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# $\exists x \forall y (P(y) \implies Q(x))$ and $\forall y P(y) \implies \exists x Q(x)$ are not logically equivalent

I am trying to show that the following are not logically equivalent (according to a practice question)

$$\exists x \forall y (P(y) \implies Q(x))$$ and $$\forall y P(y) \implies \exists x Q(x)$$

In the first case I am trying to find some kind of statement where $$x,y$$ are integers (something like $$P(x)$$ is the is even predicate and $$Q(x)$$ is odd predicate, or maybe that $$Q(x)$$ implies $$x$$ divides $$y$$).

I am imagining I need a scenario where one statement is True implies False, and the other is True, for the same values of $$x,y$$.

If $$\forall y P(y)$$ is false, then both implications will be true, so suppose that $$\forall y$$ P(y) is true. I'm not sure how to proceed from here. Hints/Clarifications appreciated.

• why are both sides true when $\forall y P(y)$ is false? – supinf Jan 20 at 18:25
• @supinf I meant both implications. – IntegrateThis Jan 20 at 18:27
• I still do not understand it. – supinf Jan 20 at 18:28
• @supinf for the statement on the right, it would imply False -> Something is True. For the first statement, it would be exists an x, for all y (P(y)=False -> Q(x)) which is also true? Maybe I am confused. – IntegrateThis Jan 20 at 18:30

hint 1:

Your statement, that the first statement is true if $$\forall y P(y)$$ is false, was a mistake. In your comment it seems like you thought that if $$\forall yP(y)$$ does not hold, then $$P(y)$$ is always false. It can also happen that $$P(y)$$ is false for some $$y$$ and true for some $$y$$.

You actually should assume, that $$\forall y P(y)$$ is false, but there is an $$y$$ such that $$P(y)$$ is true.

hint 2:

Consider the two cases for $$\exists x Q(x)$$. What happens if it is false? what if it is true?

• I have added my own overall answer, if you could check its correctness that would be appreciated. – IntegrateThis Jan 21 at 4:08

With false and true respectively $$0$$ and $$1$$, the statements are$$\max_yP(y)\le\max_xQ(x)$$ and$$\min_yP(y)=1\implies\max_xQ(x)=1.$$These statements differ in truth value if$$\min_yP(y)=0,\,\max_yP(y)=1,\,\max_xQ(x)=0.$$

Suppose that $$\exists y$$ such that $$P(y)$$ is true, but $$\forall y P(y)$$ is false. Also suppose that $$Q(x)$$ is false for any $$x$$.
Then the first logical statement is false since for such a $$y$$ where $$P(y)$$ is true, we would have there exists an $$x$$ such that $$P(y) \implies Q(x)$$ is true, which is false, and so the overall statement is false.
For the second statement, since the premise $$\forall y P(y)$$ is false, then the statement is vacuously true. Hence the statements aren't equivalent.
• This is correct. Alternatively, you could also use concrete statements for $P$ and $Q$ to show that the implications are not equivalent. – supinf Jan 21 at 8:47


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For the last 20 years Seattle's Green Pajamas have been cranking out great records. Since their debut album, Summer of Lust, in 1984, to their latest releases of Northern Gothic and Through Glass Colored Roses the Green Pajamas have maintained a strident DIY ethic while not sacrificing quality for it. I sat down to chat with Jeff Kelly and Joe Ross about their accomplishments over the years.

How did you record "Summer of Lust" in 1984?

Jeff Kelly-We recorded it on an old 7" Teac 4-track (it was a very good machine), and we had one, or (maybe)  two mics at the most.

Joe Ross- We could only do one track at a time. The basic tracks were mono.

JK-We'd do the drums and bass at Joe's  and then take it back over to my house...

JR-We could play loud at my house, but at Jeff's house we played quiet. So we would record the guitar, drums and bass, one or the other; I would play either rhythm guitar or bass depending on the song, then go down to Jeff's house and bring the deck back...

We would record 4-tracks on the reel to reel, do a mixdown to cassette and then copy the cassette back to 2 tracks of the 4-track. By that point we would have two more blank tracks on the reel to reel 4-track. So they were basically 6-track recordings.

What kind of cassette machine did you use?

JK-It was a Teac. Joe still has it too. We could edit on it without a click on the tape. Cassette decks always made a click.

JR- It wasn't just a click, there was no delay. It was powerful enough that it would start up right away. On "Summer of Lust" we do a mix from the 4-track to the  cassette deck and there would be a few times that we would be rolling tape, hit pause and then hit pause  again, the cassette would start  up again and it would be seamless. That was the beauty of that cassette deck.

What mics did you use?

JK- I had an Audio Technica $30 one. I had a stereo mic too, but I only used one side of it... JR- You put a 9 volt battery in it and it separates out. It was a Realistic with two things that separate out. It sounded really great. A little later I got a PZM mic from Radio Shack with a little battery pack on the chord. That sounded really cool. If you put it in the center of the room, it sounded natural. How long did it take? JR- Almost a month to the day. How did you do your vocals on "Summer of Lust"? JR-We did basic tracks at my parent's house (drums & guitar), but then going back to Jeff's to sing. Did you just put a mic in the center of the room and move in and out from it? JR-Most of the vocal tracks definitely. Harmony vocals were often done at the same time. If there was a tambourine, or a piece of percussion we would hold it in our hand and when it came to a chorus in order to not waste a track we'd play a tambourine for that part, quit playing the tambourine and then sing again. When you listen to the solo the tambourine comes on and when the solo quits the tambourine stops and the vocals start again. When we're mixing we gotta quickly adjust the level. How did you mix the record? JK-Just on my Advent stereo system in my bedroom, because I was living at home. JR-There was no mixing board used. The Teac 4-track had an Output volume knob for each track on the deck, because we always set the volume, but we didn't have a mixing board. There was no EQ either. There was also a switcher box with "Left/Right/Middle" switches. Everything was either hard pan left, hard pan right, or in the middle and there was no pan at all. You can definitely hear that. It has that, "Rubber Soul" sound where it was all hard right, or left. That was all The Beatles were doing. I don't think they said "Let's pan it over" they probably just had those three options; Middle, Right or Left. Did you use any effects? JR-No outboard gear at all. We didn't own any digital effects (reverbs, etc...). We could echo on the deck [using Simul-Sync & Playback function]. JK- All effects were done as it was going onto the recording. Mainly tape echo. It was setting one track on Monitor [while recording] instead of in Record and then playing the second track back to the first in Record . We would send the signal from one channel to the other and that would cause the Echo to happen and then record it on the first track. JR-If it was the last track left we couldn't do it, but if we had a blank track we could do it. It's only one speed, was [7.5 IPS or 3.75 IPS]. What were your Influences? JR-We both loved the way "Rain", the Beatles song sounded. As a bass player I've always tried to achieve somewhere between, " I am the Walrus" and "Rain". That's what turned me on. JK- That wasn't hot then, it was [the bands] The Rain Parade and the Three O'Clock. JR- The Paisley Underground. JK- There have been so many diverse influences. There is that "Baby You're a Rich Man" sound, where we go "let's get that cool groove." Joe and I were just talking about [the Band] Joy Division and I was playing the bass, trying to make it sound like Joy Division's album "Closer" when we wrote the riff to "Kim the Waitress". There is always a diverse mixture. I was really into the Paisley Underground. When I played my guitar loud, I played in a sort of primitive fashion and almost a sort of droning sound which Karl [Wilhelm] and Joe and I clicked on and led to that sort of Psychedelic Sound. JR-The Paisley Underground wasn't what we were first hearing, we loved the Paisley Underground [movement], because we could tell their influences were ours and they were doing what we were doing. We loved, "[Our] Satanic Majesties, Request" [by the Rolling Stones], which was also a big influence on the "Summer of Lust" at the time, and the "Baby You're a Rich Man"/"Rain"/"Its All Too Much" era Beatles. We were blown away when we heard the Paisley Underground [movement]. JK-REM were influenced by the 60s. I wasn't influenced by REM. I thought that first album was cool with the Rickenbacker guitars and stuff, but its not like REM influenced us. JR- I remember Jeff saying, "Wow, Listen to this!" It confirmed the concept that we could do this too. What made you want to make a record like "Summer of Lust"? JR-We were jamming a lot that Spring at my parent's house in the upstairs where my drum set was. Karl [Wilhelm], or Frank Walsh, Joe Bauer, Jeff or I both played drums too. We would jam up there. One time Karl, Jeff and I had a ghetto blaster and we just recorded this jam. Jeff was really on fire that day making up songs, and I remember playing it the next day for him and it was the skeletal works of "Katie Lied", "With a Flower in Her Hair" and "In This Castle". JK-I listened to those jam sessions on the way to work, and thought, "Those are some pretty good ideas." Joe came and said 'Let's make a tape', (because people in those days instead of making CDRs or their own CDs made cassettes and put them out themselves.) "so let's do that." "Summer of Lust" was almost already done. JR-When I met Jeff the Summer of '83, he already had his whole top dresser drawer in bedroom full of 90 minute cassettes of original tunes that he recorded on his 4-track. It was definitely my idea to market a cassette as a product and the "Summer of Lust" concept. JK-We got the inspiration. JR- Cellophane Square [a record store in Seattle] would have cases full of home made cassettes. Did you run the duplication yourselves? JR-No, we took it to Horner Audio [in Bothell Wa]. His little logo [for Horner Audio] was on the first batch of 50. The Rocket [magazine] thought we were a Bothell band. Horner listened to the tape, EQ'ed a little bit and made a production master. Any time I called him, he would run them off and I would pick them up. I did that for about three years calling him about every four or five months and picking up another batch of tapes. What's sort of Band Gear did you guys have? What kind of instruments? JK-I still have the same Telecaster! JR- My bass was an old$40 Lyle (SG copy) bass.

JK- And an old gray Supro amp.

JR-I still have that tube amp.

JK-I still have my Peavy Classic which I still use on stage now. Some of the other "weird instruments", Joe had a sitar...

JR- I had a zither and a bagpipe chanter and we used a recorder...

JK- and a Mattel® electric drum for the more exotic tracks.

JR- A lot of the time we would put the Mattel® drum down and then mic the speaker and use the kick sound from that ,  then mic the snare drum and hit the drum pad with a stick and then the snare drum so you would have the mic and the electronic drum going to the same track at the same time. Sort of, "Boom-Pap, Boom-Pap". The song, "My Mad Kitty" [off "Summer of Lust"] definitely uses that sound.

What did Tom Dyer do for you?

JK-He had a little studio in his basement in Fremont. Tom was always full of enthusiasm. That's how we came to know him. He recorded lots of singles [for his Green Monkey label], until he had to go make a real living.

JR- From 1985 to 90 almost every Pajamas recording was done at Tom's Studio [TDS sound]. The pre-Tom and the post-Tom periods definitely sound very different.

How long did it take to record "Kim the Waitress", "Peppermint Stick" and "Jennifer"? Any stories you can relate?

JR- There were about 3 sessions and probably about 16 hours for the whole thing. 6 or 7 hour sessions. Maybe 12 hours.

JK- I think it took more than 12 hours for the mixing, the re-mixing and all that.

JR- "Peppermint Stick" (the last of those three recorded) was the first vinyl on the "Monkey Business" compilation. Then "Kim the Waitress" and "Jennifer" were our first 7" single. In 1985-86 putting vinyl out  was a bigger deal. Now-a-days CDs are easier though. Putting out a cassette those days was nothin', but It took money to cut vinyl.

What "Studio Records" have you done? Where?

JR- Nothin's professional (not even the Tom stuff). "Book of Hours" was the first one (which I was not on). After that "November" was done with Jack Endino [at Reciprocal], was done really quick & live (but has never been properly released). "Ghosts of Love" which took several years to make. Those three were done with Tom Dyer.

Why did you start recording at home?

JR- Jeff had been recording at home all along. All through the period of recording at TDS, there would be a great Home Demo and we would go and re-record it properly. On some level it is technically maybe better, but another level it was maybe missing something. At home we record when the mood strikes us and Jeff has always been better when the inspiration strikes. Y'know at 3:00 a.m.

JK-The TDS recordings were a lot better sounding in most ways. When the new technology came along, I was able to mix on my own speakers and hear how it sounded in my own room. I always had a problem with that. With "Book of Hours" we always took it to different places and tried mixing it and it would never really sound that good. I get it home and I would always be disappointed. This started back when I did some recording at Crowe Recording Studio [in Seattle], pre Green Pajamas (with the Larch) and I was always disappointed with these recordings. I know how recordings sound here in my home studio and I know how I want my records to sound in reference to other ones. That is one big advantage of doing them at home. I got my  first dose of sitting and worrying about time and money at Crowe Recording. We were like, "Jesus, we are really racking up the bucks!" So we hurried and then the engineer who wanted to run the whole show tells us, "Oh you want to do this" and " You want to do that", but that's not really what I want. "I want the drums to sound different", but " No, no, no, you want them like this."

JR- Jeff spent all this money and the drums sound like generic demo band drums.

JK-I got my fill of that very early. Luckily as we got more involved with Tom [Dyer], I could spend hours there and not really worry about it. In fact, I owe Tom hundreds of hours of studio time which he fronted me. But that's because he wasn't just running the studio, he was managing the band too. He only granted all that studio time because he loved the music and he was an artist too. That's why I can' t replicate what a professional studio does. People go to recording studios, so they can have an engineer and all the modern stuff to make it sound good. In fact I don't think my records even do now.

JR- It only works if the band has a real relationship with the engineer. It doesn't work if the engineer isn't interested in the band's music.

JK- At Hanszek [Audio], Chris [Hanszek] gives input when we want it and when we want him to do something he does it. He is not like some guy who goes, "No you don't want to that..."

JR- Chris is really great to work with. The Beatles trusted George Martin [their Producer] and he trusted them. They were creating together.  It later got to the point with Tom, that we were creating together and Tom wasn't keeping track of hours, it was more like he was a part of the band, because he had an equal vested interest in the band as the manager and as Jeff's publisher.

JK- I also had big arguments with Tom where I would go home mad. At home you can do whatever you want. That's the main reason for recording at home. The technology was the reason we started to do it all at home. I got this little 8-track cassette deck and it was like I can do everything right here and have fun doing it. I am not a technical recorder, I am not a technical musician. I am not the kind of guy who could be a very good (unless it is very spontaneous) session musician, because it takes me a long time to get a track. Sometimes it will take me one take, but sometimes I will spend an hour trying to get a piano part, or two hours trying or something, over and over again. When you are in a studio, you're very self-conscious of this. I remember being in Tom's studio and being so nervous because the tape's running, or [Conrad] Uno's [Egg Studio] place. I had been at Uno's working as a session musician for Eric Lichter, it was much easier the last time, because I was more confident, but I remember feeling like, "They are all watching. They're all waiting for me to do this piano bit." My hands were shaking. It wasn't like I was on NBC TV, it was just Uno, Eric and somebody else watching.

JR- But there's money burning.

JK-Its Eric's money burning and there's something about that role. At home you roll the tape, you're by yourself all you have on is headphones, no one is waiting for you if you  do a part over and over again.

How did you learn how to record?

JR- I worked up to it when I started doing something, with Jeff especially, but I did it too before I met Jeff where I would record something, then play that tape back through speakers while playing along to those speakers and then with a new mic record onto another deck. I might do this a couple of times and pretty soon I started to figure out how to do this well.

JK-That's how I learned about tape echo. In fact we used to have the stereo turned up just slightly and we used to stick the Monitor switches up instead of in Record so we would hear the playback echo while we were playing and singing coming out of the speakers. It would make an echo/delay. I got interested in it, despite the fact that I was sort of a low-tech sort of person, I was always fascinated by pictures of the Beatles doing "Sgt. Pepper" in the studio. I could look at that for a long time and think about it all.

JR- The pictures of "Let It Be" of them and what was going on and the towels on the drum set and the microphones. We used to do that after "Summer of Lust" era, we used pillow cases (instead of towels) on our drums and realized that it makes it sound like we have compression. We didn't have any, but it makes it sound like we had. It sounded like the record when we put the towel on the drums.

JK- Currently I have 2 Alesis ADATs and a digital reverb which Eric Lichter got for me and a Mackie 1604 board. Tom Dyer came along again  when I needed him and got all that stuff for me. If fact when I was thinking about what kind of studio to set up, I was looking at this little digital things that are all in one that are either 8 or 16- track units (like my old Perception [cassette 8-track]). Jack Endino came over one day and I told him, "All I want to do is press buttons. I don't want  to think about computer stuff, I just want to hit record." He said, "If you just want to hit record then this what you need." So I trusted him and got this stuff and it's perfect. Very simple.

JR- Eric Lichter (the Green Pajamas keyboardist) got an 8 track digital porta-studio. Every little thing has got a little menu this big [gestures with hands] and everything is like [makes noises] ditk, ditk, ditk... OK here's reverb, OK here's this and it is like, "Can we pan this over?" "OK," [makes noises] ditk, ditk, ditk... "Here's the pan and pan it over this far" and then [makes noises] ditk, ditk, ditk...  We just want knobs. "OK Pan it over to this side, here ya go." We don't wanna have a little [makes noises] ditk, ditk, ditk...

JK- More than that with this system I just can just plug a mic in and hit record. It's so simple it's like an analog tape recorder. The only problem with ADAT is that they sometimes destroy my tapes! That can be very frustrating and it has happened too many times.

JR-If they eat the tape it's Game Over.

JK-It's frustrating. Jack [Endino] told me, "Make sure you back everything up these things eat tapes!" but I don't always have \$20 to get a back-up tape for every recording. So I'm living dangerously and thinking, "God I hope it doesn't eat the tape!"

Have you ever lost anything on the verge of completion?

JK-Most stuff that was damaged or lost has been salvaged. There was one song, spontaneously recorded and the recording was wonderful. I did something I rarely ever do,  because I wanted to listen to it. I did a rough mix onto a cassette. I just let everything run. I did nothing to it. It was the song, "Lost Girls Song" [off of "Northern Gothic"]. Just let it run through onto a cassette. Well the next day the tape got ruined. I put it in and it goes [makes a noise] "Crunch!" You go "OH NO!" We tried to get it out and it made it worse. As it turned out that cassette mix was used on that album. That song was written about a book and the author of the book actually contacted us (through Tony Dale of Camera Obscura Records) saying how much he liked the song. A friend of his gave him the album.  So this song has this miraculous life to it. I love that mix and it was rough mix, but  a rough mix can kinda grow on you.

JR-So the actual CD version was mastered off the cassette.

JK- And it's perfect.

What is a typical Recording Session with the Green Pajamas like?

JR- There is no typical recording session with the Pajamas. Really in the last 10 years Jeff has recorded a song and will call us to come and enhance it with something he wants.

JK-Like "Northern Gothic" [their latest release], I did a great deal of everything right here sitting with my keyboard or my guitar. The "live" Green Pajamas play the music. Sometimes the "live" Pajamas is almost like a cover band, because I do a lot of the stuff.  In the old days with "Summer of Lust", Joe and I were fairly equal.

JR-He had all the recording experience that I didn't have, but as a project we spent the time together.

JK- I wanted it to be a band. I wanted it to have a feel to it. With "Kim the Waitress" and "Peppermint Stick" we recorded as a band. Everybody, Karl [Wilhelm] on the drums; with "Book of Hours" and "Ghosts of Love" Karl played drums and we recorded as a band through the end of that "Ghosts of Love" era. In 1997, I started doing a lot more of it and Joe was pretty involved on the early records of that time. Joe was very involved in "All Clues Lead to Megan's Bed" and "Strung Behind the Sun" and stuff. Now-a-days if  Laura [Vanderpool] has a song that she thinks could be a Green Pajamas song, she'll come over and play guitar and sing it and I'll finish it. That's basically it. Eric [Lichter] will often record something at home (Because he primarily records everything on keyboards), bring it over here and I will put it on my ADAT system and we will mix it to stereo. Then I'll add bass or add lead guitar or something. His songs are an important element to those last studio albums, because they really put my songs in perspective.  I love the balance of having an Eric, or a Joe, or a Laura song  mixed in with mine, it gives a little more color. Eric has done some great recordings.

JR- The Home Studio has made it so anyone can record and get pretty nice sounding stuff.  A guy like Beck,  or Moby or whoever.  There is no reason to do in a studio what can be done at home. If you own a studio now you are hurtin'.

JK- If we were going to do a Green Pajamas recording with a "live" version,  it would very luxurious and nice to go to a studio to record a band proper.

JR- When we do that we are usually not happy with it. We listen back and go, "The performance is not that good, the sounds aren't that good" and when we are a home we are relaxed, we can smoke & drink and we can just let it roll when we are ready. If something is flawed, let's say that track is buzzing we say, "That's quaint, its nice. Let's keep it", but if we paid a bunch of money for it then we'd say,  "No that buzz can t go out!" When it is at home we have no one to blame but ourselves and if it works on some level we go with it.  When we performed a great take, and the engineer (who is the hired professional and is separate from our band) says "Well you'll hafta re-do it we had a buzz going on this track."  He is insisting because it is his record on the line now. We maybe would have lived with it at home, He makes us re-do it until the life is pissed out of our performance. So home is where it is at.

Special Thanks to Dave Dintenfass (for the Nagra & EV655c), Jorgen Bader, Karl Keefer (for their editing assistance) and Susanne Kelly (for her art and hospitality!)

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# Sorting in Math – Definition, Examples, Facts

## What is Sorting in Math? Introduction

Have you ever tried organizing your legos? How do you do that? You “sort” the parts either by color or shape! So, what is the concept of ‘sorting’ in math?

Sorting is the process where we are able to identify the similarities or the differences among a set of objects to arrange them in a defined sequence.

## Sorting: Definition

Sorting can be defined as the process of ordering, comparing, contrasting and systematically grouping or organizing different items. Sorting can be done easily on the basis of color, size, texture, etc.

There are many math sorting activities that also help students with daily tasks, like separating silverware, packing groceries, or doing laundry. The following image shows an example of sorting marbles based on their colors.

## Different Ways of Sorting

We can sort the objects or the shapes based on color, shape, size, texture, etc.

### Sorting by Size

Objects or shapes can be sorted according to how big or small they are.

The circles given below are sorted according to their size, from largest to smallest.

### Sort by Height

To sort the objects from shortest to longest, we follow these steps:

Step 1: Look through all the items and see which of them is the shortest of all. This item will go first on your sorted list.

Step 2: Look through the remaining items to find the shortest after that. Add it to your sorted list.

Step 3: Repeat step 2 until all items are sorted.

Let us look at some pencils and arrange them from shortest to longest.

The pencils sorted from the shortest to longest the order is: Pencil P, Pencil R, Pencil Q, Pencil S.

We can also use the comparison symbols which are $\lt$ and $\gt$ to show the sorted list of pencils.

We can say Pencil P $\lt$ Pencil R $\lt$ Pencil Q $\lt$ Pencil S.

Algorithms are like computer programs. They are a set of steps that are used to solve a problem. Such a sorting algorithm is known as “Bubble sort.”

An object is said to be the tallest if it has the biggest height in comparison to other objects and an object is said to be the shortest if it has the smallest height in comparison to other objects.

An example of sorting by height (shortest to tallest):

One more example of sorting by height (tallest to shortest):

### Sort by Colors

When we sort the objects by colors, we keep the shapes or objects of the same color in one group.

In the image given below, the shapes are sorted by colors. The green shapes are together, yellow shapes in another bunch, orange shapes in the third and purple shapes in the fourth.

Look at the figure given below; all rings of the same color are placed together.

### Sort by Shapes

When we sort by shape, we keep the same shapes together irrespective of their size or color. In the image given below, there are different shapes.

We see there are four different shapes based on the properties: triangle, rectangle, square, and circle. So, we place all triangles together, all rectangles together, all squares together and all circles together.

On sorting according to the shapes, we get:

## Sorting vs. Matching

Matching is the process of identifying two similar things based on their shared characteristics. We can match objects based on their color, size, form, quantity, etc.

Sorting is the process of finding similarities and differences among a collection of objects.

Pairing two objects based on a particular common characteristic is called matching. Grouping a number of objects together is called sorting.

## Ascending and Descending Order

We can sort the numbers in ascending or descending order. Ascending order means to sort the numbers from smallest to greatest. Descending order means to sort the numbers from greatest to smallest.

Let us take an example of four friends: Sophie with 157 cm height, Charlie with 162 cm height, Jacob with 143 cm and Mark with 139 cm height.

On sorting the heights in ascending order, we get: $139$ cm $\lt 143$ cm $\lt 157$ cm $\lt 162$ cm

On sorting the heights in descending order, we get $162$ cm $\gt 157$ cm $\gt 143$ cm $\gt 139$ cm.

## Sorting Numbers by Number of Digits

We can also sort the numbers on the basis of the number of digits.

Single-digit numbers are $0 − 9$.

Double-digit Numbers are $10 − 99$.

Three-digit Numbers are $100 − 999$.

## Conclusion

In this article, we learned about sorting. Sorting means identifying the similarities and differences between two or more objects and shapes. Let’s solve some examples for better understanding.

## Solved Examples on Sorting

1. Count and sort the fruits in separate groups.

Solution: On counting, we get there are 24 fruits.

Number of apples $= 5$

Number of guava $= 9$

Number of avocados $= 7$

Number of mangoes $= 3$

2. Sort the numbers on the basis of the number of digits.

24, 105, 146, 75, 92, 113, 178, 195, 46, 50

How many three-digit numbers are there?

Solution:

The two-digit numbers are 24, 75, 92, 46, 50

The three-digit numbers are 105, 146, 113, 178, 195

Number of digit numbers $= 5$

3. Sort the objects on the basis of color.

Solution:

Objects of red color $=$ Cherries, Crab

Objects of yellow color $=$ Banana, Yellow Capsicum

Objects of green color $=$ Whale, Butterfly

4. Michael is 23 years old, Ella is 16 years old, George is 19 years old, and Sam is 21 years old. Sort them from youngest to oldest.

Solution:

Firstly, we will sort the numbers in ascending order.

16 years $\lt$ 19 years $\lt$ 21 years $\lt$ 23 years

So, from youngest to oldest, we get: Ella, George, Sam, Michael.

5. Sort the objects from thinnest to thickest.

Solution:

The thinnest book has the minimum thickness, i.e., of book R.

The book with thickness more than Book R is Book P.

The thickest book is Book P.

So, from thinnest to thickest, we get Book R, Book P, Book Q.

## Practice Problems

1

### Numbers: 11, 22, 43, 106, 87, 97, 64, 52, 103 How many even numbers are there?

1
2
5
4
CorrectIncorrect
Correct answer is: 4
Even numbers are the numbers that are divisible by 2.
Even Numbers: 22, 106, 64, 52
There are 4 even numbers.
2

### When sorting the following plants/trees on the basis of tallest to shortest, which of the following will come in the middle?

Tree P
Tree S
Plant Q
Plant T
CorrectIncorrect
Correct answer is: Tree S
The tallest tree is R. Plant Q is the shortest.
Tree P is shorter than Tree R but taller than the Q, S and T.
Tree S is shorter than Tree P but taller than T and Q.
Plant T is shorter than Tree S but taller than plant Q.
On sorting from tallest to shortest, we get: Tree R $\gt$ Tree P $\gt$ Tree S $\gt$ Plant T $\gt$ Plant Q
3

### Sort the objects from lightest to heaviest.

Orange, Apple, Bananas
Orange, Bananas, Apple
Apple, Bananas, Orange
Apple, Orange, Bananas
CorrectIncorrect
Correct answer is: Orange, Apple, Bananas
The first figure says that the orange is lighter than the apple.
The second figure says that the apple is lighter than bananas.
So, from lightest to heaviest the order is: Orange, Apple, Bananas.
4

### On sorting the jugs based on low capacity to high capacity, which beaker has the least capacity?

Jug P
Jug R
Jug Q
Jug S
CorrectIncorrect
Correct answer is: Jug Q
Jug P can hold a maximum amount of liquid. Jug S has less capacity than Jug P but more capacity than Jug Q and R. Jug R has less capacity than Jug S but more capacity than Jug Q. Jug Q has the least capacity.

## Frequently Asked Questions

Sorting is important as it is used to arrange the items in a systematic order and group them according to their similarities.

When we group more than two objects based on a common characteristic, it is called sorting. But, when we talk about the order of the objects or numbers, we call it as ordering.

Color sorting helps in developing the cognitive visual perception and thinking skills in children. It helps them in quickening their reflexes and also helps them in their overall development.

We sort the solids based on the common characteristics. For example: ice cream cones and birthday caps have the shape of a cone. So, they can be grouped together.


text:
Algebra and Trigonometry

# Key Equations

 Formula for a factorial
 recursive formula for nth term of an arithmetic sequence $a n = a n−1 +d , n≥2 a n = a n−1 +d , n≥2$ explicit formula for nth term of an arithmetic sequence $a n = a 1 +d(n−1) a n = a 1 +d(n−1)$
 recursive formula for $nthnth$ term of a geometric sequence $a n =r a n−1 ,n≥ 2 a n =r a n−1 ,n≥ 2$ explicit formula for$nth nth$term of a geometric sequence $a n = a 1 r n−1 a n = a 1 r n−1$
 sum of the first$n n$ terms of an arithmetic series $S n = n( a 1 + a n ) 2 S n = n( a 1 + a n ) 2$ sum of the first$n n$ terms of a geometric series $S n = a 1 (1− r n ) 1−r ,r≠1 S n = a 1 (1− r n ) 1−r ,r≠1$ sum of an infinite geometric series with $S n = a 1 1−r ,r≠1 S n = a 1 1−r ,r≠1$
 number of permutations of$n n$distinct objects taken$r r$at a time $P(n,r)= n! (n−r)! P(n,r)= n! (n−r)!$ number of combinations of$n n$distinct objects taken$r r$at a time $C(n,r)= n! r!(n−r)! C(n,r)= n! r!(n−r)!$ number of permutations of$n n$non-distinct objects $n! r 1 ! r 2 !… r k ! n! r 1 ! r 2 !… r k !$
 Binomial Theorem $(x+y) n = ∑ k−0 n ( n k ) x n−k y k (x+y) n = ∑ k−0 n ( n k ) x n−k y k$ $(r+1)th (r+1)th$term of a binomial expansion $( n r ) x n−r y r ( n r ) x n−r y r$
 probability of an event with equally likely outcomes $P(E)= n(E) n(S) P(E)= n(E) n(S)$ probability of the union of two events $P(E∪F)=P(E)+P(F)−P(E∩F) P(E∪F)=P(E)+P(F)−P(E∩F)$ probability of the union of mutually exclusive events $P(E∪F)=P(E)+P(F) P(E∪F)=P(E)+P(F)$ probability of the complement of an event $P(E')=1−P(E) P(E')=1−P(E)$


text:
Last edited by Gardakora
Monday, July 27, 2020 | History

5 edition of corrosion of iron and steel found in the catalog.

# corrosion of iron and steel

## by J. Newton Friend

Published by Longmans, Green and co. in London, New York [etc.] .
Written in English

Subjects:
• Corrosion and anti-corrosives

• Edition Notes

Bibliographical footnotes.

Classifications The Physical Object Statement by J. Newton Friend .. LC Classifications TA467 .F7 Pagination xiv, 300 p. Number of Pages 300 Open Library OL176078M LC Control Number a 11002298 OCLC/WorldCa 1665005

The corrosion of structural steel is an electrochemical process that requires the simultaneous presence of moisture and oxygen. In the absence of either, corrosion does not occur. Essentially, the iron in the steel is oxidised to produce rust, which occupies approximately 6 times the volume of the original material consumed in the process. Additional Physical Format: Online version: Sang, Alfred. Corrosion of iron and steel. New York, McGraw-Hill, (OCoLC) Material Type: Internet resource.

Corrosion is usually defined as the degradation of metals due to an electrochemical process. The formation of rust on iron, tarnish on silver, and the blue-green patina that develops on copper are all examples of corrosion. The total cost of corrosion in the United States is significant, with estimates in excess of half a trillion dollars a : OpenStax. Cast iron differs from steel in that it has a higher carbon (>2%) and silicon (> 1%) content which makes it cheaper to produce than steel. Cast iron corrosion resistance is comparable to that of carbon steel or even better depending on the alloy content. In view of the high carbon content, graphite, which is cathodic compared with iron, forms.

The basic corrosion process: 2Fe2O3 + 3C → 4Fe + 3CO2 (Conversion to iron) Fe + O2 + H2O → Fe2O3.H2O (Change back to Ore-like state) Read more about cathodic protection. Metal corrosion rates. Metals corrode at different rates; they can be ranked in an order of ‘corrodibility’. It is estimated that about one-seventh of all iron production goes to replace the metal lost to corrosion. Rust is apparently a hydrated form of iron(III)oxide. The formula is approximately Fe 2 O 3 •$$\tfrac{\text{3}}{\text{2}}$$H 2 O, although the exact amount of water is variable.

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### Corrosion of iron and steel by J. Newton Friend Download PDF EPUB FB2

Corrosion tests on the copper steels as rolled. 1, w. (In Journal of the Iron and Steel Institute, v. 74, p. ) Tests show corrosion to take place much more slowly with rolled steel.

Huntly, G. Nevill. Sulphur as a cause of corrosion in steel. 1, w. (In Journal of 3/5(1). Page - Corrosion. — Metallic Structures: Corrosion and fouling, and their Prevention; a Practical Aid-Book to the safety of works in Iron and Steel, and of Ships ; and to the selection of Paints for them.

Corrosion of Iron and Steel by Industrial Waters and its Prevention (The Iron and Steel Institute Special Report No.

41) by N/A and corrosion of iron and steel book great selection of related books. Corrosion due to water is one of the most significant and complex causes of damage to metallic products.

Written from the viewpoint of physical chemistry, this authoritative and established text deals with the aqueous corrosion of metals. Available for the first time in English, Corrosion of Metal addressing engineers, metallurgists, physicists and chemists/5(2).

Corrosion is treated as a synthesis of corrosion science and corrosion engineering. This volume is comprised of nine chapters; the first of which provides an overview of the principles of corrosion and oxidation, with emphasis on the electrochemical mechanism of corrosion and how the kinetics of cathodic and anodic partial reactions control the.

Complemented with numerous examples to help illustrate important points, Corrosion and Corrosion Control, Fourth Edition enables readers to fully understand corrosion and its control and, in turn, help reduce massive economic and environmental loss. It is a must-read for advanced undergraduates and graduate students in engineering and materials.

Iron corrosion is generally characterized by the formation of rust due to an electrochemical process in the presence of water, oxygen and an electrolytic salt solution. Therefore, the remediation of one or all of these reactant sources can be used to reduce the rate of corrosion in a given metal.

Calculated Δσ BH as a function of amount of free carbon in ferrite for steel with prestrain = 4% and Ln(Dt) = À Source: From Das et al. [29] Effect of Mn on BH response as predicted by. Steel is essential in the machinery necessary for the manufacture of all our needs.

Even the words themselves have come to suggest strength. Phrases such as "iron willed," "iron fisted," "iron clad," "iron curtain," and "pumping iron" imply strength.

A "steely glance" is a stern look. "A heart of steel" refers to a very hard by: 5. A common example of galvanic corrosion occurs in galvanized iron, a sheet of iron or steel covered with a zinc coating. Even when the protective zinc coating is broken, the underlying steel is not attacked.

Instead, the zinc is corroded because it is less "noble"; only after it has been consumed can rusting of the base metal occur. Corrosion handbook page 2 Corrosion handbook Foreword Corrosion is a ubiquitous natural process.

Most of us, at some point in our everyday lives, become familiar with the effect that corrosion has on rusted steel parts. Corrosion has a huge economic impact.

About a fifth of the world’s annual steel production goes towards simply replacing. Corrosion is a natural process that converts a refined metal into a more chemically-stable form such as oxide, hydroxide, or is the gradual destruction of materials (usually a metal) by chemical and/or electrochemical reaction with their environment.

Corrosion engineering is the field dedicated to controlling and preventing corrosion. In the most common use of the word, this means. By Jim Mattzela. Stainless steel is typically specified because of its ability to resist corrosion in a variety of environments.

Unfortunately, stainless steel is not fully stain- or rust-proof, it is just more resistant to corrosion. Exposure to high salinity environments such as sea water can remove the native protective layer (chromium oxide) and allow oxidation/corrosion to occur at the. Yuxi Zhao, Weiliang Jin, in Steel Corrosion-Induced Concrete Cracking, Introduction.

Steel corrosion in any environment is an electrochemical process in which iron (Fe) is removed from the steel being corroded and is dissolved into the surrounding solution; it then appears as ferrous ions (Fe +).For steel embedded in concrete, the dissolution takes place in the limited volume of.

The corrosion process that takes place on a piece of uncoated steel is very complex. Factors such as variations in the composition/structure of the steel, presence of impurities due to the higher instance of recycled steel, uneven internal stress, and/or exposure to non-uniform environment all affect the corrosion process.

Carbon dioxide corrosion of carbon steel has been studied as a function of CO{sub 2} partial pressure ( to MPa), temperature (20, 40 60 and 90 C), pH (4 to 6) and flow velocity (, and 13 m/s) under conditions where protective iron carbonate fare do not form.

The experiments were. Corrosion. The different varieties of iron and steel will not oxidise in dry air, or when wholly immersed in fresh water free from air, but they all rust when exposed to the action of water or moisture and air alternately.

"Very thin iron oxidises more rapidly than thick iron, owing to the scales of rust on the former being thrown off as soon as formed in consequence of the expansion and. By Michael Ermann Animation by Taylor Kuehm. DIY Brick Rocket Stove - Cooking Without Power - Duration: Live Simple, Live Free - Tinyhouse Prepper Recommended for you.

Full text of "The corrosion of iron and steel" See other formats. 6 1 Basics of Corrosion Chemistry ions CrO2− 4 are the hard eionsBr − and sulfurous ions SO2− 3 stand somewhere between the soft base and the hard base. Cathodic Oxidant Reduction The cathodic current, i c, of oxidant reduction is also an exponential function of the electrode potential, E, of the metal as follows: i c = K c exp −α cE kT () For metallic iron in File Size: KB.

In this investigation, attempts have been made to study the corrosion inhibition properties of three new triazepine carboxylate compounds for mild steel in M hydrochloric acid medium.Additional Physical Format: Online version: Friend, J. Newton (John Newton), b. Corrosion of iron and steel.

London, New York [etc.] Longmans, Green and co., In air, a relative humidity of over 50% provides the necessary amount of water and at 80% or above corrosion of bare steel is worse. When a droplet of water containing a little dissolved oxygen falls on an steel pipe, the solid iron or Fe(s) under the droplet oxidizes.


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## Introduction

The Arctic Ocean winter sea ice cover has steadily declined since the 1970s, and the majority of this reduction has been observed in the Barents Sea1,2,3,4. With a warmer and increasingly ice-free Arctic Ocean, the border between the Boreal and Arctic biomes is predicted to move north5,6. While the present warming has allowed an opportunistic northwards expansion of Boreal species into the northern Barents Sea Arctic ecosystem7, little attention has been given to the effect of the recent ice retreat and variability on the ice-associated fish community. A keystone species in the ice-associated Arctic marine food web is the polar cod, one of few species linking the lower (i.e. zooplankton) and higher (e.g. other fish, mammals, seabirds) trophic levels8,9,10. The polar cod is endemic to the Arctic and their early life history is strongly adapted to the presence of ice: from spawning of eggs under the ice11; the ability of eggs to develop in sub-freezing temperatures12; larvae feeding on zooplankton specific to the seasonal ice-melt-water blooms13,14; and low mortality of larvae in the close to freezing temperatures typical of Arctic water masses15. At the same time observations made by Soviet-era researchers in the early 1960s11 remain the most complete descriptions of polar cod spawning along the Eurasian shelf, with a historical stronghold in spawning activity in the south-eastern corner of the Barents Sea (also known as the Pechora Sea) and a suggested spawning east of Svalbard. The Barents Sea polar cod stock have been monitored annually by a joint Norwegian-Russian survey since 198616, and the total stock biomass (TSB) has varied vastly in the past three decades, between a minimum of 127,000 t in 1990, up to a maximum of 1,941,000 t in 2006 yet with no clear trend17. The spatial distribution and abundance of 0-group polar cod (~6 months old) has also varied considerably in this period18, suggesting high recruitment variability both in space and time.

Given the tightly linked early life history of the polar cod with ice, a natural candidate for the large variation in polar cod recruitment and biomass is the inter-annual variation in ice cover18. The two main factors driving the observed variability in ice cover in the Barents Sea is the inflow of warm Atlantic water from the west and the inflow of cold, less saline Arctic water from the north and east4,19,20. The inflow of Atlantic water from the west and Arctic water from the northeast set up a strong temperature and salinity front across the entire Barents Sea, here termed the Polar Front21,22 (Fig. 1). North of the Polar Front the less dense Arctic water masses stabilizes the water column sufficiently to prevent upwards flux of the warmer Atlantic water, allowing ice to form4. In March-April the extent of the seasonal ice cover in the Barents Sea is at its largest, and at this time period the Polar Front and the ice edge usually coincide. During 1990–2017 the winter maximum ice cover in the Barents Sea varied between 632,304 km2 and 1,129,719 km2, with an average of 869,961 km2.

Here we investigate possible links between inter-annual variability in the Barents Sea ice cover and the variability in spawning distribution and recruitment of polar cod. To locate the spawning locations of polar cod under the ice we “back-tracked” larval drift trajectories from observed 0-group polar cod in autumn to the most probable spawning locations in spring. This was done by Lagrangian particle advection simulations within a 3D dynamical ocean model. Drifting particles were “spawned” uniformly over the parts of the Barents Sea that was encompassed by the historical marginal ice zone (i.e. where ice concentration had been at least 15% at any time in period 1990–2017), and subsequently an objective search algorithm identified drift trajectories that intersected the 0-group observations of the autumn survey. The coupled biophysical model was simulated over 28 spawning seasons (1990–2017) and was compared to the catch of 0-group polar cod in 8302 pelagic trawls distributed throughout the Barents Sea. The mechanistic interpretation of the scores from the objective search algorithm is twofold: first the backtracking from observations indicates spawning at a given release point for a given year (viz. inter-annual change in spawning location); and second the frequency of links between spawning location and the observed 0-group abundance reflects larval survival during the drift phase and indicates supply of recruits from a given spawning location (viz. recruitment variability in response to environmental conditions). We report a positive relationship between year class strength of polar cod to the inter-annual variability of ice cover, and a negative effect of maximum summer temperatures encountered in late larval phase, where a common driver is the heating of the system. We also observed a clear northward retreat of one the two identified spawning assemblages towards the end of the study period that may also be a response to warming. Understanding the physical-biological interactions during recent decades of both varying and changing ice conditions allow predictions of future development of the polar cod stock in the Barents Sea.

## Results

### Inter-annual change in spawning location

The objective search algorithm revealed two clearly separated maxima in spawning location probability, with one spawning assemblage in the Pechora Sea and one east of Svalbard. Moreover, the location of the two main spawning areas identified coincided with the high probability of 0-group occurrence downstream to the two areas (Generalized additive model (GAM) explaining 40% of the variation in 0-group presence/absence, Fig. 2). At the same time there was considerable inter-annual variability in the location of the center of the back-calculated main spawning areas. For example, in 1990 when the observed spatial distribution of 0-group polar cod was strongest correlated with spawning east of Svalbard, most of the eggs and larvae drifted with the East Spitsbergen Current into the Storfjord, further along the west coast of Svalbard, or onto the Svalbard Bank, where most of the 0-group polar cod was found that year (Fig. 3a, and see Fig. 1 for references to location). Recruitment in 1995 was very low, indicated by a record low abundance of 0-group polar cod found in the autumn cruise, coinciding with the Pechora Sea largely being ice-free during most of winter and spring (Fig. 3b) and maximum temperatures encountered in late summer exceeding 10 °C. This in contrast to 1998 when the ice cover in the south-eastern Barents Sea was at its highest in the study period, and the release points in the Pechora Sea was strongly correlated with the high abundance of 0-group polar cod found in a wide swath along Novaya Zemlya (Fig. 3c). Perhaps the most anomalous year in terms of drift patterns was in 2013 when the most probable spawning location of the eastern Svalbard spawning assemblage was situated north of Kvitøya, close to the shelf edge towards the Nansen Basin–yielding a qualitatively different dispersal pattern than previous years, where the majority of larvae initiated from this particular location were advected along the margins of the continental shelf instead of into the Barents Sea (Fig. 3d). This northern displacement of the eastern Svalbard spawning assemblage in 2013 was not an isolated instance, with a similar displacement also in 2009, 2015, and 2017. The predicted western spawning center was also located on the Great Bank in many of the years in the middle of the study period (2000s), and even near the Central Bank in some years. The average northwards displacement towards the end of the study period (2015–2017) constituted 2° of latitude (approx. 220 km) compared to the yearly 2000s (2001–2005), culminating a clear decadal northward trend (Fig. 4).

### Recruitment variability in response to ice loss and heating

Recruitment strength in the eastern Barents Sea/Pechora Sea had a significant correlation with maximum yearly ice cover (r = 0.69, t = −4.49, df = 26, p < 0.001, see Fig. 5), reiterating the importance of ice for early pelagic stages. The recruitment in the Pechora Sea was also significantly correlated with the estimated TSB two and a half years later (r = 0.61, t = 3.58, df = 22, p < 0.001), confirming the Pechora Sea spawning assemblage as the main supplier of recruits to the Barents Sea stock monitored during autumn surveys. Recruitment east of Svalbard had no correlation with either ice cover, temperature, or TSB at any lag, indicating other effects at play than those tested within the scope of this study. A linear regression of maximum ice cover over the study period showed a weak, yet significant yearly decrease of 1.1% (F = 4.93 on 1 and 26 df, p = 0.035, R2 = 0.16). Moreover, the 10-day mean-filtered maximum temperature encountered by larvae in late summer was significantly different between the two main spawning areas (t = −7.46, df = 47, p< 0.001), with maximum temperatures encountered in the eastern Barents Sea on average between 2.1 °C and 3.6 °C degrees warmer (95% CI). The average yearly maximum temperature encountered in the Pechora/eastern Barents Sea was 5.7 °C but in extreme years more than 9 °C (95% CI: 2.4–9.0 °C), well beyond the thermal tolerance limit of the larvae (≈100% mortality at extended periods above 7 °C15). A linear regression model of eastern Barents Sea recruitment strength as function of covariates Barents Sea ice cover, maximum temperature encountered by larvae, and TSB, explained 72% of the variation in recruitment strength (F = 20.7 on 3 and 23 df, p < 0.001). Ice and temperature explained the majority of the variance, and a smaller portion explained by TSB $$({\mathrm{R}}_{{\mathrm{ice}}\,{\mathrm{cover}}}^2 = 48\% ,{\mathrm{R}}_{{\mathrm{max}}\,{\mathrm{temp}}}^2 = 44\% ,{\mathrm{R}}_{{\mathrm{TSB}}}^2 = 16\% )$$. However, the two variables ice cover and temperature where found to share a high portion of the explained variance $$({\mathrm{R}}_{{\mathrm{ice}}\,{\mathrm{cover}}}^2 \cap {\mathrm{R}}_{{\mathrm{max}}\,{\mathrm{temp}}}^2 = 25\% )$$, albeit being significantly negatively correlated (r = −0.38, t = −2.09, df = 26, p = 0.045).

## Discussion

We suggest two plausible mechanisms driving the large recruitment variability observed in the south-eastern parts of the Barents Sea, mediated in turn by both heat and ice. First, the presence/absence of ice likely causes a “match/mis-match” scenario23,24 specific to the seasonally ice-covered Arctic food web. The primary food item for the first-feeding larval phase is the naupliar stages of Arctic copepod Calanus glacialis that are nourished by the algae bloom confined to the meltwater plume13,25,26. A mis-match most likely arises in years with reduced or no ice cover in the Pechora Sea when the phenology of the spring bloom follows Atlantic-like bloom dynamics initiated by thermal stratification rather than ice-melt25,27,28. However, if larvae indeed survive that first critical ice-associated phase all three major Calanus species found in the Barents Sea are eaten indiscriminately13. This includes the highly abundant C. finmarchicus and the larger (but rarer) C. hyperboreus, both advected from the west with the warmer Atlantic water masses, and all three Calanus species having exceptionally high energy content29. The ample food supply available to the later larval stages in the Atlantic-influenced water masses found in the south-eastern parts of the Barents Sea thus represents a near ideal nursery habitat. At the same time, the high summer temperatures sometimes encountered there may also be associated with high risk by pushing larvae beyond thermal thresholds15. For example, the seven warmest years in terms of maximum summer temperatures encountered (as observed in 1990, 1995, 2000, 2012–2013, 2016–2017), resulted in the five years of lowest recruitment. However, due to the collinearity of ice cover and high temperatures encountered in south-eastern Barents Sea during drift in late summer, where a common driver is the wind-driven advection of warmer water masses from the west20, we were not able to disentangle which of the two identified mechanisms that affected recruitment in the Barents Sea the most.

Regional downscaling of future climate change scenarios predicts an increased frequency of years with reduced or no ice cover in the south-eastern Barents Sea towards 205030,31. However, due to the strong topographic steering of the incoming Atlantic water masses away from the northern parts of the Barents Sea (cf. Figure 1), the cold and icy winter-conditions currently found east of Svalbard is likely to persist into the foreseeable future19,32,33. The predicted deterioration of the south-eastern Barents Sea nursery habitat will thus increase the relative importance of the supply of recruits coming from the eastern Svalbard spawning assemblage in the future. At the same time the observed northward displacement of the eastern Svalbard spawning assemblage towards the end of the study period may qualitatively alter the dispersal pathway of eggs and larvae. The advection of the surface layer in the far northern Barents Sea area is mainly influenced by the prevalent wind field, where the meltwater and pack ice generally follow a net westward direction towards the Fram Strait19,20. Other potential dispersal pathways that may be realized in the northernmost Barents Sea include getting entrained with the Atlantic boundary current flowing along the Nansen Basin34, or lateral transport towards the Arctic Ocean interior with eddies detaching from the boundary current itself35. In any case, polar cod larvae and juveniles drifting with the pack ice in the Arctic Ocean interior is not an uncommon phenomenon36, although the fate of these drifting individuals is not known.

The biophysical links highlighted here represent fundamental knowledge of how a keystone species of the Arctic food web respond to contemporary climatic forcing. As the frequency of years with low or no ice cover in the south-eastern Barents Sea is predicted to increase towards the middle of the millennium, we expect recruitment of polar cod to the south-eastern Barents Sea to become even more variable or diminished, most likely having dramatic consequences for the entire food web. And ultimately if summer temperatures continue to exceed the thermal tolerance of larvae, the south-eastern Barents Sea as a nursery area will be unsuitable altogether. Conversely, the topographical steering of Atlantic water masses away from the north-western Barents Sea will most likely facilitate ice and spawning east of Svalbard even in a warmer climate than today.

## Methods

### Recruitment and spawning stock biomass indices

The 0-group polar cod data were sampled on annual surveys run between late August and early October in the period 1990 and 2017, covering almost the entire Barents Sea within a regular grid of ~65 km. At each station the upper water layer (0–60 m) was sampled by three pelagic trawls with a 20 × 20 m opening, keeping the headlines at 0 m, 20 m, and 40 m. The pelagic trawls were towed at a speed of 3 knots over a time interval of 10 min, corresponding to a tow length of 0.5 nautical miles (≈0.93 km). If dense concentrations of fish appeared on the echo-sounder deeper than 40 m, additional tows were performed at 60 and 80 m. During the study period of 27 years 8302 of these depth-integrated trawl hauls were done. Due to the selectivity of the gear37, the catches were adjusted for capture efficiency using a stratified sample mean method38,39. As a proxy for total stock biomass (TSB) we estimated the total mass of polar cod found in echo-sounder transects and pelagic trawls throughout the Barents Sea17, identical to the method used for estimating capelin (Mallotus villosus) stock size in the Barents Sea40.

### Ocean circulation model and ice module

The hydrodynamic model used to represent the currents and oceanographic conditions (i.e. temperature, salinity, and ice concentration/cover) in the study area was based on the Regional Ocean Modeling System (ROMS, http://myroms.org), a free-surface, hydrostatic, primitive equation ocean general circulation model41,42. The ROMS model was run with a horizontal resolution of 4 × 4 km in an orthogonal, curvilinear grid covering parts of the North Atlantic and all the Nordic and Barents seas (see inset in Fig. 1 for extent of ROMS model) over the time period 1960–201720,43,44,45. The output from ROMS contained velocity fields, ice concentration, temperature, and salinity in 32 terrain following vertical layers, and a temporal resolution of 24 h.

### Drift simulations and search algorithm

The advection of particles in the horizontal plane was modelled by the Runge-Kutta fourth order scheme LADIM46,47. As early life stages of polar cod are usually found close to the surface13, particles were uniformly distributed in the upper 10 meters with a fixed depth throughout the drift phase from 1 January to 30 September. In an exhaustive search for potential spawning areas of polar cod in the Barents Sea, particles were released in a regular grid (≈40 km equidistance, 537 positions in total) across the entire Barents Sea shelf shallower than 400 m that had been covered by an ice concentration of more than 15% in the period 1990–2017 (see extent of release grid in Fig. 2). A new ensemble of 100 particles were released at every point in the grid, every day from 1 January to 30 April, repeated for every year between 1990 and 2017 (yielding a total of 639,030 particles each year). Subsequently, an objective search algorithm identified drift trajectories that intersected the 0-group observations of the autumn survey within a three-week period of the surveys. The ability of the drift trajectories to explain the observed 0-group abundance and distribution was thus interpreted as a confirmation of spawning at a given release point and a high larval survival integrated over the drift phase. To allow a direct comparison between number of simulated drift intersections and 0-group abundance, both indices were log-transformed and scaled between 0 and 1. In line with the hypothesis of ice as a prerequisite for spawning, the drift trajectories’ ability to predict the observed 0-group abundance was weighted by ice concentration at drift start point (i.e. at spawning area). To elucidate on the possible effects of heating on recruitment we extracted temperature profiles from all individual drift trajectories, and to decrease the effect of minor cold spells or heat waves on the subsequent analysis we applied a 10-day moving average filter on the temperature profiles.

### Statistical modelling

A probabilistic map of 0-group polar cod presence was calculated by using a two-dimensional binomial GAM smoother, based on the geographical coordinates of pelagic trawls, presence-absence of polar cod larvae in the pelagic trawls, and using the logit-link function as implemented in R-package “mgvc”48. Moreover, to quantify the effect of environmental conditions on larval survival/recruitment strength, we fitted a linear regression model with recruitment strength as independent variable with the covariates Barents Sea ice cover (area of the Barents Sea covered by ice concentration higher than 15%, extracted from the ROMS model), maximum temperature encountered by larvae (10-day mean-filtered over 100 larvae released from the most likely spawning area for a given year), and estimated TSB. This regression model was fitted separately for the north-western (Svalbard) and south-eastern (Pechora Sea) spawning areas as implemented in the base R-package “stats”49. In the model selection phase, we applied a stepwise model selection scheme with the initial inclusion of all relevant variables, where only the variables deemed significant was included in the final model. Due to the high degree of collinearity between some of the variables, we also did a variance partitioning analysis to disentangle the separate and/or common effects of the variables50.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.


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# Constructing Massive Scattering Amplitudes for Electroweak Effective Field Theory

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日時：2021年10月29日(金)　午前10時

タイトル：Constructing Massive Scattering Amplitudes for Electroweak Effective Field Theory

アブストラクト： Usual calculation for quantum field theory relies on Lagrangian under a certain symmetry. Then, one can obtain amplitudes by taking Feynman rules and diagrams. In contrast, a method called ''scattering amplitudes'' (also called on-shell
amplitudes, modern amplitude method, or spinor-helicity formalism) directly provides the amplitudes from symmetries without relying on Lagrangian. For example, calculations of gluon n-point scattering amplitudes can be greatly reduced in
this method. This approach is expected to extract some essences in quantum field theory, which are not obvious in the usual Feynman methods. Basically, conventional scattering amplitudes are methods for the massless theory. From 2017,
this method has been generalized to involve any-spin massive particles by Nima Arkani-Hamed group. In this talk, first, I will provide a brief review of the massless and massive scattering amplitudes. Next, I will introduce the scattering amplitude
calculations connected with electroweak symmetry breaking, which are related to masses, vev, and longitudinal waves. In particular, we do not use the standard model Lagrangian. We derived (strictly speaking, we re-derived) several equations
for electroweak symmetry breaking that are supposed to be inherent in field theory. Also, in such a bottom-up approach, we present the systematic derivation of independent three-point and four-point contact terms. This talk is based on arXiv:1909.10551 and arXiv:2008.09652.


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# How can I avoid compiling twice?

Sometimes I need to run compile twice (eg when I use lastpage). Is there a way to set things so TeX knows that there is need for second run and do it automatically? I use Texmaker.

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EditUpdated: I just noticed I slightly misunderstood your question. There are tools that compile the required number of times automatically, e.g. latexmk. I don't use it though, but there are certainly users who can tell you all about it. I've been using latexmk for a little while and it works really well (haven't tried it with bibliographies though), even with biblatex and Biber, and the support is great, the author even answers questions on here.

As for my initial answer below, it might still be of help in that it can eliminate the need of compiling twice.

hyperref is a package that frequently requires compiling twice . You can avoid this by loading the package bookmark after hyperref:

\usepackage{hyperref}
\usepackage{bookmark}

Generally, these two should be the last packages you load in your preamble, unless there's a package that specifically asks to be loaded after hyperref (see Which packages should be loaded after hyperref instead of before?).

bookmark provides some new possibilities, but if you've been using hyperref and you're not really looking for anything new, you can just load bookmark -- according to my experience, it won't change anything (besides the number of necessary compilations).

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It's not just about the order you load packages in; ANYTHING at all that involves a reference — \cite, \ref, page numbers even, all require double compilation, as the correct references are added in a second pass. – Mikael Vejdemo-Johansson Oct 3 '11 at 20:21

You could set up TeXmaker to use latexmk to compile. latexmk is a Perl script that you can set up to automatically run pdflatex (or other compilers), bibtex and/or makeindex if necessary.

As a very simple, example you could go to Options -> Configure TeXmaker -> Quick Build, and set up a custom Quick Build command such as

latexmk -pdf %.tex

When running Quick Build, this will run pdflatex on the file as many times as necessary. The % is TeXmaker's way of representing the filename of the current file (without extension).

See the latexmk manual (texdoc latexmk in a terminal) for all the possible options.

There are other similar options, such as texify in MikTeX, mentioned by Werner, and Rubber. Paulo Cereda wrote a blog post about Rubber, and there was a question concerning the difference between this and latexmk.

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Can be latexmk used with -shell-escape option of pdflatex? – Ichibann Nov 21 '11 at 19:59
@Ichibann Yes. The following is copied directly from page seven of the manual: latex -pdf -pdflatex=’pdflatex --shell-escape %O %S’ foo.tex Edit: I suppose the first word should really be latexmk, not just latex. – Torbjørn T. Nov 21 '11 at 20:25
It should indeed be latexmk. If you have a slightly older version of latexmk, you can also use the -e option to execute some Perl code that sets the option: latexmk -pdf -e '\$pdflatex="pdflatex --shell-escape %O %S"' foo.tex. – Elmar Zander Jun 20 '13 at 19:21

With respect to Seamus' recent blog post on overkill being a good thing, here's another approach.

The MiKTeX distribution includes texify (and pdftexify). These are different from the standard counterparts latex (->dvips) and pdflatex routines since they automatically recompile a document if needed.

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Nice, but I can't see a texify in my TexLive, which is up-to-date with everything 24 hours ago... – Brent.Longborough Oct 3 '11 at 19:48
@Brent.Longborough: texify is only distributed with MiKTeX (see my answer edits). TeX Live should use latexmk, I guess. Also see Replacement for MikTeX's texify in TeX Live. – Werner Oct 4 '11 at 6:20


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# Area of a triangle in a unit circle

What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle?

A bad first step is to put the center at the origin, one point at (1,0) , and one point at (sin x, cos x).

A start is the area of a triangle with included angle expression, $${a \times b \times \sin \theta} \over {2}$$

Assuming $\theta$ in radians. If theta is $\pi/2$ then we have a right triangle. Let a=b=1. Area expression is $$A=(\sin \theta) / 2$$ This is maximum for $\theta = \pi/2$.

Answer is maximum area for a right triangle.

• My guess would have been the origin, $(1,0)$, and $(0,1)$ which obviously has area of 1/2. Is the area of yours greater? – Shane Nov 5 '14 at 20:43

The area of a triangle with side lengths $a, b, c$ and angles $A, B, C$ can be expressed as $$\left|\triangle ABC\right| = \frac{1}{2}ab \sin C = \frac{1}{2}bc \sin A = \frac{1}{2}ca \sin B.$$ Thus the area for a fixed $a = b = 1$ and variable central angle $C$ is maximized when $\sin C$ attains its maximum; i.e., $\angle C = \pi/2$ and $\sin C = 1$. Thus the triangle is right.
You have $$S=\frac{r^2}{2}\sin\theta=\frac{\sin\theta}{2},$$where $\theta$ is the angle at the center. The rest shouldn't be too hard.
Let the centre of the circle be $O$, let one of the vertices be $A$, and let the other vertex $X$ vary. The area is maximized when the height of the triangle with respect to the base $OB$ is maximal. This happens when $XO$ is perpendicular to $OB$.


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Density Zero Exploration post by Alex Mennen 185 days ago | Abram Demski, Paul Christiano and Scott Garrabrant like this | discuss The idea here is due to Scott Garrabrant. All I did was write it. Let’s say a logical induction-based agent is making an infinite sequence of decisions, and is using $$\varepsilon$$-exploration on each decision. There are two desirable criteria, which are somewhat in conflict: First, we want there to be enough exporation that traders attempting to bet that good strategies would have bad outcomes (and thus prevent the good strategies from being tried, so that the bet never gets settled) will lose arbitrarily large amounts of money if they try doing that every time. This requires that in total, there is an infinite amount of exploration. For example, if the agent $$2^{-n}$$-explores on step $$n$$, then it is possible for a sufficiently wealthy malicious trader to bet against a good strategy by enough that the agent will avoid it every time, without the trader losing all its money, because the actions it is discouraging only are taken anyway finitely many times. But if the agent $$\varepsilon$$-explores on step $$n$$ for some fixed $$\varepsilon>0$$, then this is not possible, because each action is taken infinitely many times no matter what any of the traders do, so no trader can consistently make some good action appear bad without losing all its money. Second, we want there to be sufficiently little exploration that the agent does not sacrifice a nontrivial amount of value to it. If actions only have short-term effects, then it is enough for the probability of exploration to approach $$0$$ as $$n\rightarrow\infty$$, in order for the agent to behave optimally in the limit (if actions can have lasting consequences, then this is not enough; for instance, if there is an action that destroys all value forever if it is ever taken, then that action needs to never be taken; this directly conflicts with the first criterion). For example, if the agent $$\varepsilon$$-explores on step $$n$$ for some fixed $$\varepsilon>0$$, then it never gets any closer to acting optimally, but if it $$2^{-n}$$-explores on step $$n$$, then its probability of acting optimally will approach $$1$$ as $$n\rightarrow\infty$$. Fortunately, there are sequences that converge to $$0$$ but whose sum diverges, like $$\frac{1}{n}$$, so it is possible to satisfy both of these criteria. However, if there are important differences among what the agent should do for different steps, then this might not be enough. For example, if the agent $$\frac{1}{n}$$-explores on step $$n$$, and it is particularly important what action the agent takes on steps that are powers of $$2$$, then a wealthy malicious trader could bet against good actions on every step that is a power of $$2$$, then on the $$k$$th time the malicious trader does this, the agent $$2^{-k}$$-explores, and the malicious trader will only lose a finite amount of money doing this. Thus we should strengthen the first criterion to ensure that a wealthy malicious trader cannot bet against a good strategy infinitely many times, rather than that it cannot bet against a good strategy every time. Thus, for every efficiently computable infinite subset $$X\subseteq\mathbb{N}$$, we want an infinite amount of exploration to occur on steps in $$X$$, so that no malicious trader can bet against a good strategy on every step in $$X$$ without running out of money (we only need to consider efficiently computable subsets because only efficiently computable traders participate in the market, and they cannot pick out sets that are not efficiently computable). To do this, pick some computable probability distribution $$p$$ over all efficiently computable subsets of $$\mathbb{N}$$, which does not assign probability $$0$$ to any of them. This can be done computably by picking a probability distribution over programs that provably run in polynomial time. For each efficiently computable set $$X\subseteq\mathbb{N}$$, we want to explore with probability at least $$\frac{p\left(X\right)}{k}$$ on the $$k$$th element of $$X$$. Formally, let $$X_{n}$$ be $$\frac{1}{k}$$ if $$n$$ is the $$k$$th element of $$X$$, and $$0$$ if $$n\notin X$$. On step $$n$$, we explore with probability $$\sum_{X}p\left(X\right)X_{n}$$. Since we explore with probability at least $$\frac{p\left(X\right)}{k}$$ on the $$k$$th element of $$X$$, this satisfies the strengthened version of the first condition. Since $$\sum_{X}p\left(X\right)=1$$, $$X_{n}\leq1$$ for every $$X$$ and $$n$$, and $$X_{n}\rightarrow0$$ as $$n\rightarrow\infty$$ for every $$X$$, $$\sum_{X}p\left(X\right)X_{n}\rightarrow0$$ as $$n\rightarrow\infty$$, so the second condition is also satisfied.

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## WUSTL team receives $3.9M to engineer bacteria to produce drop-in fuels from lignin ##### 29 December 2017 Researchers at Washington University in St. Louis have received a$3.9-million grant from the Department of Energy (DOE) to develop bacteria that manufacture renewable biofuels.

Building on prior funding from the DOE Office of Biological and Environmental Research (BER) for systems biology work on Rhodococcus opacus, this project will apply innovative combinations of -omics and mass spectrometry data analysis, flux analysis, and machine learning to develop improved genome-scale models that will guide multiplex genome engineering of R. opacus strains with branched chain fatty acid esters (BCFAE) biosynthetic capacity and improved tolerance to lignin degradation products.

The project will take advantage of this organism’s capability to use both polysaccharides and lignin as carbon sources as well as its natural tolerance to phenolic compounds such as those produced by lignin breakdown.

The grant supports research in five Washington University labs, including those led by co-principal investigators Gautam Dantas, an associate professor of pathology and immunology; Tae Seok Moon, an assistant professor of energy, environment and chemical engineering; Marcus B. Foston, an assistant professor of energy, environment and chemical engineering; Yinjie Tang, an associate professor of energy, environment and chemical engineering; and Fuzhong Zhang, an associate professor of energy, environment and chemical engineering. Hector Garcia Martin of Lawrence Berkeley National Laboratory is another collaborator.

Rhodococcus opacus was originally discovered growing on toxic compounds outside a chemical plant. These bacteria thrive on these toxic compounds, using them as a source of food for the production of biofuels.

We are taking advantage of the fact that the compounds that R. opacus grows on are not just random toxins. These compounds are related closely to lignin, complex polymers that make up roughly 30 percent of plant matter. Our team is using a combination of chemistry, systems biology and synthetic biology to try to process lignin plant matter into biofuels that can be added directly to current petroleum-based engines.

—Gautam Dantas

Millions of tons of lignin are generated yearly from papermaking and lignocellulose-based biofuel industries. Currently, the value of lignin is restricted to its application as a fuel for on-site boiler operations. This project seeks to expand its uses.

The Foston lab is focused on developing chemical transformation processes that extract lignin from biomass, breaking apart that lignin into a form that can be “fed” to the bacteria. Lignin is a large and complex molecule designed to give plants mechanical stability and protect them from attack. Therefore, special conditions such as the presence of a catalyst and the application of heat and pressure are required to break apart lignin. The goal is to create a flexible process that generates compounds R. opacus essentially likes to eat.

The Moon lab is focused on optimizing the different strains of R. opacus through adaptive evolution and the use of synthetic biology tools. The researchers are growing the bacteria in these lignin-based compounds, making the compounds progressively more toxic with each generation, selecting for microbes that are becoming increasingly efficient at processing the compounds. More importantly, according to Moon, his team is developing the synthetic biology tools needed to engineer and improve R. opacus strains for fuel production.

The Dantas lab is focused on sequencing the DNA and RNA of R. opacus as it adapts to more efficiently process the raw (and toxic) biomass. As the microbes evolve, the researchers will identify patterns in genes and metabolic pathways that are the most important to the processing of the compounds and that are the easiest to genetically engineer so that in the future these pathways could be “dialed up” without waiting for generations of the bacteria to adapt and evolve.

The Tang and Garcia Martin labs will use computer models and machine learning to map the interconnecting networks of genes and metabolism and predict what the microbe will do under a given set of circumstances. One little change could propagate through the network and result in multiple changes to the resulting compounds that the microbe produces. Understanding these networks could allow for fine-tuning of the bacteria to make use of changing sources of biomass.

The Zhang lab is taking the compounds made by the bacteria and processing them into refined forms of fuel that so closely resemble petroleum or diesel, they could be pumped into vehicles that are on the road today.

While the process to manufacture this biofuel likely would be energy-intensive, at least initially, the researchers said that it would not require drilling and that the burning of the fuel itself would be carbon-neutral.


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# Graphing Curves with limits

1. Dec 1, 2013

### kald13

1. The problem statement, all variables and given/known data

I am to explain all intercepts, critical numbers, extrema, inflection points, and asymptotes of the function f(x)=(x-4)/x^3.

2. The attempt at a solution

a) The y-intercept does not exist, as the domain of the function is all real numbers except x=0. Solving the numerator for 0 yields an x-intercept at x=4
b) f'(x) is easily calculated to be (12-2x)/x^4; solving the numerator for 0 yields a critical number of x=6, and x is undefined at 0, yielding our second critical number; Using values from the intervals shows f'(x)>0 on (-infinity, 0] and [0, 6); f'(x)<0 on (6, infinity), indicating what appears to be a local maximum at x=6. Plugging this into f(x):

f(6)=(6-4)/6^3
f(6)=2/216
or 1/108

I note that this is a positive number, where f(2) is negative, meaning the line has crossed the x axis between 2 and 6 (specifically at x=4, as predicted)

f(2)=(2-4)/2^3
f(2)=-2/8
or -1/4

Inflection points and second derivative information aside, I now come to the point where I am trying to predict the behavior of the function as x approaches positive and negative infinity.
c) I know there is a vertical asymptote at x=0 due to the fact that the limit of the function does not exist as x approaches 0, but what about horizontal asymptotes?

limit as x approaches infinity of (x-4)/x^3

This is indeterminate, infinity divided by infinity, so using L'Hospital's rule I take the derivative of the numerator and denominator for

limit as x approaches infinity of 1/(3x^2)

I factor out constants:

(1/3) limit as x approaches infinity of 1/x^2

And I know that the limit of 1/x is 0. But here's my problem: I already know that my function has crossed the x axis at 4, so how can there be a horizontal asymptote at y=0?

2. Dec 1, 2013

### scurty

Asymptotic behavior is based on how the distance between the function curve and the asymptote line approaches zero as they branch out to infinity. How the function behaves before then is irrelevant. Take for example the graph of $f(x) = \sin(1/x)$. As x values approach 0 from either side, the function crosses the x-axis an infinite number of times. Yet $\displaystyle\lim_{x \rightarrow \pm\infty} \sin(1/x) = 0$.

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# How does the math work for Perception checks?

Please forgive me, it's been over 40+ years since I've played D&D and just now trying to reboot the changes. Can someone briefly explain Perception check math?

Assume I'm a 5 year old or an idiot, whichever comes first in your mind.

If a character has a Wisdom of 10 and the DM says "roll a Perception check"... The character then rolls a 1d20 with the hopes of what number? Higher than or lower than 10? Or is it a number only the DM knows about the area/monster/etc.?

Follow-up question:
If a character has a high Wisdom, wouldn't the P.C. be easier to achieve? Assuming a lower roll is easier than a higher roll (yes?).

• Welcome! When you get a chance you can edit the edition either into the body of the post or as a tag, by writing something like adnd-2e or dnd-5e into the list of tags. If you can't find anywhere on the books where it says that (like I couldn't when I came back to D&D after a 20+ year hiatus) feel free to pop into Role-playing Games Chat and people there should be able to give you a hand. Or this visual guide to the editions' corebooks might help. – nitsua60 Mar 22 '19 at 22:20
• Welcome back. I started with 3 little brown books in 1975. Here's a link to the free basic rules if that helps. I also "get" your chosen site name. :) Please ping me in chat whenever, I'll be happy to help. – KorvinStarmast Mar 23 '19 at 0:26

You may wish to refer to the introductory sections of the PHB, also contained in the freely available Basic Rules, which describe the core gameplay mechanics - which are rather different to the D&D you're used to if you last played forty years ago. Tempting as it may be to dive in at the deep end, you might be better off setting aside everything you remember from the games you played before and starting at the very beginning with fresh eyes.

As the How To Play section of the Basic Rules describes making ability checks, attack rolls, and saving throws (the three main kinds of d20 roll, of which making a "Perception check" would be a wisdom-based ability check):

1. Roll the die and add a modifier.

Roll a d20 and add the relevant modifier. This is typically the modifier derived from one of the six ability scores, and it sometimes includes a proficiency bonus to reflect a character’s particular skill. (See "Step-By-Step Characters" for details on each ability and how to determine an ability’s modifier.)

2. Apply circumstantial bonuses and penalties.

A class feature, a spell, a particular circumstance, or some other effect might give a bonus or penalty to the check.

3. Compare the total to a target number.

If the total equals or exceeds the target number, the ability check, attack roll, or saving throw is a success. Otherwise, it’s a failure. The DM is usually the one who determines target numbers and tells players whether their ability checks, attack rolls, and saving throws succeed or fail.

The target number for an ability check or a saving throw is called a Difficulty Class (DC). The target number for an attack roll is called an Armor Class (AC).

This simple rule governs the resolution of most tasks in D&D play. "Using Ability Scores" provides more detailed rules for using the d20 in the game.

# The DM determines the DC, the PC adds their Wisdom bonus and sometimes Proficiency bonus

Step by step:

• DM calls for a Wisdom (Perception) check (this is an ability check).

• The Difficulty Class (DC) is the minimum number that the PC needs to get in order to pass the check.
• An example medium DC = 10, a hard DC = 15, a very hard DC = 20, etc.
• The DM should already know what the DC of the check is. They may tell the PC or keep it secret, at their discretion.
• Player Character (PC) rolls 1d20.

• PC adds their ability modifier (Wisdom for Perception), in this case 0 due to their Wisdom score of 10.

• If the PC is proficient in Perception, they add their Proficiency bonus (+2 at level 1, it increases as the PC gains levels).

• The total result is told to the DM, who tells the PC if they pass or fail, and what happens next.

The calculation is: 1d20 + Ability Score Modifier (Wisdom) + Proficiency bonus (if proficient)

Since a higher Wisdom score will give you a higher modifier it makes the check easier. Likewise, if the DC is lower the odds of passing it are higher.

For any skill based ability check the player tells the DM what he wants to try to do, the DM then decides on the appropriate skill or ability and she sets a (usually secret) number, the DC. The player then rolls a 20-sided die to meet or exceed that number, adding his character’s ability based modifier and (if you are proficient in the skill) proficiency based modifier for that skill to the number he rolls.

So you add d20 roll + ability modifier + proficiency modifier (if you are proficient in the skill)

You will usually have the complete modifier for a skill written on your character sheet so that you don't have to think about it too much while playing and just need to adjust it when you reach certain character levels or increase ability scores. So it should really just be a matter of rolling the die, adding a number to it, and telling the DM what you got.

## The Number (DC)

Based on how the DM evaluates the difficulty, she sets a number called the DC. Because this is a game where you can do anything and the DM has to adjust things on the fly, this is usually a fairly arbitrary number based on “what feels reasonable” to them, so you just have to trust that your DM is picking a reasonable one as fairly and carefully as they can. The most typical numbers set by most DMs are probably 5, 10, 15, 20, and 25.

Some DMs have more elaborate systems for determining the number. If you are playing a published campaign it is also possible that the authors of that have specified DCs for some situations. There are also times when it will be a “contested roll” against another creature’s skill based ability roll. For perception this is often the case when there is a creature using stealth that you might perceive; your roll would have to beat theirs.

After you succeed or fail on the roll the DM may or may not tell you what the DC was (or whether you succeeded or failed) at her discretion.

## Roll and Modifier

The DM calls for a roll and specifies what skill you should be using. Each skill has an associated ability (for perception it is Wisdom) and you will get a modifier from this based on whether your ability score is lower, equal to, or higher than ten. For every 2 below ten you get a -1 modifier and for each 2 above you get a +1 modifier. If you have a 10 there is no modifier. So if you have a Wisdom score of 14, for example you would have a +2 modifier.

Each character also has proficiencies in certain skills based on character creation decisions (background, class, and race). If you are proficient in a skill you get to add your proficiency modifier as well. The proficiency modifier is based on your character level. At level 1 your proficiency is 2, and it increases by 1 every few levels.

So if a level 1 character with a Wisdom of 12 and proficiency in the perception skill made a perception roll they would add 1 to the roll for ability and 2 for skill proficiency to their total score. So if they rolled an 11 it would count as 14. If the DC was 14 or lower they succeed. If it was 15 or higher they fail.

Occasionally the DM will have you make a “straight ability check” when no skill quite fits for what you are doing, in which case you just roll and add the ability modifier without worrying about skills.

This is all much easier once you have your character sheet set up correctly as it will tell you the total to add for each skill.

Various classes have special rules about skill rolls (particularly Rogues and Bards), but you’ll learn those as you go if they apply.

## 20s and 1s

A very common house rule (possibly the most common of any kind) is that a natural (before adding modifiers) roll of 20 on the die is an automatic success, and a roll of 1 is an automatic failure. At many tables 20 often means a particularly spectacular success and 1 a particularly spectacular failure. But, technically, according to official rules 1s and 20s only have effects like this for attacks and death saving throws.

## Passive Perception

In some situations the DM will use your “passive perception” for whether you notice something without actively looking for things. This number will equal 10 + the ability and proficiency modifiers. As long as your DM knows what your passive perception is you shouldn’t have to worry about it.

• Youve got a lot of excellent and accurate information, but you need to cite your references for each point. – NautArch Mar 23 '19 at 0:02
• @NautArch In this case that seems roughly on the level of needing to cite page 1 of the PHB for the proposition that "Dungeons and Dragons is a roleplaying game", but I'll see what I can do. – Benjamin Olson Mar 23 '19 at 0:08