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# Lighting in the Academic/Educational World

## What is more important?

• Total voters
34

#### icewolf08

##### CBMod
CB Mods
I am starting this hread because there were a bunch of things in the short hijack of the Jands Event 416 thread that got me thinking, and instead of further hijacking the thread, this seemed like a better idea.

In any event, if you didn't read that thread, there was a little debate on what types of equipment is best suited to the educational market, industry standard or not (to be blunt).

I come from a background where I learned on all industry standard equipment for the most part. My first lighting console in middle school (I don't remember the brand) was an analog 2 scene preset that connected to the wall with a cable as thick as my arm. After that we upgraded to an ETC Acclaim 2-scene and from then till college I was on ETC gear. I have to admit, that in terms of ease of use and teach/learnability it is a good way to go. But this is not to say that some other consoles/equipment wouldn't be the same.

Once I got to college I met people that came from all different backgrounds, from people who went to school practically across the street from High End Systems so they had all HES gear, to people who never really touched a memory console. At college we had an Obsession II in the proscenium space, and a Strand 300 in the theatre in the round. The thought was that they could teach students both industry standard consoles. This actually is what turned me into a "Strand Guy." It was quite impressive the number of ways we found to break/crash/piss-off the Obsession, plus for me, the Strand OS just seemed more intuitive/user friendly (EOS might convert me back...). We also had 6 Martin moving heads, and after I graduated thy bought a bunch of VL1000s. This is in addition to the 500+ conventional unit inventory which includes units older than myself and modern units. We had plenty of effects devices too, from scrollers to atmospherics to fiber optics.

Now, chances are, it would have been quite beneficial to me as a student focusing in lighting tech to have exposure to other gear, because in reality there aren't a lot of non-commercial theatres that are outfitted like my college was. I worked on a cruise ship for a while where they have lots of high tech gear, but it isn't always high end equipment. I had never worked with Clay Paky fixtures before, they aren't considered top quality as it were, and people at my college would have frowned on them. Now I work in a large regional theatre with very good budgets, but when I got here, half the inventory was still old Strand radial units (they were calling them "Top Loaders"). The theatre had a bunch of Chroma-Q scrollers, Meteor Elipscans, we even have some GAM Scene Machines (which are pretty cool). I had never touched equipment like this once again because at college it was considered sub par. But, making the transition was not hard for me because one of the biggest pushes at school was in principle, not just with the equipment we had.

I personally believe that it doesn't matter if you learn on low end or out of date equipment as long as you learn the principles. I am not saying that schools should have a multi-scene preset board to teach, but the people who only learned on one are not less qualified. "Channel 6 at 20" whether recorded as a cue on a disk or on paper is the same thing. If you learn how to load a gel string in a Wybron scroller, it is almost exactly the same as loading a gel string in the original Revolution scroller, which is similar to loading a Chroma-Q.

The following quote is from the aforementioned thread:
Sean said:
I know it isn't always good to "follow the herd," but for those of you tasked with selecting gear for educational institutions: buy the gear that's being used. It will make the students' experiences that much more useful to them when they leave, and make supporting the gear easier for you.

And whoever out there is still requesting 360Q's (yes, they are still being sold) ought to be shot.
I can't say that I agree with statements like this. I think that it is more important to get the job done then to buy the newest gear. A 360Q, in functionality is almost exactly the same as a Source 4. If you teach someone how to change a lamp in a 360Q they will have no trouble doing the same for a Source 4, the principle is the same. If s school is outfitted with a NSI console, chances are the student who learned on it will be able to sit down in front of an ETC or Strand and pick up the basics in a few minutes. Besides, if you learned on an Express(ion) then sitting down in front of say a GrandMA is not going to be a walk in the park.

This is theatre, in other words, it is make do with what you have. Our industry has been this way since the beginning. If you learn how to put on the show with the equipment that you have, then moving to better/ higher end equipment should make life easier. If you start with the easy life in school you may end up being just as unprepared for your first job as the graduate who came from the low budget school.

What do the rest of you think?

#### soundlight

##### Well-Known Member
I was about to post a long rebuttle, and carefully explain my theory, but I just wanna see how this one plays out - keep in mind that by not industry standard, I was saying not industry standard, but of comparable quality, price and features.

#### ship

##### Senior Team Emeritus
Won't vote on it because I think both are important but more so in the latter for a basis of knowledge. By the time you get out of what ever school and get to use the gear, most likely it will be obsolete. What was it just like eight years ago the Mac 600 was the main light for the industry? Highly agree with the above concepts in learning a base of knowledge rather than emphisis on modern. Also teaches you a respect, and say when a beam projector is on a show, knowing what the heck it is.

"And whoever out there is still requesting 360Q's (yes, they are still being sold) ought to be shot."
Gee, me in specifying to a customer that they should buy these fixtures for their low budget theater in getting more bang for their buck should than be on your hit list. Should be shot... and you should be stuck in a place without modern gear and be expected to do your job in making art anyway. A 360Q with a HPR 575/115v lamp still out punches and has a more even flat field of beam spred than that of a S-4 fixture.

#### Sean

##### Active Member
Alex,

It seems I'm the first to vote here, and I have the same opinion as you seem to (about learning to "make do and get the job done"). I wholeheartedly support that view.
My comments made in the other thread are about buying new gear. Yes, the 360Q is a decent light. However, why should one specify it in a new project? There is better gear out there.

Also,
A 360Q, in functionality is almost exactly the same as a Source 4. If you teach someone how to change a lamp in a 360Q they will have no trouble doing the same for a Source 4
True. But it won't teach them how to rotate the gate. Or to use a drop-in iris. Or about a gel clip. That's the TEACHING part. I think it's actually easier to work with gear that is less complicated than you know already. So teach a S4, and they'll figure out a 360Q faster than the other way around. Anybody else think this way?

I don't suggest that every school out there pitch everything they own and buy all new. I suppose it would be great if they could, but what school could do that? I do think that those responsible for academic inventories should keep as much gear as they can safely operational. When money becomes available they should buy what most of the rest of the professional world is buying. When you buy new buy the most current, widely used technology that you can afford.

This is theatre, in other words, it is make do with what you have. Our industry has been this way since the beginning. If you learn how to put on the show with the equipment that you have, then moving to better/ higher end equipment should make life easier. If you start with the easy life in school you may end up being just as unprepared for your first job as the graduate who came from the low budget school.
I counter: Would you buy gear for work that isn't being purchased by any other large LORT theatre? Planning on buying some new 360Qs? How about some American DJ-type scanners (as a legit moving light, not as a "throwaway" effect)?

Again, one vote for "get it done"....but a very well argued post!

#### Sean

##### Active Member
Gee, me in specifying to a customer that they should buy these fixtures for their low budget theater in getting more bang for their buck should than be on your hit list. Should be shot... and you should be stuck in a place without modern gear and be expected to do your job in making art anyway. A 360Q with a HPR 575/115v lamp still out punches and has a more even flat field of beam spred than that of a S-4 fixture.

Ship,
First, I do apologize if I offended. Perhaps I did choose language in the too-strong range.

If it's a low budget theatre, why not have them buy used S4's? If/when they get more money in the future then their additional gear would have a better chance of matching what they have already.

Also, just to put this whole conversation back in its original context, the post that started this discussion was about a proposed list of equipment for a school potentially getting a $30,000 grant. --Sean #### Footer ##### Senior Team Senior Team Premium Member Well, I didnt read all the post above me because frankly, I'm tired (we open tomorrow/today I guess...). My personal feeling is that in the education realm teaching the technology is very important. I feel that students should get to expierience the full gambet of technology, from 2 scene preset for your basic level classes to inteligent lighting and all that good stuff for their capstone. But, I fall on the end of "the basics" more then anything else. Look at full sail, they have the best gear money can buy, but their students get out with the attitude that if they don't have that gear around, they can't do anything. I have made lighting that looks like it was done with inteligents with a rather simple rig. You get the basics, you can apply those to anything. ....now my rant...As far as consoles go, any institution that is teaching their students on a console that does not track the way strands/obsessions do is doing a dis-service to their students. Cue only consoles should go away. period. I also feel that students need to get their hands on proper video gear... ie watchout systems or the like. My feeling is the institution does not have to own it, but bringing it in for a show here or their every 2-3 years to give a "heres whats out there" thing is great. Also, all schools should have a subscription of all the trade mags, and the students should actually READ it. This also goes for paroozing lightnetwork, here, and other ponds of knowledge. Also, they should heavily push their students to get the hell out over the summer and actually go do something, and preferably not summer stock at the university. Go work, learn, and make some mulla. #### DarSax ##### Active Member Just coming out of a high school setting where I was the student, I'd have to resoundingly vote for number two. My high school had old Electro-Controls Ellipsoidals, ancient zoom units that you'd never find anywhere else. But once I knew how to use them, I had no trouble when, at 11:30 on Wednesday night before the Thursday opening night, making the decision to rent and eventually being able to light a show using Source Fours. In fact, I'd have to say that my experience was all the better having made that jump. I learned the basics of a fixture on the Ellipsoidals, about beam angle and what looks good as far as positions, colors, intensities, etc. When the Source Four was thrust upon me, suddenly I realized just how powerful and useful the S4 really was, so I could appreciate that much more its more modern features, like the rotating barrel and interchangable lens train. True, to a point that's just pure emotion/mentality but I feel that it definitely implanted and cemented the usefulness of those features in my brain. My thought process as of now is "Well I know if I take a source four and do x and y with it, I can light an actor. But what if I do this and this and this?" I'm more encouraged to experiment, in my opinion. It's the same with consoles. I started out with an ancient Strand, then had a Maxxyz thrust upon me (rental), then an Express (rental), then a Hog 2 (rental). I wish I had been able to use the Express, then Hog, then Maxxyz, so that I could have truly used the Maxxyz to its full potential. I personally would rather learn the basics on, well, a basic instrument, and then use that foundation to move onto more "standard" equipment. Side analogy, because the jazz camp I'm working at started today. It's the same thing as playing an instrument. If you start on a two thousand dollar horn, you're going to take it for granted, at least I'd think so. If you start on a cheap horn and eventually move on to a professional grade, you're going to really appreciate and learn and study that horn. #### gafftaper ##### Senior Team Senior Team Fight Leukemia To me I feel like you are asking the wrong question. I think the question should be phrased something like this... You are a typical educational institution with a decent but not extravagant theater and lighting inventory. Suddently you are given$30,000 to spend. Should you spend it to complete your conventional lighting inventory or should you spend it on two Mac 2k's?

Personally I think it's more important to master the basics than it is to have a little experience with a variety of gear. In organizing the equipment package for our new theater, I considered purchasing Mega Handles for all the instruments. However, while that would be kind of nice for keeping hangs easy, students need to learn how to use a wrench. In a similar way while I hope to have a few cool movers in my inventory, the priority is to first have a full inventory of Elipsoidals, Pars, Fresnels, Strips, Scoops, and Cyc Lights. If you can provide that well, then great, break out the big toys.

#### Charc

##### Well-Known Member
Yes, the 360Q is a decent light. However, why should one specify it in a new project? There is better gear out there.
Also,
True. But it won't teach them how to rotate the gate. Or to use a drop-in iris. Or about a gel clip. That's the TEACHING part. I think it's actually easier to work with gear that is less complicated than you know already. So teach a S4, and they'll figure out a 360Q faster than the other way around. Anybody else think this way?

Though I don't really want to get involved in the verbal fray here, I have to agree about learning on something more complicated first. It took me, another student, and the dept. head about 10 minutes to figure out how to put a gel frame into a source 4... It goes to the side then up?!

#### icewolf08

##### CBMod
CB Mods
First off, it has been quite interesting to read everyone's comments, and I hope that you all find it interesting too. The point of this thread was not to start an argument, but a friendly discussion. I hope that no one has taken any offense or felt attacked by my post or any others. It seemed to me that this could be a very relevant topic to a lot of people.

I am sure that I could have worded some of the things that I said better, especially the poll, but as I was finishing my original post the s**t was hitting the fan during Les Mis and I didn't take the time to re-read things.

I do like Gafftaper's idea on possible revisions to the poll/topic question about how an educational facility would best spend \$30K.

I just got up this morning, so I will try to post a more intelligent response later.

#### gafftaper

##### Senior Team
Senior Team
Fight Leukemia
First off, it has been quite interesting to read everyone's comments, and I hope that you all find it interesting too. The point of this thread was not to start an argument, but a friendly discussion. I hope that no one has taken any offense or felt attacked by my post or any others. It seemed to me that this could be a very relevant topic to a lot of people.

I fully agree. This is an interesting philosophical debate. Unfortunately few of us have the budget to do it all so we have to choose a strategy that we feel works best for our theater, our students, and our skills as teachers. We are all going to have feelings about what is most important based on what we know the best. I'm sure some who read this forum who work in touring rock and roll lighting would have a very different perspective than those who work in a theater. And that's how diverse our the industry is.

I'm currently taking a summer class at a university to pick up a few more credits to renew my teaching certification. They have a very nice theater program here. They have a 60+ year old proscenium space with a recent remodel and a 60X70 black box. There are about a dozen people on staff here, including one who specializes in just teaching and overseeing all the lighting. It's a good size program. The recently purchased a full set of S4 Jr. Zooms for the black box (50+?) and 30+ full size S4's for the main stage. the rest of the inventory probably contains 100 older 360Q's, and another 100+ Fresnels and pars. They are using an Expression in the black box and an Express with Emphasis system in the proscenium stage. It's a big well equipped theater program. They own 4 Technobeams that appear to be 5-10 years old.

So you can certainly see what their philosophy is. They are clearly focusing on spending their money to upgrade their conventional inventory. On the other hand, this is a 4 year University with a grad school program. A very solid argument can be made that they are not fully serving the needs of their graduates by only having 4 somewhat basic moving lights in the entire inventory.

Like I said, it's an interesting philosophical debate.

#### CLEFFEL

##### Member

I'd have to say, for me it breaks out like this:
High School: Focus on the basics...there's really no reason for a high school student to be worrying about programming 30 movers and utilizing an inventory of 400 fixtures. Give them a good base of knowledge in theory, basic drafting, different fixture types (pars, ERS, fresnels, etc.), basic memory console programming (ala Express(ion) line). Give them a base in technology with a little side note about the art.

College: This is where students should gain exposure to higher level fixtures. They should learn how to program with at least some moving lights and a higher level console (EOS, Hog, MA.) They should learn more about the art of design and how to apply the technical base they learned in high school.

But that's just my two cents.

Keep this thread going...I love it.

-Chris

#### Charc

##### Well-Known Member
I'd have to say, for me it breaks out like this:
High School: Focus on the basics...there's really no reason for a high school student to be worrying about programming 30 movers and utilizing an inventory of 400 fixtures. Give them a good base of knowledge in theory, basic drafting, different fixture types (pars, ERS, fresnels, etc.), basic memory console programming (ala Express(ion) line). Give them a base in technology with a little side note about the art.
College: This is where students should gain exposure to higher level fixtures. They should learn how to program with at least some moving lights and a higher level console (EOS, Hog, MA.) They should learn more about the art of design and how to apply the technical base they learned in high school.
But that's just my two cents.
Keep this thread going...I love it.
-Chris

Being in High School, I'd have to agree that a strong base in the technical knowledge seems to me to be the most important. I do dabble in design, but we don't draft or anything, but it's basically just me going "hmm. that looks good; well better than not having any light there."

I would, however, like an education in DMX controlled devices, basic moving light / scroller type stuff. To me that means like some mirrors, some scrollers, some gobo rotators, and maybe rent (hire, for you aussies) a few VL1000TI/S once a year. However, it'd all be used on a basic memory console, Strand 300 series / ETC Express.

##### Member
I agree a great thread. BASIC'S, BASIC'S,BASIC'S
I'm constantly amazed by some of the young people that come to work at our theatre. There lack of knowledge of the basic aspects of our craft. How to properly coil a cable (a pet pev of mine), treating the gear they use daily with respect/care. I could go on and on. I left the industry for a number of years. When I left there was no DMX and Altman fixtures where the mainstay and if you wanted moving lights you had to have Vari-lite's and a operator from them. I returned to find S4 fixtures and DMX control and any number of cool moving lights and control desks. It didn't take me long to pick up on these great new toys, but it still came down to basic's. How to read a plot, cable a show and focus. Skills I learned in high school and working after school at a small lighting rental company. When it comes time to get a show hung and ready to set cue's, I rather have one person who knows the basic's than four HIGH TECH know it all's.

#### Charc

##### Well-Known Member
I agree a great thread. BASIC'S, BASIC'S,BASIC'S
I'm constantly amazed by some of the young people that come to work at our theatre. There lack of knowledge of the basic aspects of our craft. How to properly coil a cable (a pet pev of mine), treating the gear they use daily with respect/care. I could go on and on. I left the industry for a number of years. When I left there was no DMX and Altman fixtures where the mainstay and if you wanted moving lights you had to have Vari-lite's and a operator from them. I returned to find S4 fixtures and DMX control and any number of cool moving lights and control desks. It didn't take me long to pick up on these great new toys, but it still came down to basic's. How to read a plot, cable a show and focus. Skills I learned in high school and working after school at a small lighting rental company. When it comes time to get a show hung and ready to set cue's, I rather have one person who knows the basic's than four HIGH TECH know it all's.

(This post is making me think. Maybe I don't know how to correctly coil a cable. Instead of potentially embarrassing myself, can I simply ask what the correct technique is?)

#### icewolf08

##### CBMod
CB Mods
(This post is making me think. Maybe I don't know how to correctly coil a cable. Instead of potentially embarrassing myself, can I simply ask what the correct technique is?)

That is probably a question for it's own thread, as different people prefer different coiling methods...

#### Charc

##### Well-Known Member
That is probably a question for it's own thread, as different people prefer different coiling methods...

So is this discussion going to get started again? Or has a consensus been reached?

#### Jezza

##### Active Member
First off, I'm on the basics side of the argument, but led me tell you all a story for a second.

the highschool where I previously held the positions of TD and LD, I was oh so blessed to be endowed with an underclassman named, well lets call him X for now. Well X came in as a freshman while I was still a sophomore and thought he knew everything. His parents were both NYC theater people, very successful, and he felt that he knew everything there was to know about lighting.

First day I met him, I was running a dance performance on our main stage, just some simple color washes, a few chases, nothing too exciting on an Express 48/96. We got to talking when he came up to the console and started running off the names of some big designers he knew and all the new moving lights he was interested in working with. I knew he was full of hot air so I stood him in front of the console and said, "Ok, build me a chase of channels 1, 2, 3, and 4." He sort of stared at me dumb-founded and then started fumbling around, until he succeeded in creating one hell of a mess one of my macros. Now, I was not doing this to be cruel, but I wanted to prove a point, which was "yes, you might know some big people and you might know about some of the latests and greatest gear out there, but that isn't what matters. The basics matter."

Over the past two years we have been forced to work together, for better and for worse, on every production my high school has put on and I have continually become more and more frustrated with his inability, reluctance to, and resentment towards learning the BASICS of theater lighting. How to coil a cable properly, how to tie it up on a pipe, how to hang a fixture, proper ladder safety, proper handling of instruments, simple electrical theory, you know, the basics! Sure, now he can program in circles on our Express and maybe even write in some cues on a HOG now, but he doesn't know a THING about what really matters.

So recently, we did a concert and rented 4 MAC 700s to fill in the 80+ instrument conventional rig. Our school has two Express 48/96s so I decided the best way to run the show would be one for conventionals and one for movers. Due to his whinning and complaining and me just wanting him to shut up, I let him program the mover board. He spent 3 days programming this one off, and cursed every day minute of it screaming his head off that he wanted a HOG iPC and that it would make his life so much easier and that you can't program movers on an Express.

As well all know, you CAN program movers on an Express. Sure its not the most streamlined way do it, nor the fastest, but its completely doable and can be executed very well if you have your wits about you and understand how the console is working. Well my friend X really didn't. He has been getting used to working with high end gear with high end friends and didn't learn his basics.

Needless to say, the show happened, although the moving light cues were NOT what I had wanted them to be. Now, I'm being forced out of the position by him and his get this, mother, to make way for their "rising designer". Please. No matter, I've get enough work with local theaters and production companies. My position is this: Your shouldn't be given the opportunity to design, until you've demonstrated good knowledge of the basics.

I guess the point I'm trying to make with all of this here is high school should be where you learn the BASICSSSS. Its plenty nice when we get to bring in some other gear for our big shows, but it doesn't help at all if half of my crew doesn't know how to hang it properly, or cable it properly, or understand how the standard S4 works.

High School should be the basics-rigging, electrical, basic programming, design fundamentals. College should be more design based, but with an emphasis on larger conventional inventories and more opportunities to rent/have access to more powerful control desks and equipment. Personally, I think that only juniors and seniors in college should be able to spec the newer, higher end gear for their shows. This will force students for the first two years to accomplish their design goals with conventional gear, as we all know is certainly possible.

There's my .02, although written way too late and with way too many mistakes I'm sure.

#### len

##### Well-Known Member
I didn't vote for either. Simply because in the real (i.e., the one where you get paid or you get fired) world the show goes with the equipment you have, not the equipment you want. And there are egos, there are budgets, there are availability ... you name it. Better to teach 3 ways to do something than "this is the only way"

#### Logos

##### Well-Known Member
Yeah, Having done small and medium scale theatre touring It's simply a matter of doing what you can with what you've got. Which I guess puts me with the learn the basics group.
Also down here, the gap between Music touring and top end musicals and grass roots theatre seems a lot wider than you guys talk about. I often find myself using the sort of gear we are talking about on the old lighting thread to light quite big shows.


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1
Numerical

### JEE Main 2021 (Online) 27th August Morning Shift

The number of distinct real roots of the equation 3x4 + 4x3 $-$ 12x2 + 4 = 0 is _____________.

Correct Answer is 4

## Explanation

3x4 + 4x3 $-$ 12x2 + 4 = 0

So, let f(x) = 3x4 + 4x3 $-$ 12x2 + 4

$\therefore$ f'(x) = 12x(x2 + x $-$ 2)

= 12x (x + 2) (x $-$ 1)

$\therefore$ f'(x) = 12x3 + 12x2 – 24x = 12x(x + 2) (x – 1)

Points of extrema are at x = 0, –2, 1

f(0) = 4

f(–2) = –28

f(1) = –1

So, 4 Real Roots
2
Numerical

### JEE Main 2021 (Online) 26th August Morning Shift

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then $\left( {{4 \over \pi } + 1} \right)k$ is equal to _____________.

Correct Answer is 36

## Explanation

Let x + y = 36

x is perimeter of square and y is perimeter of circle side of square = x/4

radius of circle = ${y \over {2\pi }}$

Sum Areas = ${\left( {{x \over 4}} \right)^2} + \pi {\left( {{y \over {2\pi }}} \right)^2}$

$= {{{x^2}} \over {16}} + {{{{(36 - x)}^2}} \over {4\pi }}$

For min Area :

$x = {{144} \over {\pi + 4}}$

$\Rightarrow$ Radius = y = 36 $-$ ${{144} \over {\pi + 4}}$

$\Rightarrow$ k = ${{36\pi } \over {\pi + 4}}$

$\left( {{4 \over \pi } + 1} \right)k$ = 36
3
Numerical

### JEE Main 2021 (Online) 17th March Evening Shift

Let f : [$-$1, 1] $\to$ R be defined as f(x) = ax2 + bx + c for all x$\in$[$-$1, 1], where a, b, c$\in$R such that f($-$1) = 2, f'($-$1) = 1 for x$\in$($-$1, 1) the maximum value of f ''(x) is ${{1 \over 2}}$. If f(x) $\le$ $\alpha$, x$\in$[$-$1, 1], then the least value of $\alpha$ is equal to _________.

Correct Answer is 5

## Explanation

$f(x) = a{x^2} + bx + c$

$f'(x) = 2ax + b,$

$f''(x) = 2a$

Given, $f''( - 1) = {1 \over 2}$

$\Rightarrow a = {1 \over 4}$

$f'( - 1) = 1 \Rightarrow b - 2a = 1$

$\Rightarrow b = {3 \over 2}$

$f( - 1) = a - b + c = 2$

$\Rightarrow c = {{13} \over 4}$

Now, $f(x) = {1 \over 4}({x^2} + 6x + 13),x \in [ - 1,1]$

$f'(x) = {1 \over 4}(2x + 6) = 0$

$\Rightarrow x = - 3 \notin [ - 1,1]$

$f(1) = 5,f( - 1) = 2$

$f(x) \le 5$

So, $\alpha$minimum = 5
4
Numerical

### JEE Main 2021 (Online) 26th February Evening Shift

Let a be an integer such that all the real roots of the polynomial
2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ___________.

Correct Answer is 2

## Explanation

Let, $f(x) = 2{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 10x + 10$

$\Rightarrow f'(x) = 10({x^4} + 2{x^3} + 3{x^2} + 2x + 1)$

$= 10\left( {{x^2} + {1 \over {{x^2}}} + 2\left( {x + {1 \over x}} \right) + 3} \right)$

$= 10\left( {{{\left( {x + {1 \over x}} \right)}^2} + 2\left( {x + {1 \over x}} \right) + 1} \right)$

$= 10{\left( {\left( {x + {1 \over x}} \right) + 1} \right)^2} > 0;\forall x \in R$

$\therefore$ f(x) is strictly increasing function. Since, it is an odd degree polynomial it will have exactly one real root.

Now, by observation.

$f( - 1) = 3 > 0$

$f( - 2) = - 64 + 80 - 80 + 40 - 20 + 10$

$= - 34 < 0$

$\Rightarrow f(x)$ has at least one root in $( - 2, - 1) \equiv (a,a + 1)$

$\Rightarrow a = - 2$

$\Rightarrow$ |a| = - 2

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# How to rewrite $7-\sqrt 5$ in root form without a minus sign?

How to rewrite $7-\sqrt 5$ in root form without a minus sign ? For clarity "root form " means an expression that only contains a finite amount of positive integers , additions , substractions , multiplications and root extractions (sqrt, cuberoot etc).

For example some quintic equations cannot be solved in root form.

A " root form without a minus sign " means an expression that only contains a finite amount of positive integers , additions , multiplications and root extractions (sqrt , cuberoot etc).

So the solution could look something like this :

$$7-\sqrt5 = \sqrt{...+1+(...)^{\frac{2}{3}}}+\sqrt{...+2(...)^{\frac{1}{3}}}$$

How to solve such problems ?

EDIT

Warning: $\dfrac{44}{7+\sqrt 5}$ is not a solution, since no divisions are allowed!

I got that answer 3 times now so I put it in the OP as a warning , not just the comments.

• Is $\frac{44}{7+\sqrt{5}} \,$ valid? Jan 7, 2014 at 23:17
• No it contains a division ! @K.Rmth
– mick
Jan 7, 2014 at 23:21
• Do you have some reason to suspect that an answer exists? I am skeptical. Jan 7, 2014 at 23:44
• $\displaystyle{\large -\left(-7 + \sqrt{5}\right)}$. Jan 8, 2014 at 0:23
• I don't understand the downvotes on nor closure of this question. The idea of expressions formed recursively using positive integers, addition, multiplication and root extractions is perfectly sensible and clear. I can't think of Galois or algebraic number theory machinery that keeps track of negative signs "internal" to expressions in order to tackle this problem, so I find it interesting. Upvoted and voted to reopen. Also, as the question was clear from the very beginning, it's strange there are three incorrect answers (one now deleted by others) that assumed it is simpler than it is.
– anon
Jan 12, 2014 at 4:10

$$7-\sqrt{5}$$ cannot be written in root form without a minus sign. I'm not exactly sure why you want to do this, but Galois theory proves you can't. My original answer was very brief, just quickly quoting Galois theory, but my friend convinced me it would be a good idea to add more details. So let me explain the key concept from Galois theory that we need:

We will say a number $$\alpha$$ is "Galois-maximal" if $$\alpha$$ is a positive real and there is some polynomial $$f(x)$$, with rational coefficients, such that $$\alpha$$ is a root of $$f(x)$$, and all roots $$\beta$$ of $$f(x)$$ satisfy $$|\beta| \leq \alpha$$.

There are two facts that allow us to show $$7- \sqrt{5}$$ cannot be written in root form without a minus sign. The first is that any number written in root form without a minus sign is Galois-maximal. The second is that $$7- \sqrt{5}$$ is not Galois-maximal. From these two, the conclusion follows trivially.

To prove the first claim, we do induction on the number of positive integers, additions, multiplications, and root extractions. The base case is that positive integers are Galois-maximal. Indeed, $$n$$ is the unique root of the polynomial $$x-n$$. For the induction step, we need to check that if $$\alpha$$ is Galois-maximal, then $$\alpha^{1/n}$$ is, and that if $$\alpha$$ and $$\beta$$ are Galois-maximal, then $$\alpha+\beta$$ and $$\alpha \beta$$ are.

For $$\alpha^{1/n}$$, take a polynomial $$f(x)$$ with rational coefficients whose largest root is $$\alpha$$, and consider the polynomial $$f(x^n)$$. The roots of this polynomial are the $$n$$th roots of the roots of $$f(x)$$, so $$\alpha^{1/n}$$ is a root, and all other roots are the $$n$$th root of something with absolute value at least $$\alpha$$, hence have absolute value at most $$\alpha$$.

For $$\alpha+\beta$$ and $$\alpha \beta$$, take polynomials $$f$$ and $$g$$ with rational coefficients such that $$\alpha$$ is a root of $$f$$, $$\beta$$ is a root of $$g$$, and they are each at least as large as the absolute value of all other roots. The key fact is that there are polynomials $$h_1$$ and $$h_2$$, each with rational coefficients, such that the roots of $$h_1$$ are exactly the sums of a root of $$f$$ and a root of $$g$$ and the roots of $$h_2$$ are the products of a root of $$f$$ and a root of $$g$$.

If we believe this fact, then we're done: certainly $$\alpha+\beta$$ is a root of $$h_1$$ and $$\alpha \beta$$ is a root of $$h_1$$. Moreover, any root of $$h_1+h_2$$ has the form $$x+y$$ with $$x$$ a root of $$h_1$$ so $$|x| \leq \alpha$$ and $$y$$ a root of $$h_2$$ so $$|y| \leq \beta$$ hence $$|x+y| \leq |x|+|y| \leq \alpha+\beta$$, and the same for $$h_2$$ except with multiplication.

It is easy to construct a polynomial with a given set of roots, but not completely obvious why, for this particular set of roots, it has rational coefficients. The slickest proof I know uses linear algebra, specifically, the companion matrix. This allows us to write matrices with rational entries $$M_1$$ and $$M_2$$ whose eigenvalues are the roots of $$f$$ and $$g$$ respectively. Then the eigenvalues of $$M_1 \otimes M_2$$ are the roots of $$f$$ times the roots of $$g$$, and the eigenvalues of $$M_1 \otimes I + I \otimes M_2$$ are the roots of $$f$$ plus the roots of $$g$$. Because $$M_1$$ and $$M_2$$ have rational entries, their characteristic polynomials have rational coefficients, and the eigenvalues are exactly the roots of the characteristic polynomial. This finishes the induction step.

Finally, we need to check $$7- \sqrt{5}$$ is not Galois-maximal. To do this, its sufficient to check every polynomial with rational coefficients that has $$7- \sqrt{5}$$ as a root also has $$7 + \sqrt{5}$$. By expanding $$f(7 - \sqrt{5})$$ out, we obtain a polynomial in $$-\sqrt{5}$$, which still has rational coefficients, so we have a polynomial with rational coefficients that has $$-\sqrt{5}$$ as a root, and we wish to show it has $$\sqrt{5}$$ as a root. Any time we see $$x^2$$ appearing in the polynomial, we can replace it with $$5$$ and the value of the polynomial at $$-\sqrt{5}$$ and $$\sqrt{5}$$ will not change. Once we have simplified in this way, we have a polynomial of degree at most one with rational coefficients. So it is of the form $$a - b\sqrt{5}=0$$ for $$a,b$$ rational. If $$a$$ or $$b$$ is nonzero, this implies $$\sqrt{5}= a/b$$, which is impossible, so then $$a$$ and $$b$$ are both zero, but then $$a + b\sqrt{5}$$ is zero as well, so $$\sqrt{5}$$ is indeed a root, as desired. QED.

Using the same argument, it is possible to produce many other examples of positive real numbers that can be written in root form with a minus sign but can't be written in root form without it.

• Im not sure what you meant by Galois conjugate. I assume by conjugate you meant $7+\sqrt 5$. If $x>y$ then $x^{1/n}>y^{1/n}$ So I do not know how those powers change anything. What is clear to me is that $x>1$ is neccessary since we cannot expect a "root form without a MINUS" to be smaller than 1. ( notice there are no divisions ! ). Im not an expert in Galois theory.
– mick
Jan 10, 2014 at 20:55
• This is an enticing answer, but it needs more justification than a single sentence. You need a rigorous proof for why the set A of numbers larger than all their Galois conjugates is (i) closed under addition, (ii) closed under multiplication, and (iii) closed under $n$th roots. (Of course, the positive integers are all contained in $A$, so that part is fine.)
– 6005
Dec 30, 2014 at 6:16
• It's closed under addition because the galois conjugates of a sum are the sum of the galois conjugates, and addition is monotonic. Same with multiplication. For $n$th roots, the galois conjugates of the $n$th roots are all of the $n$th roots of the galois conjugates. The largest is the principal root, and the principal root is a monotonic function, so again we are fine. Dec 30, 2014 at 12:44
• To be technical: you mean the real part of the principal root function is monotinc with respect to the real part of the argument, if the imaginary part of the argument is held fixed. One doesn't speak of complex numbers being "larger" than other complex numbers, except perhaps in terms of modulus (which is not how you're arguing, I don't think).
– anon
Jan 2, 2015 at 8:45
• @Goos since products can be expanded I think this (ii) can be ignored.
– mick
Jan 10, 2015 at 23:18

$$7 - \sqrt{5} = (7 - \sqrt{5}) \bigg(\frac{7 + \sqrt{5}}{7 + \sqrt{5}} \bigg) = \frac{44}{7 + \sqrt{5}}$$

• see the edit I made.
– mick
Jan 7, 2014 at 23:25

Note: the following argument appears to fail. For example, $\sqrt{1+\sqrt{5}}$ cannot be written in the form I suggest.

Assuming that we are taking principal roots, I don't believe that this is possible. Inductively, every such expression can be written in the form $\alpha+\beta\sqrt{5}$, where $\alpha$ and $\beta$ are nonnegative real algebraic numbers fixed by some fixed Galois action $\sigma:\sqrt{5}\mapsto -\sqrt{5}$ (as achille hui points out, it is necessary to choose $\sigma$ to fix $\mathbb{R}$, or at least, to fix the algebraic reals in some radical extension, which is a subtle point, possibly a major flaw in the argument).

First, note that such a representation (with $\alpha,\beta$ algebraic, not necessarily nonnegative reals) always exists and is unique: if $x=\alpha+\beta\sqrt{5}$ is algebraic, then $\sigma x = \alpha-\beta\sqrt{5}$, so $\alpha=\frac{x+\sigma x}{2}$ and $\beta=\frac{x-\sigma x}{2\sqrt{5}}$.

The base case for our induction is the positive integers. The inductive steps of addition and multiplication are easy, so it remains to look at $n$-th roots, say $(\alpha+\beta\sqrt{5})^{1/n} = \gamma + \delta\sqrt{5}$ with $\alpha$ and $\beta$ as above. Then $\gamma$ and $\delta$ are certainly real, by the above formulas and the fact that $(\alpha+\beta\sqrt{5})^{1/n}$ is real. By expanding out $(\gamma+\delta\sqrt{5})^n$ and comparing signs, we can see that $\gamma$ and $\delta$ are both nonnegative.

It follows that any expression involving only addition, multiplication, principal roots, and positive integers, cannot produce $7-\sqrt{5}$ or anything like it, since the $\sqrt{5}$ coefficient is negative and both coefficients are rational (hence fixed by any choice of $\sigma$).

On the other hand, if we are not taking principal roots, then the problem is almost trivial, because we can write $\sqrt{-1} = 1^{1/4}$ for a suitable fourth root of $1$, and then write $7-\sqrt{5}=7+(\sqrt{-1})^2\sqrt{5}$.

• +1 even though I'm still not sure about the part "$\gamma$ and $\delta$ are certainty real". Jan 8, 2014 at 1:02
• @achillehui $z=\gamma+\delta\sqrt{5}$ is real (since I am always taking the principal root, and we started with a nonnegative real), and $\sigma$ fixes $\mathbb{R}$. And $\gamma$ and $\delta$ can be written in terms of $z$ and $\sigma z$. Jan 8, 2014 at 1:07
• Okay, if $\sigma$ can be chosen to fix $\mathbb{R}$. But what happens if $\alpha - \beta\sigma < 0$ and $n$ is even? Shouldn't we have $(\alpha - \beta\sqrt{5})^{1/n} = \gamma - \delta\sqrt{5}$? that's what bothering me. Jan 8, 2014 at 1:20
• @achillehui You're right, I do need to pick $\sigma$ as such. But what you're saying can't happen by inductive hypothesis, if I understand correctly. The thing we're taking the root of already has both coefficients nonnegative by assumption. Jan 8, 2014 at 1:23
• @achillehui It may be that $\sigma$ cannot be chosen to fix $\mathbb{R}$; I have to think about it for a bit. But we only need to work in a finite extension, and it's true there. Jan 8, 2014 at 1:31

Does multiplying by $\frac{7 +\sqrt 5}{7+ \sqrt 5}$ do what you want?

• see the edit I made. Sorry to confuse.
– mick
Jan 7, 2014 at 23:26

Let $n$ be some positive integer.

$(7-\sqrt 5)^n = r$ where $r$ is in the desired form but there is no root symbol over the entire expression on the RHS.

Now every root symbol in $r$ denotes a principal root.

The number of ways that we can write the same identity $r$ but replace one or more of its roots by a nonprincipal is given by the product $a_1 a_2 ... a_m = k$ where the amount of roots symbols are $m$ and the roots are $*^{1/a_i}$. We denote the Original $r$ and the nonprincipal ones as

$r_0 =r$

$r_i$

( i is just an index )

Therefore (by very basic galois theory) there exists an integer polynomial $P$ of degree $k$ such that $P(x) = (x-r_0)(x-r_1)(x-r_2)...(x-r_i)...(x-r_k).$ ( notice that $P$ is completely factored and hence we have unique factorization )

Now clearly $r = r_0$ is the max in absolute value of all the $r_i$. Call this PROPERTY 1.

It is easy to show with some elementary ring theory or number theory that if

$(7-\sqrt 5)^n$ is a zero of an integer polynomial then so is $(7+\sqrt 5)^n$.

Therefore some $r_i$ must be equal to $(7+\sqrt 5)^n$. This is PROPERTY 2.

Now notice that |$(7+\sqrt 5)^n$| > |$(7-\sqrt 5)^n$|. This is PROPERTY 3.

Combining property 1,2 and 3 we see that $r$ must be both the largest in absolute value of all the $r_i$ and NOT the largest in absolute value of all the $r_i$.

• This is a good idea, but poorly expressed. You'll want to clean up the exposition big time (you seem to have a big problem with communication). The key idea is what you call proposition 1; since it's so fundamental to the argument I recommend justifying it, even if it is intuitive. Here's a hint how: use the fact that root functions are monotonic increasing on positive reals, and show that if $r,s$ vary over the roots of two polynomials and $|r|$ and $|s|$ are both maximized when $r$ and $s$ are positive reals, then $|r+s|$ is maximized at those same values of $r$ and $s$. Induct.
• @anon I think the main issue is that $(x-r_0)(x-r_1)...$ must indeed be an integer polynomial. For a fixed small degree polynomial this is easy to prove. Is there an easy way to prove that regardless of the structure ( amount of additions , roots etc ) of the roots $r_n$ for any degree ?


text:
# Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs

Craig Gentry, Jens Groth, Yuval Ishai, Chris Peikert, Amit Sahai, Adam Smith

Research output: Contribution to journalArticle

26 Citations (Scopus)

### Abstract

A non-interactive zero-knowledge (NIZK) proof can be used to demonstrate the truth of a statement without revealing anything else. It has been shown under standard cryptographic assumptions that NIZK proofs of membership exist for all languages in NP. While there is evidence that such proofs cannot be much shorter than the corresponding membership witnesses, all known NIZK proofs for NP languages are considerably longer than the witnesses. Soon after Gentry’s construction of fully homomorphic encryption, several groups independently contemplated the use of hybrid encryption to optimize the size of NIZK proofs and discussed this idea within the cryptographic community. This article formally explores this idea of using fully homomorphic hybrid encryption to optimize NIZK proofs and other related cryptographic primitives. We investigate the question of minimizing the communication overhead of NIZK proofs for NP and show that if fully homomorphic encryption exists then it is possible to get proofs that are roughly of the same size as the witnesses. Our technique consists in constructing a fully homomorphic hybrid encryption scheme with ciphertext size $$|m|+{\mathrm {poly}}(k)$$|m|+poly(k), where $$m$$m is the plaintext and $$k$$k is the security parameter. Encrypting the witness for an NP-statement allows us to evaluate the NP-relation in a communication-efficient manner. We apply this technique to both standard non-interactive zero-knowledge proofs and to universally composable non-interactive zero-knowledge proofs. The technique can also be applied outside the realm of non-interactive zero-knowledge proofs, for instance to get witness-size interactive zero-knowledge proofs in the plain model without any setup or to minimize the communication in secure computation protocols.

Original language English (US) 820-843 24 Journal of Cryptology 28 4 https://doi.org/10.1007/s00145-014-9184-y Published - Oct 30 2015

### Fingerprint

Zero-knowledge Proof
Homomorphic
Encryption
Cryptography
Minimise
Communication
Homomorphic Encryption
Optimise
Interactive Proofs
Secure Computation

### All Science Journal Classification (ASJC) codes

• Software
• Computer Science Applications
• Applied Mathematics

### Cite this

Gentry, Craig ; Groth, Jens ; Ishai, Yuval ; Peikert, Chris ; Sahai, Amit ; Smith, Adam. / Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs. In: Journal of Cryptology. 2015 ; Vol. 28, No. 4. pp. 820-843.
title = "Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs",
abstract = "A non-interactive zero-knowledge (NIZK) proof can be used to demonstrate the truth of a statement without revealing anything else. It has been shown under standard cryptographic assumptions that NIZK proofs of membership exist for all languages in NP. While there is evidence that such proofs cannot be much shorter than the corresponding membership witnesses, all known NIZK proofs for NP languages are considerably longer than the witnesses. Soon after Gentry’s construction of fully homomorphic encryption, several groups independently contemplated the use of hybrid encryption to optimize the size of NIZK proofs and discussed this idea within the cryptographic community. This article formally explores this idea of using fully homomorphic hybrid encryption to optimize NIZK proofs and other related cryptographic primitives. We investigate the question of minimizing the communication overhead of NIZK proofs for NP and show that if fully homomorphic encryption exists then it is possible to get proofs that are roughly of the same size as the witnesses. Our technique consists in constructing a fully homomorphic hybrid encryption scheme with ciphertext size $$|m|+{\mathrm {poly}}(k)$$|m|+poly(k), where $$m$$m is the plaintext and $$k$$k is the security parameter. Encrypting the witness for an NP-statement allows us to evaluate the NP-relation in a communication-efficient manner. We apply this technique to both standard non-interactive zero-knowledge proofs and to universally composable non-interactive zero-knowledge proofs. The technique can also be applied outside the realm of non-interactive zero-knowledge proofs, for instance to get witness-size interactive zero-knowledge proofs in the plain model without any setup or to minimize the communication in secure computation protocols.",
author = "Craig Gentry and Jens Groth and Yuval Ishai and Chris Peikert and Amit Sahai and Adam Smith",
year = "2015",
month = "10",
day = "30",
doi = "10.1007/s00145-014-9184-y",
language = "English (US)",
volume = "28",
pages = "820--843",
journal = "Journal of Cryptology",
issn = "0933-2790",
publisher = "Springer New York",
number = "4",

}

Gentry, C, Groth, J, Ishai, Y, Peikert, C, Sahai, A & Smith, A 2015, 'Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs', Journal of Cryptology, vol. 28, no. 4, pp. 820-843. https://doi.org/10.1007/s00145-014-9184-y

Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs. / Gentry, Craig; Groth, Jens; Ishai, Yuval; Peikert, Chris; Sahai, Amit; Smith, Adam.

In: Journal of Cryptology, Vol. 28, No. 4, 30.10.2015, p. 820-843.

Research output: Contribution to journalArticle

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N2 - A non-interactive zero-knowledge (NIZK) proof can be used to demonstrate the truth of a statement without revealing anything else. It has been shown under standard cryptographic assumptions that NIZK proofs of membership exist for all languages in NP. While there is evidence that such proofs cannot be much shorter than the corresponding membership witnesses, all known NIZK proofs for NP languages are considerably longer than the witnesses. Soon after Gentry’s construction of fully homomorphic encryption, several groups independently contemplated the use of hybrid encryption to optimize the size of NIZK proofs and discussed this idea within the cryptographic community. This article formally explores this idea of using fully homomorphic hybrid encryption to optimize NIZK proofs and other related cryptographic primitives. We investigate the question of minimizing the communication overhead of NIZK proofs for NP and show that if fully homomorphic encryption exists then it is possible to get proofs that are roughly of the same size as the witnesses. Our technique consists in constructing a fully homomorphic hybrid encryption scheme with ciphertext size $$|m|+{\mathrm {poly}}(k)$$|m|+poly(k), where $$m$$m is the plaintext and $$k$$k is the security parameter. Encrypting the witness for an NP-statement allows us to evaluate the NP-relation in a communication-efficient manner. We apply this technique to both standard non-interactive zero-knowledge proofs and to universally composable non-interactive zero-knowledge proofs. The technique can also be applied outside the realm of non-interactive zero-knowledge proofs, for instance to get witness-size interactive zero-knowledge proofs in the plain model without any setup or to minimize the communication in secure computation protocols.

AB - A non-interactive zero-knowledge (NIZK) proof can be used to demonstrate the truth of a statement without revealing anything else. It has been shown under standard cryptographic assumptions that NIZK proofs of membership exist for all languages in NP. While there is evidence that such proofs cannot be much shorter than the corresponding membership witnesses, all known NIZK proofs for NP languages are considerably longer than the witnesses. Soon after Gentry’s construction of fully homomorphic encryption, several groups independently contemplated the use of hybrid encryption to optimize the size of NIZK proofs and discussed this idea within the cryptographic community. This article formally explores this idea of using fully homomorphic hybrid encryption to optimize NIZK proofs and other related cryptographic primitives. We investigate the question of minimizing the communication overhead of NIZK proofs for NP and show that if fully homomorphic encryption exists then it is possible to get proofs that are roughly of the same size as the witnesses. Our technique consists in constructing a fully homomorphic hybrid encryption scheme with ciphertext size $$|m|+{\mathrm {poly}}(k)$$|m|+poly(k), where $$m$$m is the plaintext and $$k$$k is the security parameter. Encrypting the witness for an NP-statement allows us to evaluate the NP-relation in a communication-efficient manner. We apply this technique to both standard non-interactive zero-knowledge proofs and to universally composable non-interactive zero-knowledge proofs. The technique can also be applied outside the realm of non-interactive zero-knowledge proofs, for instance to get witness-size interactive zero-knowledge proofs in the plain model without any setup or to minimize the communication in secure computation protocols.

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text:
Prove in two different ways-“If the centre of $G$ is of index $n$,prove that every conjugacy class has atmost $n$ elements .”

Solution:By Orbit-Stabilizer theorem,$\vert G:G_g\vert$.$\vert G_g:Z(G) \vert$=$\vert G:Z(G)\vert =n \implies \vert G:G_g\vert$ divides $n$.

When the action is by conjugation,the no. of conjugates of $g$ is $\vert G:C_G(g)\vert$.

Thus,$\vert G:C_G(g)\vert$ divides $n$,which implies that every conjugacy class has atmost $n$ elements.

I'm not sure whether this is correct,please tell me if i'm missing some detail.

Also,i don't know how to prove this in another way.

• It is not really a different way: If $C$ is a conjugacy class, then $G/Z(G)$ acts on $C$ (by conjugation) and the action is transitive. – Hagen von Eitzen Jul 30 '16 at 12:39


text:
# The product of the perpendiculars from the two points (±4, 0) to the line 3x cos ϕ + 5y sin ϕ = 15 is

This question was previously asked in
NDA (Held On: 16 Dec 2015) Maths Previous Year paper
View all NDA Papers >
1. 25
2. 16
3. 9
4. 8

Option 3 : 9
Free
Electric charges and coulomb's law (Basic)
38063
10 Questions 10 Marks 10 Mins

## Detailed Solution

Concept:

Perpendicular Distance of a Point from a Line:

Let us consider a plane given by the Cartesian equation, Ax + By + C = 0

And a point whose coordinate is, (x1, y1)

Now, distance = $$\left| {\frac{{{\rm{A}}{{\rm{x}}_1}{\rm{\;}} + {\rm{\;B}}{{\rm{y}}_1}{\rm{\;}} + {\rm{\;c}}}}{{\sqrt {{{\rm{A}}^2}{\rm{\;}} + {\rm{\;\;}}{{\rm{B}}^2}{\rm{\;}}} }}} \right|$$

Calculation:

Given equation of line is 3x cos ϕ + 5y sin ϕ = 15

⇒ 3x cos ϕ + 5y sin ϕ – 15 = 0

Let perpendiculars distance from the two points (±4, 0) are p1 and p2

Lengths of perpendicular from the point (+ 4, 0),

$${{\rm{p}}_1} = {\rm{\;}}\left| {\frac{{12{\rm{\;}}\cos \phi - 15}}{{\sqrt {9{\rm{\;}}{{\cos }^2}\phi + 25{\rm{\;}}{{\sin }^2}\phi } }}} \right|$$       …. (1)

Lengths of perpendicular from the point (- 4, 0),

$${{\rm{p}}_2} = {\rm{\;}}\left| {\frac{{ - 12{\rm{\;}}\cos \phi - 15}}{{\sqrt {9{\rm{\;}}{{\cos }^2}\phi + 25{\rm{\;}}{{\sin }^2}\phi } }}} \right|$$       …. (2)

Multiplying equation 1 and 2,

$$\Rightarrow {{\rm{p}}_1}{{\rm{p}}_2} = {\rm{\;}}\left| {\frac{{\left( {12{\rm{\;}}\cos \phi - 15} \right)\left( { - 12{\rm{\;}}\cos \phi - 15} \right)}}{{{{\left( {\sqrt {9{\rm{\;}}{{\cos }^2}\phi + 25{\rm{\;}}{{\sin }^2}\phi } } \right)}^2}}}} \right|$$

$$\Rightarrow {{\rm{p}}_1}{{\rm{p}}_2} = {\rm{\;}}\left| {\frac{{ - \left( {12{\rm{\;}}\cos \phi - 15} \right)\left( {12{\rm{\;}}\cos \phi + 15} \right)}}{{9{\rm{\;}}{{\cos }^2}\phi + 25{\rm{\;}}{{\sin }^2}\phi }}} \right|$$

$$\Rightarrow {{\rm{p}}_1}{{\rm{p}}_2} = {\rm{\;}}\left| {\frac{{\left( {144{\rm{\;}}\left( {1 - {\rm{\;}}{{\sin }^2}\phi } \right) - 225} \right)}}{{9{\rm{\;}}\left( {1 - {\rm{\;}}{{\sin }^2}\phi } \right) + 25{\rm{\;}}{{\sin }^2}\phi }}} \right|$$

$$\Rightarrow {{\rm{p}}_1}{{\rm{p}}_2} = {\rm{\;}}\left| {\frac{{ - 9\;\left( {9 + 16{\rm{\;}}{{\sin }^2}\phi } \right)}}{{9 + 16{\rm{\;}}{{\sin }^2}\phi }}} \right| = 9$$


text:
# Evaluate e^-0.7-e^-1.4

Rewrite the expression using the negative exponent rule .
Rewrite the expression using the negative exponent rule .
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Evaluate e^-0.7-e^-1.4


text:
# Synopsis: An all-organic almost one-dimensional ferromagnet

#### Magnetic order in the purely organic quasi-one-dimensional ferromagnet 2-benzimidazolyl nitronyl nitroxide

Tadashi Sugano, Stephen J. Blundell, Tom Lancaster, Francis L. Pratt, and Hatsumi Mori

Published November 4, 2010

Organic magnets provide a fascinating playground for testing models of magnetism, as rich carbon chemistry allows one to tune the magnetic, optical, and other properties of these materials. Nitronyl nitroxides are a family of particularly versatile organic radicals, which provided the first example of a purely organic ferromagnet p-NPNN, formed by chains of weakly interacting molecules.

In a Rapid Communication published in Physical Review B, Tadashi Sugano and collaborators from Japan and the UK use muon-spin rotation and electron spin resonance techniques to study $2$-benzimidazolyl nitronyl nitroxide ($2$-BIMNN), another member of the nitronyl nitroxides family. They find that $2$-BIMMN displays long-range ferromagnetic ordering, with a transition temperature of ${T}_{\text{c}}=1\phantom{\rule{0.333em}{0ex}}\text{K}$. In the short list of quasi-one-dimensional purely organic ferromagnets, $2$-BIMMN turns out to be especially interesting, as the ratio of the transition temperature to the exchange energy J that describes intrachain spin-spin interactions is small, indicating that the coupling between different chains is extremely weak, and therefore $2$-BIMMN can serve as a nearly ideal model of a one-dimensional ferromagnetically coupled spin chain. – Ashot Melikyan


text:
# 1 number

The tag <number> allows the creation of numeric data type of different sizes and precision. The package <number> provides functions for the localized conversion of string formats to numeric and vice versa.

Typically, the builder <number> will be used to generate numbers automatically, so the resulting type has an efficient form for the indicated data. By default, the integer values will use the integer type, while the floating point numbers will use the double type.

Optionally, the destination type can be indicated through the attribute type. In this cases, the resulting type will be indicated. For example:

Examples of construction of numeric types
<number>0</number> 0 integer java.lang.Integer
<number type='smallint'>0</number> 0 smallint java.lang.Short
<number>0.15</number> 0.15 double java.lang.Double
<number type='decimal'>0.15</number> 0.15 decimal java.math.BigDecimal
<number type='decimal' scale='3'>0.15</number> 0.150 decimal java.math.BigDecimal

#### Notas

The decimal type can accept the attribute scale to indicate the number of decimals.

<number    type='type'    scale='scale'>    <value /> !</number>

#### Exceptions

The function needs a entry value.

The scale is not well specified.

unsupported number type [...]

Type of numeric value not admitted.

unable to convert [...]

Impossible conversion.

#### Remarks

It is possible to obtain a numeric value of smallint, integer, long, float, double or decimal type. The type of numeric value obtained depends of the argument which receives the script-tag number or the specified attribute "type".If you want to indicate the amount of decimal which supports a numeric value, this should be of a type which supports decimals, otherwise it would not make sense. If the obtained number has an amount of decimals greater than the specified in the attribute "scale", these decimals will be rounded up.

Example

Define a numeric variable.

Copy
<xsql-script name='number_sample1'>
<body>
<set name='number'>
<number>13.4</number>
</set>
<println><number /></println>
<!-- The value shown is 13.4 -->
</body>
</xsql-script>
Example

Define a numeric variable with type and scale predefined.

Copy
<xsql-script name='number_sample2'>
<body>
<set name='number'>
<number type='decimal' scale='2'>13.4655</number>
</set>
<println><number /></println>
<!-- The value shown is 13.47, cuts and round up. -->

<set name='number'>
<number type='decimal' scale='2'>13.4635</number>
</set>
<println><number /></println>
<!-- The value shown is 13.46, cuts and round up. -->
</body>

</xsql-script>


text:
# Difference between dynamic pooling and static pooling in convolutional neural networks

Since yesterday I was thinking that pooling layer in CNN has fixed size(e.g. 2 by 2). Then I saw in this paper: http://phd.nal.co/papers/Kalchbrenner_DCNN_ACL14

We define a convolutional neural network architecture and apply it to the semantic modelling of sentences. The network handles input sequences of varying length. The layers in the network interleave one-dimensional convolutional layers and dynamic k-max pooling layers. Dynamic k-max pooling is a generalisation of the max pooling operator. The max pooling operator is a non-linear subsampling function that returns the maximum of a set of values (LeCun et al., 1998). The operator is generalised in two respects. First, k-max pooling over a linear sequence of values returns the subsequence of k maximum values in the sequence, instead of the single maximum value. Secondly, the pooling parameter k can be dynamically chosen by making k a function of other aspects of the network or the input

Am I correct thinking, that this dynamic pooling is for trim sentences(of variable lengths of words), to have vector(vector for supervised learning - see picture on page 4) of constant size?


text:
## Q. 4.6

Indicate the type and number of orbitals in each of the following energy levels or sublevels:
a.  $3p \space sublevel$             b.  $n = 2$           c.  $n = 3$           d.  $4d \space sublevel$

## Verified Solution

a.  The $3p$ sublevel contains three $3p$ orbitals.
b.  The $n = 2$ energy level consists of one $2s$ and three $2p$ orbitals.
c.  The $n = 3$ energy level consists of one $3s$, three $3p$, and five $3d$ orbitals.
d.  The $4d$ sublevel contains five $4d$ orbitals.

 ANALYZE THE PROBLEM Given Need Connect energy level, sublevel orbitals n, orbitals in s, p, d, f


text:
# Super tangent bundles

## Idea

The tangent bundle of an ordinary manifold is a vector bundle whose sheaf of sections? is given by the derivations of the structure sheaf. The same idea applies to a supermanifold to produce a super vector bundle.

## Definition

The super tangent bundle $T X$ of a supermanifold $X$ is given by the sheaf $U \mapsto Der O_X(U)$.

So a super tangent vector is a global section of this sheaf of derivations.

## Example

On the supermanifold $\mathbb{R}^{1|1}$ with its canonical coordinates

$t \in C^\infty(\mathbb{R}^{1|1})^{ev}$
$\theta \in C^\infty(\mathbb{R}^{1|1})^{odd}$

there is the odd vector field

$D \coloneqq \partial_\theta + \theta \cdot \partial_{t}$

whose super Lie bracket with itself vanishes

$[D, D] = 0 \,.$

=–

###### Claim

This odd vector field $D$ is left invariant with respect to the super translation group structure on $\mathbb{R}^{1|1}$.

This means that $Lie(\mathbb{R}^{1|1})$ is free on one odd generator.

Revised on December 7, 2011 02:06:32 by Toby Bartels (64.89.53.157)


text:
### Multi-Instance Security and its Application to Password-Based Cryptography

Mihir Bellare, Thomas Ristenpart, and Stefano Tessaro

##### Abstract

This paper develops a theory of multi-instance (mi) security and applies it to provide the first proof-based support for the classical practice of salting in password-based cryptography. Mi-security comes into play in settings (like password-based cryptography) where it is computationally feasible to compromise a single instance, and provides a second line of defense, aiming to ensure (in the case of passwords, via salting) that the effort to compromise all of some large number $m$ of instances grows linearly with m. The first challenge is definitions, where we suggest LORX-security as a good metric for mi security of encryption and support this claim by showing it implies other natural metrics, illustrating in the process that even lifting simple results from the si setting to the mi one calls for new techniques. Next we provide a composition-based framework to transfer standard single-instance (si) security to mi-security with the aid of a key-derivation function. Analyzing password-based KDFs from the PKCS#5 standard to show that they meet our indifferentiability-style mi-security definition for KDFs, we are able to conclude with the first proof that per password salts amplify mi-security as hoped in practice. We believe that mi-security is of interest in other domains and that this work provides the foundation for its further theoretical development and practical application.

Available format(s)
Category
Secret-key cryptography
Publication info
Published elsewhere. Preliminary version in CRYPTO 2012. This is the full version.
Keywords
Contact author(s)
mihir @ eng ucsd edu
History
2013-05-31: last of 2 revisions
See all versions
Short URL
https://ia.cr/2012/196

CC BY

BibTeX

@misc{cryptoeprint:2012/196,
author = {Mihir Bellare and Thomas Ristenpart and Stefano Tessaro},
title = {Multi-Instance Security and its Application to Password-Based Cryptography},
howpublished = {Cryptology ePrint Archive, Paper 2012/196},
year = {2012},
note = {\url{https://eprint.iacr.org/2012/196}},
url = {https://eprint.iacr.org/2012/196}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.


text:
### - Art Gallery -

The Kac–Bernstein theorem is one of the first characterization theorems of mathematical statistics. It is easy to see that if the random variables ξ {\displaystyle \xi } \xi and η {\displaystyle \eta } \eta are independent and normally distributed, then their sum and difference are also independent. The Kac–Bernstein theorem states that the independence of the sum and difference of two independent random variables characterizes the normal distribution (the Gauss distribution). This theorem was proved independently by Polish-American mathematician Mark Kac and Soviet mathematician Sergei Bernstein.

Formulation

Let $$\xi$$ and $$\eta$$ are independent random variables. If $${\displaystyle \xi +\eta }$$ and $${\displaystyle \xi -\eta }$$ are independent then $$\xi$$ and $$\eta$$ have normal distributions (the Gaussian distribution).

Generalization

A generalization of the Kac–Bernstein theorem is the Darmois–Skitovich theorem, in which instead of sum and difference linear forms from n independent random variables are considered.

References

Kac M. "On a characterization of the normal distribution," American Journal of Mathematics. 1939. 61. pp. 726—728.
Bernstein S. N. "On a property which characterizes a Gaussian distribution," Proceedings of the Leningrad Polytechnic Institute. 1941. V. 217, No 3. pp. 21—22.


text:
# LHC explores what LEP hinted at: CP-violating type-I 2HDM

The Large Hadron Collider is shown to have great scope for a light charged Higgs discovery, in the context of the CP-violating type-I two Higgs doublet model. This scenario with similar masses of $H^\pm$ and W was suggested by the puzzling departure from charged current lepton universality found in the LEP data. With the lightest neutral Higgs mass set to 125 GeV, the charged-neutral Higgs associated production mechanism can cause a significant excess in the $\tau u b \bar{b}$ events over a vast range of tan beta as long as the Higgs mixing pattern avoids a few limiting cases. Thanks to the low $H^\pm$ mass, the charged Higgs loop can play a striking role in neutral Higgs decays into $\gamma \gamma$, thereby compensating for a suppressed gluon-gluon fusion rate. The effect of scalar-pseudo-scalar mixing on loop-induced Higgs signals is also discussed.

Author: Wolfgang Mader; Jae-hyeon Park; Giovanni Marco Pruna; Dominik Stöckinger; Arno Straessner

Source: https://archive.org/


text:
# 3.2: Important Description Logics

There are very many DLs, of which some are used more often than others. In this section, we will first look at $$\mathcal{ALC}$$, for it is typically one of the languages used in DL courses, a basis to add various language features to, and it is much easier for showing how the principles of tableau work for DLs. Subsequently, we list the more expressive $$\mathcal{SROIQ}$$, and finally comment on leaner fragments that are computationally better behaved.

## A Basic DL to Start With: $$\mathcal{ALC}$$

The DL language $$\mathcal{ALC}$$—which stands for Attributive Language with Concept negation—contains the following elements:

- Concepts denoting entity types/classes/unary predicates/universals, including top $$\top$$ and bottom $$\bot$$;

- Roles denoting relationships/associations/binary predicates/properties;

- Constructors: and $$\sqcap$$, or $$\sqcup$$, and not $$\neg$$; quantifiers $$\forall$$ and $$\exists$$;

- Complex concepts using constructors: Let $$C$$ and $$D$$ be concept names, $$R$$ a role name, then

– $$\neg C, C\sqcap D,$$ and $$C\sqcup D$$ are concepts, and

– $$\forall R.C$$ and $$\exists R.C$$ are concepts

- Individuals

Some examples that can be represented in $$\mathcal{ALC}$$ are, respectively:

- Concepts (primitive, atomic); e.g., Book, Course

- Roles; e.g., enrolled, reads

- Complex concepts; e.g.,

– $$\texttt{Student}\sqcap\exists\texttt{enrolled.(Course}\sqcup\texttt{DegreeProgramme)}$$ (this is a primitive concept)

– $$\texttt{Mother}\sqsubseteq\texttt{Woman}\sqcap\exists\texttt{ParentOf.Person}$$

– $$\texttt{Parent}\equiv\texttt{(Male}\sqcup\texttt{Female)}\sqcap\exists\texttt{ParentOf.Mammal}\sqcap\exists\texttt{caresFor.Mammal}$$ (this is a defined concept)

- Individuals; e.g., Student(Andile), Mother(Katniss), $$\neg\texttt{Student(Katniss)}$$, enrolled(Andile, COMP101)

As usual, the meaning is defined by the semantics of $$\mathcal{ALC}$$, and it follows the same approach as we have seen for the other languages that have passed the revue (recollect FOL and model-theoretic semantics from "2.1: First Order Logic Syntax and Semantics"). First, there is a domain of interpretation, and an interpretation, where:

– Domain $$\Delta$$ is a non-empty set of objects

– Interpretation: $$\cdot^{\mathcal{I}}$$ is the interpretation function, domain $$\Delta^{\mathcal{I}}$$

– $$\cdot^{\mathcal{I}}$$ maps every concept name $$A$$ to a subset $$A^{\mathcal{I}}\subseteq\Delta^{\mathcal{I}}$$

– $$\cdot^{\mathcal{I}}$$ maps every role name $$R$$ to a subset $$R^{\mathcal{I}}\subseteq\Delta^{\mathcal{I}}\times\Delta^{\mathcal{I}}$$

– $$\cdot^{\mathcal{I}}$$ maps every individual name $$a$$ to elements of $$\Delta^{\mathcal{I}} :a^{\mathcal{I}}\in\Delta^{\mathcal{I}}$$

Note $$\PageIndex{1}$$

$$\top^{\mathcal{I}}=\Delta^{\mathcal{I}}$$ and $$\bot^{\mathcal{I}}=\emptyset$$

Using the typical notation where $$C$$ and $$D$$ are concepts, $$R$$ a role, and $$a$$ and $$b$$ are individuals, then they have the following meaning, with on the left-hand side of the “$$=$$” the syntax of $$\mathcal{ALC}$$ under an interpretation and on the right-hand side its semantics:

- $$(\neg C)^{\mathcal{I}}=\Delta^{\mathcal{I}}\backslash C^{\mathcal{I}}$$

- $$(C\sqcap D)^{\mathcal{I}}=C^{\mathcal{I}}\cap D^{\mathcal{I}}$$

- $$(C\sqcup D)^{\mathcal{I}}=C^{\mathcal{I}}\cup D^{\mathcal{I}}$$

- $$(\forall R.C)^{\mathcal{I}}=\{ x|\forall y.R^{\mathcal{I}} (x,y)\to C^{\mathcal{I}} (y)\}$$

- $$(\exists R.C)^{\mathcal{I}}=\{ x|\exists y.R^{\mathcal{I}} (x,y)\wedge C^{\mathcal{I}} (y)\}$$

Observe that this list is a subset of those listed in Table 3.1.1, as there are fewer features in $$\mathcal{ALC}$$ cf. $$\mathcal{SROIQ}$$.

One also can specify the notion of satisfaction:

- An interpretation $$\mathcal{I}$$ satisfies the statement $$C\sqsubseteq D$$ if $$C^{\mathcal{I}}\subseteq D^{\mathcal{I}}$$

- An interpretation $$\mathcal{I}$$ satisfies the statement $$C\equiv D$$ if $$C^{\mathcal{I}}= D^{\mathcal{I}}$$

- $$C(a)$$ is satisfied by $$\mathcal{I}$$ if $$a^{\mathcal{I}}\in C^{\mathcal{I}}$$

- $$R(a, b)$$ is satisfied by $$\mathcal{I}$$ if $$(a^{\mathcal{I}}, b^{\mathcal{I}} )\in R^{\mathcal{I}}$$

- An interpretation $$\mathcal{I}= (\Delta^{\mathcal{I}},\cdot^{\mathcal{I}} )$$ is a model of a knowledge base $$\mathcal{KB}$$ if every axiom of $$\mathcal{KB}$$ is satisfied by $$\mathcal{I}$$

- A knowledge base $$\mathcal{KB}$$ is said to be satisfiable if it admits a model

Many DLs have be defined over the past 25 years and their complexity proved. For instance, one could add $$\mathcal{I}$$nverses to $$\mathcal{ALC}$$, giving $$\mathcal{ALCI}$$, or a $$\mathcal{H}$$ierarchy of roles, $$\mathcal{ALCH}$$, or $$\mathcal{Q}$$ualified cardinality restrictions; the appendix of the DL Handbook [BCM+08] has the full list of letters and the features they denote. You also may like to have a look at the DL Complexity Navigator8. In the next chapter about OWL 2, we shall introduce a few more expressive languages, whereas ontologybased data access in Chapter 8 introduces DL-Lite that is less expressive than $$\mathcal{ALC}$$.

## The DL $$\mathcal{SROIQ}$$

In this section, we summarise the various features that have been introduced informally above to provide a comprehensive definition of DL syntax. Doing so yields the description logic called $$\mathcal{SROIQ}$$, which is one of the most expressive DLs commonly considered today. It also largely agrees in expressivity with the ontology language OWL 2 DL, though there are still some differences as will be discussed in Chapter 4.

Formally, every DL ontology is based on three finite sets of signature symbols: a set $$\mathrm{N}_{I}$$ of individual names, a set $$\mathrm{N}_{C}$$ of concept names and a set $$\mathrm{N}_{R}$$ of role names. Usually these sets are assumed to be fixed for some application and are therefore not mentioned explicitly. Now the set of $$\mathcal{SROIQ}$$ role expressions $$\textbf{R}$$ (over this signature) is defined by the following grammar:

$\textbf{R}\::= U|\mathrm{N}_{R} |\mathrm{N}_{R}^{-}$

where $$U$$ is the universal role. Based on this, the set of $$\mathcal{SROIQ}$$ concept expressions $$\textbf{C}$$ is defined as:

$\textbf{C} ::=\mathrm{N}_{C} | (\textbf{C}\sqcap\textbf{C} ) | (\textbf{C}\sqcup\textbf{C} ) |\neg\textbf{C} |\top |\bot |\exists\textbf{R.C} |\forall\textbf{R.C} |\geqslant n\textbf{R.C} |\leqslant n\textbf{R.C} |\exists\textbf{R.} Self|\{\mathrm{N}_{I}\}$

where $$n$$ is a non-negative integer. As usual, expressions like $$(\textbf{C}\sqcap\textbf{C} )$$ represent any expression of the form $$(C\sqcap D)$$ with $$C$$, $$D$$ $$\in$$ $$\textbf{C}$$. It is common to omit parentheses if this cannot lead to confusion with expressions of different semantics. For example, parentheses do not matter for $$A\sqcup B\sqcup C$$ whereas the expressions $$A\sqcap B\sqcup C$$ and $$\exists R.A\sqcap B$$ are ambiguous.

Using the above sets of individual names, roles and concepts, the axioms of $$\mathcal{SROIQ}$$ can be defined to be of the following basic forms:

ABox: $$\textbf{C} (\mathrm{N}_{I} )$$ $$\textbf{R} (\mathrm{N}_{I} ,\mathrm{N}_{I} )$$ $$\mathrm{N}_{I}\approx\mathrm{N}_{I}$$ $$\mathrm{N}_{I}\not\approx\mathrm{N}_{I}$$

TBox: $$\textbf{C}\sqsubseteq\textbf{C}$$ $$\textbf{C}\equiv\textbf{C}$$

RBox: $$\textbf{R}\sqsubseteq\textbf{R}$$ $$\textbf{R}\equiv\textbf{R}$$ $$\textbf{R}\circ\textbf{R}\sqsubseteq\textbf{R}$$ $$Disjoint(\textbf{R}, \textbf{R} )$$

with the intuitive meanings as explained in 3.1: DL Primer.

Roughly speaking, a $$\mathcal{SROIQ}$$ ontology (or knowledge base) is simply a set of such axioms. To ensure the existence of reasoning algorithms that are correct and terminating, however, additional syntactic restrictions must be imposed on ontologies. These restrictions refer not to single axioms but to the structure of the ontology as a whole, hence they are called structural restrictions. The two such conditions relevant for $$\mathcal{SROIQ}$$ are based on the notions of simplicity and regularity. Notably, both are automatically satisfied for ontologies that do not contain complex role inclusion axioms.

A role $$R$$ in an ontology $$\mathcal{O}$$ is called non-simple if some complex role inclusion axiom (i.e., one that uses role composition $$\circ$$) in $$\mathcal{O}$$ implies instances of $$R$$; otherwise it is called simple. A more precise definition of the non-simple role expressions of the ontology $$\mathcal{O}$$ is given by the following rules:

• if $$\mathcal{O}$$ contains an axiom $$S\circ T\sqsubseteq R$$, then $$R$$ is non-simple,
• if $$R$$ is non-simple, then its inverse $$R{−}$$ is also non-simple,9
• if $$R$$ is non-simple and $$\mathcal{O}$$ contains any of the axioms $$R\sqsubseteq S$$, $$S\equiv R$$ or $$R\equiv S$$, then $$S$$ is also non-simple.

All other roles are called simple.10 Now for a $$\mathcal{SROIQ}$$ ontology it is required that the following axioms and concepts contain simple roles only:

Restricted axioms: $$Disjoint(\textbf{R} ,\textbf{R} )$$

Restricted concept expressions: $$\exists\textbf{R} .Self$$ $$\geqslant n\textbf{R}.\textbf{C}$$ $$\leqslant n\textbf{R}.\textbf{C}$$.

The other structural restriction that is relevant for $$\mathcal{SROIQ}$$ is called regularity and is concerned with RBox axioms only. Roughly speaking, the restriction ensures that cyclic dependencies between complex role inclusion axioms occur only in a limited form. For details, please see [HKS06]. For the introductory treatment in this paper, it suffices to note that regularity, just like simplicity, is a property of the ontology as a whole that cannot be checked for each axiom individually. An important practical consequence is that the union of two regular ontologies may no longer be regular. This must be taken into account when merging ontologies in practice.

The semantics of $$\mathcal{SROIQ}$$ is shown in Tables 3.1.1 and 3.1.2.

## Important Fragments of $$\mathcal{SROIQ}$$

Many different description logics have been introduced in the literature. Typically, they can be characterised by the types of constructors and axioms that they allow, which are often a subset of the constructors in $$\mathcal{SROIQ}$$. For example, the description logic $$\mathcal{ALC}$$ is the fragment of $$\mathcal{SROIQ}$$ that allows no RBox axioms and only $$\sqcap$$, $$\sqcup$$, $$\neg$$, $$\exists$$ and $$\forall$$ as its concept constructors. It is often considered the most basic DL. The extension of $$\mathcal{ALC}$$ with transitive roles is traditionally denoted by the letter $$\mathcal{S}$$. Some other letters used in DL names hint at a particular constructor, such as inverse roles $$\mathcal{I}$$, nominals $$\mathcal{O}$$, qualified number restrictions $$\mathcal{Q}$$, and role hierarchies (role inclusion axioms without composition) $$\mathcal{H}$$. So, for example, the DL named $$\mathcal{ALCHIQ}$$ extends $$\mathcal{ALC}$$ with role hierarchies, inverse roles and qualified number restrictions. The letter $$\mathcal{R}$$ most commonly refers to the presence of role inclusions, local reflexivity Self, and the universal role $$U$$, as well as the additional role characteristics of transitivity, symmetry, asymmetry, role disjointness, reflexivity, and irreflexivity. This naming scheme explains the name $$\mathcal{SROIQ}$$.

In recent years, fragments of DLs have been specifically developed in order to obtain favourable computational properties. For this purpose, $$\mathcal{ALC}$$ is already too large, since it only admits reasoning algorithms that run in worst-case exponential time. More light-weight DLs can be obtained by further restricting expressivity, while at the same time a number of additional $$\mathcal{SROIQ}$$ features can be added without loosing the good computational properties. The three main approaches for obtaining light-weight DLs are $$\mathcal{EL}$$, $$DLP$$ and $$DL-Lite$$, which also correspond to language fragments OWL EL, OWL RL and OWL QL of the Web Ontology Language.

The $$\mathcal{EL}$$ family of description logics is characterised by allowing unlimited use of existential quantifiers and concept intersection. The original description logic $$\mathcal{EL}$$ allows only those features and $$\top$$ but no unions, complements or universal quantifiers, and no RBox axioms. Further extensions of this language are known as $$\mathcal{EL}^{+}$$ and $$\mathcal{EL}^{++}$$. The largest such extension allows the constructors $$\sqcap$$, $$\top$$, $$\bot$$, $$\exists$$, Self , nominals and the universal role, and it supports all types of axioms other than role symmetry, asymmetry and irreflexivity. Interestingly, all standard reasoning tasks for this DL can still be solved in worst-case polynomial time. One can even drop the structural restriction of regularity that is important for $$\mathcal{SROIQ}$$. $$\mathcal{EL}$$-type ontologies have been used to model large but light-weight ontologies that consist mainly of terminological data, in particular in the life sciences. A number of reasoners are specifically optimised for handling $$\mathcal{EL}$$-type ontologies, the most recent of which is the ELK reasoner11 for OWL 2 EL.

DLP is short for Description Logic Programs and comprises various DLs that are syntactically restricted in such a way that axioms could also be read as rules in first-order Horn logic without function symbols. Due to this, DLP-type logics can be considered as kinds of rule languages (hence the name OWL 2 RL) contained in DLs. To accomplish this, one has to allow different syntactic forms for subconcepts and superconcepts in concept inclusion axioms. We do not provide the details here. While DLs in general may require us to consider domain elements that are not denoted by individual names, for DLP one can always restrict attention to models in which all domain elements are denoted by individual names. This is why DLP is often used to augment databases (interpreted as sets of ABox axioms), e.g., in an implementation of OWL 2 RL in the Oracle 11g database management system.

DL-Lite is a family of DLs that is also used in combination with large data collections and existing databases, in particular to augment the expressivity of a query language that retrieves such data. This approach, known as Ontology Based Data Access, considers ontologies as a language for constructing views or mapping rules on top of existing data. The core feature of DL-Lite is that data access can be realised with standard query languages such as SQL that are not aware of the DL semantics. Ontological information is merely used in a query preprocessing step. Like DLP, DL-Lite requires different syntactic restrictions for subconcepts and superconcepts. It is the basis for the OWL 2 QL species of OWL ontology languages, and we will present its DL definition and its use in Chapter 8.

## Footnotes

9If $$R=S^{−}$$ already is an inverse role, then $$R^{−}$$ should be read as $$S$$. We do not allow expressions like $$S^{−−}$$.

10Whether the universal role $$U$$ is simple or not is a matter of preference that does not affect the computational properties of the logic [RKH08b]. However, the universal role in OWL 2 is considered non-simple.


text:
Generalized Susceptibilities for a Perfect Quantum Gas

#### P. Briet, H. Cornean, D. Louis

2005, v.11, Issue 2, 177-188

ABSTRACT

The system we consider here is a charged fermions gas in the effective mass approximation, and in grand-canonical conditions. We assume that the particles are confined in a three dimensional cubic box $\Lambda$ with side $L\geq 1$, and subjected to a constant magnetic field of intensity $B \geq 0$. Define the grand canonical generalized susceptibilities $\chi_L^N$, $N\geq 1$, as successive partial derivatives with respect to $B$ of the grand canonical pressure $P_L$. Denote by $P_{\infty}$ the thermodynamic limit of $P_L$. Our main result is that $\chi_L^N$ admit as thermodynamic limit the corresponding partial derivatives with respect to $B$ of $P_{\infty}$. In this paper we only give the main steps of the proofs, technical details will be given elsewhere.

Keywords: quantum gas,magnetic field,thermodynamic limit


text:
## Stream: general

### Topic: filter map

#### Patrick Massot (May 10 2018 at 18:38):

Is this already somewhere in core or mathlib?

lemma filter_map_comm {I : Type*} {J : Type*} (f : I → J) (P : J → Prop) (r: list I) [decidable_pred P] :
filter P (map f r) = map f (filter (P ∘ f) r) :=
begin
induction r with h _ IH,
{ simp },
{ by_cases H : P (f h) ; simp [filter_cons_of_pos, filter_cons_of_neg, H, IH] }
end

#### Simon Hudon (May 10 2018 at 19:08):

try:

theorem filter_map_eq_map (f : α → β) : filter_map (some ∘ f) = map f :=
theorem filter_map_eq_filter (p : α → Prop) [decidable_pred p] :
filter_map (option.guard p) = filter p :=
theorem filter_map_filter_map (f : α → option β) (g : β → option γ) (l : list α) :
filter_map g (filter_map f l) = filter_map (λ x, (f x).bind g) l :=

from data.list.basic

#### Patrick Massot (May 10 2018 at 19:10):

What do you mean? The lemma is not there but there may be a shorter proof using those results?

#### Simon Hudon (May 10 2018 at 19:11):

Yes exactly. Sorry, I was overly concise

#### Patrick Massot (May 10 2018 at 19:12):

It looks at least as complicated as what I already have

#### Patrick Massot (May 10 2018 at 19:13):

And I don't want to frustrate Kenny by stealing a golfing challenge from him

#### Simon Hudon (May 10 2018 at 19:14):

That's very considerate :)

#### Patrick Massot (May 10 2018 at 19:15):

Actually I'd rather use Kenny (or anyone else) to help me fighting nat substraction

For instance:

#### Simon Hudon (May 10 2018 at 19:15):

I like the resulting proof because it avoids induction

#### Patrick Massot (May 10 2018 at 19:15):

example (a b k : ℕ) : b + k - (a + k) = b - a

#### Patrick Massot (May 10 2018 at 19:16):

What do you mean avoid induction? map and filter are defined inductively

#### Patrick Massot (May 10 2018 at 19:16):

You can at best hide induction

#### Patrick Massot (May 10 2018 at 19:16):

Note that both sides are zero is b is larger than a

#### Patrick Massot (May 10 2018 at 19:17):

So it looks like a "false" result but this one is actually true

I think

#### Patrick Massot (May 10 2018 at 19:18):

Do you have a solution with filter_map? Actually it could be useful to learn what filter_map is good for

#### Simon Hudon (May 10 2018 at 19:20):

What do you mean avoid induction? map and filter are defined inductively

They're recursively defined. But yeah, you can never get around using induction / recursion directly or indirectly but I feel hiding induction produces nicer interfaces. The laws about filter_map seem like you can prove a lot about filter and map without induction.

#### Simon Hudon (May 10 2018 at 19:21):

It's a generalization of both map and filter.

#### Patrick Massot (May 10 2018 at 19:22):

I still don't see how it could help me here

#### Reid Barton (May 10 2018 at 19:29):

example (a b k : ℕ) : b + k - (a + k) = b - a

example (a b k : ℕ) : b + k - (a + k) = b - a :=

:astonished:

#### Simon Hudon (May 10 2018 at 19:30):

I think I overlooked a detail. I thought just doing a rw would work but here is what I get:

  rw [← filter_map_eq_map
,← filter_map_eq_filter
,← filter_map_eq_filter
,filter_map_filter_map
,filter_map_filter_map],
congr, funext,
-- ⊢ option.bind ((some ∘ f) x) (option.guard P) = option.bind (option.guard (P ∘ f) x) (some ∘ f)

That should be hard either, but it makes the proof longer than expected

That's crazy

#### Patrick Massot (May 10 2018 at 19:30):

That's also crazy

Thanks Reid

#### Patrick Massot (May 10 2018 at 19:31):

I wasn't hoping for this to be in Lean core...

#### Patrick Massot (May 10 2018 at 19:43):

I'm becoming better and better at proof obfuscation. If I ever need to read those proofs, I'll hate myself.

#### Reid Barton (May 10 2018 at 19:44):

it's too hard to resist the golf

#### Patrick Massot (May 10 2018 at 19:46):

Right now I'm staring at line saying congr_n 1 ; funext ; simp only [nat.add_sub_cancel, nat.add_sub_add_right]

#### Patrick Massot (May 10 2018 at 19:46):

I wrote two minutes ago

#### Patrick Massot (May 10 2018 at 19:46):

And I already have no clue what it does

#### Patrick Massot (May 10 2018 at 19:46):

Mario and Johannes will be so proud

#### Patrick Massot (May 10 2018 at 19:48):

Because it's half the proof of some trivial statement, and trivial statement must have obfuscated proof according to mathlib style guide

#### Patrick Massot (May 10 2018 at 19:48):

lemma big.shift (P : ℕ → Prop) [decidable_pred P] (F : ℕ → R) (a b k : ℕ) :
(big[(◆)/nil]_(i=a..b | (P i)) (F i)) = (big[(◆)/nil]_(i=(a+k)..(b+k) | (P (i-k))) (F (i-k))) :=
begin
end

#### Patrick Massot (May 10 2018 at 19:49):

(for instance big[(◆)/nil] could be a \Sum operator

#### Kevin Buzzard (May 10 2018 at 19:51):

This is just the sort of stuff which I really needed last November and wasn't there

#### Kevin Buzzard (May 10 2018 at 19:51):

I remember having to give up on a 1st year problem sheet question because I couldn't prove that sum from 1 to n was equal to sum from n to 1 :-)

[at the time]

#### Kevin Buzzard (May 10 2018 at 19:52):

I don't even want to think about that

#### Kevin Buzzard (May 10 2018 at 19:52):

how much time did Gauss and Euler waste working out examples etc

#### Kevin Buzzard (May 10 2018 at 19:53):

Maybe they would have just played minecraft all day

#### Kevin Buzzard (May 10 2018 at 19:53):

and we'd be worse off

yep

#### Patrick Massot (May 10 2018 at 19:53):

or watched lol cats on youtube

#### Sean Leather (May 11 2018 at 07:07):

If they were functional programmers (esp. in Haskell), I think they would have enjoyed Lambdacats.

#### Moses Schönfinkel (May 11 2018 at 08:00):

Hehe, we're slowly eroding formality of this forum :).

#### Moses Schönfinkel (May 11 2018 at 08:06):

I believe cat pictures are new tho.

Dank memes next.

#### Sean Leather (May 11 2018 at 08:33):

Hehe, we're slowly eroding formality of this forum :smiley:.

Given that theorem proving is a formal method, what does an eroded formal method look like? :thinking_face:

Ah ha! :light_bulb: This must be why the name “lean” was chosen: the methodological mountain was so eroded, it couldn't stand up straight.

Last updated: May 14 2021 at 07:19 UTC


text:
# Acoular 20.10 documentation

## Sector

«  MergeGrid   ::   grids   ::   RectSector  »

# Sector¶

class acoular.grids.Sector

Bases: traits.has_traits.HasPrivateTraits

Base class for sector types.

Defines the common interface for all sector classes. This class may be used as a base for diverse sector implementaions. If used directly, it implements a sector encompassing the whole grid.

include_border = Bool(True, …

Boolean flag, if ‘True’ (default), grid points lying on the sector border are included.

abs_tol = Float(1e-12, …

Absolute tolerance for sector border

default_nearest = Bool(True, …

Boolean flag, if ‘True’ (default), the nearest grid point is returned if none is inside the sector.

contains(pos)

Queries whether the coordinates in a given array lie within the defined sector. For this sector type, any position is valid.

Parameters
posarray of floats

Array with the shape 3x[number of gridpoints] containing the grid positions

Returns
array of bools with as many entries as columns in pos

Array indicating which of the given positions lie within the given sector

«  MergeGrid   ::   grids   ::   RectSector  »


text:
Mathematics

# In figure, write down each linear pair.

##### SOLUTION
As per the figure, there are 8 pair of linear angles.
1. $\angle1$ and $\angle 2$
2. $\angle1$ and $\angle 3$
3. $\angle4$ and $\angle 2$
4. $\angle 4$ and $\angle 3$
5. $\angle5$ and $\angle 6$
6. $\angle5$ and $\angle 7$
7. $\angle8$ and $\angle 6$ and
8. $\angle 8$ and $\angle 7$

You're just one step away

Subjective Medium Published on 09th 09, 2020
Questions 120418
Subjects 10
Chapters 88
Enrolled Students 85

#### Realted Questions

Q1 Subjective Medium
Classify the following angle as right, straight, acute, obtuse or reflex.

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q2 Subjective Hard
In the given alongside :
If $\angle AOB=45^{o},\angle BOC=30^{o}$ and $\angle AOD=110^{o}$, find angles $COD$ and $BOD$.

Asked in: Mathematics - Straight Lines

1 Verified Answer | Published on 17th 08, 2020

Q3 TRUE/FALSE Medium
State whether following statement is true or false:
A line perpendicular to one of two parallel lines is perpendicular to the other.
• A. False
• B. True

Asked in: Mathematics - Straight Lines

1 Verified Answer | Published on 17th 08, 2020

Q4 Single Correct Medium
In the given figure, if $r \parallel s,p\parallel q$ and $u \parallel t$, then $c$ equals
• A. $a-b$
• B. $2a+b$
• C. $b-a$
• D. $a+b$

Asked in: Mathematics - Straight Lines

1 Verified Answer | Published on 17th 08, 2020

Q5 Subjective Medium
$POQ$ is a line ray $OR$ is a perpendicular to line $PQ$. $OS$ is another ray lying between ray $OP$ and $OR$. Prove that  $\angle ROS=\dfrac {1}{2}(\angle QOS -\angle POS)$

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020


text:
#### 选择题

##### 选择题 1

• hash函数应该“展开”大多数（即“非病理”）数据集（跨散列表的桶/槽）

• hash函数应该易于计算（恒定时间或接近它）

• hash函数应该易于存储（恒定空间或接近它）

• hash函数应该“展开”每个数据集（跨散列表的桶/槽）

##### 选择题 2

• $m/(2n)$
• $1/n$
• $n/m$
• $1/m$
• $n/(2m)$
• $m/n$

##### 选择题 3

• $1/m^2$
• $1/n^2$
• $1/(m-1)$
• $1/n$
• $1/m$

$x$的选择是任意的，$x,y$冲突当且仅当$y$落在$x$的桶中，所以概率为

##### 选择题 4

• $n/m^2$
• $n^2/m$
• $n/m$
• $n(n-1)/m$
• $n(n-1)/2m$

##### 选择题 5

• 小于$1\%$
• 小于$0.1\%$
• 小于$0.01\%$
• 小于$0.001\%$

#### 思考题

Recall that a set $H$ of hash functions (mapping the elements of a universe $U$ to the buckets $\{0,1,2,\ldots,n-1\}$) is universal if for every distinct $x,y∈U$, the probability $Prob[h(x) = h(y)]$ that $x$ and $y$ collide, assuming that the hash function $h$ is chosen uniformly at random from $H$, is at most $1/n$. In this problem you will prove that a collision probability of $1/n$ is essentially the best possible. Precisely, suppose that $H$ is a family of hash functions mapping $U$ to $\{0,1,2,\ldots,n-1\}$, as above. Show that there must be a pair $x,y∈U$ of distinct elements such that, if $h$ is chosen uniformly at random from $H$, then $\operatorname{Prob}[h(x)=h(y)] \geq \frac{1}{n}-\frac{1}{|U|}$.


text:
## Algebra 1: Common Core (15th Edition)

$p^3q^3$, sextic monomial
Since this polynomial has only one term, it is already in standard form. To name a polynomial, we must look at the degree and the number of terms. Since this polynomial has $1$ term with a combined power of $3+3=6$, we call it a sextic monomial.


text:
# Anti-concentration inequalities: lower bound on realized second moment

Let $$X:(\Omega,\Sigma)\rightarrow (\mathbb{R}^d;\mathbb{B}(\mathbb{R}^d))$$ be a Borel-measurable random vector. What are some general classes of such random vector for which one can give a "lower concentration inequality" of the form: $$\mathbb{P}(\|X\|^2>\lambda) \geq \mbox{(insert non-trivial lower bound)}$$ where $$\lambda>0$$.

Suppose e.g. that $$X=X_1+\dots+X_n$$, where the $$X_i$$' are independent zero-mean random vectors with $$\|X_i\|\le1$$ for all $$i$$. Then the Hoeffding--Azuma inequality (see e.g. Wikipedia) yields $$P(\|X\|>u)\ge1- e^{-(E\|X\|-u)^2/(2n)}$$ for $$u\le E\|X\|$$.

A number of lower bounds on $$P(\|X\|\ge u)$$ were obtained by de Acosta A. and Samur J.D. (Infinitely divisible probability measures and the converse Kolmogorov inequality in Banach spaces. Studia Math. 1979. V. 66, 143--160). For instance, a special case, for $$p=2$$, of their Corollary 3.1 on p. 151 is the following: $$P(\|X\|>u)\ge \frac14\,\Big(1-\frac{(u+1)^2+u^2/2}{E\|X\|^2}\Big)$$ for $$u>0$$, where $$X$$ is just as above.

Also, in the proof of their Lemma 2.3, de Acosta and Samur showed that, if the $$X_i$$'s are also symmetric, then $$P(\|X\|>u)\ge\frac12\,P(\max_i\|X_i\|>u) \ge\frac12\,\Big(1-\exp\Big\{-\sum_i P(\|X_i\|>u)\Big\}\Big)$$ for $$u>0$$.

• Hmm. The Hoeffding-Azuma inequality you quote is not the one in the reference, and is actually wrong as the example $n=1$ and $u=E|X|$ shows. I believe there is a typo also in the reference you give (Bartlett's notes) - it should be an upper bound, not a lower bound, as the example t=0 (and the proof given there) show. Feb 19 '20 at 20:07
• @oferzeitouni : Thank you for your comment. This is now corrected. Feb 20 '20 at 0:30

Well, in your case notice that $$\|X\|^2>0$$ and so $$\mu\triangleq \mathbb{E}[\|X\|^2]>0$$. Thus, for any $$\lambda \in \left(0,\mu\right)$$ the Cantelli Inequality gives $$\Pr(X\ge\lambda) \ge 1 - \frac{\sigma^2}{\sigma^2 + \lambda^2},$$ where $$\sigma\triangleq \mathbb{E}\left[\left(\|X\|^2 - \mu\right)^2\right]$$.

• This is interesting, but it only holds for small $\lambda$...not sure I'm I'm asking for too much...
– BLBA
Feb 19 '20 at 17:18
• $0<\lambda<\mu$? Feb 19 '20 at 21:27