coundn't agree more. RicJac (talk) 10:19, 15 May 2012 (UTC)

## Style?

What does this parameter mean? There is little explanation or guidance. That is, when the readers look at the infobox in the article, they see "Style" instead of "Manner of Address". Accordingly, the input would be "your highness", but other editing inputs such as "tyrant", "elected", "constitutional" might be appropriate. As the template itself does not have explanation, the input line should be removed or the "Style" label should be changed. (Someone with more technical expertise needs to fix/remove.) Thanks. --S. Rich (talk) 18:55, 7 February 2011 (UTC)19:00, 7 February 2011 (UTC)

I think the best solution would be to simply add a new parameter for "Manner of Address" below "Style". For someone entitled Excellency it would be; Style - His/Her Excellency, and Manner of Address - Your Excellency. RicJac (talk) 10:16, 15 May 2012 (UTC)

## Predecessor

How about also including in the box, a mention to the person who previously held the office before the incumbant by puttting in a predecesso option? The C of E. God Save The Queen! (talk) 21:22, 25 April 2011 (UTC)

## Footnotes

This infobox should have a footnotes parameter, similar to Template:Infobox countryJosh Gorand (talk) 20:46, 25 March 2012 (UTC)

## Caption

I've put a draft template together for the Mayor of Thames page, and the draft is in my Sandbox. The issue that makes this template unusable for my purpose is that in order to produce a photo caption, I appear to be forced to use the field 'incumbent', and it adds "incumbent" to the caption. In my case, however, I am not showing an incumbent. It won't always be appropriate to have incumbent, as this template also has the fields 'last' and 'abolished', and in those cases, there isn't an incumbent.

Can I suggest that we have an alternate parameter 'caption' that produces text underneath the photo in normal text (not bold). Any thoughts? Schwede66 02:00, 19 May 2012 (UTC)

Just drawing attention to the fact that this issue is unresolved. Schwede66 21:16, 29 January 2013 (UTC)

## Help

Can anybody help me to improve the Finnish version of the template fi:Malline:Poliittinen virka? I don't know how to use parameters or how it is done.--82.128.222.232 (talk) 16:53, 21 June 2012 (UTC)

## Immediate superior

Shouldn't there be a field for the immediate superior since the appointer is not always the office to whom an appointee directly reports. Examples of which are: U.S. ambassadors who while appointed by the President works directly for the U.S. Secretary of State and the service secretaries who works directly for the U.S. Secretary of Defense. RicJac (talk) 20:57, 2 November 2012 (UTC)

## nativename and department

Shouldn't the nativename field be in italics or smaller, I mean I understand when the nativename is in a non-Latin alphabet like the President of the People's Republic of China, its very clear what is the native name, but when it is in a Latin alphabet it looks like a run-off name like in Leader of the Opposition (Ireland) or Ambassador of Colombia to the United States. Some articles have chosen to manually add the italics to this field like in Chancellor of Germany (Federal Republic of Germany) or Prime Minister of Sweden, some made it both small and in italics like in President of Croatia or Speaker of the Lok Sabha, I think there should be changes to this field to regulate present and/or future articles.

Also, can anyone add a field to change the default "department" field. in the US departments are used, but in most of the rest of the world ministries are used, so saying that the United States Secretary of State's department is the United States Department of State works, but the Minister of Foreign Affairs of XXX's department cannot be the Ministry of Foreign Affairs, specially cause in many countries a department ranks below a ministry or is at the very least not the same thing. At least a ministry field should be added that reflects when this one is used.mijotoba (talk) 02:19, 16 December 2012 (UTC)

I think that organisation might be the most neutral term. RicJac (talk) 16:41, 8 June 2013 (UTC)

## Is this infobox appropriate for sinecure posts?

I am having a dispute over the use of this infobox in these two articles. They are posts in the British system that, while notionally a 'paid office of the Crown', don't exist except on paper. They're only used as a procedural device to allow MPs to resign, so while there is an 'appointer' (the Chancellor), as they have no roles or duties I cannot see how any of the other fields make any sense, and there is little to no point in what is then left. I've tried to note this in 'departemnet', but even that looks stupid. As this is probably the same for all such posts, I think it best to settle this at this level. I've notified the two people who disagree, The C of E and Keith D. Gruesome Foursome (talk) 18:40, 11 April 2013 (UTC)

I think that a number of the other parts could be filled if needs be. The allegence, date of formation and pay parts could be filled on those articles. 18:48, 11 April 2013 (UTC)
Nonsense. The purpose of infoboxes is not to confuse readers. Can you even source any of those for these articles? Each one of those would need long explanatory notes before they would make any sense in the context of a sinecure post, thereby defeating the whole purpose of an infobox. Gruesome Foursome (talk) 13:22, 12 April 2013 (UTC)
The most misleading part of this is the name and photo of the "incumbent": for all real purposes the significance of appointment ends on the day it's announced. The House of commons web site says "The Member retains the position until the Chancellor appoints another applicant or until the holder applies for release from it." so it is perhaps technically true that Huhne/Miliband are the incumbents as of today's date. But ideally there would be an official source that confirms that they are still in post. I suggest if the infobox is to stay, the "incumbent" and especially the photo should be removed. Sussexonian (talk) 19:03, 12 May 2013 (UTC)

## Superfluous parameters

I'm going to remove the parameters for "allegiance" and "commands held" as they make no sense in this template. RicJac (talk) 08:18, 9 June 2013 (UTC)

## Requested move

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: page moved. Plastikspork ―Œ(talk) 21:22, 12 October 2013 (UTC)

Template:Infobox political postTemplate:Infobox official post – Per template's documentation: "This template is intended to be used for a variety of official posts. Although the template name contains the words ”political post”, it is not meant only for ”political offices” per se, but can also be used for civil service, diplomatic, ecclesiastical, judicial, law enforcement, military and other posts of an official and permanent nature." eh bien mon prince (talk) 11:23, 1 October 2013 (UTC)

• support as per nom; it appears that the template name defines a scope that is too narrow. Schwede66 18:46, 1 October 2013 (UTC)
The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.


text:
Vector Algebra – JEE Main Previous Year Question with Solutions
JEE Main Previous Year Question of Math with Solutions are available at eSaral. Practicing JEE Main Previous Year Papers Questions of mathematics will help the JEE aspirants in realizing the question pattern as well as help in analyzing weak & strong areas. eSaral helps the students in clearing and understanding each topic in a better way. eSaral is providing complete chapter-wise notes of Class 11th and 12th both for all subjects. Besides this, eSaral also offers NCERT Solutions, Previous year questions for JEE Main and Advance, Practice questions, Test Series for JEE Main, JEE Advanced and NEET, Important questions of Physics, Chemistry, Math, and Biology and many more. Download eSaral app for free study material and video tutorials.
Q. If $\overrightarrow{\mathrm{u}}, \overrightarrow{\mathrm{v}}, \overrightarrow{\mathrm{w}}$ are non-coplanar vectors and $\mathrm{p}, \mathrm{q}$ are real numbers, then the equality $[3 \vec{u} p \vec{v} p \vec{w}]-[p \vec{v} \vec{w} q \vec{u}]-[2 \vec{w} q \vec{v} q \vec{u}]=0$ holds for $:-$ (1) More than two but not all values of (p,q) (2) All values of (p, q) (3) Exactly one value of (p, q) (4) Exactly two values of (p, q) [AIEEE-2009]

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Sol. (3)

Q. Let $\vec{a}=\hat{j}-\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k} .$ Then the vector $\vec{b}$ satisfying $\vec{a} \times \vec{b}+\vec{c}=\overrightarrow{0}$ and $\vec{a} . \vec{b}=3$ is : (1) $-\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ (2) $2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ (3) $\hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ (4) $\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ [AIEEE-2010]

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Sol. (1) $(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})+\overrightarrow{\mathrm{c}}=0$ $(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})=-\overrightarrow{\mathrm{c}}$ $\Rightarrow \overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})=-\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}} \Rightarrow(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}) \overrightarrow{\mathrm{a}}-|\overrightarrow{\mathrm{a}}|^{2} \overrightarrow{\mathrm{b}}=-\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}$ $\Rightarrow 3(\overrightarrow{\mathrm{j}}-\overrightarrow{\mathrm{k}})-2 \overrightarrow{\mathrm{b}}=-(-2 \mathrm{i}-\mathrm{j}-\mathrm{k})$ $(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=-2 \mathrm{i}-\mathrm{j}-\mathrm{k})$ $\Rightarrow 2 \overrightarrow{\mathrm{b}}=(-2 \mathrm{i}+2 \mathrm{j}-4 \mathrm{k}) \quad \Rightarrow \overrightarrow{\mathrm{b}}=-\mathrm{i}+\mathrm{j}-2 \mathrm{k}$

Q. The vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ are not perpendicular and $\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{d}}$ are two vectors satisfying: $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}$ and $\overrightarrow{\mathrm{a} .} \overrightarrow{\mathrm{d}}=0 .$ Then the vector $\overrightarrow{\mathrm{d}}$ is equal to :- $(1) \overrightarrow{\mathrm{b}}+\left(\frac{\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}}{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}\right) \overrightarrow{\mathrm{c}}$ (2) $\overrightarrow{\mathrm{c}}-\left(\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}}{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}\right) \overrightarrow{\mathrm{b}}$ (3) $\overrightarrow{\mathrm{b}}-\left(\frac{\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}}{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}\right) \overrightarrow{\mathrm{c}}$ (4) $\overrightarrow{\mathrm{c}}+\left(\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}}{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}\right) \overrightarrow{\mathrm{b}}$ [AIEEE-2011]

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Sol. (2)

Q. If $\overrightarrow{\mathrm{a}}=\frac{1}{\sqrt{10}}(3 \hat{\mathrm{i}}+\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=\frac{1}{7}(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}),$ then the value of $(2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}) \cdot[(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times(\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}})]$ is 1: (1) 5                    (2) 3                   (3) – 5                   (4) – 3 [AIEEE-2011]

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Sol. (3) $=-4 a^{2}-b^{2}+4 \vec{a} \cdot \vec{b}=-5$

Q. Let $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors which are pairwise non-collinear. If $\vec{a}+3 \vec{b}$ is collinear with $\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{b}}+2 \overrightarrow{\mathrm{c}}$ is colliner with $\overrightarrow{\mathrm{a}},$ then $\overrightarrow{\mathrm{a}}+3 \overrightarrow{\mathrm{b}}+6 \overrightarrow{\mathrm{c}}$ is: (1) $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}}$ ( 2)$\quad \vec{a}$ ( 3)$\quad \overrightarrow{\mathrm{c}}$ ( 4) $\overrightarrow{0}$ [AIEEE-2011]

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Sol. (4)

Q. Let $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ be two unit vectors. If the vectors $\overrightarrow{\mathbf{c}}=\hat{\mathbf{a}}+2 \hat{\mathbf{b}}$ and $\overrightarrow{\mathbf{d}}=5 \hat{\mathbf{a}}-4 \hat{\mathbf{b}}$ are perpendicular to each other, then the angle between $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ is : (1) $\frac{\pi}{4}$ ( 2)$\frac{\pi}{6}$ (3) $\frac{\pi}{2}$ ( 4)$\frac{\pi}{3}$ [AIEEE-2012]

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Sol. (4) $\overline{\mathrm{c}} \cdot \overline{\mathrm{d}}=0 \quad \Rightarrow(\hat{\mathrm{a}}+2 \hat{\mathrm{b}}) \cdot(5 \hat{\mathrm{a}}-4 \hat{\mathrm{b}})=0$ $\Rightarrow 5-8+6 \hat{a} \cdot \hat{b}=0$ $\Rightarrow \hat{\mathrm{a}} \cdot \hat{\mathrm{b}}=1 / 2 \Rightarrow \cos \theta=1 / 2$ $\Rightarrow \theta=\frac{\pi}{3}$

Q. Let ABCD be a parallelogram such that $\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{q}}, \overrightarrow{\mathrm{AD}}=\overrightarrow{\mathrm{p}}$ and $\angle \mathrm{BAD}$ be an acute angle. If $\overrightarrow{\mathrm{r}}$ is the vector that coincides with the altitude directed from the vertex $B$ to the side $A D,$ then $\overrightarrow{\mathrm{r}}$ is given by : [AIEEE-2012]

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Sol. (3)

Q. If the vectors $\overrightarrow{\mathrm{AB}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{AC}}=5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ are the sides of a triangle ABC, then the length of the median through A is : (1) $\sqrt{18}$ (2) $\sqrt{72}$ (3) $\sqrt{33}$ (4) $\sqrt{45}$ [JEE-MAINS 2013]

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Sol. (3) $\overrightarrow{\mathrm{AD}}=\frac{\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AC}}}{2}=4 \hat{\mathrm{j}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ $|\overrightarrow{\mathrm{AD}}|=\sqrt{33}$

Q. Let $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ be three vectors. A vectors. A vectors of the type $\overrightarrow{\mathrm{b}}+\lambda \overrightarrow{\mathrm{c}}$ for some scalar $\lambda,$ whose projection on $\overrightarrow{\mathrm{a}}$ is of magnitude $\sqrt{\frac{2}{3}},$ is : (1) $2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ (2) $2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ (3) $2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ (4) $2 \hat{i}+3 \hat{j}+3 \hat{k}$ [JEE-MAINS Online 2013]

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Sol. (1) $\frac{|(\overline{\mathrm{b}}+\lambda \overline{\mathrm{c}}) \cdot \overline{\mathrm{a}}|}{|\overline{\mathrm{a}}|}=\sqrt{\frac{2}{3}}$ $\left|[(1+\lambda) \hat{\mathrm{i}}+(2+\lambda) \hat{\mathrm{j}}+(-1-2 \lambda) \hat{\mathrm{k}}] \cdot \frac{2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{6}}\right|=\sqrt{\frac{2}{3}}$ $|(2+2 \lambda-2-\lambda-1-2 \lambda)|=2$ $|\lambda+1|=2$ $\lambda+1=\pm 2$ $\lambda=1,-3$ Vect $=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ or $-2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}}$

Q. Let $\vec{a}=2 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=\hat{i}+\hat{j} .$ If $\vec{c}$ is a vector such that $\vec{a} \bullet \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2 \sqrt{2}$ and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $30^{\circ},$ then $|(\vec{a} \times \vec{b}) \times \vec{c}|$ equals : ( 1)$\frac{3}{2}$ (2) 3 ( 3)$\frac{1}{2}$ (4) $\frac{3 \sqrt{3}}{2}$ [JEE-MAINS Online 2013]

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Sol. (1) $|\bar{c}|^{2}+|\bar{a}|^{2}-2 \bar{a} \bar{c}=8$ $|\bar{c}|^{2}+9-2|\bar{c}|=8$ $|\overline{\mathrm{c}}|=1$ $|(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}|=|\overline{\mathrm{a}} \times \overline{\mathrm{b}}||\overline{\mathrm{c}}| \sin 30^{\circ}$ $=\frac{|\overline{\mathrm{a}} \times \overline{\mathrm{b}}|}{2}$ $|\overline{\mathrm{a}} \times \overline{\mathrm{b}}|=\left|\begin{array}{ccc}{\hat{\mathrm{i}}} & {\hat{\mathrm{j}}} & {\hat{\mathrm{k}}} \\ {2} & {1} & {-2} \\ {1} & {1} & {0}\end{array}\right|$ $=|2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}|=3$

Q. If $[\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}]=\lambda[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]^{2}$ then $\lambda$ is equal to : (1) 2 (2) 3 (3) 0 (4) 1 [JEE(Main)-2014]

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Sol. (4) $[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]=(\vec{a} \times \vec{b}) \cdot((\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a}))$ $=(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \cdot((\overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}) \overrightarrow{\mathrm{c}}-[\overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{c}}] \overrightarrow{\mathrm{a}})=[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]^{2}$

Q. Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear and $(\vec{a}, \times \vec{b}) \times \vec{c}=\frac{1}{3}|\vec{b}||\vec{c}| \vec{a} \cdot$ If $\theta$ is the angle between vectors $\vec{b}$ and $\vec{c},$ then a value of $\sin \theta$ is : [JEE(Main)-2015]

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Sol. (3) $(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}) \overrightarrow{\mathrm{b}}-(\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}) \overrightarrow{\mathrm{a}}=\frac{1}{3}|\overrightarrow{\mathrm{b}}||\overrightarrow{\mathrm{c}}| \overrightarrow{\mathrm{a}}$ $\Rightarrow\left[\begin{array}{c}{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=0} \\ {-\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=\frac{1}{3}|\overrightarrow{\mathrm{b}} \| \overrightarrow{\mathrm{c}}|}\end{array}\right.$ $\Rightarrow \cos \theta=-\frac{1}{3} \Rightarrow \sin \theta=\frac{2 \sqrt{2}}{3}$

Q. Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ be three unit vectors such that $\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\frac{\sqrt{3}}{2}(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}) .$ If $\overrightarrow{\mathrm{b}}$ is not parallel to $\overrightarrow{\mathrm{c}}$ then the angle between a and $\overrightarrow{\mathrm{b}}$ is :- (1) $\frac{5 \pi}{6}$                 (2) $\frac{3 \pi}{4}$              (3) $\frac{\pi}{2}$              (4) $\frac{2 \pi}{3}$ [JEE(Main)-2016]

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Sol. (1) $\left(\overrightarrow{\mathrm{a} .} \overrightarrow{\mathrm{c}}-\frac{\sqrt{3}}{2}\right) \overrightarrow{\mathrm{b}}-\left(\overrightarrow{\mathrm{a} .} \overrightarrow{\mathrm{b}}+\frac{\sqrt{3}}{2}\right) \overrightarrow{\mathrm{c}}=0$ $\Rightarrow \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\cos \theta=-\sqrt{3} / 2 \Rightarrow \theta=5 \pi / 6$

Q. Let $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}} .$ Let $\overrightarrow{\mathrm{c}}$ be a vector such that $|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=3,|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}|=3$ and the angle between $\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ be $30^{\circ} .$ Then $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}$ is equal to : ( 1)$\frac{1}{8}$                     (2) $\frac{25}{8}$                (3) 2                    (4) 5 [JEE(Main)-2018]

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Sol. (3) $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and $|\overrightarrow{\mathrm{a}}|=3$ $\therefore \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ $|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=3$ Now $:(\vec{a} \times \vec{b}) \times \vec{c}=|\vec{a} \times \vec{b}||\vec{c}| \sin 30 \hat{n}$ $|(\vec{a} \times \vec{b}) \times \vec{c}|=3 \cdot|\vec{c}| \cdot \frac{1}{2}$ $3=3|\overrightarrow{\mathrm{c}}| \cdot \frac{1}{2}$ $\therefore|\overrightarrow{\mathrm{c}}|=2$ Now $:|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=3$ $c^{2}+a^{2}-2 \vec{c} \cdot \vec{a}=9$ $4+9-2 \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=9$ $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=2$

Q. If the position vectors of the vertices $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ of a $\Delta \mathrm{ABC}$ are respectively $4 \hat{i}+7 \hat{j}+8 \hat{k}, 2 \hat{i}+3 \hat{j}+4 \hat{k}$ and $2 \hat{i}+5 \hat{j}+7 \hat{k},$ then the position vector of the point, where the bisector of $\angle \mathrm{A}$ meets $\mathrm{BC}$ is : (1) $\frac{1}{2}(4 \hat{i}+8 \hat{j}+11 \hat{k})$ (2) $\frac{1}{3}(6 \hat{i}+13 \hat{j}+18 \hat{k})$ (3) $\frac{1}{4}(8 \hat{\mathrm{i}}+14 \hat{\mathrm{j}}+19 \hat{\mathrm{k}})$ (4) $\frac{1}{3}(6 \hat{i}+11 \hat{j}+15 \hat{k})$ [JEE(Main)-2018]

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Sol. (2)

Q. If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}},$ and $\overrightarrow{\mathrm{c}}$ are unit vectors such that $\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}+2 \overrightarrow{\mathrm{c}}=\overrightarrow{0},$ then $|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}|$ is equal to : [JEE(Main)-2018]

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Sol. (4)

Q. If the angle between the lines, $\frac{x}{2}=\frac{y}{2}=\frac{z}{1}$ and $\frac{5-x}{-2}=\frac{7 y-14}{p}=\frac{z-3}{4}$ is $\cos ^{-1}\left(\frac{2}{3}\right),$ then $p$ is equal to: [JEE(Main)-2018]

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Sol. (2)

Q. Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{c}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and a vector $\overrightarrow{\mathrm{b}}$ be such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=3 .$ Then $|\overrightarrow{\mathrm{b}}|$ equals : (1) $\sqrt{\frac{11}{3}}$ (2) $\frac{11}{\sqrt{3}}$ (2) $\frac{11}{\sqrt{3}}$ ( 4)$\frac{11}{3}$ [JEE(Main)-2018]

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Sol. (1)

Q. Let $\vec{u}$ be a vector coplanar with the vectors $\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\vec{b}=\hat{j}+\hat{k} .$ If $\vec{u}$ is perpendicular to $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{u} . \overrightarrow{\mathrm{b}}}=24,$ then $|\overrightarrow{\mathrm{u}}|^{2}$ is equal to- (1) 315               (2) 256                  (3) 84                 (4) 336 [JEE(Main)-2018]

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Sol. (4)

• August 11, 2021 at 3:35 pm

Useful

0
• July 11, 2021 at 12:50 pm

Sol. Plz

15
• March 13, 2021 at 12:03 pm

2
• February 14, 2021 at 5:55 am

👌

6
• October 11, 2020 at 7:54 am

nice probles but easy to solve

14
• December 23, 2020 at 1:15 am

go for jee advanced ques then….those ques might be challenging for you.. ^_^

4
• September 30, 2020 at 4:16 pm

Last ques soln pls

6
• September 21, 2020 at 1:06 pm

Lalalllalllalallalalllla

2
• September 10, 2020 at 6:16 pm

Awesome 😀😇

1
• September 3, 2020 at 12:20 pm

Good

5
• August 30, 2020 at 6:38 pm

Last question solution pls

2
• August 28, 2020 at 7:05 am

Helped a lot

1
• August 1, 2020 at 12:05 pm

1
• June 20, 2020 at 11:05 am

Last question solution pls

2
• May 21, 2020 at 4:52 am

happy with this

2
• May 20, 2020 at 4:28 pm

Saaho

1


text:
2 TeXed up

I am a string theorist who has encountered the following number theory problem in my research.

Consider the sums

Z_+(s)

$$Z_+(s) = \sum_{p=1}^\infty p^{-s} S(1,1; p)p)$$

and

Z_-(s)

$$Z_-(s) = \sum_{p=1}^\infty p^{-s} S(1,-1; p)p)$$

where S(n,m;p) $S(n,m;p)$ is the Kloosterman sum. This sum converges when s $s$ has a large real part and can be analytically continued to define Z_{+/-}(s) $Z_{+/-}(s)$ on the whole complex s $s$ plane. This is a special case of the Kloosterman zeta function; it contains data about the spectra of certain Riemann surfaces.

I am interested in the behaviour of this analytically continued function when s is an integer. In particular I would like to know whether Z_{+/-}(s) $Z_{+/-}(s)$ has poles for any integer values of s, $s$, and if so what the residues are.

I've found some literature on the subject of Kloosterman Zeta functions but I have not found quite what I'm looking for. Most treatments focus on the general properties of the Kloosterman Zeta function, which are quite complicated.
My suspicion is that because I am only interested in integer values the answer might be relatively simple.

Any help would be appreciated, either direct answers or specific pointers to the literature.

1

# Poles of Kloosterman Zeta Function

I am a string theorist who has encountered the following number theory problem in my research.

Consider the sums

Z_+(s) = \sum_{p=1}^\infty p^{-s} S(1,1; p)

and

Z_-(s) = \sum_{p=1}^\infty p^{-s} S(1,-1; p)

where S(n,m;p) is the Kloosterman sum. This sum converges when s has a large real part and can be analytically continued to define Z_{+/-}(s) on the whole complex s plane. This is a special case of the Kloosterman zeta function; it contains data about the spectra of certain Riemann surfaces.

I am interested in the behaviour of this analytically continued function when s is an integer. In particular I would like to know whether Z_{+/-}(s) has poles for any integer values of s, and if so what the residues are.

I've found some literature on the subject of Kloosterman Zeta functions but I have not found quite what I'm looking for. Most treatments focus on the general properties of the Kloosterman Zeta function, which are quite complicated.
My suspicion is that because I am only interested in integer values the answer might be relatively simple.

Any help would be appreciated, either direct answers or specific pointers to the literature.


text:
Articles written in Journal of Chemical Sciences

• Hydrogen bonding in organometallic and metal-organic crystals. Transition metal amido complexes

• Assembling one-dimensional coordination polymers into threedimensional architectures via hydrogen bonds

The reactions of bis(pyridylcarboxamido)alkanes (amides) and bis(3-pyridyl)alkanediamides (reverse amides) with copper(II) and zinc(II) in the presence of various anions resulted in onedimensional polymeric crystalline complexes with or without guest inclusion. The crystal structure analyses of these complexes reveal that the one-dimensional networks observed here are of three types: simple linear chain, chains with wavy nature and chains containing cavities. The self-complementary amide groups of the ligands assembled these coordination networks into higher dimensional architectures via N-H$\cdots$O hydrogen bonds.

• 3D, 2D and 1D networks via N-H…O and N-H…N hydrogen bonding by the bis-amide analogues: Effect of chain lengths and odd-even spacers

The synthesis, crystal structures and hydrogen bonding networks of four members of the bis(pyridinecarboxamido)alkane and bis(pyridyl)alkanediamides series (1 ≤ 𝑛 ≤ 8), where the amide moieties are separated by alkyl chain (-(CH2)$_n$-) having even or odd number of -(CH2)-groups are explored and correlated with the previously reported structures. The odd members (n= odd) of both the series are found to adopt three-dimensional networks in contrast to the 1D or 2D structures of the even members (n= even). This odd-even effect on the dimensionality of the networks however disappears with increase in chain length.

• # Journal of Chemical Sciences

Volume 134, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
# Let M be a 2 x 2 symmetric matrix with integer entries. Then M is invertible if A. The first column of M is the transpose of the second row of MB. The second row of M is the transpose of the first column of M C. M is a diagonal matrix with non-zero entries in the main diagonal D. The product of entries in the main diagonal of M is not the square of an integer

Verified
119.4k+ views
Hint:
We recall that definition of symmetric matrix and the condition for invertible matrices. We take $M=\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right]$ and find the condition for symmetric matrix as $b=c$ and the condition for invertible is $ad-bc\ne 0$ .We check each option and see if we obtain the conditions $b=c,ad-bc\ne 0$ to find their truth values.

We know that a symmetric matrix is a square matrix whose transpose is the matrix itself. Let us denote the four integral entries of the given matrix $M$ as $a,b,c,d$ as
$M=\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right]$
We take the transpose of the above matrix and have
${{M}^{T}}=\left[ \begin{matrix} a & c \\ b & d \\ \end{matrix} \right]$
Since we are give in the question $M$ is symmetric we have;
\begin{align} & M={{M}^{T}} \\ & \Rightarrow \left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right]=\left[ \begin{matrix} a & c \\ b & d \\ \end{matrix} \right] \\ & \Rightarrow b=c \\ \end{align}

So we can write the matrix as
$M=\left[ \begin{matrix} a & b \\ b & d \\ \end{matrix} \right]$

We know that we call matrix invertible when the determinant of the matrix is not zero. So we have;
\begin{align} & \det \left( M \right)\ne 0 \\ & \Rightarrow \det \left( \left| \begin{matrix} a & b \\ b & d \\ \end{matrix} \right| \right)\ne 0 \\ & \Rightarrow ad-{{b}^{2}}\ne 0 \\ \end{align}
Let us check option A. The first column of $M$ is $\left[ \begin{matrix} a \\ c \\ \end{matrix} \right]$ and the transpose of the second row of $M$ is ${{\left[ \begin{matrix} b & d \\ \end{matrix} \right]}^{T}}=\left[ \begin{matrix} b \\ d \\ \end{matrix} \right]$ . We are given that
\begin{align} & \left[ \begin{matrix} a \\ b \\ \end{matrix} \right]=\left[ \begin{matrix} b \\ d \\ \end{matrix} \right] \\ & \Rightarrow a=b=d \\ & \Rightarrow ad-{{b}^{2}}=0 \\ \end{align}
So option A is incorrect. Let us check option B. The second row of $M$ is $\left[ \begin{matrix} b & d \\ \end{matrix} \right]$ and the transpose of the first column of $M$ is ${{\left[ \begin{matrix} a \\ b \\ \end{matrix} \right]}^{T}}=\left[ \begin{matrix} a & b \\ \end{matrix} \right]$ . We are given
\begin{align} & \left[ \begin{matrix} b & d \\ \end{matrix} \right]=\left[ \begin{matrix} a & b \\ \end{matrix} \right] \\ & \Rightarrow b=a,d=b \\ & \Rightarrow a=b=d \\ & \Rightarrow ad-{{b}^{2}}=0 \\ \end{align}
So option B is incorrect. Let us check option C. Since $M$ is diagonal matrix with non-zero entries in the main diagonal we have $b=c=0$ . So we have
$\det \left( M \right)=ad-{{b}^{2}}=ad-0=ad\ne 0\left( \because a,d\in Z \right)$
So option C is correct. Let us check option D. The product of entries in the main diagonal of $M$ is $a\times d=ad$ . We are given $ad$ is not perfect square and hence $ad\ne {{b}^{2}}$ for any integer $b$ . So we have
$\det \left( M \right)=ad-{{b}^{2}}\ne 0$
So option D is correct. The correct options are C and D.

Note:
We note that we only have equality between matrices $A=B$ when they have the same order and corresponding elements in the positions are equal which means ${{a}_{ij}}={{b}_{ij}}$ . We use invertible matrices to solve a system of equations. We find the transpose by writing rows of an original matrix as columns or writing columns of the original matrix as rows. $A$ is skew-symmetric matrix if $A=-{{A}^{T}}$ .


text:
# How do you prove corresponding angles are equal?

Dec 24, 2015

Suppose you have two parallel lines cut by a transversal.

Due to the straight angle (linear pair) theorem, we know that

{(mangle2+mangle3=180˚),(mangle5+mangle6=180˚):}

Thought the transitive property, we can say that

$m \angle 2 + m \angle 3 = m \angle 5 + m \angle 6 \textcolor{w h i t e}{\times \times}$ (1)

Though the alternate interior angles theorem, we know that

$m \angle 3 = m \angle 5$

Use substitution in (1):

$m \angle 2 + m \angle 3 = m \angle 3 + m \angle 6$

Subtract $m \angle 3$ from both sides of the equation

$m \angle 2 = m \angle 6$

$\therefore \angle 2 \cong \angle 6$

Thus $\angle 2$ and $\angle 6$ are corresponding angles and have proven to be congruent.


text:
# When to use Inclusion Exclusion principle?

I'm a bit confused as to when you should use the Incl/Excl principle. Take for example the following problem:

How many 13-card hands can be selected from the standard 52-card deck with exactly four spades or exactly four diamonds?

My university tutor solved this problem by using the principle. My theory would be that you need to use it in this case because a 13-card hand with exactly four spades can also contain four diamonds, and a hand with exactly four diamond can also contain four spades - so you end up counting too many times.

Is there a trick or a general rule for when you should use it?

Sub-question: My tutor wrote this for choosing exactly four spades, and I believe there might be a mistake:

$c(13,4) * c(9,39)$

$c(13,4)$ is the number of ways to choose four positions for the spades
$c(9,39)$ is the number of ways to choose the remaining 9 cards that are not spades

But isn't it missing another $c(13,4)$ to choose the four spades out of 13 spades?

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Have you tried it also with looking at the Vann Diagram of the sets? – Arjang May 22 '11 at 0:25
I'm not sure what the venn diagram would look like but would it be something like this? img851.imageshack.us/img851/9831/capturehj.jpg – Arvin May 22 '11 at 0:48
Yes I suppose looking at the venn diagram I made there, it does make sense. I guess venn diagrams are one way to tell you need to use the principle. – Arvin May 22 '11 at 0:56
If you want an example where Inclusion-Exclusion can be used but may not be the easiest approach, see math.stackexchange.com/questions/23685/… – Henry May 22 '11 at 1:40

First and foremost, your notation $c(9,39)$ should read $c(39,9)$. That is presumably a typo. Now:

The inclusion-exclusion principle helps you find the cardinality of the sets $A \cup B$, $A \cup B \cup C$, $A \cup B \cup C \cup D$, etc. In your case, you have to find the cardinality of the set $A \cup B$, where $A$ is the event that you draw exactly four spades and $B$ is the event that you draw exactly four diamonds. If $|A|$ denotes the cardinality (or number of elements) in the set $A$, then

$$|A \cup B| = |A| + |B| - |A \cap B|.$$

This makes intuitive sense, since if you want to count the number of elements in the set $A \cup B$, you count the number of elements in $A$, you count the number of elements in $B$ and you subtract the number of elements in $A \cap B$, since they were counted twice.

To find $|A|$ or the number of $13$ card hands which have exactly four spades, you need to choose $4$ spades from the possible $13$ and $9$ non spades from the rest of the deck. The number of ways of doing this is

$$c(13,4) \cdot c(39,9) = 151519319380.$$

-
Thanks DJC, that confirms what I thought as well. This explanation works for my particular problem, but I was looking for a general rule/trick to figure out when best to use the principle and when not to use it. I suppose it's different for every problem so there aren't any rules or tricks – Arvin May 22 '11 at 1:51
Hmm, how would you find the cardinality of A intersect B? – Arvin May 22 '11 at 2:04
Well, what is $A \cap B$ here? – JavaMan May 22 '11 at 2:19
It would be the set of all card hands which contain Exactly 4 Spades AND 4 Diamonds. I suppose then it would be c(13,4) * c(13,4) * c(26, 5) ? – Arvin May 22 '11 at 2:50
@Arvin. Right on. – JavaMan May 22 '11 at 2:52

Don't really like to give general rules or tricks, since then looking for them can interfere with the analysis of a problem. But the presence of or means we are trying to count the union of two sets $A$ and $B$. And sometimes the best way to count a union is to count $A$, count $B$, count what they have in common, and do the obvious thing. Your jntuitive explanation of the "why" in this case was clear and correct.

Your tutor's answer is right, but your description of it is not. We are counting the number of hands, order is irrelevant. The $c(13,4)$ in the answer comes not from the location of the spades. It actually counts the number of ways of choosing $4$ spades from the $13$. For every way of choosing the spades, there are $c(39,9)$ ways to choose the remaining cards (you kind of typed it backwards), so the number of $4$-spade hands is $$C(13,4)C(39,9)$$

-
Thanks for that. Of course the order doesn't matter since it's a card hand - what was I thinking? :) – Arvin May 22 '11 at 1:50


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2428 views

[font=verdana]

# The inception

[/font]
[font=verdana]A while back, after slowly becoming mad tweaking animations and movement behavior on our avatar, I decided to have some simple fun with Unity. See where the limit was and what's possible in a certain domain. Since I was obsessed with the character's animation, my attention was driven towards improving what our animator gave us in terms of natural movement. Even though the animations were quite good, I wanted some physics madness in there. So I went on a quest to mess with the bones of mecanim.
It turns out it's rather straightforward to activate the so called "jiggly bones". A few changes in your model before export does the trick. It does however require a great deal of value tweaking and physics engine know-how. You may follow this tutorial if you want to try it for yourself.
Here's what I could do as a quick 'n dirty test. This will *not* be in the game. Even if the effect could be nice, It is at the bottomest bottom of our priorities.[/font]

[font=verdana][/font]

[font=verdana]But doing this as a playful "what if" helped me learn a lot about how Unity works with animations and physics. Even better, I would never have thought of what I'm about to show if I hadn't gamed the system.
A few days later, the movement and animation finally polished, I found one last thing I could do, turn the head of the fox in the direction of where the avatar will go next. Providing a subconscious feedback on the movement. Using the head bone and overriding the animation, it should be possible. Right?[/font]

[font=verdana][/font]

[font=verdana]It turns out it is. This gif was taken when a bug prevented the fox to move or turn so you can clearly see that the head orientation is overridden (and a bit disturbing, sorry).

# How to do it

[/font]

[indent=1][font=verdana]

## [color=rgb(51,51,51)]First, you have to get a reference to your head bone:[/color]

[/font]
private void Awake(){ HeadBone = /*find the headbone in the hierarchy*/.transform;}
[indent=1][font=verdana]

## [color=rgb(51,51,51)][color=rgb(51,51,51)]There's nothing to do to your model. All you have to do is use the LateUpdate function which is called afterthe internal update of the animations.[/color][/color]

[/font]
private void LateUpdate(){ Quaternion look_at = Quaternion.LookRotation(Direction, new Vector3(0.0f, 1.0f, 0.0f)); look_at *= Quaternion.Euler(-90.0f, 0.0f, 0.0f); // Our models are rotated HeadBone.rotation = look_at;}
[indent=1][font=verdana][color=rgb(51,51,51)]You can do all sorts of crazy stuff with this.[/color][/font]

[font=verdana][color=rgb(51,51,51)]

### # The result

[/color][/color][/color][/font]

[font=verdana][color=rgb(51,51,51)]

### [color=rgb(51,51,51)]

[/color][/color][/color][/font]

[font=verdana]

## [color=rgb(51,51,51)][color=rgb(51,51,51)][color=rgb(51,51,51)]Here you can see the result in game. Slowed down on purpose to make sure it's seeable in a gif. The effect is very subtle but you can see that the head will always point to the right direction (almost instantly) and the body follows soon after.The next time you are frustrated with a particular task or simply bored out of your mind after a day of repetitive tweaks. Do yourself a favor and just play with the tools on your workbench. Who knows what kind of gem you'll unearth.[/color][/color][/color]

[/font]

### Images

View the entire Epistory album

Another nice entry. I like where this is going.

"(and a bit disturbing, sorry)" Disturbing is my middle name!

This looks very nice. Keep up the good work.

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text:
# A polynomial determined by two values

Prove or disprove: a non constant polynomial $P$ with non-negative integer coefficients is uniquely determined by its values $P(2)$ and $P(P(2))$.

-

True. If $P(x)=a_nx^n+\cdots+a_1x+a_0$ then each of the coefficients are less then $b\equiv P(2)$. Each of these coefficients can then be read off from the base-b expansion of $P(b)=P(P(2))$.

-
Several people seem to be answering this one at the same time (how to select a single accepted answer?). Actually, it's a common question that keeps popping up on various maths forums. –  George Lowther Aug 26 '10 at 22:40
For what it's worth, Steve got there a few seconds before me. –  George Lowther Aug 26 '10 at 22:43
Perhaps you can wait for the 3rd upvote and then delete. You will get a badge :-) –  Aryabhata Aug 26 '10 at 22:48
Yeah, I saw that you deleted your answer, but I don't see any issue with having two the same. I just voted Steve up and pointed out that he did get there before before me in the comment, to be fair to him. –  George Lowther Aug 26 '10 at 22:50
Oh, I like badges :) –  George Lowther Aug 26 '10 at 22:51

Look at $P(P(2))$ in base $P(2)$. The nth place is the coefficient of $x^n$.

Steve

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You don't have to sign your answers, BTW. –  ShreevatsaR Aug 27 '10 at 5:57
I don't "have" to, I want to :). I do it on all the other forums, so force of habit. –  user641 Aug 27 '10 at 19:08


text:
# Physics Events

• Monday, November 12, 2018
High Energy Theory Seminar
“Energy is Entanglement”

Location: Bloomberg Lecture Hall
Time: 2:30 PM
Speaker(s): Arvin Shahbazi Moghaddam, University of California, Berkeley

Description: We compute the local second variation of the von Neumann entropy of a region in theories with a gravity dual. For null variations our formula says that the diagonal part of the Quantum Null Energy Condition is saturated in every state, thus providing an equivalence between energy and entropy. We conjecture that the QNEC is saturated in all interacting theories and discuss aspects of the proof of this statement in general interacting CFTs. We also discuss the special case of free theories, and the implications of our formula for the Averaged Null Energy Condition, Quantum Focusing Conjecture, and gravitational equations of motion.

• Tuesday, November 13, 2018
Pheno & Vino Seminar
“Conformal Field Theory, Hamiltonian Truncation, and You”

Location: Jadwin Hall, Room 303
Time: 4:00 PM
Speaker(s): Liam Fitzpatrick, Boston University

Description: Conformal Field Theories (CFTs) have a highly constrained structure that make them much more tractable, especially at strong coupling, than generic QFTs.  An appealing picture for more general QFTs is as points along the RG flow from a UV CFT.  We discuss conformal truncation as a method for nonperturbatively following such RG flows.  We describe some examples of past and potential applications, and in particular some of the advantages of the method for studying gauge theories, as well as remaining challenges.

• Wednesday, November 14, 2018
Physics Group Meeting
“Nonsupersymmetric D-branes”

Location: Bloomberg Hall Physics Library
Time: 1:45 PM
Speaker(s): Edward Witten, Faculty, School of Natural Sciences, IAS

Description: These were originally discovered by Sen (see hep-th/9808141, hep-th/9809111, hep-th/9904207, for example).   Another interesting paper is by Kraus and Larsen, hep-th/0012198.
I will describe a new perspective based on anomalies.

• Thursday, November 15, 2018
Special Condensed Matter Seminar
“Exact Strong-ETH Violating Eigenstates in the Rydberg-blockaded Atom Chain”

Location: PCTS Seminar Room 407
Time: 2:15 PM
Speaker(s): Cheng Ju Lin, Caltech

Description: A recent experiment in the Rydberg atom chain observed unusual oscillatory quench dynamics with a charge density wave initial state, and theoretical works identified a set of many-body scar states'' in the Hamiltonian as potentially responsible for the atypical dynamics. In the same nonintegrable Hamiltonian, we discover several eigenstates at infinite temperature that can be represented exactly as matrix product states with finite bond dimension, for both periodic boundary conditions (two exact E = 0 states) and open boundary conditions (two E = 0 states and one each E = 2). This discovery explicitly demonstrates violation of strong eigenstate thermalization hypothesis in this model. These states show signatures of translational symmetry breaking with period-2 bond-centered pattern, despite being in 1D at infinite temperature. We show that the nearby many-body scar states with energies E ~ 1.33 andE ~ 2.66 can be well approximated as quasiparticle excitations" on top of our exact E = 0 states, and propose a quasiparticle explanation of the strong oscillations observed in experiments.

• Thursday, November 15, 2018
Princeton University Department of Physics Donald R. Hamilton Colloquium Series
“Wonders of Viscous Electronics”

Location: Jadwin Hall, Room A10
Time: 4:00 PM
Speaker(s): Gregory Falkovich, Weizmann Institute of Science

Description: Quantum-critical strongly correlated systems feature universal collision-dominated collective transport. Viscous electronics is an emerging field dealing with systems in which strongly interacting electrons flow like a fluid. Such flows have some remarkable properties never seen before. I shall describe recent theoretical and experimental works devoted, in particular, to a striking macroscopic DC transport behavior: viscous friction can drive electric current against an applied field, resulting in a negative resistance, recently measured experimentally in graphene. I shall also describe conductance exceeding the fundamental quantum-ballistic limit, field-theoretical anomalies and other wonders of viscous electronics. Strongly interacting electron-hole plasma in high-mobility graphene affords a unique link between quantum-critical electron transport and the wealth of fluid mechanics phenomena.
https://phy.princeton.edu/events/donald-r-hamilton-colloquium-series

• Friday, November 16, 2018
High Energy Theory Seminar
“M-theory in 4D, N=1 Superspace”

Location: Bloomberg Lecture Hall
Time: 1:45 PM
Speaker(s): Katrin Becker, Member, School of Natural Sciences / Texas A&M University

Description: I report on progress in formulating eleven-dimensional supergravity in a framework in which four supersymmetries are manifest.

• Monday, November 19, 2018
High Energy Theory Seminar
“Boundary Dual of the Bulk Symplectic Form and the Volume of Maximal Slices”

Location: Jadwin Hall, PCTS Seminar Room, 4th Floor
Time: 2:30 PM
Speaker(s): Gabor Sarosi, University of Pennsylvania & University of Brussels

Description: In this talk, I will explain how to understand the symplectic structure of the gravitational phase space in AdS in terms of the quantum overlap in the holographically dual CFT. As an application, I will use this to study the boundary description of the volume of maximal Cauchy slices. I will construct the boundary deformation that is conjugate to the volume in empty AdS, and the thermofield double at infinite time. I will explain the bulk interpretation of this deformation and speculate on its boundary interpretation. This will motivate a concrete version of the complexity equals volume conjecture, where the boundary complexity is defined as the energy of geodesics in a Kahler geometry of half sided Euclidean sources. I will calculate this complexity for states dual to Banados geometries that are close to the vacuum, and discuss a mini superspace approximation in the case of the time dependent thermofield double.

• Wednesday, November 21, 2018
Physics Group Meeting
“TBA”

Location: Bloomberg Hall Physics Library
Time: 1:45 PM
Speaker(s): Tom Rudelius, Member, School of Natural Sciences, IAS

• Monday, November 26, 2018
High Energy Theory Seminar
“Stringy ER=EPR”

Location: Bloomberg Lecture Hall
Time: 2:30 PM
Speaker(s): Daniel Jafferis, Harvard University

Description:

• Tuesday, November 27, 2018
Pheno & Vino Seminar
“From the Table-Top to the Cosmos: Searching High and Low for Dark Matter”

Location: Jadwin Hall, Room 303
Time: 4:00 PM
Speaker(s): Hongwan Liu, MIT

Description: In this talk, I will explore two recent developments in the search for dark matter. First, I will discuss how the 21-cm global signal measurement constrains the potential impact of dark matter annihilation/decay on the thermal history of the universe. I will also explain how to accurately compute such modifications to the baryon temperature in the early universe using a soon-to-be-released public code called DarkHistory. Second, I will discuss the Axion Detection with Birefringent Cavities (ADBC) experiment, which aims to observe rotations induced by axion dark matter in the polarization of light using optical interferometry techniques.

• Wednesday, November 28, 2018
Physics Group Meeting
“TBA”

Location: Bloomberg Hall Physics Library
Time: 1:45 PM
Speaker(s): Nima Arkani-Hamed, Faculty, School of Natural Sciences, IAS

• Thursday, November 29, 2018
Princeton University Department of Physics Donald R. Hamilton Colloquium Series
“The Chiral Anomaly in Dirac Semimetals”

Location: Jadwin Hall, Room A10
Time: 4:00 PM
Speaker(s): Nai Phuan Ong, Princeton University

Description: I will talk about recent experiments on the chiral anomaly in Dirac semimetals. In field theory, Dirac fermions of zero mass must segregate into left- and right-handed populations that do not ever mix. In this limit, chiral symmetry (handedness) is a global symmetry of the Lagrangian. However, quantum effects induced by coupling to a vector gauge field kill the symmetry. This is known as the chiral (or axial) anomaly. The first example appeared (1968) in the decay of neutral pions into 2 photons (the Adler-Bell-Jackiw anomaly). Since then, anomalies have been implicated in many important problems, e.g. the $U(1)_A$ problem, renormalizability of the electroweak theory, and the fermion-doubling problem in lattice QCD. It is also directly related to the Atiyah-Singer index theorem. In 1983, Nielsen and Ninomiya predicted that the chiral anomaly should be observable as well in bulk semimetals that feature protected 3D Dirac cones. Breaking of time-reversal invariance in a magnetic field B converts the Dirac electrons in a semimetal to Weyl fermions. The chiral nature of the Weyl fermions become manifest in the lowest Landau level (in intense B). An electric field applied parallel to B shifts the left and right-moving branches to produce an axial current. I will describe experiments on $Na_3Bi$ and GdPtBi which show the dramatic emergence of the anomaly, and focus on issues peculiar to the anomaly in crystals. I will describe a litmus test that sharply distinguishes this quantum effect from (classical) artifacts caused by current jetting.
https://phy.princeton.edu/events/donald-r-hamilton-colloquium-series

• Friday, November 30, 2018
High Energy Theory Seminar
“TBA”

Location: Bloomberg Lecture Hall
Time: 1:45 PM
Speaker(s): Anna Karlsson, Member, School of Natural Sciences, IAS

Description:

• Monday, December 3, 2018
PCTS Workshop
“Fracton Phases of Matter and Topologial Crystalline Order”

Location: Jadwin Hall, PCTS Seminar Room
Time: 8:30 AM
Speaker(s): http://www.physics.princeton.edu/pcts/Fracton2018/posterFracton.pdf
Note: Registration is now open!

Description: Program Organizers: Fiona Burnell, Biao Lian, Abhinav Prem, Shivaji Sondhi, and Yizhi You
Topological quantum phases of matter remain one of the most exciting playgrounds for exploring the interplay between strong correlations, symmetry, and topology in many-body quantum systems. Recently, there has been growing interest in novel quantum phases of matterso-called fracton phases. These systems extend and challenge our existing notions of topological order and have attracted broad interdisciplinary interest, lying at the intersection of multiple fields including topological order, higher rank gauge theories, gravity, quantum information, and elastic descriptions of soft matter. We expect that this workshop will bring together leading experts from these various communities, with the primary aim being the stimulation of discussions and promotion of cross-disciplinary interactions amongst the participants.

http://www.physics.princeton.edu/pcts/Fracton2018/fracton2018.html

• Tuesday, December 4, 2018
Qubit Workshop
“Quantum Information and the Structure of Spacetime”

Location: Wolfensohn Hall
Time: 8:00 AM

Description: Please Note: This workshop is not open to the general public, but only to active researchers.

This workshop will focus on quantum aspects of black holes, focusing on applying ideas from quantum information theory.

This meeting is sponsored by the It from Qubit collaboration and is followed by the collaboration meeting in New York City.

Wednesday, November 28, 2018
All participants must register. https://www.sns.ias.edu/quantum-information-workshop-2018/registration
https://www.sns.ias.edu/quantum-information-workshop-2018

• Tuesday, December 4, 2018
PCTS Workshop
“Fracton Phases of Matter and Topologial Crystalline Order”

Location: Jadwin Hall, PCTS Seminar Room
Time: 8:30 AM
Speaker(s): http://www.physics.princeton.edu/pcts/Fracton2018/posterFracton.pdf
Note: Registration is now open!

Description: Program Organizers: Fiona Burnell, Biao Lian, Abhinav Prem, Shivaji Sondhi, and Yizhi You
Topological quantum phases of matter remain one of the most exciting playgrounds for exploring the interplay between strong correlations, symmetry, and topology in many-body quantum systems. Recently, there has been growing interest in novel quantum phases of matterso-called fracton phases. These systems extend and challenge our existing notions of topological order and have attracted broad interdisciplinary interest, lying at the intersection of multiple fields including topological order, higher rank gauge theories, gravity, quantum information, and elastic descriptions of soft matter. We expect that this workshop will bring together leading experts from these various communities, with the primary aim being the stimulation of discussions and promotion of cross-disciplinary interactions amongst the participants.

http://www.physics.princeton.edu/pcts/Fracton2018/fracton2018.html

• Wednesday, December 5, 2018
Qubit Workshop
“Quantum Information and the Structure of Spacetime”

Location: Wolfensohn Hall (Bloomberg Hall 4:00 - 6:00 pm)
Time: 8:00 AM

Description: Please Note: This workshop is not open to the general public, but only to active researchers.

This workshop will focus on quantum aspects of black holes, focusing on applying ideas from quantum information theory.

This meeting is sponsored by the It from Qubit collaboration and is followed by the collaboration meeting in New York City.

Wednesday, November 28, 2018
All participants must register. https://www.sns.ias.edu/quantum-information-workshop-2018/registration
https://www.sns.ias.edu/quantum-information-workshop-2018

• Wednesday, December 5, 2018
PCTS Workshop
“Fracton Phases of Matter and Topologial Crystalline Order”

Location: Jadwin Hall, PCTS Seminar Room
Time: 8:30 AM
Speaker(s): http://www.physics.princeton.edu/pcts/Fracton2018/posterFracton.pdf
Note: Registration is now open!

Description: Program Organizers: Fiona Burnell, Biao Lian, Abhinav Prem, Shivaji Sondhi, and Yizhi You
Topological quantum phases of matter remain one of the most exciting playgrounds for exploring the interplay between strong correlations, symmetry, and topology in many-body quantum systems. Recently, there has been growing interest in novel quantum phases of matterso-called fracton phases. These systems extend and challenge our existing notions of topological order and have attracted broad interdisciplinary interest, lying at the intersection of multiple fields including topological order, higher rank gauge theories, gravity, quantum information, and elastic descriptions of soft matter. We expect that this workshop will bring together leading experts from these various communities, with the primary aim being the stimulation of discussions and promotion of cross-disciplinary interactions amongst the participants.

http://www.physics.princeton.edu/pcts/Fracton2018/fracton2018.html

• Friday, December 7, 2018
High Energy Theory Seminar
“TBA”

Time: 1:45 PM
Speaker(s): TBA, TBA

• Monday, December 10, 2018
High Energy Theory Seminar
“TBA”

Location: Bloomberg Lecture Hall
Time: 2:30 PM
Speaker(s): Erich Poppitz, University of Toronto

• Friday, December 14, 2018
High Energy Theory Seminar
“TBA”

Location: Bloomberg Lecture Hall
Time: 1:45 PM
Speaker(s): TBA, TBA

• Monday, January 14, 2019
PCTS Workshop
“Novel Ideas for Dark Matter”

Location: Jadwin Hall, Fourth Floor, Room 407, PCTS Seminar Room
Time: 8:30 AM
Note: Register here: https://docs.google.com/forms/d/e/1FAIpQLSf30X77lY09RVrDcsx98KJJkRoy3akoR5TCQESS101MoT2Y5g/viewform

Description: Dark matter research is undergoing a paradigm shift. Over the last few years, many novel theories have been proposed that challenge the standard assumptions made about dark matter. At the same time, new observations and simulation results are providing valuable clues regarding the most fruitful directions moving forward.

The “Novel Ideas for Dark Matter 2019” workshop at Princeton University attempts to bridge the gap between communities working on non-standard solutions to the dark matter problem. Plenary talks on theory, observations and simulations will be structured into a three day schedule that will provide a broad overview of research avenues of this type. The workshop aims to promote conversations between research communities with ample time for open discussion.

Organizing committee:
Oren Slone - oslone@princeton.edu
Mariangela Lisanti - mlistanti@princeton.edu
Tomer Volansky - tomerv@post.tau.ac.il
http://pcts.princeton.edu/pcts/current_future_programs.html

• Tuesday, January 15, 2019
PCTS Workshop
“Novel Ideas for Dark Matter”

Location: Jadwin Hall, Fourth Floor, Room 407, PCTS Seminar Room
Time: 8:30 AM
Note: Register here: https://docs.google.com/forms/d/e/1FAIpQLSf30X77lY09RVrDcsx98KJJkRoy3akoR5TCQESS101MoT2Y5g/viewform

Description: Dark matter research is undergoing a paradigm shift. Over the last few years, many novel theories have been proposed that challenge the standard assumptions made about dark matter. At the same time, new observations and simulation results are providing valuable clues regarding the most fruitful directions moving forward.

The “Novel Ideas for Dark Matter 2019” workshop at Princeton University attempts to bridge the gap between communities working on non-standard solutions to the dark matter problem. Plenary talks on theory, observations and simulations will be structured into a three day schedule that will provide a broad overview of research avenues of this type. The workshop aims to promote conversations between research communities with ample time for open discussion.

Organizing committee:
Oren Slone - oslone@princeton.edu
Mariangela Lisanti - mlistanti@princeton.edu
Tomer Volansky - tomerv@post.tau.ac.il
http://pcts.princeton.edu/pcts/current_future_programs.html

• Tuesday, January 15, 2019
PCTS Workshop
“Novel Ideas for Dark Matter”

Location: Jadwin Hall, Fourth Floor, Room 407, PCTS Seminar Room
Time: 8:30 AM
Note: Register here: https://docs.google.com/forms/d/e/1FAIpQLSf30X77lY09RVrDcsx98KJJkRoy3akoR5TCQESS101MoT2Y5g/viewform

Description: Dark matter research is undergoing a paradigm shift. Over the last few years, many novel theories have been proposed that challenge the standard assumptions made about dark matter. At the same time, new observations and simulation results are providing valuable clues regarding the most fruitful directions moving forward.

The “Novel Ideas for Dark Matter 2019” workshop at Princeton University attempts to bridge the gap between communities working on non-standard solutions to the dark matter problem. Plenary talks on theory, observations and simulations will be structured into a three day schedule that will provide a broad overview of research avenues of this type. The workshop aims to promote conversations between research communities with ample time for open discussion.

Organizing committee:
Oren Slone - oslone@princeton.edu
Mariangela Lisanti - mlistanti@princeton.edu
Tomer Volansky - tomerv@post.tau.ac.il
http://pcts.princeton.edu/pcts/current_future_programs.html


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# Sum and product of coefficient binomials

1. Mar 30, 2014

### Jhenrique

Given two coefficient binomials $$\binom{a}{b}$$ and $$\binom{c}{d}$$ is possbile to express the sum and product those coefficient binomials as one other?

2. Mar 31, 2014

### Staff: Mentor

Why don't you try answering your own question by expanding both and adding them or multiplying them?

3. Apr 1, 2014

### Jhenrique

I didn't find an satisfactory answer, anyway, thanks!